def set(self, key, value):
key = base64.b64encode(key)
now = datetime.datetime.now()
filename = now.isoformat() + '.dat'
file_path = os.path.join(self.path, filename)
while os.path.exists(file_path):
now = now + datetime.timedelta(microseconds=1)
filename = now().isoformat()
file_path = os.path.join(self.path, filename)
fd = os.open(file_path, os.O_WRONLY | os.O_EXCL | os.O_CREAT)
try:
f = os.fdopen(fd, 'w')
dump(value, f)
file_size = f.tell()
finally:
f.close()
conn = self.conn
c = conn.cursor()
try:
c.execute("INSERT INTO entries(key, filename, size) VALUES ('{}', '{}', {})".format(key, filename, file_size))
conn.commit()
except sqlite3.IntegrityError:
conn.commit()
from pymor.core.logger import getLogger
getLogger('pymor.core.cache.SQLiteRegion').warn('Key already present in cache region, ignoring.')
os.unlink(file_path)
self.bytes_written += file_size
if self.bytes_written >= 0.1 * self.max_size:
self.housekeeping()
def set(self, key, value):
if key in self._cache:
getLogger('pymor.core.cache.MemoryRegion').warn('Key already present in cache region, ignoring.')
return
if len(self._cache) == self.max_keys:
self._cache.popitem(last=False)
self._cache[key] = value
def housekeeping(self):
self.bytes_written = 0
conn = self.conn
c = conn.cursor()
c.execute('SELECT SUM(size) FROM entries')
size = c.fetchone()
# size[0] can apparently also be None
try:
size = int(size[0]) if size is not None else 0
except TypeError:
size = 0
if size > self.max_size:
bytes_to_delete = size - self.max_size + 0.75 * self.max_size
deleted = 0
ids_to_delete = []
files_to_delete = []
c.execute('SELECT id, filename, size FROM entries ORDER BY id ASC')
while deleted < bytes_to_delete:
id_, filename, file_size = c.fetchone()
ids_to_delete.append(id_)
files_to_delete.append(filename)
deleted += file_size
c.execute('DELETE FROM entries WHERE id in ({})'.format(','.join(map(str, ids_to_delete))))
conn.commit()
path = self.path
for filename in files_to_delete:
try:
os.unlink(os.path.join(path, filename))
except OSError:
from pymor.core.logger import getLogger
getLogger('pymor.core.cache.SQLiteRegion').warn('Cannot delete cache entry ' + filename)
from pymor.core.logger import getLogger
getLogger('pymor.core.cache.SQLiteRegion').info('Removed {} old cache entries'.format(len(ids_to_delete)))
def _call_pymess_dense_nm_gmpare(A, E, B, C, R, S, trans=False, options=None, plus=False):
"""Return the solution from pymess.dense_nm_gmpare solver."""
A = to_matrix(A, format='dense')
E = to_matrix(E, format='dense') if E else None
B = B.to_numpy().T
C = C.to_numpy()
S = S.to_numpy().T if S else None
Q = B.dot(B.T) if not trans else C.T.dot(C)
pymess_trans = pymess.MESS_OP_NONE if not trans else pymess.MESS_OP_TRANSPOSE
if not trans:
RinvC = spla.solve(R, C) if R is not None else C
G = C.T.dot(RinvC)
if S is not None:
RinvST = spla.solve(R, S.T) if R is not None else S.T
if not plus:
A -= S.dot(RinvC)
Q -= S.dot(RinvST)
else:
A += S.dot(RinvC)
Q += S.dot(RinvST)
else:
RinvBT = spla.solve(R, B.T) if R is not None else B.T
G = B.dot(RinvBT)
if S is not None:
RinvST = spla.solve(R, S.T) if R is not None else S.T
if not plus:
A -= RinvBT.T.dot(S.T)
Q -= S.dot(RinvST)
else:
A += RinvBT.T.dot(S.T)
Q += S.dot(RinvST)
X, absres, relres = pymess.dense_nm_gmpare(None,
A, E, Q, G,
plus=plus, trans=pymess_trans,
linesearch=options['linesearch'],
maxit=options['maxit'],
absres_tol=options['absres_tol'],
relres_tol=options['relres_tol'],
nrm=options['nrm'])
if absres > options['absres_tol']:
logger = getLogger('pymess.dense_nm_gmpcare')
logger.warning(f'Desired absolute residual tolerance was not achieved '
f'({absres:e} > {options["absres_tol"]:e}).')
if relres > options['relres_tol']:
logger = getLogger('pymess.dense_nm_gmpcare')
logger.warning(f'Desired relative residual tolerance was not achieved '
f'({relres:e} > {options["relres_tol"]:e}).')
return X
def set(self, key, value):
key = base64.b64encode(key)
response = self.server.set(self.secret, key)
assert len(response) == 2 and isinstance(response[0], bool) and isinstance(response[1], str)
if response[0]:
with open(response[1], 'w') as f:
dump(value, f)
file_size = f.tell()
response = self.server.set_finished(self.secret, key, file_size)
assert isinstance(response, bool) and response
else:
from pymor.core.logger import getLogger
getLogger('pymor.core.network_cache.NetworkFilesystemRegion')\
.warn('Key already present in cache region, ignoring.')
def _setup(name='pymor'):
root_logger = logger.getLogger(name)
root_logger.setLevel(logging.ERROR)
test_logger = logger.getLogger(name)
test_logger.setLevel(logging.DEBUG) # config_files.append(os.path.join(os.path.dirname(pymor.__file__), '../../setup.cfg'))
# config defaults to no plugins -> specify defaults...
manager = nose.plugins.manager.DefaultPluginManager()
config_files = nose.config.all_config_files()
config = nose.config.Config(files=config_files, plugins=manager)
config.exclude = []
selector = PymorTestSelector(config=config)
loader = nose.loader.defaultTestLoader(config=config, selector=selector)
cli = [__file__, '-vv', '-d']
return cli, loader, config
def _load_all():
import pymor
ignore_playground = True
fails = []
for _, module_name, _ in pkgutil.walk_packages(pymor.__path__, pymor.__name__ + '.',
lambda n: fails.append((n, ''))):
if ignore_playground and 'playground' in module_name:
continue
try:
__import__(module_name, level=0)
except (TypeError, ImportError) as t:
fails.append((module_name, t))
if len(fails) > 0:
logger.getLogger(__name__).fatal('Failed imports: {}'.format(pprint.pformat(fails)))
raise ImportError(__name__)
def load_matrix(path, key=None):
logger = getLogger('pymor.tools.io.load_matrix')
logger.info('Loading matrix from file ' + path)
path_parts = path.split('.')
if len(path_parts[-1]) == 3:
extension = path_parts[-1].lower()
elif path_parts[-1].lower() == 'gz' and len(path_parts) >= 2 and len(path_parts[-2]) == 3:
extension = '.'.join(path_parts[-2:]).lower()
else:
extension = None
file_format_map = {'mat': ('MATLAB', _loadmat),
'mtx': ('Matrix Market', _mmread),
'mtz.gz': ('Matrix Market', _mmread),
'npy': ('NPY/NPZ', _load),
'npz': ('NPY/NPZ', _load),
'txt': ('Text', _loadtxt)}
if extension in file_format_map:
file_type, loader = file_format_map[extension]
logger.info(file_type + ' file detected.')
return loader(path, key)
logger.warning('Could not detect file format. Trying all loaders ...')
loaders = [_loadmat, _mmread, _loadtxt, _load]
for loader in loaders:
try:
return loader(path, key)
except IOError:
pass
raise IOError(f'Could not load file {path} (key = {key})')
def __init__(cls, name, bases, namespace):
'''I copy my class docstring if deriving class has none. I tell base classes when I derive
a new class from them. I publish a new contract type for each new class I create.
'''
doc = namespace.get("__doc__", None)
if not doc:
for base in cls.__mro__[1:]:
if base.__doc__:
doc = base.__doc__
break
cls.__doc__ = doc
# monkey a new contract into the decorator module so checking for that type at runtime can work
dname = (cls.__module__ + '.' + name).replace('__main__.', 'main.').replace('.', '_')
if not dname in decorators.__dict__:
decorators.__dict__[dname] = contracts.new_contract(dname, lambda x: isinstance(x, cls))
# all bases except object get the derived class' name appended
for base in [b for b in bases if b != object]:
derived = cls
# mangle the name to the base scope
attribute = '_%s__implementors' % base.__name__
if hasattr(base, attribute):
getattr(base, attribute).append(derived)
else:
setattr(base, attribute, [derived])
cls.logger = logger.getLogger('{}.{}'.format(cls.__module__.replace('__main__', 'pymor'), name))
abc.ABCMeta.__init__(cls, name, bases, namespace)
def add_class(self, cls, wrapped_cls):
self.wrapped_classes[cls] = wrapped_cls
if hasattr(cls, 'type_this'):
try:
self.wrapped_classes_by_type_this[cls.type_this()] = wrapped_cls
except TypeError:
logger = getLogger('dune.pymor.core')
logger.warn('Could not call type_this on {}. (Not a static method?)'.format(cls.__name__))
def clear(self):
# Try to safely delete all cache entries, even if another process
# accesses the same region.
self.bytes_written = 0
conn = self.conn
c = conn.cursor()
c.execute('SELECT id, filename FROM entries ORDER BY id ASC')
entries = c.fetchall()
if entries:
ids_to_delete, files_to_delete = zip(*entries)
c.execute('DELETE FROM entries WHERE id in ({})'.format(','.join(map(str, ids_to_delete))))
conn.commit()
path = self.path
for filename in files_to_delete:
try:
os.unlink(os.path.join(path, filename))
except OSError:
from pymor.core.logger import getLogger
getLogger('pymor.core.cache.SQLiteRegion').warn('Cannot delete cache entry ' + filename)
def update(self, defaults, type='user'):
assert type in ('user', 'file')
import pymor.core.interfaces
if pymor.core.interfaces.ImmutableMeta.sids_created:
from pymor.core.logger import getLogger
getLogger('pymor.core.defaults').warn(
'Changing defaults after calculation of the first state id. '
+ '(see pymor.core.defaults for more information.)')
for k, v in defaults.iteritems():
self._data[k][type] = v
func = self._data[k].get('func', None)
if func:
argname = k.split('.')[-1]
func._defaultsdict[argname] = v
argspec = inspect.getargspec(func)
argind = argspec.args.index(argname) - len(argspec.args)
defaults = list(argspec.defaults)
defaults[argind] = v
func.__defaults__ = tuple(defaults)
self._calc_sid()
def set(self, key, value):
fd, file_path = tempfile.mkstemp('.dat', _safe_filename(datetime.datetime.now().isoformat()[:-7]) + '-', self.path)
filename = os.path.basename(file_path)
with os.fdopen(fd, 'wb') as f:
dump(value, f)
file_size = f.tell()
conn = self.conn
c = conn.cursor()
try:
c.execute("INSERT INTO entries(key, filename, size) VALUES ('{}', '{}', {})"
.format(key, filename, file_size))
conn.commit()
except sqlite3.IntegrityError:
conn.commit()
from pymor.core.logger import getLogger
getLogger('pymor.core.cache.SQLiteRegion').warn('Key already present in cache region, ignoring.')
os.unlink(file_path)
self.bytes_written += file_size
if self.bytes_written >= 0.1 * self.max_size:
self.housekeeping()
def colormap_texture(name='viridis'):
resolution = min(gl.GL_MAX_TEXTURE_SIZE, 1024)
colormap_id = gl.glGenTextures(1)
gl.glBindTexture(gl.GL_TEXTURE_1D, colormap_id)
gl.glTexParameteri(gl.GL_TEXTURE_1D, gl.GL_TEXTURE_MAG_FILTER, gl.GL_NEAREST)
gl.glTexParameteri(gl.GL_TEXTURE_1D, gl.GL_TEXTURE_MIN_FILTER, gl.GL_NEAREST)
gl.glTexParameteri(gl.GL_TEXTURE_1D, gl.GL_TEXTURE_WRAP_S, gl.GL_CLAMP_TO_EDGE)
colormap = np.empty((resolution, 4), dtype='f4')
from matplotlib.pyplot import get_cmap
try:
cmap = get_cmap(name)
except ValueError:
from pymor.core.logger import getLogger
# this is our default which might not exist for older matplotlib so a warning would be annoying
if name != 'viridis':
getLogger('pymor.gui.gl.colormap_texture').warn('Unknown colormap {}, using default colormap'.format(name))
cmap = get_cmap()
colormap[:] = cmap(np.linspace(0., 1., resolution))
gl.glTexImage1D(gl.GL_TEXTURE_1D, 0, gl.GL_RGBA, resolution, 0, gl.GL_RGBA, gl.GL_FLOAT, colormap)
gl.glBindTexture(gl.GL_TEXTURE_1D, 0)
return colormap_id
def __new__(cls, classname, bases, classdict):
c = UberMeta.__new__(cls, classname, bases, classdict)
init_arguments = c._init_arguments
try:
for a in c.sid_ignore:
if a not in init_arguments and a not in ImmutableMeta.init_arguments_never_warn:
raise ValueError(a)
except ValueError as e:
# The _logger attribute of our new class has not been initialized yet, so create
# our own logger.
l = logger.getLogger('{}.{}'.format(c.__module__.replace('__main__', 'pymor'), classname))
l.warn('sid_ignore contains "{}" which is not an __init__ argument!'.format(e))
return c
def _import_all(package_name='pymor'):
package = __import__(package_name)
if hasattr(package, '__path__'):
def onerror(name):
from pymor.core.logger import getLogger
logger = getLogger('pymor.core.defaults._import_all')
logger.warn('Failed to import ' + name)
for p in pkgutil.walk_packages(package.__path__, package_name + '.', onerror=onerror):
try:
__import__(p[1])
except ImportError:
from pymor.core.logger import getLogger
logger = getLogger('pymor.core.defaults._import_all')
logger.warn('Failed to import ' + p[1])
def check_consistency(self, delete=False):
self.import_all()
from pymor.core.logger import getLogger
logger = getLogger('pymor.core.defaults')
keys_to_delete = []
for k, v in self._data.iteritems():
if ('user' in v or 'file' in v) and 'code' not in v:
keys_to_delete.append(k)
if delete:
for k in keys_to_delete:
del self._data[k]
for k in keys_to_delete:
logger.warn('Undefined default value: ' + k + (' (deleted)' if delete else ''))
return len(keys_to_delete) > 0
def __init__(cls, name, bases, namespace):
"""Metaclass of :class:`BasicInterface`.
I tell base classes when I derive a new class from them. I create a logger
for each class I create. I add an `init_args` attribute to the class.
"""
# all bases except object get the derived class' name appended
for base in [b for b in bases if b != object]:
derived = cls
# mangle the name to the base scope
attribute = '_%s__implementors' % base.__name__
if hasattr(base, attribute):
getattr(base, attribute).append(derived)
else:
setattr(base, attribute, [derived])
cls._logger = logger.getLogger('{}.{}'.format(cls.__module__.replace('__main__', 'pymor'), name))
abc.ABCMeta.__init__(cls, name, bases, namespace)
def testDump(basicinterface_subclass):
try:
obj = basicinterface_subclass()
assert isinstance(obj, basicinterface_subclass)
if issubclass(basicinterface_subclass, core.Unpicklable):
return
except TypeError as e:
logger = getLogger('pymortests.core.pickling')
logger.debug('PicklingError: Not testing {} because its init failed: {}'.format(basicinterface_subclass,
str(e)))
return
with tempfile.NamedTemporaryFile(mode='wb', delete=False) as dump_file:
core.dump(obj, dump_file)
dump_file.close()
f = open(dump_file.name, 'rb')
unpickled = core.load(f)
assert obj.__class__ == unpickled.__class__
os.unlink(dump_file.name)
def _import_all(package_name="pymor"):
package = importlib.import_module(package_name)
if hasattr(package, "__path__"):
def onerror(name):
from pymor.core.logger import getLogger
logger = getLogger("pymor.core.defaults._import_all")
logger.warn("Failed to import " + name)
for p in pkgutil.walk_packages(package.__path__, package_name + ".", onerror=onerror):
try:
importlib.import_module(p[1])
except ImportError:
from pymor.core.logger import getLogger
logger = getLogger("pymor.core.defaults._import_all")
logger.warn("Failed to import " + p[1])
def _parse_options(options, default_options, default_solver, default_least_squares_solver, least_squares):
if options is None:
options = default_options[default_least_squares_solver] if least_squares else default_options[default_solver]
elif isinstance(options, str):
options = default_options[options]
else:
assert 'type' in options and options['type'] in default_options \
and options.keys() <= default_options[options['type']].keys()
user_options = options
options = default_options[user_options['type']]
options.update(user_options)
if least_squares != ('least_squares' in options['type']):
logger = getLogger('foo')
if least_squares:
logger.warning('Non-least squares solver selected for least squares problem.')
else:
logger.warning('Least squares solver selected for non-least squares probelm.')
return options
def load_gmsh(gmsh_file):
"""Parse a Gmsh file and create a corresponding :class:`GmshGrid` and :class:`GmshBoundaryInfo`.
Parameters
----------
gmsh_file
File handle of the Gmsh MSH-file.
Returns
-------
grid
The generated :class:`GmshGrid`.
boundary_info
The generated :class:`GmshBoundaryInfo`.
"""
logger = getLogger("pymor.grids.gmsh.load_gmsh")
logger.info("Parsing gmsh file ...")
tic = time.time()
sections = _parse_gmsh_file(gmsh_file)
toc = time.time()
t_parse = toc - tic
logger.info("Create GmshGrid ...")
tic = time.time()
grid = GmshGrid(sections)
toc = time.time()
t_grid = toc - tic
logger.info("Create GmshBoundaryInfo ...")
tic = time.time()
bi = GmshBoundaryInfo(grid, sections)
toc = time.time()
t_bi = toc - tic
logger.info(
"Parsing took {} s; Grid creation took {} s; BoundaryInfo creation took {} s".format(t_parse, t_grid, t_bi)
)
return grid, bi
def new_parallel_pool(ipython_num_engines=None, ipython_profile=None, allow_mpi=True):
"""Creates a new default |WorkerPool|.
If `ipython_num_engines` or `ipython_profile` is provided as an argument or set as
a |default|, an :class:`~pymor.parallel.ipython.IPythonPool` |WorkerPool| will
be created using the given parameters via the `ipcluster` script.
Otherwise, when `allow_mpi` is `True` and an MPI parallel run is detected,
an :class:`~pymor.parallel.mpi.MPIPool` |WorkerPool| will be created.
Otherwise, a sequential run is assumed and
:attr:`pymor.parallel.dummy.dummy_pool <pymor.parallel.dummy.DummyPool>`
is returned.
"""
global _pool
if _pool:
logger = getLogger('pymor.parallel.default.new_parallel_pool')
logger.warn('new_parallel_pool already called; returning old pool (this might not be what you want).')
return _pool[1]
if ipython_num_engines or ipython_profile:
from pymor.parallel.ipython import new_ipcluster_pool
nip = new_ipcluster_pool(profile=ipython_profile, num_engines=ipython_num_engines)
pool = nip.__enter__()
_pool = ('ipython', pool, nip)
return pool
elif allow_mpi:
from pymor.tools import mpi
if mpi.parallel:
from pymor.parallel.mpi import MPIPool
pool = MPIPool()
_pool = ('mpi', pool)
return pool
else:
_pool = ('dummy', dummy_pool)
return dummy_pool
else:
_pool = ('dummy', dummy_pool)
return dummy_pool
def __init__(cls, name, bases, namespace):
'''Metaclass of :class:`BasicInterface`.
I tell base classes when I derive a new class from them. I publish
a new contract type for each new class I create. I create a logger
for each class I create. I add an `init_args` attribute to the class.
'''
# monkey a new contract into the decorator module so checking for that type at runtime can work
if HAVE_CONTRACTS:
dname = (cls.__module__ + '.' + name).replace('__main__.', 'main.').replace('.', '_')
if not dname in decorators.__dict__:
decorators.__dict__[dname] = contracts.new_contract(dname, lambda x: isinstance(x, cls))
# all bases except object get the derived class' name appended
for base in [b for b in bases if b != object]:
derived = cls
# mangle the name to the base scope
attribute = '_%s__implementors' % base.__name__
if hasattr(base, attribute):
getattr(base, attribute).append(derived)
else:
setattr(base, attribute, [derived])
cls._logger = logger.getLogger('{}.{}'.format(cls.__module__.replace('__main__', 'pymor'), name))
abc.ABCMeta.__init__(cls, name, bases, namespace)
def onerror(name):
from pymor.core.logger import getLogger
logger = getLogger('pymor.core.defaults._import_all')
logger.warn('Failed to import ' + name)
def reduction_error_analysis(reduced_discretization,
discretization=None,
reconstructor=None,
test_mus=10,
basis_sizes=0,
random_seed=None,
estimator=True,
condition=False,
error_norms=(),
error_norm_names=None,
estimator_norm_index=0,
custom=(),
plot=False,
plot_custom_logarithmic=True,
pool=dummy_pool):
"""Analyze the model reduction error.
The maximum model reduction error is estimated by solving the reduced
|Discretization| for given random |Parameters|.
Parameters
----------
reduced_discretization
The reduced |Discretization|.
discretization
The high-dimensional |Discretization|. Must be specified if
`error_norms` are given.
reconstructor
The reconstructor for `reduced_discretization`. Must be specified
if `error_norms` are given.
test_mus
Either a list of |Parameters| to compute the errors for, or
the number of parameters which are sampled randomly from
`parameter_space` (if given) or `reduced_discretization.parameter_space`.
basis_sizes
Either a list of reduced basis dimensions to consider, or
the number of dimensions (which are then selected equidistantly,
always including the maximum reduced space dimension).
The dimensions are input for `~pymor.reductors.basic.reduce_to_subbasis`
to quickly obtain smaller reduced |Discretizations| from
`rb_discretization`.
random_seed
If `test_mus` is a number, use this value as random seed
for drawing the |Parameters|.
estimator
If `True` evalute the error estimator of `reduced_discretization`
on the test |Parameters|.
condition
If `True`, compute the condition of the reduced system matrix
for the given test |Parameters|. (Can only be specified if
`rb_discretization` is an instance of |StationaryDiscretization|
and `rb_discretization.operator` is linear.
error_norms
List of norms in which to compute the model reduction error.
error_norm_names
Names of the norms given by `error_norms`. If `None`, the
`name` attributes of the given norms are used.
estimator_norm_index
When `estimator` is `True` and `error_norms` are specified,
this is the index of the norm in `error_norms` w.r.t. which
to compute the effectivity of the estimator.
custom
List of custom functions which are evaluated for each test |Parameter|
and basis size. The function must have the signature ::
def custom_value(reduced_discretization, discretization=d,
reconstructor, mu, dim):
pass
plot
If `True`, generate a plot of the computed quantities w.r.t.
the basis size.
plot_custom_logarithmic
If `True`, use a logarithmic y-axis to plot the computed custom
values.
pool
If not `None`, the |WorkerPool| to use for parallelization.
Returns
-------
Dict with the following fields:
:mus: The test |Parameters| which have been considered.
:basis_sizes: The reduced basis dimensions which have been considered.
:norms: |Array| of the norms of the high-dimensional solutions
w.r.t. all given test |Parameters|, reduced basis
dimensions and norms in `error_norms`.
(Only present when `error_norms` has been specified.)
:max_norms: Maxima of `norms` over the given test |Parameters|.
:max_norm_mus: |Parameters| corresponding to `max_norms`.
:errors: |Array| of the norms of the model reduction errors
w.r.t. all given test |Parameters|, reduced basis
dimensions and norms in `error_norms`.
(Only present when `error_norms` has been specified.)
:max_errors: Maxima of `errors` over the given test |Parameters|.
:max_error_mus: |Parameters| corresponding to `max_errors`.
:rel_errors: `errors` divided by `norms`.
(Only present when `error_norms` has been specified.)
:max_rel_errors: Maxima of `rel_errors` over the given test |Parameters|.
:max_rel_error_mus: |Parameters| corresponding to `max_rel_errors`.
:error_norm_names: Names of the the given `error_norms`.
(Only present when `error_norms` has been specified.)
:estimates: |Array| of the model reduction error estimates
w.r.t. all given test |Parameters| and reduced basis
dimensions.
(Only present when `estimator` is `True`.)
:max_estimate: Maxima of `estimates` over the given test |Parameters|.
:max_estimate_mus: |Parameters| corresponding to `max_estimates`.
:effectivities: `errors` divided by `estimates`.
(Only present when `estimator` is `True` and `error_norms`
has been specified.)
:min_effectivities: Minima of `effectivities` over the given test |Parameters|.
:min_effectivity_mus: |Parameters| corresponding to `min_effectivities`.
:max_effectivities: Maxima of `effectivities` over the given test |Parameters|.
:max_effectivity_mus: |Parameters| corresponding to `max_effectivities`.
:errors: |Array| of the reduced system matrix conditions
w.r.t. all given test |Parameters| and reduced basis
dimensions.
(Only present when `conditions` is `True`.)
:max_conditions: Maxima of `conditions` over the given test |Parameters|.
:max_condition_mus: |Parameters| corresponding to `max_conditions`.
:custom_values: |Array| of custom function evaluations
w.r.t. all given test |Parameters|, reduced basis
dimensions and functions in `custom`.
(Only present when `custom` has been specified.)
:max_custom_values: Maxima of `custom_values` over the given test |Parameters|.
:max_custom_values_mus: |Parameters| corresponding to `max_custom_values`.
:time: Time (in seconds) needed for the error analysis.
:summary: String containing a summary of all computed quantities for
the largest (last) considered basis size.
:figure: The figure containing the generated plots.
(Only present when `plot` is `True`.)
"""
assert not error_norms or (discretization and reconstructor)
assert error_norm_names is None or len(error_norm_names) == len(
error_norms)
assert not condition \
or isinstance(reduced_discretization, StationaryDiscretization) and reduced_discretization.operator.linear
logger = getLogger('pymor.algorithms.error')
if pool is None or pool is dummy_pool:
pool = dummy_pool
else:
logger.info('Using pool of {} workers for error analysis'.format(
len(pool)))
tic = time.time()
d, rd, rc = discretization, reduced_discretization, reconstructor
if isinstance(test_mus, Number):
test_mus = reduced_discretization.parameter_space.sample_randomly(
test_mus, seed=random_seed)
if isinstance(basis_sizes, Number):
if basis_sizes == 1:
basis_sizes = [rd.solution_space.dim]
else:
if basis_sizes == 0:
basis_sizes = rd.solution_space.dim + 1
basis_sizes = min(rd.solution_space.dim + 1, basis_sizes)
basis_sizes = np.linspace(0, rd.solution_space.dim,
basis_sizes).astype(int)
if error_norm_names is None:
error_norm_names = tuple(norm.name for norm in error_norms)
norms, estimates, errors, conditions, custom_values = \
list(zip(*pool.map(_compute_errors, test_mus, d=d, rd=rd, rc=rc, estimator=estimator,
error_norms=error_norms, condition=condition, custom=custom, basis_sizes=basis_sizes)))
print()
result = {}
result['mus'] = test_mus = np.array(test_mus)
result['basis_sizes'] = basis_sizes
summary = []
summary.append(('number of samples', str(len(test_mus))))
if error_norms:
result['norms'] = norms = np.array(norms)
result['max_norms'] = max_norms = np.max(norms, axis=0)
result['max_norm_mus'] = max_norm_mus = test_mus[np.argmax(norms,
axis=0)]
result['errors'] = errors = np.array(errors)
result['max_errors'] = max_errors = np.max(errors, axis=0)
result['max_error_mus'] = max_error_mus = test_mus[np.argmax(errors,
axis=0)]
result['rel_errors'] = rel_errors = errors / norms[:, :, np.newaxis]
result['max_rel_errors'] = np.max(rel_errors, axis=0)
result['max_rel_error_mus'] = test_mus[np.argmax(rel_errors, axis=0)]
for name, norm, norm_mu, error, error_mu in zip(
error_norm_names, max_norms, max_norm_mus, max_errors[:, -1],
max_error_mus[:, -1]):
summary.append(('maximum {}-norm'.format(name),
'{:.7e} (mu = {})'.format(norm, error_mu)))
summary.append(('maximum {}-error'.format(name),
'{:.7e} (mu = {})'.format(error, error_mu)))
result['error_norm_names'] = error_norm_names
if estimator:
result['estimates'] = estimates = np.array(estimates)
result['max_estimates'] = max_estimates = np.max(estimates, axis=0)
result['max_estimate_mus'] = max_estimate_mus = test_mus[np.argmax(
estimates, axis=0)]
summary.append(('maximum estimated error',
'{:.7e} (mu = {})'.format(max_estimates[-1],
max_estimate_mus[-1])))
if estimator and error_norms:
result[
'effectivities'] = effectivities = errors[:,
estimator_norm_index, :] / estimates
result['max_effectivities'] = max_effectivities = np.max(effectivities,
axis=0)
result['max_effectivity_mus'] = max_effectivity_mus = test_mus[
np.argmax(effectivities, axis=0)]
result['min_effectivities'] = min_effectivities = np.min(effectivities,
axis=0)
result['min_effectivity_mus'] = min_effectivity_mus = test_mus[
np.argmin(effectivities, axis=0)]
summary.append(('minimum estimator effectivity',
'{:.7e} (mu = {})'.format(min_effectivities[-1],
min_effectivity_mus[-1])))
summary.append(('maximum estimator effectivity',
'{:.7e} (mu = {})'.format(max_effectivities[-1],
max_effectivity_mus[-1])))
if condition:
result['conditions'] = conditions = np.array(conditions)
result['max_conditions'] = max_conditions = np.max(conditions, axis=0)
result['max_condition_mus'] = max_condition_mus = test_mus[np.argmax(
conditions, axis=0)]
summary.append(('maximum system matrix condition',
'{:.7e} (mu = {})'.format(max_conditions[-1],
max_condition_mus[-1])))
if custom:
result['custom_values'] = custom_values = np.array(custom_values)
result['max_custom_values'] = max_custom_values = np.max(custom_values,
axis=0)
result['max_custom_values_mus'] = max_custom_values_mus = test_mus[
np.argmax(custom_values, axis=0)]
for i, (value, mu) in enumerate(
zip(max_custom_values[:, -1], max_custom_values_mus[:, -1])):
summary.append(('maximum custom value {}'.format(i),
'{:.7e} (mu = {})'.format(value, mu)))
toc = time.time()
result['time'] = toc - tic
summary.append(('elapsed time', str(toc - tic)))
summary_fields, summary_values = list(zip(*summary))
summary_field_width = np.max(list(map(len, summary_fields))) + 2
summary_lines = [
' {:{}} {}'.format(field + ':', summary_field_width, value)
for field, value in zip(summary_fields, summary_values)
]
summary = 'Stochastic error estimation:\n' + '\n'.join(summary_lines)
result['summary'] = summary
if plot:
import matplotlib.pyplot as plt
fig = plt.figure()
num_plots = (int(bool(error_norms) or estimator) +
int(bool(error_norms) and estimator) + int(condition) +
int(bool(custom)))
current_plot = 1
if bool(error_norms) or estimator:
ax = fig.add_subplot(1, num_plots, current_plot)
legend = []
if error_norms:
for name, errors in zip(error_norm_names, max_errors):
ax.semilogy(basis_sizes, errors)
legend.append(name)
if estimator:
ax.semilogy(basis_sizes, max_estimates)
legend.append('estimator')
ax.legend(legend)
ax.set_title('maximum errors')
current_plot += 1
if bool(error_norms) and estimator:
ax = fig.add_subplot(1, num_plots, current_plot)
ax.semilogy(basis_sizes, min_effectivities)
ax.semilogy(basis_sizes, max_effectivities)
ax.legend(('min', 'max'))
ax.set_title('estimator effectivities')
current_plot += 1
if condition:
ax = fig.add_subplot(1, num_plots, current_plot)
ax.semilogy(basis_sizes, max_conditions)
ax.set_title('maximum condition')
current_plot += 1
if custom:
ax = fig.add_subplot(1, num_plots, current_plot)
legend = []
for i, values in enumerate(custom_values):
if plot_custom_logarithmic:
ax.semilogy(basis_sizes, values)
else:
ax.plot(basis_sizes, values)
legend.append('value ' + str(i))
ax.legend(legend)
ax.set_title('maximum custom values')
current_plot += 1
result['figure'] = fig
return result
def solve_ricc_lrcf(A,
E,
B,
C,
R=None,
S=None,
trans=False,
options=None,
default_solver=None):
"""Compute an approximate low-rank solution of a Riccati equation.
See :func:`pymor.algorithms.riccati.solve_ricc_lrcf` for a
general description.
This function uses `pymess.dense_nm_gmpcare` and `pymess.lrnm`.
For both methods,
:meth:`~pymor.vectorarrays.interfaces.VectorArrayInterface.to_numpy`
and
:meth:`~pymor.vectorarrays.interfaces.VectorSpaceInterface.from_numpy`
need to be implemented for `A.source`.
Additionally, since `dense_nm_gmpcare` is a dense solver, it
expects :func:`~pymor.algorithms.to_matrix.to_matrix` to work
for A and E.
If the solver is not specified using the options or
default_solver arguments, `dense_nm_gmpcare` is used for small
problems (smaller than defined with
:func:`~pymor.algorithms.lyapunov.mat_eqn_sparse_min_size`) and
`lrnm` for large problems.
Parameters
----------
A
The non-parametric |Operator| A.
E
The non-parametric |Operator| E or `None`.
B
The operator B as a |VectorArray| from `A.source`.
C
The operator C as a |VectorArray| from `A.source`.
R
The operator R as a 2D |NumPy array| or `None`.
S
The operator S as a |VectorArray| from `A.source` or `None`.
trans
Whether the first |Operator| in the Riccati equation is
transposed.
options
The solver options to use (see
:func:`ricc_lrcf_solver_options`).
default_solver
Default solver to use (pymess_lrnm,
pymess_dense_nm_gmpcare).
If `None`, chose solver depending on dimension `A`.
Returns
-------
Z
Low-rank Cholesky factor of the Riccati equation solution,
|VectorArray| from `A.source`.
"""
_solve_ricc_check_args(A, E, B, C, R, S, trans)
if default_solver is None:
default_solver = 'pymess_lrnm' if A.source.dim >= mat_eqn_sparse_min_size(
) else 'pymess_dense_nm_gmpcare'
options = _parse_options(options, ricc_lrcf_solver_options(),
default_solver, None, False)
if options['type'] == 'pymess_dense_nm_gmpcare':
X = _call_pymess_dense_nm_gmpare(A,
E,
B,
C,
R,
S,
trans=trans,
options=options['opts'],
plus=False,
method_name='solve_ricc_lrcf')
Z = _chol(X)
elif options['type'] == 'pymess_lrnm':
if S is not None:
raise NotImplementedError
if R is not None:
import scipy.linalg as spla
Rc = spla.cholesky(R) # R = Rc^T * Rc
Rci = spla.solve_triangular(Rc, np.eye(
Rc.shape[0])) # R^{-1} = Rci * Rci^T
if not trans:
C = C.lincomb(Rci.T) # C <- Rci^T * C = (C^T * Rci)^T
else:
B = B.lincomb(Rci.T) # B <- B * Rci
opts = options['opts']
opts.type = pymess.MESS_OP_NONE if not trans else pymess.MESS_OP_TRANSPOSE
eqn = RiccatiEquation(opts, A, E, B, C)
Z, status = pymess.lrnm(eqn, opts)
relres = status.res2_norm / status.res2_0
if relres > opts.adi.res2_tol:
logger = getLogger('pymor.bindings.pymess.solve_ricc_lrcf')
logger.warning(
f'Desired relative residual tolerance was not achieved '
f'({relres:e} > {opts.adi.res2_tol:e}).')
else:
raise ValueError(
f'Unexpected Riccati equation solver ({options["type"]}).')
return A.source.from_numpy(Z.T)
def solve_lyap_lrcf(A, E, B, trans=False, options=None):
"""Compute an approximate low-rank solution of a Lyapunov equation.
See :func:`pymor.algorithms.lyapunov.solve_lyap_lrcf` for a
general description.
This function uses the low-rank ADI iteration as described in
Algorithm 4.3 in [PK16]_.
Parameters
----------
A
The non-parametric |Operator| A.
E
The non-parametric |Operator| E or `None`.
B
The operator B as a |VectorArray| from `A.source`.
trans
Whether the first |Operator| in the Lyapunov equation is
transposed.
options
The solver options to use (see
:func:`lyap_lrcf_solver_options`).
Returns
-------
Z
Low-rank Cholesky factor of the Lyapunov equation solution,
|VectorArray| from `A.source`.
"""
_solve_lyap_lrcf_check_args(A, E, B, trans)
options = _parse_options(options, lyap_lrcf_solver_options(), 'lradi',
None, False)
logger = getLogger('pymor.algorithms.lradi.solve_lyap_lrcf')
shift_options = options['shift_options'][options['shifts']]
if shift_options['type'] == 'projection_shifts':
init_shifts = projection_shifts_init
iteration_shifts = projection_shifts
else:
raise ValueError('Unknown lradi shift strategy.')
if E is None:
E = IdentityOperator(A.source)
Z = A.source.empty(reserve=len(B) * options['maxiter'])
W = B.copy()
j = 0
j_shift = 0
shifts = init_shifts(A, E, W, shift_options)
res = np.linalg.norm(W.gramian(), ord=2)
init_res = res
Btol = res * options['tol']
while res > Btol and j < options['maxiter']:
if shifts[j_shift].imag == 0:
AaE = A + shifts[j_shift].real * E
if not trans:
V = AaE.apply_inverse(W)
W -= E.apply(V) * (2 * shifts[j_shift].real)
else:
V = AaE.apply_inverse_adjoint(W)
W -= E.apply_adjoint(V) * (2 * shifts[j_shift].real)
Z.append(V * np.sqrt(-2 * shifts[j_shift].real))
j += 1
else:
AaE = A + shifts[j_shift] * E
gs = -4 * shifts[j_shift].real
d = shifts[j_shift].real / shifts[j_shift].imag
if not trans:
V = AaE.apply_inverse(W)
W += E.apply(V.real + V.imag * d) * gs
else:
V = AaE.apply_inverse_adjoint(W).conj()
W += E.apply_adjoint(V.real + V.imag * d) * gs
g = np.sqrt(gs)
Z.append((V.real + V.imag * d) * g)
Z.append(V.imag * (g * np.sqrt(d**2 + 1)))
j += 2
j_shift += 1
res = np.linalg.norm(W.gramian(), ord=2)
logger.info(f'Relative residual at step {j}: {res/init_res:.5e}')
if j_shift >= shifts.size:
shifts = iteration_shifts(A, E, V, shifts)
j_shift = 0
if res > Btol:
logger.warning(
f'Prescribed relative residual tolerance was not achieved '
f'({res/init_res:e} > {options["tol"]:e}) after '
f'{options["maxiter"]} ADI steps.')
return Z
def visualize_patch(grid, U, bounding_box=([0, 0], [1, 1]), codim=2, title=None, legend=None,
separate_colorbars=False, rescale_colorbars=False, backend='gl', block=False, columns=2):
"""Visualize scalar data associated to a two-dimensional |Grid| as a patch plot.
The grid's |ReferenceElement| must be the triangle or square. The data can either
be attached to the faces or vertices of the grid.
Parameters
----------
grid
The underlying |Grid|.
U
|VectorArray| of the data to visualize. If `len(U) > 1`, the data is visualized
as a time series of plots. Alternatively, a tuple of |VectorArrays| can be
provided, in which case a subplot is created for each entry of the tuple. The
lengths of all arrays have to agree.
bounding_box
A bounding box in which the grid is contained.
codim
The codimension of the entities the data in `U` is attached to (either 0 or 2).
title
Title of the plot.
legend
Description of the data that is plotted. Most useful if `U` is a tuple in which
case `legend` has to be a tuple of strings of the same length.
separate_colorbars
If `True`, use separate colorbars for each subplot.
rescale_colorbars
If `True`, rescale colorbars to data in each frame.
backend
Plot backend to use ('gl' or 'matplotlib').
block
If `True`, block execution until the plot window is closed.
columns
The number of columns in the visualizer GUI in case multiple plots are displayed
at the same time.
"""
if not config.HAVE_QT:
raise QtMissing()
assert backend in {'gl', 'matplotlib'}
if backend == 'gl':
if not config.HAVE_GL:
logger = getLogger('pymor.discretizers.builtin.gui.qt.visualize_patch')
logger.warning('import of PyOpenGL failed, falling back to matplotlib; rendering will be slow')
backend = 'matplotlib'
elif not config.HAVE_QTOPENGL:
logger = getLogger('pymor.discretizers.builtin.gui.qt.visualize_patch')
logger.warning('import of Qt.QtOpenGL failed, falling back to matplotlib; rendering will be slow')
backend = 'matplotlib'
if backend == 'matplotlib' and not config.HAVE_MATPLOTLIB:
raise ImportError('cannot visualize: import of matplotlib failed')
else:
if not config.HAVE_MATPLOTLIB:
raise ImportError('cannot visualize: import of matplotlib failed')
# TODO extract class
class MainWindow(PlotMainWindow):
def __init__(self, grid, U, bounding_box, codim, title, legend, separate_colorbars, rescale_colorbars, backend):
assert isinstance(U, VectorArray) \
or (isinstance(U, tuple) and all(isinstance(u, VectorArray) for u in U)
and all(len(u) == len(U[0]) for u in U))
U = (U.to_numpy().astype(np.float64, copy=False),) if isinstance(U, VectorArray) else \
tuple(u.to_numpy().astype(np.float64, copy=False) for u in U)
if isinstance(legend, str):
legend = (legend,)
assert legend is None or isinstance(legend, tuple) and len(legend) == len(U)
if backend == 'gl':
widget = GLPatchWidget
cbar_widget = ColorBarWidget
else:
widget = MatplotlibPatchWidget
cbar_widget = None
if not separate_colorbars and len(U) > 1:
l = getLogger('pymor.discretizers.builtin.gui.qt.visualize_patch')
l.warning('separate_colorbars=False not supported for matplotlib backend')
separate_colorbars = True
class PlotWidget(QWidget):
def __init__(self):
super().__init__()
if separate_colorbars:
if rescale_colorbars:
self.vmins = tuple(np.min(u[0]) for u in U)
self.vmaxs = tuple(np.max(u[0]) for u in U)
else:
self.vmins = tuple(np.min(u) for u in U)
self.vmaxs = tuple(np.max(u) for u in U)
else:
if rescale_colorbars:
self.vmins = (min(np.min(u[0]) for u in U),) * len(U)
self.vmaxs = (max(np.max(u[0]) for u in U),) * len(U)
else:
self.vmins = (min(np.min(u) for u in U),) * len(U)
self.vmaxs = (max(np.max(u) for u in U),) * len(U)
layout = QHBoxLayout()
plot_layout = QGridLayout()
self.colorbarwidgets = [cbar_widget(self, vmin=vmin, vmax=vmax) if cbar_widget else None
for vmin, vmax in zip(self.vmins, self.vmaxs)]
plots = [widget(self, grid, vmin=vmin, vmax=vmax, bounding_box=bounding_box, codim=codim)
for vmin, vmax in zip(self.vmins, self.vmaxs)]
if legend:
for i, plot, colorbar, l in zip(range(len(plots)), plots, self.colorbarwidgets, legend):
subplot_layout = QVBoxLayout()
caption = QLabel(l)
caption.setAlignment(Qt.AlignHCenter)
subplot_layout.addWidget(caption)
if not separate_colorbars or backend == 'matplotlib':
subplot_layout.addWidget(plot)
else:
hlayout = QHBoxLayout()
hlayout.addWidget(plot)
if colorbar:
hlayout.addWidget(colorbar)
subplot_layout.addLayout(hlayout)
plot_layout.addLayout(subplot_layout, int(i/columns), (i % columns), 1, 1)
else:
for i, plot, colorbar in zip(range(len(plots)), plots, self.colorbarwidgets):
if not separate_colorbars or backend == 'matplotlib':
plot_layout.addWidget(plot, int(i/columns), (i % columns), 1, 1)
else:
hlayout = QHBoxLayout()
hlayout.addWidget(plot)
if colorbar:
hlayout.addWidget(colorbar)
plot_layout.addLayout(hlayout, int(i/columns), (i % columns), 1, 1)
layout.addLayout(plot_layout)
if not separate_colorbars:
layout.addWidget(self.colorbarwidgets[0])
for w in self.colorbarwidgets[1:]:
w.setVisible(False)
self.setLayout(layout)
self.plots = plots
def set(self, U, ind):
if rescale_colorbars:
if separate_colorbars:
self.vmins = tuple(np.min(u[ind]) for u in U)
self.vmaxs = tuple(np.max(u[ind]) for u in U)
else:
self.vmins = (min(np.min(u[ind]) for u in U),) * len(U)
self.vmaxs = (max(np.max(u[ind]) for u in U),) * len(U)
for u, plot, colorbar, vmin, vmax in zip(U, self.plots, self.colorbarwidgets, self.vmins,
self.vmaxs):
plot.set(u[ind], vmin=vmin, vmax=vmax)
if colorbar:
colorbar.set(vmin=vmin, vmax=vmax)
super().__init__(U, PlotWidget(), title=title, length=len(U[0]))
self.grid = grid
self.codim = codim
def save(self):
if not config.HAVE_PYEVTK:
msg = QMessageBox(QMessageBox.Critical, 'Error', 'VTK output disabled. Pleas install pyvtk.')
msg.exec_()
return
filename = QFileDialog.getSaveFileName(self, 'Save as vtk file')[0]
base_name = filename.split('.vtu')[0].split('.vtk')[0].split('.pvd')[0]
if base_name:
if len(self.U) == 1:
write_vtk(self.grid, NumpyVectorSpace.make_array(self.U[0]), base_name, codim=self.codim)
else:
for i, u in enumerate(self.U):
write_vtk(self.grid, NumpyVectorSpace.make_array(u), f'{base_name}-{i}',
codim=self.codim)
_launch_qt_app(lambda: MainWindow(grid, U, bounding_box, codim, title=title, legend=legend,
separate_colorbars=separate_colorbars, rescale_colorbars=rescale_colorbars,
backend=backend),
block)
class TestInterface(object):
logger = logger.getLogger(__name__)
def solve_ricc_lrcf(A, E, B, C, R=None, S=None, trans=False, options=None):
"""Compute an approximate low-rank solution of a Riccati equation.
See :func:`pymor.algorithms.riccati.solve_ricc_lrcf` for a
general description.
This function is an implementation of Algorithm 2 in [BBKS18]_.
Parameters
----------
A
The |Operator| A.
E
The |Operator| E or `None`.
B
The operator B as a |VectorArray| from `A.source`.
C
The operator C as a |VectorArray| from `A.source`.
R
The operator R as a 2D |NumPy array| or `None`.
S
The operator S as a |VectorArray| from `A.source` or `None`.
trans
Whether the first |Operator| in the Riccati equation is
transposed.
options
The solver options to use. (see
:func:`ricc_lrcf_solver_options`)
Returns
-------
Z
Low-rank Cholesky factor of the Riccati equation solution,
|VectorArray| from `A.source`.
"""
_solve_ricc_check_args(A, E, B, C, None, None, trans)
options = _parse_options(options, ricc_lrcf_solver_options(), 'lrradi',
None, False)
logger = getLogger('pymor.algorithms.lrradi.solve_ricc_lrcf')
shift_options = options['shift_options'][options['shifts']]
if shift_options['type'] == 'hamiltonian_shifts':
init_shifts = hamiltonian_shifts_init
iteration_shifts = hamiltonian_shifts
else:
raise ValueError('Unknown lrradi shift strategy.')
if E is None:
E = IdentityOperator(A.source)
if S is not None:
raise NotImplementedError
if R is not None:
Rc = spla.cholesky(R) # R = Rc^T * Rc
Rci = spla.solve_triangular(Rc, np.eye(
Rc.shape[0])) # R^{-1} = Rci * Rci^T
if not trans:
C = C.lincomb(Rci.T) # C <- Rci^T * C = (C^T * Rci)^T
else:
B = B.lincomb(Rci.T) # B <- B * Rci
if not trans:
B, C = C, B
Z = A.source.empty(reserve=len(C) * options['maxiter'])
Y = np.empty((0, 0))
K = A.source.zeros(len(B))
RF = C.copy()
j = 0
j_shift = 0
shifts = init_shifts(A, E, B, C, shift_options)
res = np.linalg.norm(RF.gramian(), ord=2)
init_res = res
Ctol = res * options['tol']
while res > Ctol and j < options['maxiter']:
if not trans:
AsE = A + shifts[j_shift] * E
else:
AsE = A + np.conj(shifts[j_shift]) * E
if j == 0:
if not trans:
V = AsE.apply_inverse(RF) * np.sqrt(-2 * shifts[j_shift].real)
else:
V = AsE.apply_inverse_adjoint(RF) * np.sqrt(
-2 * shifts[j_shift].real)
else:
if not trans:
LN = AsE.apply_inverse(cat_arrays([RF, K]))
else:
LN = AsE.apply_inverse_adjoint(cat_arrays([RF, K]))
L = LN[:len(RF)]
N = LN[-len(K):]
ImBN = np.eye(len(K)) - B.dot(N)
ImBNKL = spla.solve(ImBN, B.dot(L))
V = (L + N.lincomb(ImBNKL.T)) * np.sqrt(-2 * shifts[j_shift].real)
if np.imag(shifts[j_shift]) == 0:
Z.append(V)
VB = V.dot(B)
Yt = np.eye(len(C)) - (VB @ VB.T) / (2 * shifts[j_shift].real)
Y = spla.block_diag(Y, Yt)
if not trans:
EVYt = E.apply(V).lincomb(np.linalg.inv(Yt))
else:
EVYt = E.apply_adjoint(V).lincomb(np.linalg.inv(Yt))
RF.axpy(np.sqrt(-2 * shifts[j_shift].real), EVYt)
K += EVYt.lincomb(VB.T)
j += 1
else:
Z.append(V.real)
Z.append(V.imag)
Vr = V.real.dot(B)
Vi = V.imag.dot(B)
sa = np.abs(shifts[j_shift])
F1 = np.vstack((-shifts[j_shift].real / sa * Vr -
shifts[j_shift].imag / sa * Vi,
shifts[j_shift].imag / sa * Vr -
shifts[j_shift].real / sa * Vi))
F2 = np.vstack((Vr, Vi))
F3 = np.vstack((shifts[j_shift].imag / sa * np.eye(len(C)),
shifts[j_shift].real / sa * np.eye(len(C))))
Yt = spla.block_diag(np.eye(len(C)), 0.5 * np.eye(len(C))) \
- (F1 @ F1.T) / (4 * shifts[j_shift].real) \
- (F2 @ F2.T) / (4 * shifts[j_shift].real) \
- (F3 @ F3.T) / 2
Y = spla.block_diag(Y, Yt)
EVYt = E.apply(cat_arrays([V.real,
V.imag])).lincomb(np.linalg.inv(Yt))
RF.axpy(np.sqrt(-2 * shifts[j_shift].real), EVYt[:len(C)])
K += EVYt.lincomb(F2.T)
j += 2
j_shift += 1
res = np.linalg.norm(RF.gramian(), ord=2)
logger.info(f'Relative residual at step {j}: {res/init_res:.5e}')
if j_shift >= shifts.size:
shifts = iteration_shifts(A, E, B, RF, K, Z, shift_options)
j_shift = 0
# transform solution to lrcf
cf = spla.cholesky(Y)
Z_cf = Z.lincomb(spla.solve_triangular(cf, np.eye(len(Z))).T)
return Z_cf
def apply_inverse(self, V, mu=None, initial_guess=None, least_squares=False,
check_finite=True, default_sparse_solver_backend='scipy'):
"""Apply the inverse operator.
Parameters
----------
V
|VectorArray| of vectors to which the inverse operator is applied.
mu
The |parameter values| for which to evaluate the inverse operator.
initial_guess
|VectorArray| with the same length as `V` containing initial guesses
for the solution. Some implementations of `apply_inverse` may
ignore this parameter. If `None` a solver-dependent default is used.
least_squares
If `True`, solve the least squares problem::
u = argmin ||op(u) - v||_2.
Since for an invertible operator the least squares solution agrees
with the result of the application of the inverse operator,
setting this option should, in general, have no effect on the result
for those operators. However, note that when no appropriate
|solver_options| are set for the operator, most implementations
will choose a least squares solver by default which may be
undesirable.
check_finite
Test if solution only contains finite values.
default_sparse_solver_backend
Default sparse solver backend to use (scipy, pyamg, generic).
Returns
-------
|VectorArray| of the inverse operator evaluations.
Raises
------
InversionError
The operator could not be inverted.
"""
assert V in self.range
assert initial_guess is None or initial_guess in self.source and len(initial_guess) == len(V)
if V.dim == 0:
if self.source.dim == 0 or least_squares:
return self.source.make_array(np.zeros((len(V), self.source.dim)))
else:
raise InversionError
if self.source.dim != self.range.dim and not least_squares:
raise InversionError
options = self.solver_options.get('inverse') if self.solver_options else None
assert self.sparse or not options
if self.sparse:
if options:
solver = options if isinstance(options, str) else options['type']
backend = solver.split('_')[0]
else:
backend = default_sparse_solver_backend
if backend == 'scipy':
from pymor.bindings.scipy import apply_inverse as apply_inverse_impl
elif backend == 'pyamg':
if not config.HAVE_PYAMG:
raise RuntimeError('PyAMG support not enabled.')
from pymor.bindings.pyamg import apply_inverse as apply_inverse_impl
elif backend == 'generic':
logger = getLogger('pymor.bindings.scipy.scipy_apply_inverse')
logger.warning('You have selected a (potentially slow) generic solver for a NumPy matrix operator!')
from pymor.algorithms.genericsolvers import apply_inverse as apply_inverse_impl
else:
raise NotImplementedError
return apply_inverse_impl(self, V, initial_guess=initial_guess, options=options,
least_squares=least_squares, check_finite=check_finite)
else:
if least_squares:
try:
R, _, _, _ = np.linalg.lstsq(self.matrix, V.to_numpy().T)
except np.linalg.LinAlgError as e:
raise InversionError(f'{str(type(e))}: {str(e)}')
R = R.T
else:
try:
R = solve(self.matrix, V.to_numpy().T).T
except np.linalg.LinAlgError as e:
raise InversionError(f'{str(type(e))}: {str(e)}')
if check_finite:
if not np.isfinite(np.sum(R)):
raise InversionError('Result contains non-finite values')
return self.source.make_array(R)
def visualize_patch(grid, U, bounding_box=([0, 0], [1, 1]), codim=2, title=None, legend=None,
separate_colorbars=False, rescale_colorbars=False, backend='gl', block=False, columns=2):
"""Visualize scalar data associated to a two-dimensional |Grid| as a patch plot.
The grid's |ReferenceElement| must be the triangle or square. The data can either
be attached to the faces or vertices of the grid.
Parameters
----------
grid
The underlying |Grid|.
U
|VectorArray| of the data to visualize. If `len(U) > 1`, the data is visualized
as a time series of plots. Alternatively, a tuple of |VectorArrays| can be
provided, in which case a subplot is created for each entry of the tuple. The
lengths of all arrays have to agree.
bounding_box
A bounding box in which the grid is contained.
codim
The codimension of the entities the data in `U` is attached to (either 0 or 2).
title
Title of the plot.
legend
Description of the data that is plotted. Most useful if `U` is a tuple in which
case `legend` has to be a tuple of strings of the same length.
separate_colorbars
If `True`, use separate colorbars for each subplot.
rescale_colorbars
If `True`, rescale colorbars to data in each frame.
backend
Plot backend to use ('gl' or 'matplotlib').
block
If `True`, block execution until the plot window is closed.
columns
The number of columns in the visualizer GUI in case multiple plots are displayed
at the same time.
"""
if not config.HAVE_QT:
raise QtMissing()
assert backend in {'gl', 'matplotlib'}
if backend == 'gl':
if not config.HAVE_GL:
logger = getLogger('pymor.gui.qt.visualize_patch')
logger.warning('import of PyOpenGL failed, falling back to matplotlib; rendering will be slow')
backend = 'matplotlib'
elif not config.HAVE_QTOPENGL:
logger = getLogger('pymor.gui.qt.visualize_patch')
logger.warning('import of Qt.QtOpenGL failed, falling back to matplotlib; rendering will be slow')
backend = 'matplotlib'
if backend == 'matplotlib' and not config.HAVE_MATPLOTLIB:
raise ImportError('cannot visualize: import of matplotlib failed')
else:
if not config.HAVE_MATPLOTLIB:
raise ImportError('cannot visualize: import of matplotlib failed')
# TODO extract class
class MainWindow(PlotMainWindow):
def __init__(self, grid, U, bounding_box, codim, title, legend, separate_colorbars, rescale_colorbars, backend):
assert isinstance(U, VectorArrayInterface) and hasattr(U, 'data') \
or (isinstance(U, tuple) and all(isinstance(u, VectorArrayInterface) and hasattr(u, 'data') for u in U)
and all(len(u) == len(U[0]) for u in U))
U = (U.data.astype(np.float64, copy=False),) if hasattr(U, 'data') else \
tuple(u.data.astype(np.float64, copy=False) for u in U)
if isinstance(legend, str):
legend = (legend,)
assert legend is None or isinstance(legend, tuple) and len(legend) == len(U)
if backend == 'gl':
widget = GLPatchWidget
cbar_widget = ColorBarWidget
else:
widget = MatplotlibPatchWidget
cbar_widget = None
if not separate_colorbars and len(U) > 1:
l = getLogger('pymor.gui.qt.visualize_patch')
l.warn('separate_colorbars=False not supported for matplotlib backend')
separate_colorbars = True
class PlotWidget(QWidget):
def __init__(self):
super().__init__()
if separate_colorbars:
if rescale_colorbars:
self.vmins = tuple(np.min(u[0]) for u in U)
self.vmaxs = tuple(np.max(u[0]) for u in U)
else:
self.vmins = tuple(np.min(u) for u in U)
self.vmaxs = tuple(np.max(u) for u in U)
else:
if rescale_colorbars:
self.vmins = (min(np.min(u[0]) for u in U),) * len(U)
self.vmaxs = (max(np.max(u[0]) for u in U),) * len(U)
else:
self.vmins = (min(np.min(u) for u in U),) * len(U)
self.vmaxs = (max(np.max(u) for u in U),) * len(U)
layout = QHBoxLayout()
plot_layout = QGridLayout()
self.colorbarwidgets = [cbar_widget(self, vmin=vmin, vmax=vmax) if cbar_widget else None
for vmin, vmax in zip(self.vmins, self.vmaxs)]
plots = [widget(self, grid, vmin=vmin, vmax=vmax, bounding_box=bounding_box, codim=codim)
for vmin, vmax in zip(self.vmins, self.vmaxs)]
if legend:
for i, plot, colorbar, l in zip(range(len(plots)), plots, self.colorbarwidgets, legend):
subplot_layout = QVBoxLayout()
caption = QLabel(l)
caption.setAlignment(Qt.AlignHCenter)
subplot_layout.addWidget(caption)
if not separate_colorbars or backend == 'matplotlib':
subplot_layout.addWidget(plot)
else:
hlayout = QHBoxLayout()
hlayout.addWidget(plot)
if colorbar:
hlayout.addWidget(colorbar)
subplot_layout.addLayout(hlayout)
plot_layout.addLayout(subplot_layout, int(i/columns), (i % columns), 1, 1)
else:
for i, plot, colorbar in zip(range(len(plots)), plots, self.colorbarwidgets):
if not separate_colorbars or backend == 'matplotlib':
plot_layout.addWidget(plot, int(i/columns), (i % columns), 1, 1)
else:
hlayout = QHBoxLayout()
hlayout.addWidget(plot)
if colorbar:
hlayout.addWidget(colorbar)
plot_layout.addLayout(hlayout, int(i/columns), (i % columns), 1, 1)
layout.addLayout(plot_layout)
if not separate_colorbars:
layout.addWidget(self.colorbarwidgets[0])
for w in self.colorbarwidgets[1:]:
w.setVisible(False)
self.setLayout(layout)
self.plots = plots
def set(self, U, ind):
if rescale_colorbars:
if separate_colorbars:
self.vmins = tuple(np.min(u[ind]) for u in U)
self.vmaxs = tuple(np.max(u[ind]) for u in U)
else:
self.vmins = (min(np.min(u[ind]) for u in U),) * len(U)
self.vmaxs = (max(np.max(u[ind]) for u in U),) * len(U)
for u, plot, colorbar, vmin, vmax in zip(U, self.plots, self.colorbarwidgets, self.vmins,
self.vmaxs):
plot.set(u[ind], vmin=vmin, vmax=vmax)
if colorbar:
colorbar.set(vmin=vmin, vmax=vmax)
super().__init__(U, PlotWidget(), title=title, length=len(U[0]))
self.grid = grid
self.codim = codim
def save(self):
if not config.HAVE_PYVTK:
msg = QMessageBox(QMessageBox.Critical, 'Error', 'VTK output disabled. Pleas install pyvtk.')
msg.exec_()
return
filename = QFileDialog.getSaveFileName(self, 'Save as vtk file')[0]
base_name = filename.split('.vtu')[0].split('.vtk')[0].split('.pvd')[0]
if base_name:
if len(self.U) == 1:
write_vtk(self.grid, NumpyVectorSpace.make_array(self.U[0]), base_name, codim=self.codim)
else:
for i, u in enumerate(self.U):
write_vtk(self.grid, NumpyVectorSpace.make_array(u), '{}-{}'.format(base_name, i),
codim=self.codim)
_launch_qt_app(lambda: MainWindow(grid, U, bounding_box, codim, title=title, legend=legend,
separate_colorbars=separate_colorbars, rescale_colorbars=rescale_colorbars,
backend=backend),
block)
def apply_inverse(op, rhs, options=None):
"""Solve linear equation system.
Applies the inverse of `op` to the vectors in `rhs`.
Parameters
----------
op
The linear, non-parametric |Operator| to invert.
rhs
|VectorArray| of right-hand sides for the equation system.
options
|invert_options| to use. (See :func:`invert_options`.)
Returns
-------
|VectorArray| of the solution vectors.
"""
default_options = invert_options()
if options is None:
options = default_options.values()[0]
elif isinstance(options, str):
if options == 'least_squares':
for k, v in default_options.iteritems():
if k.startswith('least_squares'):
options = v
break
assert not isinstance(options, str)
else:
options = default_options[options]
else:
assert 'type' in options and options['type'] in default_options \
and options.viewkeys() <= default_options[options['type']].viewkeys()
user_options = options
options = default_options[user_options['type']]
options.update(user_options)
R = op.source.empty(reserve=len(rhs))
if options['type'] == 'generic_lgmres':
for i in xrange(len(rhs)):
r, info = lgmres(op, rhs.copy(i),
tol=options['tol'],
maxiter=options['maxiter'],
inner_m=options['inner_m'],
outer_k=options['outer_k'])
if info > 0:
raise InversionError('lgmres failed to converge after {} iterations'.format(info))
assert info == 0
R.append(r)
elif options['type'] == 'least_squares_generic_lsmr':
for i in xrange(len(rhs)):
r, info, itn, _, _, _, _, _ = lsmr(op, rhs.copy(i),
damp=options['damp'],
atol=options['atol'],
btol=options['btol'],
conlim=options['conlim'],
maxiter=options['maxiter'],
show=options['show'])
assert 0 <= info <= 7
if info == 7:
raise InversionError('lsmr failed to converge after {} iterations'.format(itn))
getLogger('pymor.algorithms.genericsolvers.lsmr').info('Converged after {} iterations'.format(itn))
R.append(r)
elif options['type'] == 'least_squares_generic_lsqr':
for i in xrange(len(rhs)):
r, info, itn, _, _, _, _, _, _ = lsqr(op, rhs.copy(i),
damp=options['damp'],
atol=options['atol'],
btol=options['btol'],
conlim=options['conlim'],
iter_lim=options['iter_lim'],
show=options['show'])
assert 0 <= info <= 7
if info == 7:
raise InversionError('lsmr failed to converge after {} iterations'.format(itn))
getLogger('pymor.algorithms.genericsolvers.lsqr').info('Converged after {} iterations'.format(itn))
R.append(r)
else:
raise ValueError('Unknown solver type')
return R
def estimate_image_hierarchical(operators=(),
vectors=(),
domain=None,
extends=None,
orthonormalize=True,
product=None,
riesz_representatives=False):
"""Estimate the image of given |Operators| for all mu.
This is an extended version of :func:`estimate_image`, which calls
:func:`estimate_image` individually for each vector of `domain`.
As a result, the vectors in the returned `image` |VectorArray| will
be ordered by the `domain` vector they correspond to (starting with
vectors which correspond to the `functionals` and to |Operators| for
which the image is estimated independently from `domain`).
This function also returns an `image_dims` list, such that the first
`image_dims[i+1]` vectors of `image` correspond to the first `i`
vectors of `domain` (the first `image_dims[0]` vectors correspond
to `vectors` and to |Operators| with fixed image estimate).
Parameters
----------
operators
See :func:`estimate_image`.
vectors
See :func:`estimate_image`.
domain
See :func:`estimate_image`.
extends
When additional vectors have been appended to the `domain` |VectorArray|
after :func:`estimate_image_hierarchical` has been called, and
:func:`estimate_image_hierarchical` shall be called again for the extended
`domain` array, `extends` can be set to `(image, image_dims)`, where
`image`, `image_dims` are the return values of the last
:func:`estimate_image_hierarchical` call. The old `domain` vectors will
then be skipped during computation and `image`, `image_dims` will be
modified in-place.
orthonormalize
See :func:`estimate_image`.
product
See :func:`estimate_image`.
riesz_representatives
See :func:`estimate_image`.
Returns
-------
image
See above.
image_dims
See above.
Raises
------
ImageCollectionError
Is raised when for a given |Operator| no image estimate is possible.
"""
assert operators or vectors
domain_space = operators[0].source if operators else None
image_space = operators[0].range if operators \
else vectors[0].space if isinstance(vectors[0], VectorArrayInterface) \
else vectors[0].range
assert all(op.source == domain_space and op.range == image_space
for op in operators)
assert all(
isinstance(v, VectorArrayInterface) and (v in image_space)
or isinstance(v, OperatorInterface) and
(v.range == image_space and isinstance(v.source, NumpyVectorSpace)
and v.linear) for v in vectors)
assert domain is None or domain_space is None or domain in domain_space
assert product is None or product.source == product.range == image_space
assert extends is None or len(extends) == 2
logger = getLogger('pymor.algorithms.image.estimate_image_hierarchical')
if operators and domain is None:
domain = domain_space.empty()
if extends:
image = extends[0]
image_dims = extends[1]
ind_range = range(len(image_dims) -
1, len(domain)) if operators else range(
len(image_dims) - 1, 0)
else:
image = image_space.empty()
image_dims = []
ind_range = range(-1, len(domain)) if operators else [-1]
for i in ind_range:
logger.info('Estimating image for basis vector {} ...'.format(i))
if i == -1:
new_image = estimate_image(
operators,
vectors,
None,
extends=False,
orthonormalize=False,
product=product,
riesz_representatives=riesz_representatives)
else:
new_image = estimate_image(
operators, [],
domain[i],
extends=True,
orthonormalize=False,
product=product,
riesz_representatives=riesz_representatives)
gram_schmidt_offset = len(image)
image.append(new_image, remove_from_other=True)
if orthonormalize:
with logger.block('Orthonormalizing ...'):
gram_schmidt(image,
offset=gram_schmidt_offset,
product=product,
copy=False)
image_dims.append(len(image))
return image, image_dims
def set(self, key, value):
has_key = key in self._cache
if has_key:
getLogger('pymor.core.cache.DiskRegion').warning('Key already present in cache region, ignoring.')
return
self._cache.set(key, value)
int(size[:-1]) * 1024 ** 3 if size[-1] == 'G' else \
int(size)
if isinstance(disk_max_size, str):
disk_max_size = parse_size_string(disk_max_size)
cache_regions['disk'] = DiskRegion(path=disk_path, max_size=disk_max_size, persistent=False)
cache_regions['persistent'] = DiskRegion(path=persistent_path, max_size=persistent_max_size, persistent=True)
cache_regions['memory'] = MemoryRegion(memory_max_keys)
cache_regions = {}
_caching_disabled = int(os.environ.get('PYMOR_CACHE_DISABLE', 0)) == 1
if _caching_disabled:
getLogger('pymor.core.cache').warning('caching globally disabled by environment')
def enable_caching():
"""Globally enable caching."""
global _caching_disabled
_caching_disabled = int(os.environ.get('PYMOR_CACHE_DISABLE', 0)) == 1
def disable_caching():
"""Globally disable caching."""
global _caching_disabled
_caching_disabled = True
def clear_caches():
def _solve_check(dtype, solver, sep, ferr):
if ferr > 1e-1:
logger = getLogger(solver)
logger.warning(
f'Estimated forward relative error bound is large (ferr={ferr:e}, sep={sep:e}). '
f'Result may not be accurate.')
def __init__(self, section, log=getLogger(__name__)):
self._section = section
self._log = log
self._start = 0
def newton(operator,
rhs,
initial_guess=None,
mu=None,
error_norm=None,
least_squares=False,
miniter=0,
maxiter=100,
rtol=-1.,
atol=-1.,
stagnation_window=3,
stagnation_threshold=0.9,
return_stages=False,
return_residuals=False):
"""Basic Newton algorithm.
This method solves the nonlinear equation ::
A(U, mu) = V
for `U` using the Newton method.
Parameters
----------
operator
The |Operator| `A`. `A` must implement the
:meth:`~pymor.operators.interfaces.OperatorInterface.jacobian` interface method.
rhs
|VectorArray| of length 1 containing the vector `V`.
initial_guess
If not `None`, a |VectorArray| of length 1 containing an initial guess for the
solution `U`.
mu
The |Parameter| for which to solve the equation.
error_norm
The norm with which the norm of the residual is computed. If `None`, the
Euclidean norm is used.
least_squares
If `True`, use a least squares linear solver (e.g. for residual minimization).
miniter
Minimum amount of iterations to perform.
maxiter
Fail if the iteration count reaches this value without converging.
rtol
Finish when the residual norm has been reduced by this factor relative to the
norm of the initial residual.
atol
Finish when the residual norm is below this threshold.
stagnation_window
Finish when the residual norm has not been reduced by a factor of
`stagnation_threshold` during the last `stagnation_window` iterations.
stagnation_threshold
See `stagnation_window`.
return_stages
If `True`, return a |VectorArray| of the intermediate approximations of `U`
after each iteration.
return_residuals
If `True`, return a |VectorArray| of all residual vectors which have been computed
during the Newton iterations.
Returns
-------
U
|VectorArray| of length 1 containing the computed solution
data
Dict containing the following fields:
:error_sequence: |NumPy array| containing the residual norms after each iteration.
:stages: See `return_stages`.
:residuals: See `return_residuals`.
Raises
------
NewtonError
Raised if the Netwon algorithm failed to converge.
"""
logger = getLogger('pymor.algorithms.newton')
data = {}
if initial_guess is None:
initial_guess = operator.source.zeros()
if return_stages:
data['stages'] = operator.source.empty()
if return_residuals:
data['residuals'] = operator.range.empty()
U = initial_guess.copy()
residual = rhs - operator.apply(U, mu=mu)
err = residual.l2_norm()[0] if error_norm is None else error_norm(
residual)[0]
logger.info(f' Initial Residual: {err:5e}')
iteration = 0
error_sequence = [err]
while True:
if iteration >= miniter:
if err <= atol:
logger.info(f'Absolute limit of {atol} reached. Stopping.')
break
if err / error_sequence[0] <= rtol:
logger.info(
f'Prescribed total reduction of {rtol} reached. Stopping.')
break
if (len(error_sequence) >= stagnation_window + 1
and err / max(error_sequence[-stagnation_window - 1:]) >=
stagnation_threshold):
logger.info(
f'Error is stagnating (threshold: {stagnation_threshold:5e}, window: {stagnation_window}). '
f'Stopping.')
break
if iteration >= maxiter:
raise NewtonError('Failed to converge')
if iteration > 0 and return_stages:
data['stages'].append(U)
if return_residuals:
data['residuals'].append(residual)
iteration += 1
jacobian = operator.jacobian(U, mu=mu)
try:
correction = jacobian.apply_inverse(residual,
least_squares=least_squares)
except InversionError:
raise NewtonError('Could not invert jacobian')
U += correction
residual = rhs - operator.apply(U, mu=mu)
err = residual.l2_norm()[0] if error_norm is None else error_norm(
residual)[0]
logger.info(
f'Iteration {iteration:2}: Residual: {err:5e}, '
f'Reduction: {err / error_sequence[-1]:5e}, Total Reduction: {err / error_sequence[0]:5e}'
)
error_sequence.append(err)
if not np.isfinite(err):
raise NewtonError('Failed to converge')
data['error_sequence'] = np.array(error_sequence)
return U, data
def armijo(f,
starting_point,
direction,
grad=None,
initial_value=None,
alpha_init=1.0,
tau=0.5,
beta=0.0001,
maxiter=10):
"""Armijo line search algorithm.
This method computes a step size such that the Armijo condition (see [NW06]_, p. 33)
is fulfilled.
Parameters
----------
f
Real-valued function that can be evaluated for its value.
starting_point
A |VectorArray| of length 1 containing the starting point of the line search.
direction
Descent direction along which the line search is performed.
grad
Gradient of `f` in the point `starting_point`.
initial_value
Value of `f` in the point `starting_point`.
alpha_init
Initial step size that is gradually reduced.
tau
The fraction by which the step size is reduced in each iteration.
beta
Control parameter to adjust the required decrease of the function value of `f`.
maxiter
Use `alpha_init` as default if the iteration count reaches this value without
finding a point fulfilling the Armijo condition.
Returns
-------
alpha
Step size computed according to the Armijo condition.
"""
assert alpha_init > 0
assert 0 < tau < 1
assert maxiter > 0
# Start line search with step size of alpha_init
alpha = alpha_init
# Compute initial function value
if initial_value is None:
initial_value = f(starting_point)
iteration = 0
slope = 0.0
# Compute slope if gradient is provided
if grad:
slope = min(grad.dot(direction), 0.0)
while True:
# Compute new function value
current_value = f(starting_point + alpha * direction)
# Check the Armijo condition
if current_value < initial_value + alpha * beta * slope:
break
# Check if maxiter is reached
if iteration >= maxiter:
# Use default value as step size
alpha = alpha_init
# Log warning
logger = getLogger('pymor.algorithms.line_search.armijo')
logger.warning(
f'Reached maximum number of line search steps; using initial step size of {alpha_init} instead'
)
break
iteration += 1
# Adjust step size
alpha *= tau
return alpha
def __init__(self):
self.memo = {}
self.logger = logger.getLogger('pymor.core.interfaces')
def adaptive_greedy(discretization,
reductor,
parameter_space=None,
initial_basis=None,
use_estimator=True,
error_norm=None,
extension_algorithm=gram_schmidt_basis_extension,
target_error=None,
max_extensions=None,
validation_mus=0,
rho=1.1,
gamma=0.2,
theta=0.,
visualize=False,
visualize_vertex_size=80,
pool=dummy_pool):
"""Greedy basis generation algorithm with adaptively refined training set.
This method extends pyMOR's default :func:`~pymor.algorithms.greedy.greedy`
greedy basis generation algorithm by adaptive refinement of the
parameter training set according to [HDO11]_ to prevent overfitting
of the reduced basis to the training set. This is achieved by
estimating the reduction error on an additional validation set of
parameters. If the ratio between the estimated errors on the validation
set and the validation set is larger than `rho`, the training set
is refined using standard grid refinement techniques.
.. [HDO11] Haasdonk, B.; Dihlmann, M. & Ohlberger, M.,
A training set and multiple bases generation approach for
parameterized model reduction based on adaptive grids in
parameter space,
Math. Comput. Model. Dyn. Syst., 2011, 17, 423-442
Parameters
----------
discretization
See :func:`~pymor.algorithms.greedy.greedy`.
reductor
See :func:`~pymor.algorithms.greedy.greedy`.
parameter_space
The |ParameterSpace| for which to compute the reduced model. If `None`
the parameter space of the `discretization` is used.
initial_basis
See :func:`~pymor.algorithms.greedy.greedy`.
use_estimator
See :func:`~pymor.algorithms.greedy.greedy`.
error_norm
See :func:`~pymor.algorithms.greedy.greedy`.
extension_algorithm
See :func:`~pymor.algorithms.greedy.greedy`.
target_error
See :func:`~pymor.algorithms.greedy.greedy`.
max_extensions
See :func:`~pymor.algorithms.greedy.greedy`.
validation_mus
One of the following:
- a list of |Parameters| to use as validation set,
- a positive number indicating the number of random parameters
to use as validation set,
- a non-positive number, indicating the negative number of random
parameters to use as validation set in addition to the centers
of the elements of the adaptive training set.
rho
Maximum allowed ratio between maximum estimated error on validation
set vs. maximum estimated error on training set. If the ratio is
larger, the training set is refined.
gamma
Weight of the age penalty term of the training set refinement
indicators.
theta
Ratio of training set elements to select for refinement.
(One element is always refined.)
visualize
If `True`, visualize the refinement indicators. (Only available
for 2 and 3 dimensional parameter spaces.)
visualize_vertex_size
Size of the vertices in the visualization.
pool
See :func:`~pymor.algorithms.greedy.greedy`.
Returns
-------
Dict with the following fields:
:basis: The reduced basis.
:reduced_discretization: The reduced |Discretization| obtained for the
computed basis.
:reconstructor: Reconstructor for `reduced_discretization`.
:max_errs: Sequence of maximum errors during the greedy run.
:max_err_mus: The parameters corresponding to `max_errs`.
:max_val_errs: Sequence of maximum errors on the validation set.
:max_val_err_mus: The parameters corresponding to `max_val_errs`.
:refinements: Number of refinements made in each extension step.
:training_set_sizes: The final size of the training set in each extension step.
"""
def estimate(mus):
if use_estimator:
errors = pool.map(_estimate, mus, rd=rd)
else:
errors = pool.map(_estimate,
mus,
rd=rd,
d=d,
rc=rc,
error_norm=error_norm)
# most error_norms will return an array of length 1 instead of a number, so we extract the numbers
# if necessary
return np.array([x[0] if hasattr(x, '__len__') else x for x in errors])
logger = getLogger('pymor.algorithms.adaptivegreedy.adaptive_greedy')
if pool is None or pool is dummy_pool:
pool = dummy_pool
else:
logger.info(
'Using pool of {} workers for parallel greedy search'.format(
len(pool)))
with RemoteObjectManager() as rom:
# Push everything we need during the greedy search to the workers.
if not use_estimator:
rom.manage(pool.push(discretization))
if error_norm:
rom.manage(pool.push(error_norm))
tic = time.time()
# initial setup for main loop
d = discretization
basis = initial_basis
rd, rc, reduction_data = None, None, None
hierarchic = False
# setup training and validation sets
parameter_space = parameter_space or d.parameter_space
sample_set = AdaptiveSampleSet(parameter_space)
if validation_mus <= 0:
validation_set = sample_set.center_mus + parameter_space.sample_randomly(
-validation_mus)
else:
validation_set = parameter_space.sample_randomly(validation_mus)
if visualize and sample_set.dim not in (2, 3):
raise NotImplementedError
logger.info('Training set size: {}. Validation set size: {}'.format(
len(sample_set.vertex_mus), len(validation_set)))
extensions = 0
max_errs = []
max_err_mus = []
max_val_errs = []
max_val_err_mus = []
refinements = []
training_set_sizes = []
while True: # main loop
with logger.block('Reducing ...'):
rd, rc, reduction_data = reductor(discretization, basis) if not hierarchic \
else reductor(discretization, basis, extends=(rd, rc, reduction_data))
current_refinements = 0
while True: # estimate reduction errors and refine training set until no overfitting is detected
# estimate on training set
with logger.block('Estimating errors ...'):
errors = estimate(sample_set.vertex_mus)
max_err_ind = np.argmax(errors)
max_err, max_err_mu = errors[
max_err_ind], sample_set.vertex_mus[max_err_ind]
logger.info(
'Maximum error after {} extensions: {} (mu = {})'.format(
extensions, max_err, max_err_mu))
# estimate on validation set
val_errors = estimate(validation_set)
max_val_err_ind = np.argmax(val_errors)
max_val_err, max_val_err_mu = val_errors[
max_val_err_ind], validation_set[max_val_err_ind]
logger.info('Maximum validation error: {}'.format(max_val_err))
logger.info(
'Validation error to training error ratio: {:.3e}'.format(
max_val_err / max_err))
if max_val_err >= max_err * rho: # overfitting?
# compute element indicators for training set refinement
if current_refinements == 0:
logger.info2(
'Overfitting detected. Computing element indicators ...'
)
else:
logger.info3(
'Overfitting detected after refinement. Computing element indicators ...'
)
vertex_errors = np.max(errors[sample_set.vertex_ids],
axis=1)
center_errors = estimate(sample_set.center_mus)
indicators_age_part = (gamma * sample_set.volumes /
sample_set.total_volume *
(sample_set.refinement_count -
sample_set.creation_times))
indicators_error_part = np.max(
[vertex_errors, center_errors], axis=0) / max_err
indicators = indicators_age_part + indicators_error_part
# select elements
sorted_indicators_inds = np.argsort(indicators)[::-1]
refinement_elements = sorted_indicators_inds[:max(
int(len(sorted_indicators_inds) * theta), 1)]
logger.info('Refining {} elements: {}'.format(
len(refinement_elements), refinement_elements))
# visualization
if visualize:
from mpl_toolkits.mplot3d import Axes3D # NOQA
import matplotlib.pyplot as plt
plt.figure()
plt.subplot(
2,
2,
1,
projection=None if sample_set.dim == 2 else '3d')
plt.title('estimated errors')
sample_set.visualize(vertex_data=errors,
center_data=center_errors,
new_figure=False)
plt.subplot(
2,
2,
2,
projection=None if sample_set.dim == 2 else '3d')
plt.title('indicators_error_part')
vmax = np.max([
indicators_error_part, indicators_age_part,
indicators
])
data = {
('volume_data' if sample_set.dim == 2 else 'center_data'):
indicators_error_part
}
sample_set.visualize(vertex_size=visualize_vertex_size,
vmin=0,
vmax=vmax,
new_figure=False,
**data)
plt.subplot(
2,
2,
3,
projection=None if sample_set.dim == 2 else '3d')
plt.title('indicators_age_part')
data = {
('volume_data' if sample_set.dim == 2 else 'center_data'):
indicators_age_part
}
sample_set.visualize(vertex_size=visualize_vertex_size,
vmin=0,
vmax=vmax,
new_figure=False,
**data)
plt.subplot(
2,
2,
4,
projection=None if sample_set.dim == 2 else '3d')
if sample_set.dim == 2:
plt.title('indicators')
sample_set.visualize(
volume_data=indicators,
center_data=np.zeros(len(refinement_elements)),
center_inds=refinement_elements,
vertex_size=visualize_vertex_size,
vmin=0,
vmax=vmax,
new_figure=False)
else:
plt.title('selected cells')
sample_set.visualize(
center_data=np.zeros(len(refinement_elements)),
center_inds=refinement_elements,
vertex_size=visualize_vertex_size,
vmin=0,
vmax=vmax,
new_figure=False)
plt.show()
# refine training set
sample_set.refine(refinement_elements)
current_refinements += 1
# update validation set if needed
if validation_mus <= 0:
validation_set = sample_set.center_mus + parameter_space.sample_randomly(
-validation_mus)
logger.info(
'New training set size: {}. New validation set size: {}'
.format(len(sample_set.vertex_mus),
len(validation_set)))
logger.info('Number of refinements: {}'.format(
sample_set.refinement_count))
logger.info('')
else:
break # no overfitting, leave the refinement loop
max_errs.append(max_err)
max_err_mus.append(max_err_mu)
max_val_errs.append(max_val_err)
max_val_err_mus.append(max_val_err_mu)
refinements.append(current_refinements)
training_set_sizes.append(len(sample_set.vertex_mus))
# break if traget error reached
if target_error is not None and max_err <= target_error:
logger.info(
'Reached maximal error on snapshots of {} <= {}'.format(
max_err, target_error))
break
# basis extension
with logger.block(
'Computing solution snapshot for mu = {} ...'.format(
max_err_mu)):
U = discretization.solve(max_err_mu)
with logger.block('Extending basis with solution snapshot ...'):
try:
basis, extension_data = extension_algorithm(basis, U)
except ExtensionError:
logger.info('Extension failed. Stopping now.')
break
extensions += 1
if 'hierarchic' not in extension_data:
logger.warn(
'Extension algorithm does not report if extension was hierarchic. Assuming it was\'nt ..'
)
hierarchic = False
else:
hierarchic = extension_data['hierarchic']
logger.info('')
# break if prescribed basis size reached
if max_extensions is not None and extensions >= max_extensions:
logger.info('Maximum number of {} extensions reached.'.format(
max_extensions))
with logger.block('Reducing once more ...'):
rd, rc, reduction_data = reductor(discretization, basis) if not hierarchic \
else reductor(discretization, basis, extends=(rd, rc, reduction_data))
break
tictoc = time.time() - tic
logger.info('Greedy search took {} seconds'.format(tictoc))
return {
'basis': basis,
'reduced_discretization': rd,
'reconstructor': rc,
'max_errs': max_errs,
'max_err_mus': max_err_mus,
'extensions': extensions,
'max_val_errs': max_val_errs,
'max_val_err_mus': max_val_err_mus,
'refinements': refinements,
'training_set_sizes': training_set_sizes,
'time': tictoc,
'reduction_data': reduction_data
}
def samdp(A,
E,
B,
C,
nwanted,
init_shifts=None,
which='LR',
tol=1e-10,
imagtol=1e-6,
conjtol=1e-8,
dorqitol=1e-4,
rqitol=1e-10,
maxrestart=100,
krestart=20,
rqi_maxiter=10,
seed=0):
"""Compute the dominant pole triplets and residues of the transfer function of an LTI system.
This function uses the subspace accelerated dominant pole (SAMDP) algorithm as described in
[RM06]_ in Algorithm 2 in order to compute dominant pole triplets and residues of the transfer
function
.. math::
H(s) = C (s E - A)^{-1} B
of an LTI system. It is possible to take advantage of prior knowledge about the poles
by specifying shift parameters, which are injected after a new pole has been found.
Parameters
----------
A
The |Operator| A.
E
The |Operator| E or `None`.
B
The operator B as a |VectorArray| from `A.source`.
C
The operator C as a |VectorArray| from `A.source`.
nwanted
The number of dominant poles that should be computed.
init_shifts
A |NumPy array| containing shifts which are injected after a new pole has been found.
which
A string specifying the strategy by which the dominant poles and residues are selected.
Possible values are:
- `'LR'`: select poles with largest norm(residual) / abs(Re(pole))
- `'LS'`: select poles with largest norm(residual) / abs(pole)
- `'LM'`: select poles with largest norm(residual)
tol
Tolerance for the residual of the poles.
imagtol
Relative tolerance for imaginary parts of pairs of complex conjugate eigenvalues.
conjtol
Tolerance for the residual of the complex conjugate of a pole.
dorqitol
If the residual is smaller than dorqitol the two-sided Rayleigh quotient iteration
is executed.
rqitol
Tolerance for the relative change of a pole in the two-sided Rayleigh quotient
iteration.
maxrestart
The maximum number of restarts.
krestart
Maximum dimension of search space before performing a restart.
rqi_maxiter
Maximum number of iterations for the two-sided Rayleigh quotient iteration.
seed
Random seed which is used for computing the initial shift and random restarts.
Returns
-------
poles
A 1D |NumPy array| containing the computed dominant poles.
residues
A 3D |NumPy array| of shape `(len(poles), len(C), len(B))` containing the computed residues.
rightev
A |VectorArray| containing the right eigenvectors of the computed poles.
leftev
A |VectorArray| containing the left eigenvectors of the computed poles.
"""
logger = getLogger('pymor.algorithms.samdp.samdp')
if E is None:
E = IdentityOperator(A.source)
assert isinstance(A, Operator) and A.linear
assert not A.parametric
assert A.source == A.range
if E is not None:
assert isinstance(E, Operator) and E.linear
assert not E.parametric
assert E.source == E.range
assert E.source == A.source
assert B in A.source
assert C in A.source
B_defl = B.copy()
C_defl = C.copy()
k = 0
nrestart = 0
nr_converged = 0
np.random.seed(seed)
X = A.source.empty()
Q = A.source.empty()
Qt = A.source.empty()
Qs = A.source.empty()
Qts = A.source.empty()
AX = A.source.empty()
V = A.source.empty()
H = np.empty((0, 1))
G = np.empty((0, 1))
poles = np.empty(0)
if init_shifts is None:
st = np.random.uniform() * 10.j
shift_nr = 0
nr_shifts = 0
else:
st = init_shifts[0]
shift_nr = 1
nr_shifts = len(init_shifts)
shifts = init_shifts
while nrestart < maxrestart:
k += 1
sEmA = st * E - A
sEmAB = sEmA.apply_inverse(B_defl)
Hs = C_defl.inner(sEmAB)
y_all, _, u_all = spla.svd(Hs)
u = u_all.conj()[0]
y = y_all[:, 0]
x = sEmAB.lincomb(u)
v = sEmA.apply_inverse_adjoint(C_defl.lincomb(y.T))
X.append(x)
V.append(v)
gram_schmidt(V, atol=0, rtol=0, copy=False)
gram_schmidt(X, atol=0, rtol=0, copy=False)
AX.append(A.apply(X[k - 1]))
if k > 1:
H = np.hstack((H, V[0:k - 1].inner(AX[k - 1])))
H = np.vstack((H, V[k - 1].inner(AX)))
EX = E.apply(X)
if k > 1:
G = np.hstack((G, V[0:k - 1].inner(EX[k - 1])))
G = np.vstack((G, V[k - 1].inner(EX)))
SH, UR, URt, res = _select_max_eig(H, G, X, V, B_defl, C_defl, which)
if np.all(res < np.finfo(float).eps):
st = np.random.uniform() * 10.j
found = False
else:
found = True
do_rqi = True
while found:
theta = SH[0, 0]
schurvec = X.lincomb(UR[:, 0])
schurvec.scal(1 / schurvec.norm())
lschurvec = V.lincomb(URt[:, 0])
lschurvec.scal(1 / lschurvec.norm())
st = theta
nres = (A.apply(schurvec) - (E.apply(schurvec) * theta)).norm()[0]
logger.info(f'Step: {k}, Theta: {theta:.5e}, Residual: {nres:.5e}')
if np.abs(np.imag(theta)) / np.abs(theta) < imagtol:
rres = A.apply(
schurvec.real) - E.apply(schurvec.real) * np.real(theta)
nrr = rres.norm() / np.abs(np.real(theta))
if nrr - nres < np.finfo(float).eps:
schurvec = schurvec.real
lschurvec = lschurvec.real
theta = np.real(theta)
nres = nrr
if nres < dorqitol and do_rqi and nres >= tol:
schurvec, lschurvec, theta, nres = _twosided_rqi(
A, E, schurvec, lschurvec, theta, nres, imagtol, rqitol,
rqi_maxiter)
do_rqi = False
if np.abs(np.imag(theta)) / np.abs(theta) < imagtol:
rres = A.apply(schurvec.real) - E.apply(
schurvec.real) * np.real(theta)
nrr = rres.norm() / np.abs(np.real(theta))
if nrr - nres < np.finfo(float).eps:
schurvec = schurvec.real
lschurvec = lschurvec.real
theta = np.real(theta)
nres = nrr
if nres >= tol:
logger.warning(
'Two-sided RQI did not reach desired tolerance.')
found = nr_converged < nwanted and nres < tol
if found:
poles = np.append(poles, theta)
logger.info(f'Pole: {theta:.5e}')
Q.append(schurvec)
Qt.append(lschurvec)
Esch = E.apply(schurvec)
Qs.append(Esch)
Qts.append(E.apply_adjoint(lschurvec))
nqqt = lschurvec.inner(Esch)[0][0]
Q[-1].scal(1 / nqqt)
Qs[-1].scal(1 / nqqt)
nr_converged += 1
if k > 1:
X = X.lincomb(UR[:, 1:k].T)
V = V.lincomb(URt[:, 1:k].T)
else:
X = A.source.empty()
V = A.source.empty()
if np.abs(np.imag(theta)) / np.abs(theta) < imagtol:
gram_schmidt(V, atol=0, rtol=0, copy=False)
gram_schmidt(X, atol=0, rtol=0, copy=False)
B_defl -= E.apply(Q[-1].lincomb(Qt[-1].inner(B_defl).T))
C_defl -= E.apply_adjoint(Qt[-1].lincomb(
Q[-1].inner(C_defl).T))
k -= 1
cce = theta.conj()
if np.abs(np.imag(cce)) / np.abs(cce) >= imagtol:
ccv = schurvec.conj()
ccv.scal(1 / ccv.norm())
r = A.apply(ccv) - E.apply(ccv) * cce
if r.norm() / np.abs(cce) < conjtol:
logger.info(f'Conjugate Pole: {cce:.5e}')
poles = np.append(poles, cce)
Q.append(ccv)
ccvt = lschurvec.conj()
Qt.append(ccvt)
Esch = E.apply(ccv)
Qs.append(Esch)
Qts.append(E.apply_adjoint(ccvt))
nqqt = ccvt.inner(E.apply(ccv))[0][0]
Q[-1].scal(1 / nqqt)
Qs[-1].scal(1 / nqqt)
gram_schmidt(V, atol=0, rtol=0, copy=False)
gram_schmidt(X, atol=0, rtol=0, copy=False)
B_defl -= E.apply(Q[-1].lincomb(
Qt[-1].inner(B_defl).T))
C_defl -= E.apply_adjoint(Qt[-1].lincomb(
Q[-1].inner(C_defl).T))
AX = A.apply(X)
if k > 0:
G = V.inner(E.apply(X))
H = V.inner(AX)
SH, UR, URt, residues = _select_max_eig(
H, G, X, V, B_defl, C_defl, which)
found = np.any(res >= np.finfo(float).eps)
else:
G = np.empty((0, 1))
H = np.empty((0, 1))
found = False
if nr_converged < nwanted:
if found:
st = SH[0, 0]
else:
st = np.random.uniform() * 10.j
if shift_nr < nr_shifts:
st = shifts[shift_nr]
shift_nr += 1
elif k >= krestart:
logger.info('Perform restart...')
EX = E.apply(X)
RR = AX.lincomb(UR.T) - EX.lincomb(UR.T).lincomb(SH.T)
minidx = RR.norm().argmin()
k = 1
X = X.lincomb(UR[:, minidx])
V = V.lincomb(URt[:, minidx])
gram_schmidt(V, atol=0, rtol=0, copy=False)
gram_schmidt(X, atol=0, rtol=0, copy=False)
G = V.inner(E.apply(X))
AX = A.apply(X)
H = V.inner(AX)
nrestart += 1
if k >= krestart:
logger.info('Perform restart...')
EX = E.apply(X)
RR = AX.lincomb(UR.T) - EX.lincomb(UR.T).lincomb(SH.T)
minidx = RR.norm().argmin()
k = 1
X = X.lincomb(UR[:, minidx])
V = V.lincomb(URt[:, minidx])
gram_schmidt(V, atol=0, rtol=0, copy=False)
gram_schmidt(X, atol=0, rtol=0, copy=False)
G = V.inner(E.apply(X))
AX = A.apply(X)
H = V.inner(AX)
nrestart += 1
if nr_converged == nwanted or nrestart == maxrestart:
rightev = Q
leftev = Qt
absres = np.empty(len(poles))
residues = []
for i in range(len(poles)):
leftev[i].scal(1 / leftev[i].inner(E.apply(rightev[i]))[0][0])
residues.append(C.inner(rightev[i]) @ leftev[i].inner(B))
absres[i] = spla.norm(residues[-1], ord=2)
residues = np.array(residues)
if which == 'LR':
idx = np.argsort(-absres / np.abs(np.real(poles)))
elif which == 'LS':
idx = np.argsort(-absres / np.abs(poles))
elif which == 'LM':
idx = np.argsort(-absres)
else:
raise ValueError('Unknown SAMDP selection strategy.')
residues = residues[idx]
poles = poles[idx]
rightev = rightev[idx]
leftev = leftev[idx]
if nr_converged < nwanted:
logger.warning(
'The specified number of poles could not be computed.')
break
return poles, residues, rightev, leftev
def eigs(A,
E=None,
k=3,
which='LM',
b=None,
l=None,
maxiter=1000,
tol=1e-13,
imag_tol=1e-12,
complex_pair_tol=1e-12,
seed=0):
"""Approximate a few eigenvalues of a linear |Operator|.
Computes `k` eigenvalues `w` with corresponding eigenvectors `v` which solve
the eigenvalue problem
.. math::
A v_i = w_i v_i
or the generalized eigenvalue problem
.. math::
A v_i = w_i E v_i
if `E` is not `None`.
The implementation is based on Algorithm 4.2 in [RL95]_.
Parameters
----------
A
The real linear |Operator| for which the eigenvalues are to be computed.
E
The real linear |Operator| which defines the generalized eigenvalue problem.
k
The number of eigenvalues and eigenvectors which are to be computed.
which
A string specifying which `k` eigenvalues and eigenvectors to compute:
- `'LM'`: select eigenvalues with largest magnitude
- `'SM'`: select eigenvalues with smallest magnitude
- `'LR'`: select eigenvalues with largest real part
- `'SR'`: select eigenvalues with smallest real part
- `'LI'`: select eigenvalues with largest imaginary part
- `'SI'`: select eigenvalues with smallest imaginary part
b
Initial vector for Arnoldi iteration. Default is a random vector.
l
The size of the Arnoldi factorization. Default is `min(n - 1, max(2*k + 1, 20))`.
maxiter
The maximum number of iterations.
tol
The relative error tolerance for the Ritz estimates.
imag_tol
Relative imaginary parts below this tolerance are set to 0.
complex_pair_tol
Tolerance for detecting pairs of complex conjugate eigenvalues.
seed
Random seed which is used for computing the initial vector for the Arnoldi
iteration.
Returns
-------
w
A 1D |NumPy array| which contains the computed eigenvalues.
v
A |VectorArray| which contains the computed eigenvectors.
"""
logger = getLogger('pymor.algorithms.eigs.eigs')
assert isinstance(A, Operator) and A.linear
assert not A.parametric
assert A.source == A.range
if E is None:
E = IdentityOperator(A.source)
else:
assert isinstance(E, Operator) and E.linear
assert not E.parametric
assert E.source == E.range
assert E.source == A.source
if b is None:
b = A.source.random(seed=seed)
else:
assert b in A.source
n = A.source.dim
l_min = 20
if l is None:
l = min(n - 1, max(2 * k + 1, l_min))
assert k < n
assert l > k
V, H, f = _arnoldi(A, E, k, b)
k0 = k
i = 0
while True:
i += 1
V, H, f = _extend_arnoldi(A, E, V, H, f, l - k)
ew, ev = spla.eig(H)
# truncate small imaginary parts
ew.imag[np.abs(ew.imag) / np.abs(ew) < imag_tol] = 0
if which == 'LM':
idx = np.argsort(-np.abs(ew))
elif which == 'SM':
idx = np.argsort(np.abs(ew))
elif which == 'LR':
idx = np.argsort(-ew.real)
elif which == 'SR':
idx = np.argsort(ew.real)
elif which == 'LI':
idx = np.argsort(-np.abs(ew.imag))
elif which == 'SI':
idx = np.argsort(np.abs(ew.imag))
k = k0
ews = ew[idx]
evs = ev[:, idx]
rres = f.l2_norm()[0] * np.abs(evs[l - 1]) / np.abs(ews)
# increase k by one in order to keep complex conjugate pairs together
if ews[k - 1].imag != 0 and ews[
k - 1].imag + ews[k].imag < complex_pair_tol:
k += 1
logger.info(
f'Maximum of relative Ritz estimates at step {i}: {rres[:k].max():.5e}'
)
if np.all(rres[:k] <= tol) or i >= maxiter:
break
# increase k in order to prevent stagnation
k = min(l - 1, k + min(np.count_nonzero(rres[:k] <= tol),
(l - k) // 2))
# sort shifts for QR iteration based on their residual
shifts = ews[k:l]
srres = rres[k:l]
idx = np.argsort(-srres)
srres = srres[idx]
shifts = shifts[idx]
# don't use converged unwanted Ritz values as shifts
shifts = shifts[srres != 0]
k += np.count_nonzero(srres == 0)
if shifts[0].imag != 0 and shifts[0].imag + ews[
1].imag >= complex_pair_tol:
shifts = shifts[1:]
k += 1
H, Qs = _qr_iteration(H, shifts)
V = V.lincomb(Qs.T)
f = V[k] * H[k, k - 1] + f * Qs[l - 1, k - 1]
V = V[:k]
H = H[:k, :k]
return ews[:k0], V.lincomb(evs[:, :k0].T)
def apply_inverse(op, V, initial_guess=None, options=None, least_squares=False, check_finite=True,
default_solver='scipy_spsolve', default_least_squares_solver='scipy_least_squares_lsmr'):
"""Solve linear equation system.
Applies the inverse of `op` to the vectors in `V` using SciPy.
Parameters
----------
op
The linear, non-parametric |Operator| to invert.
V
|VectorArray| of right-hand sides for the equation system.
initial_guess
|VectorArray| with the same length as `V` containing initial guesses
for the solution. Some implementations of `apply_inverse` may
ignore this parameter. If `None` a solver-dependent default is used.
options
The |solver_options| to use (see :func:`solver_options`).
least_squares
If `True`, return least squares solution.
check_finite
Test if solution only contains finite values.
default_solver
Default solver to use (scipy_spsolve, scipy_bicgstab, scipy_bicgstab_spilu,
scipy_lgmres, scipy_least_squares_lsmr, scipy_least_squares_lsqr).
default_least_squares_solver
Default solver to use for least squares problems (scipy_least_squares_lsmr,
scipy_least_squares_lsqr).
Returns
-------
|VectorArray| of the solution vectors.
"""
assert V in op.range
assert initial_guess is None or initial_guess in op.source and len(initial_guess) == len(V)
if isinstance(op, NumpyMatrixOperator):
matrix = op.matrix
else:
from pymor.algorithms.to_matrix import to_matrix
matrix = to_matrix(op)
options = _parse_options(options, solver_options(), default_solver, default_least_squares_solver, least_squares)
V = V.to_numpy()
initial_guess = initial_guess.to_numpy() if initial_guess is not None else None
promoted_type = np.promote_types(matrix.dtype, V.dtype)
R = np.empty((len(V), matrix.shape[1]), dtype=promoted_type)
if options['type'] == 'scipy_bicgstab':
for i, VV in enumerate(V):
R[i], info = bicgstab(matrix, VV, initial_guess[i] if initial_guess is not None else None,
tol=options['tol'], maxiter=options['maxiter'])
if info != 0:
if info > 0:
raise InversionError(f'bicgstab failed to converge after {info} iterations')
else:
raise InversionError('bicgstab failed with error code {} (illegal input or breakdown)'.
format(info))
elif options['type'] == 'scipy_bicgstab_spilu':
if Version(scipy.version.version) >= Version('0.19'):
ilu = spilu(matrix, drop_tol=options['spilu_drop_tol'], fill_factor=options['spilu_fill_factor'],
drop_rule=options['spilu_drop_rule'], permc_spec=options['spilu_permc_spec'])
else:
if options['spilu_drop_rule']:
logger = getLogger('pymor.operators.numpy._apply_inverse')
logger.error("ignoring drop_rule in ilu factorization due to old SciPy")
ilu = spilu(matrix, drop_tol=options['spilu_drop_tol'], fill_factor=options['spilu_fill_factor'],
permc_spec=options['spilu_permc_spec'])
precond = LinearOperator(matrix.shape, ilu.solve)
for i, VV in enumerate(V):
R[i], info = bicgstab(matrix, VV, initial_guess[i] if initial_guess is not None else None,
tol=options['tol'], maxiter=options['maxiter'], M=precond)
if info != 0:
if info > 0:
raise InversionError(f'bicgstab failed to converge after {info} iterations')
else:
raise InversionError('bicgstab failed with error code {} (illegal input or breakdown)'.
format(info))
elif options['type'] == 'scipy_spsolve':
try:
# maybe remove unusable factorization:
if hasattr(matrix, 'factorization'):
fdtype = matrix.factorizationdtype
if not np.can_cast(V.dtype, fdtype, casting='safe'):
del matrix.factorization
if Version(scipy.version.version) >= Version('0.14'):
if hasattr(matrix, 'factorization'):
# we may use a complex factorization of a real matrix to
# apply it to a real vector. In that case, we downcast
# the result here, removing the imaginary part,
# which should be zero.
R = matrix.factorization.solve(V.T).T.astype(promoted_type, copy=False)
elif options['keep_factorization']:
# the matrix is always converted to the promoted type.
# if matrix.dtype == promoted_type, this is a no_op
matrix.factorization = splu(matrix_astype_nocopy(matrix.tocsc(), promoted_type),
permc_spec=options['permc_spec'])
matrix.factorizationdtype = promoted_type
R = matrix.factorization.solve(V.T).T
else:
# the matrix is always converted to the promoted type.
# if matrix.dtype == promoted_type, this is a no_op
R = spsolve(matrix_astype_nocopy(matrix, promoted_type), V.T, permc_spec=options['permc_spec']).T
else:
# see if-part for documentation
if hasattr(matrix, 'factorization'):
for i, VV in enumerate(V):
R[i] = matrix.factorization.solve(VV).astype(promoted_type, copy=False)
elif options['keep_factorization']:
matrix.factorization = splu(matrix_astype_nocopy(matrix.tocsc(), promoted_type),
permc_spec=options['permc_spec'])
matrix.factorizationdtype = promoted_type
for i, VV in enumerate(V):
R[i] = matrix.factorization.solve(VV)
elif len(V) > 1:
factorization = splu(matrix_astype_nocopy(matrix.tocsc(), promoted_type),
permc_spec=options['permc_spec'])
for i, VV in enumerate(V):
R[i] = factorization.solve(VV)
else:
R = spsolve(matrix_astype_nocopy(matrix, promoted_type), V.T,
permc_spec=options['permc_spec']).reshape((1, -1))
except RuntimeError as e:
raise InversionError(e)
elif options['type'] == 'scipy_lgmres':
for i, VV in enumerate(V):
R[i], info = lgmres(matrix, VV, initial_guess[i] if initial_guess is not None else None,
tol=options['tol'],
atol=options['tol'],
maxiter=options['maxiter'],
inner_m=options['inner_m'],
outer_k=options['outer_k'])
if info > 0:
raise InversionError(f'lgmres failed to converge after {info} iterations')
assert info == 0
elif options['type'] == 'scipy_least_squares_lsmr':
from scipy.sparse.linalg import lsmr
for i, VV in enumerate(V):
R[i], info, itn, _, _, _, _, _ = lsmr(matrix, VV,
damp=options['damp'],
atol=options['atol'],
btol=options['btol'],
conlim=options['conlim'],
maxiter=options['maxiter'],
show=options['show'],
x0=initial_guess[i] if initial_guess is not None else None)
assert 0 <= info <= 7
if info == 7:
raise InversionError(f'lsmr failed to converge after {itn} iterations')
elif options['type'] == 'scipy_least_squares_lsqr':
for i, VV in enumerate(V):
R[i], info, itn, _, _, _, _, _, _, _ = lsqr(matrix, VV,
damp=options['damp'],
atol=options['atol'],
btol=options['btol'],
conlim=options['conlim'],
iter_lim=options['iter_lim'],
show=options['show'],
x0=initial_guess[i] if initial_guess is not None else None)
assert 0 <= info <= 7
if info == 7:
raise InversionError(f'lsmr failed to converge after {itn} iterations')
else:
raise ValueError('Unknown solver type')
if check_finite:
if not np.isfinite(np.sum(R)):
raise InversionError('Result contains non-finite values')
return op.source.from_numpy(R)
def solve_lyap_lrcf(A,
E,
B,
trans=False,
options=None,
default_solver=None):
"""Compute an approximate low-rank solution of a Lyapunov equation.
See :func:`pymor.algorithms.lyapunov.solve_lyap_lrcf` for a
general description.
This function uses `pymess.glyap` and `pymess.lradi`.
For both methods,
:meth:`~pymor.vectorarrays.interfaces.VectorArrayInterface.to_numpy`
and
:meth:`~pymor.vectorarrays.interfaces.VectorSpaceInterface.from_numpy`
need to be implemented for `A.source`.
Additionally, since `glyap` is a dense solver, it expects
:func:`~pymor.algorithms.to_matrix.to_matrix` to work for A and
E.
If the solver is not specified using the options or
default_solver arguments, `glyap` is used for small problems
(smaller than defined with
:func:`~pymor.algorithms.lyapunov.mat_eqn_sparse_min_size`) and
`lradi` for large problems.
Parameters
----------
A
The non-parametric |Operator| A.
E
The non-parametric |Operator| E or `None`.
B
The operator B as a |VectorArray| from `A.source`.
trans
Whether the first |Operator| in the Lyapunov equation is
transposed.
options
The solver options to use (see
:func:`lyap_lrcf_solver_options`).
default_solver
Default solver to use (pymess_lradi, pymess_glyap).
If `None`, choose solver depending on the dimension of A.
Returns
-------
Z
Low-rank Cholesky factor of the Lyapunov equation solution,
|VectorArray| from `A.source`.
"""
_solve_lyap_lrcf_check_args(A, E, B, trans)
if default_solver is None:
default_solver = 'pymess_lradi' if A.source.dim >= mat_eqn_sparse_min_size(
) else 'pymess_glyap'
options = _parse_options(options, lyap_lrcf_solver_options(),
default_solver, None, False)
if options['type'] == 'pymess_glyap':
X = solve_lyap_dense(to_matrix(A, format='dense'),
to_matrix(E, format='dense') if E else None,
B.to_numpy().T if not trans else B.to_numpy(),
trans=trans,
options=options)
Z = _chol(X)
elif options['type'] == 'pymess_lradi':
opts = options['opts']
opts.type = pymess.MESS_OP_NONE if not trans else pymess.MESS_OP_TRANSPOSE
eqn = LyapunovEquation(opts, A, E, B)
Z, status = pymess.lradi(eqn, opts)
relres = status.res2_norm / status.res2_0
if relres > opts.adi.res2_tol:
logger = getLogger('pymor.bindings.pymess.solve_lyap_lrcf')
logger.warning(
f'Desired relative residual tolerance was not achieved '
f'({relres:e} > {opts.adi.res2_tol:e}).')
else:
raise ValueError(
f'Unexpected Lyapunov equation solver ({options["type"]}).')
return A.source.from_numpy(Z.T)
def __init__(self, grid, U, bounding_box, codim, title, legend,
separate_colorbars, rescale_colorbars, backend):
assert isinstance(U, VectorArrayInterface) and hasattr(U, 'data') \
or (isinstance(U, tuple) and all(isinstance(u, VectorArrayInterface) and hasattr(u, 'data') for u in U)
and all(len(u) == len(U[0]) for u in U))
U = (U.data, ) if hasattr(U, 'data') else tuple(u.data for u in U)
if isinstance(legend, str):
legend = (legend, )
assert legend is None or isinstance(
legend, tuple) and len(legend) == len(U)
if backend == 'gl':
widget = GLPatchWidget
else:
widget = MatplotlibPatchWidget
if not separate_colorbars and len(U) > 1:
l = getLogger('pymor.gui.qt.visualize_patch')
l.warn(
'separate_colorbars=False not supported for matplotlib backend'
)
separate_colorbars = True
class PlotWidget(QWidget):
def __init__(self):
super(PlotWidget, self).__init__()
if separate_colorbars:
if rescale_colorbars:
self.vmins = tuple(np.min(u[0]) for u in U)
self.vmaxs = tuple(np.max(u[0]) for u in U)
else:
self.vmins = tuple(np.min(u) for u in U)
self.vmaxs = tuple(np.max(u) for u in U)
else:
if rescale_colorbars:
self.vmins = (min(np.min(u[0])
for u in U), ) * len(U)
self.vmaxs = (max(np.max(u[0])
for u in U), ) * len(U)
else:
self.vmins = (min(np.min(u) for u in U), ) * len(U)
self.vmaxs = (max(np.max(u) for u in U), ) * len(U)
layout = QHBoxLayout()
plot_layout = QGridLayout()
self.colorbarwidgets = [
ColorBarWidget(self, vmin=vmin, vmax=vmax)
for vmin, vmax in izip(self.vmins, self.vmaxs)
]
plots = [
widget(self,
grid,
vmin=vmin,
vmax=vmax,
bounding_box=bounding_box,
codim=codim)
for vmin, vmax in izip(self.vmins, self.vmaxs)
]
if legend:
for i, plot, colorbar, l in izip(
xrange(len(plots)), plots,
self.colorbarwidgets, legend):
subplot_layout = QVBoxLayout()
caption = QLabel(l)
caption.setAlignment(Qt.AlignHCenter)
subplot_layout.addWidget(caption)
if not separate_colorbars or backend == 'matplotlib':
subplot_layout.addWidget(plot)
else:
hlayout = QHBoxLayout()
hlayout.addWidget(plot)
hlayout.addWidget(colorbar)
subplot_layout.addLayout(hlayout)
plot_layout.addLayout(subplot_layout,
int(i / columns),
(i % columns), 1, 1)
else:
for i, plot, colorbar in izip(xrange(len(plots)),
plots,
self.colorbarwidgets):
if not separate_colorbars or backend == 'matplotlib':
plot_layout.addWidget(plot, int(i / columns),
(i % columns), 1, 1)
else:
hlayout = QHBoxLayout()
hlayout.addWidget(plot)
hlayout.addWidget(colorbar)
plot_layout.addLayout(hlayout,
int(i / columns),
(i % columns), 1, 1)
layout.addLayout(plot_layout)
if not separate_colorbars:
layout.addWidget(self.colorbarwidgets[0])
for w in self.colorbarwidgets[1:]:
w.setVisible(False)
self.setLayout(layout)
self.plots = plots
def set(self, U, ind):
if rescale_colorbars:
if separate_colorbars:
self.vmins = tuple(np.min(u[ind]) for u in U)
self.vmaxs = tuple(np.max(u[ind]) for u in U)
else:
self.vmins = (min(np.min(u[ind])
for u in U), ) * len(U)
self.vmaxs = (max(np.max(u[ind])
for u in U), ) * len(U)
for u, plot, colorbar, vmin, vmax in izip(
U, self.plots, self.colorbarwidgets, self.vmins,
self.vmaxs):
plot.set(u[ind], vmin=vmin, vmax=vmax)
colorbar.set(vmin=vmin, vmax=vmax)
super(MainWindow, self).__init__(U,
PlotWidget(),
title=title,
length=len(U[0]))
self.grid = grid
self.codim = codim
int(size[:-1]) * 1024 ** 3 if size[-1] == 'G' else \
int(size)
if isinstance(disk_max_size, str):
disk_max_size = parse_size_string(disk_max_size)
cache_regions['disk'] = SQLiteRegion(path=disk_path, max_size=disk_max_size, persistent=False)
cache_regions['persistent'] = SQLiteRegion(path=persistent_path, max_size=persistent_max_size, persistent=True)
cache_regions['memory'] = MemoryRegion(memory_max_keys)
cache_regions = {}
_caching_disabled = int(os.environ.get('PYMOR_CACHE_DISABLE', 0)) == 1
if _caching_disabled:
from pymor.core.logger import getLogger
getLogger('pymor.core.cache').warn('caching globally disabled by environment')
def enable_caching():
"""Globally enable caching."""
global _caching_disabled
_caching_disabled = int(os.environ.get('PYMOR_CACHE_DISABLE', 0)) == 1
def disable_caching():
"""Globally disable caching."""
global _caching_disabled
_caching_disabled = True
def clear_caches():
def _ricc_rcond_check(solver, rcond):
if rcond < np.finfo(np.float64).eps:
logger = getLogger(solver)
logger.warning(
f'Estimated reciprocal condition number is small (rcond={rcond:e}). '
f'Result may not be accurate.')
def _call_pymess_dense_nm_gmpare(A,
E,
B,
C,
R,
S,
trans=False,
options=None,
plus=False,
method_name=''):
"""Return the solution from pymess.dense_nm_gmpare solver."""
A = to_matrix(A, format='dense')
E = to_matrix(E, format='dense') if E else None
B = B.to_numpy().T
C = C.to_numpy()
S = S.to_numpy().T if S else None
Q = B.dot(B.T) if not trans else C.T.dot(C)
pymess_trans = pymess.MESS_OP_NONE if not trans else pymess.MESS_OP_TRANSPOSE
if not trans:
RinvC = spla.solve(R, C) if R is not None else C
G = C.T.dot(RinvC)
if S is not None:
RinvST = spla.solve(R, S.T) if R is not None else S.T
if not plus:
A -= S.dot(RinvC)
Q -= S.dot(RinvST)
else:
A += S.dot(RinvC)
Q += S.dot(RinvST)
else:
RinvBT = spla.solve(R, B.T) if R is not None else B.T
G = B.dot(RinvBT)
if S is not None:
RinvST = spla.solve(R, S.T) if R is not None else S.T
if not plus:
A -= RinvBT.T.dot(S.T)
Q -= S.dot(RinvST)
else:
A += RinvBT.T.dot(S.T)
Q += S.dot(RinvST)
X, absres, relres = pymess.dense_nm_gmpare(
None,
A,
E,
Q,
G,
plus=plus,
trans=pymess_trans,
linesearch=options['linesearch'],
maxit=options['maxit'],
absres_tol=options['absres_tol'],
relres_tol=options['relres_tol'],
nrm=options['nrm'])
if absres > options['absres_tol']:
logger = getLogger('pymor.bindings.pymess.' + method_name)
logger.warning(
f'Desired absolute residual tolerance was not achieved '
f'({absres:e} > {options["absres_tol"]:e}).')
if relres > options['relres_tol']:
logger = getLogger('pymor.bindings.pymess.' + method_name)
logger.warning(
f'Desired relative residual tolerance was not achieved '
f'({relres:e} > {options["relres_tol"]:e}).')
return X
from pymor.playground.operators.block import BlockOperator
from pymor.playground.reductors import GenericBlockRBReconstructor
from pymor.reductors.basic import reduce_generic_rb
from pymor.vectorarrays.block import BlockVectorArray
from pymor.vectorarrays.list import ListVectorArray
from pymor.vectorarrays.numpy import NumpyVectorArray
import pymor.core.logger
from dune.pymor.la.container import make_listvectorarray
from simdb.run import new_dataset, add_values, add_logfile
logfile = NamedTemporaryFile(delete=False).name
pymor.core.logger.FILENAME = logfile
logger = getLogger('.ORS2016__3_3.main')
logger.setLevel('INFO')
class InstationaryDuneVisualizer(object):
def __init__(self, disc, prefix):
self.disc = disc
self.prefix = prefix
def visualize(self, U, *args, **kwargs):
import numpy as np
dune_disc = self.disc._impl
assert isinstance(U, ListVectorArray)
filename = kwargs['filename'] if 'filename' in kwargs else self.prefix
size = len(U)
pad = len(str(size))
def gram_schmidt(A,
product=None,
atol=1e-13,
rtol=1e-13,
offset=0,
find_duplicates=True,
reiterate=True,
reiteration_threshold=1e-1,
check=True,
check_tol=1e-3,
copy=True):
"""Orthonormalize a |VectorArray| using the stabilized Gram-Schmidt algorithm.
Parameters
----------
A
The |VectorArray| which is to be orthonormalized.
product
The inner product |Operator| w.r.t. which to orthonormalize.
If `None`, the Euclidean product is used.
atol
Vectors of norm smaller than `atol` are removed from the array.
rtol
Relative tolerance used to detect linear dependent vectors
(which are then removed from the array).
offset
Assume that the first `offset` vectors are already orthonormal and start the
algorithm at the `offset + 1`-th vector.
reiterate
If `True`, orthonormalize again if the norm of the orthogonalized vector is
much smaller than the norm of the original vector.
reiteration_threshold
If `reiterate` is `True`, re-orthonormalize if the ratio between the norms of
the orthogonalized vector and the original vector is smaller than this value.
check
If `True`, check if the resulting |VectorArray| is really orthonormal.
check_tol
Tolerance for the check.
copy
If `True`, create a copy of `A` instead of modifying `A` in-place.
find_duplicates
unused
Returns
-------
The orthonormalized |VectorArray|.
"""
logger = getLogger('pymor.algorithms.gram_schmidt.gram_schmidt')
if copy:
A = A.copy()
# main loop
remove = []
for i in range(offset, len(A)):
# first calculate norm
if product is None:
initial_norm = A[i].l2_norm()[0]
else:
initial_norm = np.sqrt(product.pairwise_apply2(A[i], A[i]))[0]
if initial_norm < atol:
logger.info("Removing vector {} of norm {}".format(
i, initial_norm))
remove.append(i)
continue
if i == 0:
A[0].scal(1 / initial_norm)
else:
first_iteration = True
norm = initial_norm
# If reiterate is True, reiterate as long as the norm of the vector changes
# strongly during orthonormalization (due to Andreas Buhr).
while first_iteration or reiterate and norm / old_norm < reiteration_threshold:
if first_iteration:
first_iteration = False
else:
logger.info('Orthonormalizing vector {} again'.format(i))
# orthogonalize to all vectors left
for j in range(i):
if j in remove:
continue
if product is None:
p = A[i].pairwise_dot(A[j])[0]
else:
p = product.pairwise_apply2(A[i], A[j])[0]
A[i].axpy(-p, A[j])
# calculate new norm
if product is None:
old_norm, norm = norm, A[i].l2_norm()[0]
else:
old_norm, norm = norm, np.sqrt(
product.pairwise_apply2(A[i], A[i])[0])
# remove vector if it got too small:
if norm / initial_norm < rtol:
logger.info(
"Removing linear dependent vector {}".format(i))
remove.append(i)
break
if norm > 0:
A[i].scal(1 / norm)
if remove:
del A[remove]
if check:
if product:
error_matrix = product.apply2(A[offset:len(A)], A)
else:
error_matrix = A[offset:len(A)].dot(A)
error_matrix[:len(A) - offset,
offset:len(A)] -= np.eye(len(A) - offset)
if error_matrix.size > 0:
err = np.max(np.abs(error_matrix))
if err >= check_tol:
raise AccuracyError(
'result not orthogonal (max err={})'.format(err))
return A
'estimator_return': 'eta_red',
'num_test_samples': 10,
'estimate_some_errors': True,
'uniform_enrichment_factor': 10,
'local_indicators': 'eta_red',
'marking_strategy': 'doerfler_and_age',
'marking_max_age': 4,
'doerfler_marking_theta': 0.33,
'local_boundary_values': 'dirichlet',
'online_target_error': 0.05,
'online_max_extensions': 20
}
DATASET_ID = 'OS2015_SISC__6_2__academic_example'
pymor.core.logger.MAX_HIERACHY_LEVEL = 2
getLogger('pymor.WrappedDiscretization').setLevel('WARN')
getLogger('pymor.algorithms').setLevel('INFO')
getLogger('dune.pymor.discretizations').setLevel('WARN')
if __name__ == '__main__':
logfile = NamedTemporaryFile(delete=False).name
pymor.core.logger.FILENAME = logfile
new_dataset(DATASET_ID, **config)
detailed_data = prepare(config)
print('')
offline_data = offline_phase(config, detailed_data)
print('')
_ = online_phase(config, detailed_data, offline_data)
def lgmres(A, b, x0=None, tol=1e-5, maxiter=1000, M=None, callback=None,
inner_m=30, outer_k=3, outer_v=None, store_outer_Av=True):
if A.source != A.range:
raise InversionError
from scipy.linalg.basic import lstsq
x = A.source.zeros() if x0 is None else x0.copy()
# psolve = M.matvec
if outer_v is None:
outer_v = []
b_norm = b.l2_norm()[0]
if b_norm == 0:
b_norm = 1
for k_outer in xrange(maxiter):
r_outer = A.apply(x) - b
# -- callback
if callback is not None:
callback(x)
# -- check stopping condition
r_norm = r_outer.l2_norm()[0]
if r_norm < tol * b_norm or r_norm < tol:
break
# -- inner LGMRES iteration
vs0 = -r_outer # -psolve(r_outer)
inner_res_0 = vs0.l2_norm()[0]
if inner_res_0 == 0:
rnorm = r_outer.l2_norm()[0]
raise RuntimeError("Preconditioner returned a zero vector; "
"|v| ~ %.1g, |M v| = 0" % rnorm)
vs0.scal(1.0/inner_res_0)
hs = []
vs = [vs0]
ws = []
y = None
for j in xrange(1, 1 + inner_m + len(outer_v)):
# -- Arnoldi process:
#
# Build an orthonormal basis V and matrices W and H such that
# A W = V H
# Columns of W, V, and H are stored in `ws`, `vs` and `hs`.
#
# The first column of V is always the residual vector, `vs0`;
# V has *one more column* than the other of the three matrices.
#
# The other columns in V are built by feeding in, one
# by one, some vectors `z` and orthonormalizing them
# against the basis so far. The trick here is to
# feed in first some augmentation vectors, before
# starting to construct the Krylov basis on `v0`.
#
# It was shown in [BJM]_ that a good choice (the LGMRES choice)
# for these augmentation vectors are the `dx` vectors obtained
# from a couple of the previous restart cycles.
#
# Note especially that while `vs0` is always the first
# column in V, there is no reason why it should also be
# the first column in W. (In fact, below `vs0` comes in
# W only after the augmentation vectors.)
#
# The rest of the algorithm then goes as in GMRES, one
# solves a minimization problem in the smaller subspace
# spanned by W (range) and V (image).
#
# XXX: Below, I'm lazy and use `lstsq` to solve the
# small least squares problem. Performance-wise, this
# is in practice acceptable, but it could be nice to do
# it on the fly with Givens etc.
#
# ++ evaluate
v_new = None
if j < len(outer_v) + 1:
z, v_new = outer_v[j-1]
elif j == len(outer_v) + 1:
z = vs0
else:
z = vs[-1]
if v_new is None:
v_new = A.apply(z) # psolve(matvec(z))
else:
# Note: v_new is modified in-place below. Must make a
# copy to ensure that the outer_v vectors are not
# clobbered.
v_new = v_new.copy()
# ++ orthogonalize
hcur = []
for v in vs:
alpha = v.dot(v_new)[0, 0]
hcur.append(alpha)
v_new.axpy(-alpha, v) # v_new -= alpha*v
hcur.append(v_new.l2_norm()[0])
if hcur[-1] == 0:
# Exact solution found; bail out.
# Zero basis vector (v_new) in the least-squares problem
# does no harm, so we can just use the same code as usually;
# it will give zero (inner) residual as a result.
bailout = True
else:
bailout = False
v_new.scal(1.0/hcur[-1])
vs.append(v_new)
hs.append(hcur)
ws.append(z)
# XXX: Ugly: should implement the GMRES iteration properly,
# with Givens rotations and not using lstsq. Instead, we
# spare some work by solving the LSQ problem only every 5
# iterations.
if not bailout and j % 5 != 1 and j < inner_m + len(outer_v) - 1:
continue
# -- GMRES optimization problem
hess = np.zeros((j+1, j))
e1 = np.zeros((j+1,))
e1[0] = inner_res_0
for q in xrange(j):
hess[:(q+2), q] = hs[q]
y, resids, rank, s = lstsq(hess, e1)
inner_res = np.linalg.norm(np.dot(hess, y) - e1)
# -- check for termination
if inner_res < tol * inner_res_0:
break
# -- GMRES terminated: eval solution
dx = ws[0]*y[0]
for w, yc in zip(ws[1:], y[1:]):
dx.axpy(yc, w) # dx += w*yc
# -- Store LGMRES augmentation vectors
nx = dx.l2_norm()[0]
if store_outer_Av:
q = np.dot(hess, y)
ax = vs[0]*q[0]
for v, qc in zip(vs[1:], q[1:]):
ax.axpy(qc, v)
outer_v.append((dx * (1./nx), ax * (1./nx)))
else:
outer_v.append((dx * (1./nx), None))
# -- Retain only a finite number of augmentation vectors
while len(outer_v) > outer_k:
del outer_v[0]
# -- Apply step
x += dx
else:
# didn't converge ...
return x, maxiter
getLogger('pymor.algorithms.genericsolvers.lgmres').info('Converged after {} iterations'.format(k_outer + 1))
return x, 0
def gram_schmidt_biorth(V,
W,
product=None,
reiterate=True,
reiteration_threshold=1e-1,
check=True,
check_tol=1e-3,
copy=True):
"""Biorthonormalize a pair of |VectorArrays| using the biorthonormal Gram-Schmidt process.
See Algorithm 1 in [BKS11]_.
.. [BKS11] P. Benner, M. Köhler, J. Saak,
Sparse-Dense Sylvester Equations in :math:`\mathcal{H}_2`-Model Order Reduction,
Max Planck Institute Magdeburg Preprint, available from http://www.mpi-magdeburg.mpg.de/preprints/,
2011.
Parameters
----------
V, W
The |VectorArrays| which are to be biorthonormalized.
product
The inner product |Operator| w.r.t. which to biorthonormalize.
If `None`, the Euclidean product is used.
reiterate
If `True`, orthonormalize again if the norm of the orthogonalized vector is
much smaller than the norm of the original vector.
reiteration_threshold
If `reiterate` is `True`, re-orthonormalize if the ratio between the norms of
the orthogonalized vector and the original vector is smaller than this value.
check
If `True`, check if the resulting |VectorArray| is really orthonormal.
check_tol
Tolerance for the check.
copy
If `True`, create a copy of `V` and `W` instead of modifying `V` and `W` in-place.
Returns
-------
The biorthonormalized |VectorArrays|.
"""
assert V.space == W.space
assert len(V) == len(W)
logger = getLogger('pymor.algorithms.gram_schmidt.gram_schmidt_biorth')
if copy:
V = V.copy()
W = W.copy()
# main loop
for i in range(len(V)):
# calculate norm of V[i]
if product is None:
initial_norm = V[i].l2_norm()[0]
else:
initial_norm = np.sqrt(product.pairwise_apply2(V[i], V[i]))[0]
# project V[i]
if i == 0:
V[0].scal(1 / initial_norm)
else:
first_iteration = True
norm = initial_norm
# If reiterate is True, reiterate as long as the norm of the vector changes
# strongly during projection.
while first_iteration or reiterate and norm / old_norm < reiteration_threshold:
if first_iteration:
first_iteration = False
else:
logger.info('Projecting vector V[{}] again'.format(i))
for j in range(i):
# project by (I - V[j] * W[j]^T * E)
if product is None:
p = W[j].pairwise_dot(V[i])[0]
else:
p = product.pairwise_apply2(W[j], V[i])[0]
V[i].axpy(-p, V[j])
# calculate new norm
if product is None:
old_norm, norm = norm, V[i].l2_norm()[0]
else:
old_norm, norm = norm, np.sqrt(
product.pairwise_apply2(V[i], V[i])[0])
if norm > 0:
V[i].scal(1 / norm)
# calculate norm of W[i]
if product is None:
initial_norm = W[i].l2_norm()[0]
else:
initial_norm = np.sqrt(product.pairwise_apply2(W[i], W[i]))[0]
# project W[i]
if i == 0:
W[0].scal(1 / initial_norm)
else:
first_iteration = True
norm = initial_norm
# If reiterate is True, reiterate as long as the norm of the vector changes
# strongly during projection.
while first_iteration or reiterate and norm / old_norm < reiteration_threshold:
if first_iteration:
first_iteration = False
else:
logger.info('Projecting vector W[{}] again'.format(i))
for j in range(i):
# project by (I - W[j] * V[j]^T * E)
if product is None:
p = V[j].pairwise_dot(W[i])[0]
else:
p = product.pairwise_apply2(V[j], W[i])[0]
W[i].axpy(-p, W[j])
# calculate new norm
if product is None:
old_norm, norm = norm, W[i].l2_norm()[0]
else:
old_norm, norm = norm, np.sqrt(
product.pairwise_apply2(W[i], W[i])[0])
if norm > 0:
W[i].scal(1 / norm)
# rescale V[i]
if product is None:
p = W[i].pairwise_dot(V[i])[0]
else:
p = product.pairwise_apply2(W[i], V[i])[0]
V[i].scal(1 / p)
if check:
if product:
error_matrix = product.apply2(W, V)
else:
error_matrix = W.dot(V)
error_matrix -= np.eye(len(V))
if error_matrix.size > 0:
err = np.max(np.abs(error_matrix))
if err >= check_tol:
raise AccuracyError(
'Result not biorthogonal (max err={})'.format(err))
return V, W
def __init__(self, grid, U, bounding_box, codim, title, legend, separate_colorbars, rescale_colorbars, backend):
assert isinstance(U, VectorArrayInterface) and hasattr(U, 'data') \
or (isinstance(U, tuple) and all(isinstance(u, VectorArrayInterface) and hasattr(u, 'data') for u in U)
and all(len(u) == len(U[0]) for u in U))
U = (U.data.astype(np.float64, copy=False),) if hasattr(U, 'data') else \
tuple(u.data.astype(np.float64, copy=False) for u in U)
if isinstance(legend, str):
legend = (legend,)
assert legend is None or isinstance(legend, tuple) and len(legend) == len(U)
if backend == 'gl':
widget = GLPatchWidget
cbar_widget = ColorBarWidget
else:
widget = MatplotlibPatchWidget
cbar_widget = None
if not separate_colorbars and len(U) > 1:
l = getLogger('pymor.gui.qt.visualize_patch')
l.warn('separate_colorbars=False not supported for matplotlib backend')
separate_colorbars = True
class PlotWidget(QWidget):
def __init__(self):
super().__init__()
if separate_colorbars:
if rescale_colorbars:
self.vmins = tuple(np.min(u[0]) for u in U)
self.vmaxs = tuple(np.max(u[0]) for u in U)
else:
self.vmins = tuple(np.min(u) for u in U)
self.vmaxs = tuple(np.max(u) for u in U)
else:
if rescale_colorbars:
self.vmins = (min(np.min(u[0]) for u in U),) * len(U)
self.vmaxs = (max(np.max(u[0]) for u in U),) * len(U)
else:
self.vmins = (min(np.min(u) for u in U),) * len(U)
self.vmaxs = (max(np.max(u) for u in U),) * len(U)
layout = QHBoxLayout()
plot_layout = QGridLayout()
self.colorbarwidgets = [cbar_widget(self, vmin=vmin, vmax=vmax) if cbar_widget else None
for vmin, vmax in zip(self.vmins, self.vmaxs)]
plots = [widget(self, grid, vmin=vmin, vmax=vmax, bounding_box=bounding_box, codim=codim)
for vmin, vmax in zip(self.vmins, self.vmaxs)]
if legend:
for i, plot, colorbar, l in zip(range(len(plots)), plots, self.colorbarwidgets, legend):
subplot_layout = QVBoxLayout()
caption = QLabel(l)
caption.setAlignment(Qt.AlignHCenter)
subplot_layout.addWidget(caption)
if not separate_colorbars or backend == 'matplotlib':
subplot_layout.addWidget(plot)
else:
hlayout = QHBoxLayout()
hlayout.addWidget(plot)
if colorbar:
hlayout.addWidget(colorbar)
subplot_layout.addLayout(hlayout)
plot_layout.addLayout(subplot_layout, int(i/columns), (i % columns), 1, 1)
else:
for i, plot, colorbar in zip(range(len(plots)), plots, self.colorbarwidgets):
if not separate_colorbars or backend == 'matplotlib':
plot_layout.addWidget(plot, int(i/columns), (i % columns), 1, 1)
else:
hlayout = QHBoxLayout()
hlayout.addWidget(plot)
if colorbar:
hlayout.addWidget(colorbar)
plot_layout.addLayout(hlayout, int(i/columns), (i % columns), 1, 1)
layout.addLayout(plot_layout)
if not separate_colorbars:
layout.addWidget(self.colorbarwidgets[0])
for w in self.colorbarwidgets[1:]:
w.setVisible(False)
self.setLayout(layout)
self.plots = plots
def set(self, U, ind):
if rescale_colorbars:
if separate_colorbars:
self.vmins = tuple(np.min(u[ind]) for u in U)
self.vmaxs = tuple(np.max(u[ind]) for u in U)
else:
self.vmins = (min(np.min(u[ind]) for u in U),) * len(U)
self.vmaxs = (max(np.max(u[ind]) for u in U),) * len(U)
for u, plot, colorbar, vmin, vmax in zip(U, self.plots, self.colorbarwidgets, self.vmins,
self.vmaxs):
plot.set(u[ind], vmin=vmin, vmax=vmax)
if colorbar:
colorbar.set(vmin=vmin, vmax=vmax)
super().__init__(U, PlotWidget(), title=title, length=len(U[0]))
self.grid = grid
self.codim = codim
# This file is part of the pyMOR project (http://www.pymor.org).
# Copyright Holders: Rene Milk, Stephan Rave, Felix Schindler
# License: BSD 2-Clause License (http://opensource.org/licenses/BSD-2-Clause)
from __future__ import absolute_import, division, print_function
import numpy as np
from pymor.core.logger import getLogger
logger = getLogger(__name__)
def inverse_relation(R, size_rhs=None, with_indices=False):
"""Computes the inverse relation of a relation.
If `r` is a relation, then the inverse relation `ri` is defined by
x ri y <=> y r x
Parameters
----------
R
2D |NumPy array| of integers representing a relation r on the
natural numbers via ::
x r y <=> (x < R.size[0] and y in R[x]).
Rows of `R` which are to short are padded with -1.
size_rhs
Can be provided for speedup. Has to be greater than `R.max()`.
with_indices
def greedy(discretization, reductor, samples, initial_basis=None, use_estimator=True, error_norm=None,
extension_algorithm=gram_schmidt_basis_extension, atol=None, rtol=None, max_extensions=None,
pool=None):
"""Greedy basis generation algorithm.
This algorithm generates a reduced basis by iteratively adding the
worst approximated solution snapshot for a given training set to the
reduced basis. The approximation error is computed either by directly
comparing the reduced solution to the detailed solution or by using
an error estimator (`use_estimator == True`). The reduction and basis
extension steps are performed by calling the methods provided by the
`reductor` and `extension_algorithm` arguments.
Parameters
----------
discretization
The |Discretization| to reduce.
reductor
Reductor for reducing the given |Discretization|. This has to be a
function of the form `reductor(discretization, basis, extends=None)`.
If your reductor takes more arguments, use, e.g., :func:`functools.partial`.
The method has to return a tuple
`(reduced_discretization, reconstructor, reduction_data)`.
In case the last basis extension was `hierarchic` (see
`extension_algorithm`), the extends argument is set to
`(last_reduced_discretization, last_reconstructor, last_reduction_data)`
which can be used by the reductor to speed up the reduction
process. For an example see
:func:`~pymor.reductors.coercive.reduce_coercive`.
samples
The set of |Parameter| samples on which to perform the greedy search.
initial_basis
The initial reduced basis with which the algorithm starts. If `None`,
an empty basis is used as initial basis.
use_estimator
If `True`, use `reduced_discretization.estimate()` to estimate the
errors on the sample set. Otherwise a detailed simulation is
performed to calculate the error.
error_norm
If `use_estimator == False`, use this function to calculate the
norm of the error. If `None`, the Euclidean norm is used.
extension_algorithm
The extension algorithm to be used to extend the current reduced
basis with the maximum error snapshot. This has to be a function
of the form `extension_algorithm(old_basis, new_vector)`, which
returns a tuple `(new_basis, extension_data)`, where
`extension_data` is a dict at least containing the key
`hierarchic`. `hierarchic` should be set to `True` if `new_basis`
contains `old_basis` as its first vectors.
atol
If not `None`, stop the algorithm if the maximum (estimated) error
on the sample set drops below this value.
rtol
If not `None`, stop the algorithm if the maximum (estimated)
relative error on the sample set drops below this value.
max_extensions
If not `None`, stop the algorithm after `max_extensions` extension
steps.
pool
If not `None`, the |WorkerPool| to use for parallelization.
Returns
-------
Dict with the following fields:
:basis: The reduced basis.
:reduced_discretization: The reduced |Discretization| obtained for the
computed basis.
:reconstructor: Reconstructor for `reduced_discretization`.
:max_errs: Sequence of maximum errors during the greedy run.
:max_err_mus: The parameters corresponding to `max_errs`.
"""
logger = getLogger('pymor.algorithms.greedy.greedy')
samples = list(samples)
sample_count = len(samples)
logger.info('Started greedy search on {} samples'.format(sample_count))
if pool is None or pool is dummy_pool:
pool = dummy_pool
else:
logger.info('Using pool of {} workers for parallel greedy search'.format(len(pool)))
with RemoteObjectManager() as rom:
# Push everything we need during the greedy search to the workers.
# Distribute the training set evenly among the workes.
if not use_estimator:
rom.manage(pool.push(discretization))
if error_norm:
rom.manage(pool.push(error_norm))
samples = rom.manage(pool.scatter_list(samples))
basis = initial_basis
tic = time.time()
extensions = 0
max_errs = []
max_err_mus = []
hierarchic = False
rd, rc, reduction_data = None, None, None
while True:
with logger.block('Reducing ...'):
rd, rc, reduction_data = reductor(discretization, basis) if not hierarchic \
else reductor(discretization, basis, extends=(rd, rc, reduction_data))
if sample_count == 0:
logger.info('There is nothing else to do for empty samples.')
return {'basis': basis, 'reduced_discretization': rd, 'reconstructor': rc,
'max_errs': [], 'max_err_mus': [], 'extensions': 0,
'time': time.time() - tic, 'reduction_data': reduction_data}
with logger.block('Estimating errors ...'):
if use_estimator:
errors, mus = list(zip(*pool.apply(_estimate, rd=rd, d=None, rc=None, samples=samples, error_norm=None)))
else:
# FIXME: Always communicating rc may become a bottleneck in some use cases.
# Add special treatment for GenericRBReconstructor?
errors, mus = list(zip(*pool.apply(_estimate, rd=rd, d=discretization, rc=rc,
samples=samples, error_norm=error_norm)))
max_err_ind = np.argmax(errors)
max_err, max_err_mu = errors[max_err_ind], mus[max_err_ind]
max_errs.append(max_err)
max_err_mus.append(max_err_mu)
logger.info('Maximum error after {} extensions: {} (mu = {})'.format(extensions, max_err, max_err_mu))
if atol is not None and max_err <= atol:
logger.info('Absolute error tolerance ({}) reached! Stoping extension loop.'.format(atol))
break
if rtol is not None and max_err / max_errs[0] <= rtol:
logger.info('Relative error tolerance ({}) reached! Stoping extension loop.'.format(rtol))
break
with logger.block('Computing solution snapshot for mu = {} ...'.format(max_err_mu)):
U = discretization.solve(max_err_mu)
with logger.block('Extending basis with solution snapshot ...'):
try:
basis, extension_data = extension_algorithm(basis, U)
except ExtensionError:
logger.info('Extension failed. Stopping now.')
break
extensions += 1
if 'hierarchic' not in extension_data:
logger.warn('Extension algorithm does not report if extension was hierarchic. Assuming it was\'nt ..')
hierarchic = False
else:
hierarchic = extension_data['hierarchic']
logger.info('')
if max_extensions is not None and extensions >= max_extensions:
logger.info('Maximum number of {} extensions reached.'.format(max_extensions))
with logger.block('Reducing once more ...'):
rd, rc, reduction_data = reductor(discretization, basis) if not hierarchic \
else reductor(discretization, basis, extends=(rd, rc, reduction_data))
break
tictoc = time.time() - tic
logger.info('Greedy search took {} seconds'.format(tictoc))
return {'basis': basis, 'reduced_discretization': rd, 'reconstructor': rc,
'max_errs': max_errs, 'max_err_mus': max_err_mus, 'extensions': extensions,
'time': tictoc, 'reduction_data': reduction_data}
def init_grid_and_problem(config, mu_bar=1, mu_hat=1, mpi_comm=MPI.COMM_WORLD):
# assert mpi_comm.Get_size() < MPI.COMM_WORLD.Get_size() or mpi_comm.Get_size() == 1
logger = getLogger('OS2015_academic_problem.OS2015_academic_problem')
logger.info('initializing grid and problem ... ')
lower_left = [-1, -1]
upper_right = [1, 1]
inner_boundary_id = 18446744073709551573
grid = make_grid((lower_left, upper_right),
config['num_subdomains'],
config['half_num_fine_elements_per_subdomain_and_dim'],
inner_boundary_id,
mpi_comm=mpi_comm)
grid_info(logger.error, grid, mpi_comm)
all_dirichlet_boundary_info = make_boundary_info(
grid, {'type': 'xt.grid.boundaryinfo.alldirichlet'})
cos = '(cos(0.5*pi*x[0])*cos(0.5*pi*x[1]))'
diffusion_functions = [
make_expression_function_1x1(grid,
'x',
'1+{}'.format(cos),
order=2,
name='lambda_0'),
make_expression_function_1x1(grid,
'x',
'-1*{}'.format(cos),
order=2,
name='lambda_1')
]
# diffusion_functions = [make_expression_function_1x1(
# grid, 'x', '1', order=2, name='lambda_0'),
# make_expression_function_1x1(grid, 'x', 'x[0]', order=2, name='lambda_1')]
parameter_type = {'diffusion': (1, )}
coefficients = [
ExpressionParameterFunctional('1.', parameter_type),
ExpressionParameterFunctional('diffusion', parameter_type)
]
kappa = make_constant_function_2x2(grid, [[1., 0.], [0., 1.]],
name='kappa')
f = make_expression_function_1x1(grid,
'x',
'0.5*pi*pi*{}'.format(cos),
order=2,
name='f')
mbc = '1+(1-{})*{}'.format(mu_bar, cos)
lambda_bar = make_expression_function_1x1(grid,
'x',
mbc,
order=2,
name='lambda_bar')
lambda_hat = make_expression_function_1x1(grid,
'x',
mbc,
order=2,
name='lambda_hat')
return {
'grid': grid,
'mpi_comm': mpi_comm,
'boundary_info': all_dirichlet_boundary_info,
'inner_boundary_id': inner_boundary_id,
'lambda': {
'functions': diffusion_functions,
'coefficients': coefficients
},
'lambda_bar': lambda_bar,
'lambda_hat': lambda_hat,
'kappa': kappa,
'f': f,
'parameter_type': parameter_type,
'mu_bar': (mu_bar, ),
'mu_hat': (mu_hat, ),
'mu_min': (min(0.1, mu_bar, mu_hat), ),
'mu_max': (max(1, mu_bar, mu_hat), ),
'parameter_range': (min(0.1, mu_bar, mu_hat), max(1, mu_bar, mu_hat))
}
