def extend_basis(U, basis, product=None, method='gram_schmidt', pod_modes=1, pod_orthonormalize=True, copy_U=True):
assert method in ('trivial', 'gram_schmidt', 'pod')
basis_length = len(basis)
if method == 'trivial':
remove = set()
for i in range(len(U)):
if np.any(almost_equal(U[i], basis)):
remove.add(i)
basis.append(U[[i for i in range(len(U)) if i not in remove]],
remove_from_other=(not copy_U))
elif method == 'gram_schmidt':
basis.append(U, remove_from_other=(not copy_U))
gram_schmidt(basis, offset=basis_length, product=product, copy=False, check=False)
elif method == 'pod':
U_proj_err = U - basis.lincomb(U.inner(basis, product))
basis.append(pod(U_proj_err, modes=pod_modes, product=product, orth_tol=np.inf)[0])
if pod_orthonormalize:
gram_schmidt(basis, offset=basis_length, product=product, copy=False, check=False)
if len(basis) <= basis_length:
raise ExtensionError
def build_basis(self):
"""Compute a reduced basis using proper orthogonal decomposition."""
self.training_data = []
with self.logger.block('Building reduced basis ...'):
# compute snapshots for POD and training of neural networks
with self.logger.block('Computing training snapshots ...'):
U = self.fom.solution_space.empty()
for mu in self.training_set:
U.append(self.fom.solve(mu))
# compute reduced basis via POD
reduced_basis, svals = pod(U,
modes=self.basis_size,
rtol=self.rtol / 2.,
atol=self.atol / 2.,
l2_err=self.l2_err / 2.,
**(self.pod_params or {}))
# determine the coefficients of the full-order solutions in the reduced basis to obtain the
# training data; convert everything into tensors that are compatible with PyTorch
for mu, u in zip(self.training_set, U):
mu_tensor = torch.DoubleTensor(mu.to_numpy())
u_tensor = torch.DoubleTensor(reduced_basis.inner(u)[:, 0])
self.training_data.append((mu_tensor, u_tensor))
# compute mean square loss
mean_square_loss = (sum(U.norm2()) - sum(svals**2)) / len(U)
return reduced_basis, mean_square_loss
def build_basis(self):
"""Compute a reduced basis using proper orthogonal decomposition."""
with self.logger.block('Building reduced basis ...'):
# compute snapshots for POD and training of neural networks
with self.logger.block('Computing training snapshots ...'):
U = self.fom.solution_space.empty()
for mu in self.training_set:
u = self.fom.solve(mu)
if hasattr(self, 'nt'):
assert self.nt == len(u)
else:
self.nt = len(u)
U.append(u)
# compute reduced basis via POD
reduced_basis, svals = pod(U,
modes=self.basis_size,
rtol=self.rtol / 2.,
atol=self.atol / 2.,
l2_err=self.l2_err / 2.,
**(self.pod_params or {}))
self.training_data = []
for i, mu in enumerate(self.training_set):
sample = self._compute_sample(
mu, U[i * self.nt:(i + 1) * self.nt], reduced_basis)
self.training_data.extend(sample)
# compute mean square loss
mean_square_loss = (sum(U.norm2()) - sum(svals**2)) / len(U)
return reduced_basis, mean_square_loss
def reduce_pod(d, reductor, snapshots_per_block, basis_size):
from pymor.algorithms.pod import pod
tic = time.time()
training_set = d.parameter_space.sample_uniformly(snapshots_per_block)
print('Solving on training set ...')
snapshots = d.operator.source.empty(reserve=len(training_set))
for mu in training_set:
snapshots.append(d.solve(mu))
print('Performing POD ...')
basis, singular_values = pod(snapshots, modes=basis_size, product=reductor.product)
print('Reducing ...')
reductor.extend_basis(basis, 'trivial')
rd = reductor.reduce()
elapsed_time = time.time() - tic
# generate summary
real_rb_size = rd.solution_space.dim
training_set_size = len(training_set)
summary = '''POD basis generation:
size of training set: {training_set_size}
prescribed basis size: {basis_size}
actual basis size: {real_rb_size}
elapsed time: {elapsed_time}
'''.format(**locals())
return rd, summary
def extend_basis(U, basis, product=None, method='gram_schmidt', pod_modes=1, pod_orthonormalize=True, copy_U=True):
assert method in ('trivial', 'gram_schmidt', 'pod')
basis_length = len(basis)
if method == 'trivial':
remove = set()
for i in range(len(U)):
if np.any(almost_equal(U[i], basis)):
remove.add(i)
basis.append(U[[i for i in range(len(U)) if i not in remove]],
remove_from_other=(not copy_U))
elif method == 'gram_schmidt':
basis.append(U, remove_from_other=(not copy_U))
gram_schmidt(basis, offset=basis_length, product=product, copy=False, check=False)
elif method == 'pod':
U_proj_err = U - basis.lincomb(U.inner(basis, product))
basis.append(pod(U_proj_err, modes=pod_modes, product=product, orthonormalize=False)[0])
if pod_orthonormalize:
gram_schmidt(basis, offset=basis_length, product=product, copy=False, check=False)
if len(basis) <= basis_length:
raise ExtensionError
def reduce_pod(fom, reductor, snapshots_per_block, basis_size):
from pymor.algorithms.pod import pod
tic = time.time()
training_set = fom.parameter_space.sample_uniformly(snapshots_per_block)
print('Solving on training set ...')
snapshots = fom.operator.source.empty(reserve=len(training_set))
for mu in training_set:
snapshots.append(fom.solve(mu))
print('Performing POD ...')
basis, singular_values = pod(snapshots, modes=basis_size, product=reductor.products['RB'])
print('Reducing ...')
reductor.extend_basis(basis, method='trivial')
rom = reductor.reduce()
elapsed_time = time.time() - tic
# generate summary
real_rb_size = rom.solution_space.dim
training_set_size = len(training_set)
summary = f'''POD basis generation:
size of training set: {training_set_size}
prescribed basis size: {basis_size}
actual basis size: {real_rb_size}
elapsed time: {elapsed_time}
'''
return rom, summary
def default_pod_method(U, eps, is_root_node, product):
return pod(U,
atol=0.,
rtol=0.,
l2_err=eps,
product=product,
orth_tol=None if is_root_node else np.inf)
def default_pod_method(U, eps, is_root_node, product):
return pod(U,
atol=0.,
rtol=0.,
l2_err=eps,
product=product,
orthonormalize=is_root_node,
check=False)
def test_pod_with_product(operator_with_arrays_and_products, method):
_, _, A, _, p, _ = operator_with_arrays_and_products
B = A.copy()
with log_levels({"pymor.algorithms": "ERROR"}):
U, s = pod(A, product=p, method=method)
assert np.all(almost_equal(A, B))
assert len(U) == len(s)
assert np.allclose(U.gramian(p), np.eye(len(s)))
def test_pod_with_product(operator_with_arrays_and_products, method):
_, _, A, _, p, _ = operator_with_arrays_and_products
print(type(A))
print(A.dim, len(A))
B = A.copy()
U, s = pod(A, product=p, method=method)
assert np.all(almost_equal(A, B))
assert len(U) == len(s)
assert np.allclose(U.gramian(p), np.eye(len(s)))
def test_pod(vector_array, method):
A = vector_array
print(type(A))
print(A.dim, len(A))
B = A.copy()
U, s = pod(A, method=method)
assert np.all(almost_equal(A, B))
assert len(U) == len(s)
assert np.allclose(U.gramian(), np.eye(len(s)))
def test_pod(vector_array, method):
A = vector_array
# TODO assumption here masks a potential issue with the algorithm
# where it fails in internal lapack instead of a proper error
assume(len(A) > 1 or A.dim > 1)
assume(not contains_zero_vector(A, rtol=1e-13, atol=1e-13))
B = A.copy()
orth_tol = 1e-10
U, s = pod(A, method=method, orth_tol=orth_tol)
assert np.all(almost_equal(A, B))
assert len(U) == len(s)
assert np.allclose(U.gramian(), np.eye(len(s)), atol=orth_tol)
def hapod_demo(args):
args['--grid'] = int(args['--grid'])
args['--nt'] = int(args['--nt'])
args['--omega'] = float(args['--omega'])
args['--procs'] = int(args['--procs'])
args['--snap'] = int(args['--snap'])
args['--threads'] = int(args['--threads'])
args['TOL'] = float(args['TOL'])
args['DIST'] = int(args['DIST'])
args['INC'] = int(args['INC'])
assert args['--procs'] == 0 or args['--threads'] == 0
tol = args['TOL']
omega = args['--omega']
executor = ProcessPoolExecutor(args['--procs']) if args['--procs'] > 0 else \
ThreadPoolExecutor(args['--threads']) if args['--threads'] > 0 else \
None
p = burgers_problem_2d()
d, data = discretize_instationary_fv(p, grid_type=RectGrid, diameter=np.sqrt(2)/args['--grid'], nt=args['--nt'])
U = d.solution_space.empty()
for mu in d.parameter_space.sample_randomly(args['--snap']):
U.append(d.solve(mu))
tic = time()
pod_modes = pod(U, l2_err=tol * np.sqrt(len(U)), product=d.l2_product, check=False)[0]
pod_time = time() - tic
tic = time()
dist_modes = dist_vectorarray_hapod(args['DIST'], U, tol, omega, product=d.l2_product, executor=executor)[0]
dist_time = time() - tic
tic = time()
inc_modes = inc_vectorarray_hapod(args['INC'], U, tol, omega, product=d.l2_product)[0]
inc_time = time() - tic
print('Snapshot matrix: {} x {}'.format(U.dim, len(U)))
print(format_table([
['Method', 'Error', 'Modes', 'Time'],
['POD', np.linalg.norm(d.l2_norm(U-pod_modes.lincomb(d.l2_product.apply2(U, pod_modes)))/np.sqrt(len(U))),
len(pod_modes), pod_time],
['DIST HAPOD', np.linalg.norm(d.l2_norm(U-dist_modes.lincomb(d.l2_product.apply2(U, dist_modes)))/np.sqrt(len(U))),
len(dist_modes), dist_time],
['INC HAPOD', np.linalg.norm(d.l2_norm(U-inc_modes.lincomb(d.l2_product.apply2(U, inc_modes)))/np.sqrt(len(U))),
len(inc_modes), inc_time]]
))
def reduce_pod(d, reductor, snapshots_per_block, product_name, basis_size):
from pymor.algorithms.pod import pod
tic = time.time()
training_set = d.parameter_space.sample_uniformly(snapshots_per_block)
print("Solving on training set ...")
snapshots = d.operator.source.empty(reserve=len(training_set))
for mu in training_set:
snapshots.append(d.solve(mu))
if product_name == "h1":
pod_product = d.h1_0_semi_product
elif product_name == "euclidean":
pod_product = None
else:
assert False
print("Performing POD ...")
basis, singular_values = pod(snapshots, modes=basis_size, product=pod_product)
print("Reducing ...")
rd, rc, _ = reductor(d, basis)
elapsed_time = time.time() - tic
# generate summary
real_rb_size = rd.solution_space.dim
training_set_size = len(training_set)
summary = """POD basis generation:
size of training set: {training_set_size}
inner product for POD: {product_name}
prescribed basis size: {basis_size}
actual basis size: {real_rb_size}
elapsed time: {elapsed_time}
""".format(
**locals()
)
return rd, rc, summary
def reduce_pod(d, reductor, snapshots_per_block, product_name, basis_size):
from pymor.algorithms.pod import pod
tic = time.time()
training_set = d.parameter_space.sample_uniformly(snapshots_per_block)
print('Solving on training set ...')
snapshots = d.operator.source.empty(reserve=len(training_set))
for mu in training_set:
snapshots.append(d.solve(mu))
if product_name == 'h1':
pod_product = d.h1_0_semi_product
elif product_name == 'euclidean':
pod_product = None
else:
assert False
print('Performing POD ...')
basis, singular_values = pod(snapshots,
modes=basis_size,
product=pod_product)
print('Reducing ...')
rd, rc, _ = reductor(d, basis)
elapsed_time = time.time() - tic
# generate summary
real_rb_size = rd.solution_space.dim
training_set_size = len(training_set)
summary = '''POD basis generation:
size of training set: {training_set_size}
inner product for POD: {product_name}
prescribed basis size: {basis_size}
actual basis size: {real_rb_size}
elapsed time: {elapsed_time}
'''.format(**locals())
return rd, rc, summary
def thermalblock_demo(args):
args['XBLOCKS'] = int(args['XBLOCKS'])
args['YBLOCKS'] = int(args['YBLOCKS'])
args['--grid'] = int(args['--grid'])
args['SNAPSHOTS'] = int(args['SNAPSHOTS'])
args['RBSIZE'] = int(args['RBSIZE'])
args['--test'] = int(args['--test'])
args['--pod-norm'] = args['--pod-norm'].lower()
assert args['--pod-norm'] in {'trivial', 'h1'}
print('Solving on TriaGrid(({0},{0}))'.format(args['--grid']))
print('Setup Problem ...')
problem = ThermalBlockProblem(num_blocks=(args['XBLOCKS'], args['YBLOCKS']))
print('Discretize ...')
discretization, _ = discretize_elliptic_cg(problem, diameter=1. / args['--grid'])
print('The parameter type is {}'.format(discretization.parameter_type))
if args['--plot-solutions']:
print('Showing some solutions')
Us = tuple()
legend = tuple()
for mu in discretization.parameter_space.sample_randomly(2):
print('Solving for diffusion = \n{} ... '.format(mu['diffusion']))
sys.stdout.flush()
Us = Us + (discretization.solve(mu),)
legend = legend + (str(mu['diffusion']),)
discretization.visualize(Us, legend=legend, title='Detailed Solutions for different parameters', block=True)
print('RB generation ...')
tic = time.time()
print('Solving on training set ...')
S_train = list(discretization.parameter_space.sample_uniformly(args['SNAPSHOTS']))
snapshots = discretization.operator.source.empty(reserve=len(S_train))
for mu in S_train:
snapshots.append(discretization.solve(mu))
print('Performing POD ...')
pod_product = discretization.h1_product if args['--pod-norm'] == 'h1' else None
rb = pod(snapshots, modes=args['RBSIZE'], product=pod_product)[0]
print('Reducing ...')
reductor = reduce_generic_rb
rb_discretization, reconstructor, _ = reductor(discretization, rb)
toc = time.time()
t_offline = toc - tic
print('\nSearching for maximum error on random snapshots ...')
tic = time.time()
h1_err_max = -1
cond_max = -1
for mu in discretization.parameter_space.sample_randomly(args['--test']):
print('Solving RB-Scheme for mu = {} ... '.format(mu), end='')
URB = reconstructor.reconstruct(rb_discretization.solve(mu))
U = discretization.solve(mu)
h1_err = discretization.h1_norm(U - URB)[0]
cond = np.linalg.cond(rb_discretization.operator.assemble(mu)._matrix)
if h1_err > h1_err_max:
h1_err_max = h1_err
mumax = mu
if cond > cond_max:
cond_max = cond
cond_max_mu = mu
print('H1-error = {}, condition = {}'.format(h1_err, cond))
toc = time.time()
t_est = toc - tic
real_rb_size = len(rb)
print('''
*** RESULTS ***
Problem:
number of blocks: {args[XBLOCKS]}x{args[YBLOCKS]}
h: sqrt(2)/{args[--grid]}
POD basis generation:
number of snapshots: {args[SNAPSHOTS]}^({args[XBLOCKS]}x{args[YBLOCKS]})
pod norm: {args[--pod-norm]}
prescribed basis size: {args[RBSIZE]}
actual basis size: {real_rb_size}
elapsed time: {t_offline}
Stochastic error estimation:
number of samples: {args[--test]}
maximal H1-error: {h1_err_max} (mu = {mumax})
maximal condition of system matrix: {cond_max} (mu = {cond_max_mu})
elapsed time: {t_est}
'''.format(**locals()))
sys.stdout.flush()
if args['--plot-err']:
discretization.visualize((U, URB, U - URB), legend=('Detailed Solution', 'Reduced Solution', 'Error'),
title='Maximum Error Solution', separate_colorbars=True, block=True)
def reduce_pod(fom, reductor, snapshots, basis_size):
basis, singular_values = pod(snapshots, modes=basis_size)
reductor.extend_basis(basis, method='trivial')
rom = reductor.reduce()
return rom
fom = InstationaryModel(T=120.,
initial_data=q0_array,
operator=op,
rhs=None,
mass=None,
visualizer=vis,
time_stepper=ts,
num_values=20)
# Constant Timesteps
dt = fom.T / fom.time_stepper.nt
problem.claw.solver.dt = dt
U = fom.solve(mu=None) # Solve
modes, svals = pod(U)
print(np.shape(modes.to_numpy()))
print(svals)
## Write ROM to File
# writeROMtoFile(U, modes, outDir)
# Plot the singular values
plt.figure()
plt.semilogy(svals, linewidth=2, label='Singular Values')
plt.legend()
plt.show()
fom.visualize(U) # plot the solution
num_output_times = fom.time_stepper.nt
def hapod_demo(args):
args['--grid'] = int(args['--grid'])
args['--nt'] = int(args['--nt'])
args['--omega'] = float(args['--omega'])
args['--procs'] = int(args['--procs'])
args['--snap'] = int(args['--snap'])
args['--threads'] = int(args['--threads'])
args['TOL'] = float(args['TOL'])
args['DIST'] = int(args['DIST'])
args['INC'] = int(args['INC'])
assert args['--procs'] == 0 or args['--threads'] == 0
tol = args['TOL']
omega = args['--omega']
executor = ProcessPoolExecutor(args['--procs']) if args['--procs'] > 0 else \
ThreadPoolExecutor(args['--threads']) if args['--threads'] > 0 else \
None
p = burgers_problem_2d()
m, data = discretize_instationary_fv(p,
grid_type=RectGrid,
diameter=np.sqrt(2) / args['--grid'],
nt=args['--nt'])
U = m.solution_space.empty()
for mu in m.parameter_space.sample_randomly(args['--snap']):
U.append(m.solve(mu))
tic = time()
pod_modes = pod(U,
l2_err=tol * np.sqrt(len(U)),
product=m.l2_product,
check=False)[0]
pod_time = time() - tic
tic = time()
dist_modes = dist_vectorarray_hapod(args['DIST'],
U,
tol,
omega,
product=m.l2_product,
executor=executor)[0]
dist_time = time() - tic
tic = time()
inc_modes = inc_vectorarray_hapod(args['INC'],
U,
tol,
omega,
product=m.l2_product)[0]
inc_time = time() - tic
print(f'Snapshot matrix: {U.dim} x {len(U)}')
print(
format_table([
['Method', 'Error', 'Modes', 'Time'],
[
'POD',
np.linalg.norm(
m.l2_norm(U - pod_modes.lincomb(
m.l2_product.apply2(U, pod_modes))) / np.sqrt(len(U))),
len(pod_modes), pod_time
],
[
'DIST HAPOD',
np.linalg.norm(
m.l2_norm(U - dist_modes.lincomb(
m.l2_product.apply2(U, dist_modes))) /
np.sqrt(len(U))),
len(dist_modes), dist_time
],
[
'INC HAPOD',
np.linalg.norm(
m.l2_norm(U - inc_modes.lincomb(
m.l2_product.apply2(U, inc_modes))) / np.sqrt(len(U))),
len(inc_modes), inc_time
]
]))
def default_pod_method(U, eps, is_root_node, product):
return pod(U, atol=0., rtol=0.,
l2_err=eps, product=product,
orthonormalize=is_root_node, check=False)
visualizer=visualizer,
name='C++-Discretization',
cache_region=None)
return d
# discretize
d = discretize(50, 10000, 4)
# generate solution snapshots
snapshots = d.solution_space.empty()
for mu in d.parameter_space.sample_uniformly(2):
snapshots.append(d.solve(mu))
# apply POD
reduced_basis = pod(snapshots, 4)[0]
# reduce the model
rd, rc, _ = reduce_generic_rb(d, reduced_basis)
# stochastic error estimation
mu_max = None
err_max = -1.
for mu in d.parameter_space.sample_randomly(10):
U_RB = (rc.reconstruct(rd.solve(mu)))
U = d.solve(mu)
err = np.max((U_RB - U).l2_norm())
if err > err_max:
err_max = err
mu_max = mu
def deim(U, modes=None, error_norm=None, product=None):
"""Generate data for empirical interpolation using DEIM algorithm.
Given a |VectorArray| `U`, this method generates a collateral basis and
interpolation DOFs for empirical interpolation of the vectors contained in `U`.
The returned objects can also be used to instantiate an |EmpiricalInterpolatedOperator|.
The collateral basis is determined by the first :func:`~pymor.algorithms.pod.pod` modes of `U`.
Parameters
----------
U
A |VectorArray| of vectors to interpolate.
modes
Dimension of the collateral basis i.e. number of POD modes of the vectors in `U`.
error_norm
Norm w.r.t. which to calculate the interpolation error. If `None`, the Euclidean norm
is used.
product
Product |Operator| used for POD.
Returns
-------
interpolation_dofs
|NumPy array| of the DOFs at which the vectors are interpolated.
collateral_basis
|VectorArray| containing the generated collateral basis.
data
Dict containing the following fields:
:errors: Sequence of maximum approximation errors during greedy search.
"""
assert isinstance(U, VectorArrayInterface)
logger = getLogger("pymor.algorithms.ei.deim")
logger.info("Generating Interpolation Data ...")
collateral_basis = pod(U, modes, product=product)[0]
interpolation_dofs = np.zeros((0,), dtype=np.int32)
interpolation_matrix = np.zeros((0, 0))
errs = []
for i in xrange(len(collateral_basis)):
if len(interpolation_dofs) > 0:
coefficients = np.linalg.solve(
interpolation_matrix, collateral_basis.components(interpolation_dofs, ind=i).T
).T
U_interpolated = collateral_basis.lincomb(coefficients, ind=range(len(interpolation_dofs)))
ERR = collateral_basis.copy(ind=i)
ERR -= U_interpolated
else:
ERR = collateral_basis.copy(ind=i)
err = ERR.l2_norm() if error_norm is None else error_norm(ERR)
logger.info("Interpolation error for basis vector {}: {}".format(i, err))
# compute new interpolation dof and collateral basis vector
new_dof = ERR.amax()[0][0]
if new_dof in interpolation_dofs:
logger.info("DOF {} selected twice for interplation! Stopping extension loop.".format(new_dof))
break
interpolation_dofs = np.hstack((interpolation_dofs, new_dof))
interpolation_matrix = collateral_basis.components(interpolation_dofs, ind=range(len(interpolation_dofs))).T
errs.append(err)
logger.info("")
if len(interpolation_dofs) < len(collateral_basis):
collateral_basis.remove(ind=range(len(interpolation_dofs), len(collateral_basis)))
logger.info("Finished.".format(new_dof))
data = {"errors": errs}
return interpolation_dofs, collateral_basis, data
def fenics_nonlinear_demo(args):
DIM = int(args['DIM'])
N = int(args['N'])
ORDER = int(args['ORDER'])
from pymor.tools import mpi
if mpi.parallel:
from pymor.models.mpi import mpi_wrap_model
local_models = mpi.call(mpi.function_call_manage, discretize, DIM, N,
ORDER)
fom = mpi_wrap_model(local_models,
use_with=True,
pickle_local_spaces=False)
else:
fom = discretize(DIM, N, ORDER)
# ### ROM generation (POD/DEIM)
from pymor.algorithms.ei import ei_greedy
from pymor.algorithms.newton import newton
from pymor.algorithms.pod import pod
from pymor.operators.ei import EmpiricalInterpolatedOperator
from pymor.reductors.basic import StationaryRBReductor
U = fom.solution_space.empty()
residuals = fom.solution_space.empty()
for mu in fom.parameter_space.sample_uniformly(10):
UU, data = newton(fom.operator,
fom.rhs.as_vector(),
mu=mu,
rtol=1e-6,
return_residuals=True)
U.append(UU)
residuals.append(data['residuals'])
dofs, cb, _ = ei_greedy(residuals, rtol=1e-7)
ei_op = EmpiricalInterpolatedOperator(fom.operator,
collateral_basis=cb,
interpolation_dofs=dofs,
triangular=True)
rb, svals = pod(U, rtol=1e-7)
fom_ei = fom.with_(operator=ei_op)
reductor = StationaryRBReductor(fom_ei, rb)
rom = reductor.reduce()
# the reductor currently removes all solver_options so we need to add them again
rom = rom.with_(operator=rom.operator.with_(
solver_options=fom.operator.solver_options))
# ### ROM validation
import time
import numpy as np
# ensure that FFC is not called during runtime measurements
rom.solve(1)
errs = []
speedups = []
for mu in fom.parameter_space.sample_randomly(10):
tic = time.time()
U = fom.solve(mu)
t_fom = time.time() - tic
tic = time.time()
u_red = rom.solve(mu)
t_rom = time.time() - tic
U_red = reductor.reconstruct(u_red)
errs.append(((U - U_red).l2_norm() / U.l2_norm())[0])
speedups.append(t_fom / t_rom)
print(f'Maximum relative ROM error: {max(errs)}')
print(f'Median of ROM speedup: {np.median(speedups)}')
def deim(U, modes=None, error_norm=None, product=None):
"""Generate data for empirical interpolation using DEIM algorithm.
Given a |VectorArray| `U`, this method generates a collateral basis and
interpolation DOFs for empirical interpolation of the vectors contained in `U`.
The returned objects can be used to instantiate an |EmpiricalInterpolatedOperator|
(with `triangular=False`).
The collateral basis is determined by the first :func:`~pymor.algorithms.pod.pod` modes of `U`.
Parameters
----------
U
A |VectorArray| of vectors to interpolate.
modes
Dimension of the collateral basis i.e. number of POD modes of the vectors in `U`.
error_norm
Norm w.r.t. which to calculate the interpolation error. If `None`, the Euclidean norm
is used.
product
Inner product |Operator| used for the POD.
Returns
-------
interpolation_dofs
|NumPy array| of the DOFs at which the vectors are interpolated.
collateral_basis
|VectorArray| containing the generated collateral basis.
data
Dict containing the following fields:
:errors: Sequence of maximum approximation errors during greedy search.
"""
assert isinstance(U, VectorArrayInterface)
logger = getLogger('pymor.algorithms.ei.deim')
logger.info('Generating Interpolation Data ...')
collateral_basis = pod(U, modes, product=product)[0]
interpolation_dofs = np.zeros((0,), dtype=np.int32)
interpolation_matrix = np.zeros((0, 0))
errs = []
for i in range(len(collateral_basis)):
if len(interpolation_dofs) > 0:
coefficients = np.linalg.solve(interpolation_matrix,
collateral_basis[i].dofs(interpolation_dofs).T).T
U_interpolated = collateral_basis[:len(interpolation_dofs)].lincomb(coefficients)
ERR = collateral_basis[i].copy()
ERR -= U_interpolated
else:
ERR = collateral_basis[i].copy()
err = np.max(ERR.l2_norm() if error_norm is None else error_norm(ERR))
logger.info('Interpolation error for basis vector {}: {}'.format(i, err))
# compute new interpolation dof and collateral basis vector
new_dof = ERR.amax()[0][0]
if new_dof in interpolation_dofs:
logger.info('DOF {} selected twice for interplation! Stopping extension loop.'.format(new_dof))
break
interpolation_dofs = np.hstack((interpolation_dofs, new_dof))
interpolation_matrix = collateral_basis[:len(interpolation_dofs)].dofs(interpolation_dofs).T
errs.append(err)
logger.info('')
if len(interpolation_dofs) < len(collateral_basis):
del collateral_basis[len(interpolation_dofs):len(collateral_basis)]
logger.info('Finished.')
data = {'errors': errs}
return interpolation_dofs, collateral_basis, data
def main(
dim: int = Argument(..., help='Spatial dimension of the problem.'),
n: int = Argument(
..., help='Number of mesh intervals per spatial dimension.'),
order: int = Argument(..., help='Finite element order.'),
):
"""Reduces a FEniCS-based nonlinear diffusion problem using POD/DEIM."""
from pymor.tools import mpi
if mpi.parallel:
from pymor.models.mpi import mpi_wrap_model
local_models = mpi.call(mpi.function_call_manage, discretize, dim, n,
order)
fom = mpi_wrap_model(local_models,
use_with=True,
pickle_local_spaces=False)
else:
fom = discretize(dim, n, order)
parameter_space = fom.parameters.space((0, 1000.))
# ### ROM generation (POD/DEIM)
from pymor.algorithms.ei import ei_greedy
from pymor.algorithms.newton import newton
from pymor.algorithms.pod import pod
from pymor.operators.ei import EmpiricalInterpolatedOperator
from pymor.reductors.basic import StationaryRBReductor
U = fom.solution_space.empty()
residuals = fom.solution_space.empty()
for mu in parameter_space.sample_uniformly(10):
UU, data = newton(fom.operator,
fom.rhs.as_vector(),
mu=mu,
rtol=1e-6,
return_residuals=True)
U.append(UU)
residuals.append(data['residuals'])
dofs, cb, _ = ei_greedy(residuals, rtol=1e-7)
ei_op = EmpiricalInterpolatedOperator(fom.operator,
collateral_basis=cb,
interpolation_dofs=dofs,
triangular=True)
rb, svals = pod(U, rtol=1e-7)
fom_ei = fom.with_(operator=ei_op)
reductor = StationaryRBReductor(fom_ei, rb)
rom = reductor.reduce()
# the reductor currently removes all solver_options so we need to add them again
rom = rom.with_(operator=rom.operator.with_(
solver_options=fom.operator.solver_options))
# ### ROM validation
import time
import numpy as np
# ensure that FFC is not called during runtime measurements
rom.solve(1)
errs = []
speedups = []
for mu in parameter_space.sample_randomly(10):
tic = time.perf_counter()
U = fom.solve(mu)
t_fom = time.perf_counter() - tic
tic = time.perf_counter()
u_red = rom.solve(mu)
t_rom = time.perf_counter() - tic
U_red = reductor.reconstruct(u_red)
errs.append(((U - U_red).norm() / U.norm())[0])
speedups.append(t_fom / t_rom)
print(f'Maximum relative ROM error: {max(errs)}')
print(f'Median of ROM speedup: {np.median(speedups)}')
time_stepper=time_stepper, num_values=20,
visualizer=visualizer, name='C++-Model')
return fom
# discretize
fom = discretize(50, 10000, 4)
parameter_space = fom.parameters.space(0.1, 1)
# generate solution snapshots
snapshots = fom.solution_space.empty()
for mu in parameter_space.sample_uniformly(2):
snapshots.append(fom.solve(mu))
# apply POD
reduced_basis = pod(snapshots, modes=4)[0]
# reduce the model
reductor = InstationaryRBReductor(fom, reduced_basis, check_orthonormality=True)
rom = reductor.reduce()
# stochastic error estimation
mu_max = None
err_max = -1.
for mu in parameter_space.sample_randomly(10):
U_RB = (reductor.reconstruct(rom.solve(mu)))
U = fom.solve(mu)
err = np.max((U_RB-U).norm())
if err > err_max:
err_max = err
mu_max = mu
def extend_basis(self, U, method='gram_schmidt', pod_modes=1, pod_orthonormalize=True, copy_U=True):
"""Extend basis by new vectors.
Parameters
----------
U
|VectorArray| containing the new basis vectors.
method
Basis extension method to use. The following methods are available:
:trivial: Vectors in `U` are appended to the basis. Duplicate vectors
in the sense of :func:`~pymor.algorithms.basic.almost_equal`
are removed.
:gram_schmidt: New basis vectors are orthonormalized w.r.t. to the old
basis using the :func:`~pymor.algorithms.gram_schmidt.gram_schmidt`
algorithm.
:pod: Append the first POD modes of the defects of the projections
of the vectors in U onto the existing basis
(e.g. for use in POD-Greedy algorithm).
.. warning::
In case of the `'gram_schmidt'` and `'pod'` extension methods, the existing reduced
basis is assumed to be orthonormal w.r.t. the given inner product.
pod_modes
In case `method == 'pod'`, the number of POD modes that shall be appended to
the basis.
pod_orthonormalize
If `True` and `method == 'pod'`, re-orthonormalize the new basis vectors obtained
by the POD in order to improve numerical accuracy.
copy_U
If `copy_U` is `False`, the new basis vectors might be removed from `U`.
Raises
------
ExtensionError
Raised when the selected extension method does not yield a basis of increased
dimension.
"""
assert method in ('trivial', 'gram_schmidt', 'pod')
basis_length = len(self.RB)
if method == 'trivial':
remove = set()
for i in range(len(U)):
if np.any(almost_equal(U[i], self.RB)):
remove.add(i)
self.RB.append(U[[i for i in range(len(U)) if i not in remove]],
remove_from_other=(not copy_U))
elif method == 'gram_schmidt':
self.RB.append(U, remove_from_other=(not copy_U))
gram_schmidt(self.RB, offset=basis_length, product=self.product, copy=False)
elif method == 'pod':
if self.product is None:
U_proj_err = U - self.RB.lincomb(U.dot(self.RB))
else:
U_proj_err = U - self.RB.lincomb(self.product.apply2(U, self.RB))
self.RB.append(pod(U_proj_err, modes=pod_modes, product=self.product, orthonormalize=False)[0])
if pod_orthonormalize:
gram_schmidt(self.RB, offset=basis_length, product=self.product, copy=False)
if len(self.RB) <= basis_length:
raise ExtensionError
def deim(U, modes=None, atol=None, rtol=None, product=None, pod_options={}):
"""Generate data for empirical interpolation using DEIM algorithm.
Given a |VectorArray| `U`, this method generates a collateral basis and
interpolation DOFs for empirical interpolation of the vectors contained in `U`.
The returned objects can be used to instantiate an |EmpiricalInterpolatedOperator|
(with `triangular=False`).
The collateral basis is determined by the first :func:`~pymor.algorithms.pod.pod` modes of `U`.
Parameters
----------
U
A |VectorArray| of vectors to interpolate.
modes
Dimension of the collateral basis i.e. number of POD modes of the vectors in `U`.
atol
Absolute POD tolerance.
rtol
Relative POD tolerance.
product
Inner product |Operator| used for the POD.
pod_options
Dictionary of additional options to pass to the :func:`~pymor.algorithms.pod.pod` algorithm.
Returns
-------
interpolation_dofs
|NumPy array| of the DOFs at which the vectors are interpolated.
collateral_basis
|VectorArray| containing the generated collateral basis.
data
Dict containing the following fields:
:svals: POD singular values.
"""
assert isinstance(U, VectorArrayInterface)
logger = getLogger('pymor.algorithms.ei.deim')
logger.info('Generating Interpolation Data ...')
collateral_basis, svals = pod(U,
modes=modes,
atol=atol,
rtol=rtol,
product=product,
**pod_options)
interpolation_dofs = np.zeros((0, ), dtype=np.int32)
interpolation_matrix = np.zeros((0, 0))
for i in range(len(collateral_basis)):
logger.info(f'Choosing interpolation point for basis vector {i}.')
if len(interpolation_dofs) > 0:
coefficients = solve(
interpolation_matrix,
collateral_basis[i].dofs(interpolation_dofs).T).T
U_interpolated = collateral_basis[:len(interpolation_dofs
)].lincomb(coefficients)
ERR = collateral_basis[i].copy()
ERR -= U_interpolated
else:
ERR = collateral_basis[i].copy()
# compute new interpolation dof and collateral basis vector
new_dof = ERR.amax()[0][0]
if new_dof in interpolation_dofs:
logger.info(
f'DOF {new_dof} selected twice for interplation! Stopping extension loop.'
)
break
interpolation_dofs = np.hstack((interpolation_dofs, new_dof))
interpolation_matrix = collateral_basis[:len(interpolation_dofs)].dofs(
interpolation_dofs).T
if len(interpolation_dofs) < len(collateral_basis):
del collateral_basis[len(interpolation_dofs):len(collateral_basis)]
logger.info('Finished.')
data = {'svals': svals}
return interpolation_dofs, collateral_basis, data
def pod_basis_extension(basis, U, count=1, copy_basis=True, product=None, orthonormalize=True):
"""Extend basis with the first `count` POD modes of the projection of `U` onto the
orthogonal complement of the basis.
Note that the provided basis is assumed to be orthonormal w.r.t. the provided
scalar product!
Parameters
----------
basis
|VectorArray| containing the basis to extend. The basis is expected to be
orthonormal w.r.t. `product`.
U
|VectorArray| containing the vectors to which the POD is applied.
count
Number of POD modes that are to be appended to the basis.
product
The scalar product w.r.t. which to orthonormalize; if `None`, the Euclidean
product is used.
copy_basis
If `copy_basis` is `False`, the old basis is extended in-place.
orthonormalize
If `True`, re-orthonormalize the new basis vectors obtained by the POD
in order to improve numerical accuracy.
Returns
-------
new_basis
The extended basis.
extension_data
Dict containing the following fields:
:hierarchic: `True` if `new_basis` contains `basis` as its first vectors.
Raises
------
ExtensionError
POD produces no new vectors. This is the case when no vector in `U`
is linearly independent from the basis.
"""
if basis is None:
return pod(U, modes=count, product=product)[0], {'hierarchic': True}
basis_length = len(basis)
new_basis = basis.copy() if copy_basis else basis
if product is None:
U_proj_err = U - basis.lincomb(U.dot(basis))
else:
U_proj_err = U - basis.lincomb(product.apply2(U, basis))
new_basis.append(pod(U_proj_err, modes=count, product=product, orthonormalize=False)[0])
if orthonormalize:
gram_schmidt(new_basis, offset=len(basis), product=product, copy=False)
if len(new_basis) <= basis_length:
raise ExtensionError
return new_basis, {'hierarchic': True}
def deim(U, modes=None, atol=None, rtol=None, product=None, pod_options={}):
"""Generate data for empirical interpolation using DEIM algorithm.
Given a |VectorArray| `U`, this method generates a collateral basis and
interpolation DOFs for empirical interpolation of the vectors contained in `U`.
The returned objects can be used to instantiate an |EmpiricalInterpolatedOperator|
(with `triangular=False`).
The collateral basis is determined by the first :func:`~pymor.algorithms.pod.pod` modes of `U`.
Parameters
----------
U
A |VectorArray| of vectors to interpolate.
modes
Dimension of the collateral basis i.e. number of POD modes of the vectors in `U`.
atol
Absolute POD tolerance.
rtol
Relative POD tolerance.
product
Inner product |Operator| used for the POD.
pod_options
Dictionary of additional options to pass to the :func:`~pymor.algorithms.pod.pod` algorithm.
Returns
-------
interpolation_dofs
|NumPy array| of the DOFs at which the vectors are interpolated.
collateral_basis
|VectorArray| containing the generated collateral basis.
data
Dict containing the following fields:
:svals: POD singular values.
"""
assert isinstance(U, VectorArrayInterface)
logger = getLogger('pymor.algorithms.ei.deim')
logger.info('Generating Interpolation Data ...')
collateral_basis, svals = pod(U, modes=modes, atol=atol, rtol=rtol, product=product, **pod_options)
interpolation_dofs = np.zeros((0,), dtype=np.int32)
interpolation_matrix = np.zeros((0, 0))
for i in range(len(collateral_basis)):
logger.info('Choosing interpolation point for basis vector {}.'.format(i))
if len(interpolation_dofs) > 0:
coefficients = np.linalg.solve(interpolation_matrix,
collateral_basis[i].dofs(interpolation_dofs).T).T
U_interpolated = collateral_basis[:len(interpolation_dofs)].lincomb(coefficients)
ERR = collateral_basis[i].copy()
ERR -= U_interpolated
else:
ERR = collateral_basis[i].copy()
# compute new interpolation dof and collateral basis vector
new_dof = ERR.amax()[0][0]
if new_dof in interpolation_dofs:
logger.info('DOF {} selected twice for interplation! Stopping extension loop.'.format(new_dof))
break
interpolation_dofs = np.hstack((interpolation_dofs, new_dof))
interpolation_matrix = collateral_basis[:len(interpolation_dofs)].dofs(interpolation_dofs).T
if len(interpolation_dofs) < len(collateral_basis):
del collateral_basis[len(interpolation_dofs):len(collateral_basis)]
logger.info('Finished.')
data = {'svals': svals}
return interpolation_dofs, collateral_basis, data
def extend_basis(self,
U,
method='gram_schmidt',
pod_modes=1,
pod_orthonormalize=True,
copy_U=True):
"""Extend basis by new vectors.
Parameters
----------
U
|VectorArray| containing the new basis vectors.
method
Basis extension method to use. The following methods are available:
:trivial: Vectors in `U` are appended to the basis. Duplicate vectors
in the sense of :func:`~pymor.algorithms.basic.almost_equal`
are removed.
:gram_schmidt: New basis vectors are orthonormalized w.r.t. to the old
basis using the :func:`~pymor.algorithms.gram_schmidt.gram_schmidt`
algorithm.
:pod: Append the first POD modes of the defects of the projections
of the vectors in U onto the existing basis
(e.g. for use in POD-Greedy algorithm).
.. warning::
In case of the `'gram_schmidt'` and `'pod'` extension methods, the existing reduced
basis is assumed to be orthonormal w.r.t. the given inner product.
pod_modes
In case `method == 'pod'`, the number of POD modes that shall be appended to
the basis.
pod_orthonormalize
If `True` and `method == 'pod'`, re-orthonormalize the new basis vectors obtained
by the POD in order to improve numerical accuracy.
copy_U
If `copy_U` is `False`, the new basis vectors might be removed from `U`.
Raises
------
ExtensionError
Raised when the selected extension method does not yield a basis of increased
dimension.
"""
assert method in ('trivial', 'gram_schmidt', 'pod')
basis_length = len(self.RB)
if method == 'trivial':
remove = set()
for i in range(len(U)):
if np.any(almost_equal(U[i], self.RB)):
remove.add(i)
self.RB.append(U[[i for i in range(len(U)) if i not in remove]],
remove_from_other=(not copy_U))
elif method == 'gram_schmidt':
self.RB.append(U, remove_from_other=(not copy_U))
gram_schmidt(self.RB,
offset=basis_length,
product=self.product,
copy=False)
elif method == 'pod':
if self.product is None:
U_proj_err = U - self.RB.lincomb(U.dot(self.RB))
else:
U_proj_err = U - self.RB.lincomb(
self.product.apply2(U, self.RB))
self.RB.append(
pod(U_proj_err,
modes=pod_modes,
product=self.product,
orthonormalize=False)[0])
if pod_orthonormalize:
gram_schmidt(self.RB,
offset=basis_length,
product=self.product,
copy=False)
if len(self.RB) <= basis_length:
raise ExtensionError
d = InstationaryDiscretization(T=1e-0, operator=operator, rhs=rhs, initial_data=initial_data,
time_stepper=time_stepper, num_values=20, parameter_space=parameter_space,
visualizer=visualizer, name='C++-Discretization', cache_region=None)
return d
# discretize
d = discretize(50, 10000, 4)
# generate solution snapshots
snapshots = d.solution_space.empty()
for mu in d.parameter_space.sample_uniformly(2):
snapshots.append(d.solve(mu))
# apply POD
reduced_basis = pod(snapshots, 4)[0]
# reduce the model
rd, rc, _ = reduce_generic_rb(d, reduced_basis)
# stochastic error estimation
mu_max = None
err_max = -1.
for mu in d.parameter_space.sample_randomly(10):
U_RB = (rc.reconstruct(rd.solve(mu)))
U = d.solve(mu)
err = np.max((U_RB-U).l2_norm())
if err > err_max:
err_max = err
mu_max = mu
def main(
tol: float = Argument(..., help='Prescribed mean l2 approximation error.'),
dist: int = Argument(..., help='Number of slices for distributed HAPOD.'),
inc: int = Argument(..., help='Number of steps for incremental HAPOD.'),
grid: int = Option(60, help='Use grid with (2*NI)*NI elements.'),
nt: int = Option(100, help='Number of time steps.'),
omega: float = Option(0.9, help='Parameter omega from HAPOD algorithm.'),
procs: int = Option(
0, help='Number of processes to use for parallelization.'),
snap: int = Option(20, help='Number of snapshot trajectories to compute.'),
threads: int = Option(
0, help='Number of threads to use for parallelization.'),
):
"""Compression of snapshot data with the HAPOD algorithm from [HLR18]."""
assert procs == 0 or threads == 0
executor = ProcessPoolExecutor(procs) if procs > 0 else \
ThreadPoolExecutor(threads) if threads > 0 else \
None
p = burgers_problem_2d()
m, data = discretize_instationary_fv(p,
grid_type=RectGrid,
diameter=np.sqrt(2) / grid,
nt=nt)
U = m.solution_space.empty()
for mu in p.parameter_space.sample_randomly(snap):
U.append(m.solve(mu))
tic = time.perf_counter()
pod_modes = pod(U, l2_err=tol * np.sqrt(len(U)), product=m.l2_product)[0]
pod_time = time.perf_counter() - tic
tic = time.perf_counter()
dist_modes = dist_vectorarray_hapod(dist,
U,
tol,
omega,
product=m.l2_product,
executor=executor)[0]
dist_time = time.perf_counter() - tic
tic = time.perf_counter()
inc_modes = inc_vectorarray_hapod(inc, U, tol, omega,
product=m.l2_product)[0]
inc_time = time.perf_counter() - tic
print(f'Snapshot matrix: {U.dim} x {len(U)}')
print(
format_table([
['Method', 'Error', 'Modes', 'Time'],
[
'POD',
np.linalg.norm(
m.l2_norm(U - pod_modes.lincomb(
m.l2_product.apply2(U, pod_modes))) / np.sqrt(len(U))),
len(pod_modes), pod_time
],
[
'DIST HAPOD',
np.linalg.norm(
m.l2_norm(U - dist_modes.lincomb(
m.l2_product.apply2(U, dist_modes))) /
np.sqrt(len(U))),
len(dist_modes), dist_time
],
[
'INC HAPOD',
np.linalg.norm(
m.l2_norm(U - inc_modes.lincomb(
m.l2_product.apply2(U, inc_modes))) / np.sqrt(len(U))),
len(inc_modes), inc_time
]
]))