def helmholtz_problem(domain=RectDomain(), rhs=None, parameter_range=(0., 100.),
dirichlet_data=None, neumann_data=None):
"""Helmholtz equation problem.
This problem is to solve the Helmholtz equation ::
- ∆ u(x, k) - k^2 u(x, k) = f(x, k)
on a given domain.
Parameters
----------
domain
A |DomainDescription| of the domain the problem is posed on.
rhs
The |Function| f(x, μ).
parameter_range
A tuple `(k_min, k_max)` describing the interval in which k is allowd to vary.
dirichlet_data
|Function| providing the Dirichlet boundary values.
neumann_data
|Function| providing the Neumann boundary values.
"""
return StationaryProblem(
domain=domain,
rhs=rhs or ConstantFunction(1., dim_domain=domain.dim),
dirichlet_data=dirichlet_data,
neumann_data=neumann_data,
diffusion=ConstantFunction(1., dim_domain=domain.dim),
reaction=LincombFunction([ConstantFunction(1., dim_domain=domain.dim)],
[ExpressionParameterFunctional('-k[0]**2', {'k': 1})]),
parameter_ranges={'k': parameter_range},
name='helmholtz_problem'
)
def main(
problem_number: int = Argument(
..., min=0, max=1, help='Selects the problem to solve [0 or 1].'),
n: int = Argument(..., help='Grid interval count.'),
fv: bool = Option(
False,
help='Use finite volume discretization instead of finite elements.'),
):
"""Solves the Poisson equation in 1D using pyMOR's builtin discreization toolkit."""
rhss = [
ExpressionFunction('ones(x.shape[:-1]) * 10', 1, ()),
ExpressionFunction('(x - 0.5)**2 * 1000', 1, ())
]
rhs = rhss[problem_number]
d0 = ExpressionFunction('1 - x', 1, ())
d1 = ExpressionFunction('x', 1, ())
f0 = ProjectionParameterFunctional('diffusionl')
f1 = 1.
problem = StationaryProblem(domain=LineDomain(),
rhs=rhs,
diffusion=LincombFunction([d0, d1], [f0, f1]),
dirichlet_data=ConstantFunction(value=0,
dim_domain=1),
name='1DProblem')
parameter_space = problem.parameters.space(0.1, 1)
print('Discretize ...')
discretizer = discretize_stationary_fv if fv else discretize_stationary_cg
m, data = discretizer(problem, diameter=1 / n)
print(data['grid'])
print()
print('Solve ...')
U = m.solution_space.empty()
for mu in parameter_space.sample_uniformly(10):
U.append(m.solve(mu))
m.visualize(U, title='Solution for diffusionl in [0.1, 1]')
def elliptic_oned_demo(args):
args['PROBLEM-NUMBER'] = int(args['PROBLEM-NUMBER'])
assert 0 <= args['PROBLEM-NUMBER'] <= 1, ValueError(
'Invalid problem number.')
args['N'] = int(args['N'])
rhss = [
ExpressionFunction('ones(x.shape[:-1]) * 10', 1, ()),
ExpressionFunction('(x - 0.5)**2 * 1000', 1, ())
]
rhs = rhss[args['PROBLEM-NUMBER']]
d0 = ExpressionFunction('1 - x', 1, ())
d1 = ExpressionFunction('x', 1, ())
f0 = ProjectionParameterFunctional('diffusionl')
f1 = 1.
problem = StationaryProblem(domain=LineDomain(),
rhs=rhs,
diffusion=LincombFunction([d0, d1], [f0, f1]),
dirichlet_data=ConstantFunction(value=0,
dim_domain=1),
name='1DProblem')
parameter_space = problem.parameters.space(0.1, 1)
print('Discretize ...')
discretizer = discretize_stationary_fv if args[
'--fv'] else discretize_stationary_cg
m, data = discretizer(problem, diameter=1 / args['N'])
print(data['grid'])
print()
print('Solve ...')
U = m.solution_space.empty()
for mu in parameter_space.sample_uniformly(10):
U.append(m.solve(mu))
m.visualize(U, title='Solution for diffusionl in [0.1, 1]')
def thermal_block_problem(num_blocks=(3, 3), parameter_range=(0.1, 1)):
"""Analytical description of a 2D 'thermal block' diffusion problem.
The problem is to solve the elliptic equation ::
- ∇ ⋅ [ d(x, μ) ∇ u(x, μ) ] = f(x, μ)
on the domain [0,1]^2 with Dirichlet zero boundary values. The domain is
partitioned into nx x ny blocks and the diffusion function d(x, μ) is
constant on each such block (i,j) with value μ_ij. ::
----------------------------
| | | |
| μ_11 | μ_12 | μ_13 |
| | | |
|---------------------------
| | | |
| μ_21 | μ_22 | μ_23 |
| | | |
----------------------------
Parameters
----------
num_blocks
The tuple `(nx, ny)`
parameter_range
A tuple `(μ_min, μ_max)`. Each |Parameter| component μ_ij is allowed
to lie in the interval [μ_min, μ_max].
"""
def parameter_functional_factory(ix, iy):
return ProjectionParameterFunctional(component_name='diffusion',
component_shape=(num_blocks[1], num_blocks[0]),
index=(num_blocks[1] - iy - 1, ix),
name=f'diffusion_{ix}_{iy}')
def diffusion_function_factory(ix, iy):
if ix + 1 < num_blocks[0]:
X = '(x[..., 0] >= ix * dx) * (x[..., 0] < (ix + 1) * dx)'
else:
X = '(x[..., 0] >= ix * dx)'
if iy + 1 < num_blocks[1]:
Y = '(x[..., 1] >= iy * dy) * (x[..., 1] < (iy + 1) * dy)'
else:
Y = '(x[..., 1] >= iy * dy)'
return ExpressionFunction(f'{X} * {Y} * 1.',
2, (), {}, {'ix': ix, 'iy': iy, 'dx': 1. / num_blocks[0], 'dy': 1. / num_blocks[1]},
name=f'diffusion_{ix}_{iy}')
return StationaryProblem(
domain=RectDomain(),
rhs=ConstantFunction(dim_domain=2, value=1.),
diffusion=LincombFunction([diffusion_function_factory(ix, iy)
for ix, iy in product(range(num_blocks[0]), range(num_blocks[1]))],
[parameter_functional_factory(ix, iy)
for ix, iy in product(range(num_blocks[0]), range(num_blocks[1]))],
name='diffusion'),
parameter_space=CubicParameterSpace({'diffusion': (num_blocks[1], num_blocks[0])}, *parameter_range),
name=f'ThermalBlock({num_blocks})'
)
def elliptic2_demo(args):
args['PROBLEM-NUMBER'] = int(args['PROBLEM-NUMBER'])
assert 0 <= args['PROBLEM-NUMBER'] <= 1, ValueError(
'Invalid problem number.')
args['N'] = int(args['N'])
rhss = [
ExpressionFunction('ones(x.shape[:-1]) * 10', 2, ()),
LincombFunction([
ExpressionFunction('ones(x.shape[:-1]) * 10', 2, ()),
ConstantFunction(1., 2)
], [ProjectionParameterFunctional('mu'), 0.1])
]
dirichlets = [
ExpressionFunction('zeros(x.shape[:-1])', 2, ()),
LincombFunction([
ExpressionFunction('2 * x[..., 0]', 2, ()),
ConstantFunction(1., 2)
], [ProjectionParameterFunctional('mu'), 0.5])
]
neumanns = [
None,
LincombFunction([
ExpressionFunction('1 - x[..., 1]', 2, ()),
ConstantFunction(1., 2)
], [ProjectionParameterFunctional('mu'), 0.5**2])
]
robins = [
None,
(LincombFunction(
[ExpressionFunction('x[..., 1]', 2, ()),
ConstantFunction(1., 2)],
[ProjectionParameterFunctional('mu'), 1]), ConstantFunction(1., 2))
]
domains = [
RectDomain(),
RectDomain(right='neumann', top='dirichlet', bottom='robin')
]
rhs = rhss[args['PROBLEM-NUMBER']]
dirichlet = dirichlets[args['PROBLEM-NUMBER']]
neumann = neumanns[args['PROBLEM-NUMBER']]
domain = domains[args['PROBLEM-NUMBER']]
robin = robins[args['PROBLEM-NUMBER']]
problem = StationaryProblem(domain=domain,
rhs=rhs,
diffusion=LincombFunction([
ExpressionFunction('1 - x[..., 0]', 2, ()),
ExpressionFunction('x[..., 0]', 2, ())
], [ProjectionParameterFunctional('mu'), 1]),
dirichlet_data=dirichlet,
neumann_data=neumann,
robin_data=robin,
parameter_ranges=(0.1, 1),
name='2DProblem')
print('Discretize ...')
discretizer = discretize_stationary_fv if args[
'--fv'] else discretize_stationary_cg
m, data = discretizer(problem, diameter=1. / args['N'])
print(data['grid'])
print()
print('Solve ...')
U = m.solution_space.empty()
for mu in problem.parameter_space.sample_uniformly(10):
U.append(m.solve(mu))
m.visualize(U, title='Solution for mu in [0.1, 1]')
burgers_problem(parameter_range=(1., 1.3)),
burgers_problem_2d(),
burgers_problem_2d(torus=False, initial_data_type='bump', parameter_range=(1.3, 1.5))]
picklable_elliptic_problems = \
[StationaryProblem(domain=RectDomain(), rhs=ConstantFunction(dim_domain=2, value=1.)),
helmholtz_problem()]
non_picklable_elliptic_problems = \
[StationaryProblem(domain=RectDomain(),
rhs=ConstantFunction(dim_domain=2, value=21.),
diffusion=LincombFunction(
[GenericFunction(dim_domain=2, mapping=lambda X, p=p: X[..., 0]**p)
for p in range(5)],
[ExpressionParameterFunctional(f'max(mu["exp"], {m})', parameters={'exp': 1})
for m in range(5)]
))]
elliptic_problems = picklable_thermalblock_problems + non_picklable_elliptic_problems
@pytest.fixture(params=elliptic_problems + thermalblock_problems + burgers_problems)
def analytical_problem(request):
return request.param
@pytest.fixture(params=picklable_elliptic_problems + picklable_thermalblock_problems + burgers_problems)
def picklable_analytical_problem(request):
return request.param
def elliptic2_demo(args):
args['PROBLEM-NUMBER'] = int(args['PROBLEM-NUMBER'])
assert 0 <= args['PROBLEM-NUMBER'] <= 1, ValueError('Invalid problem number.')
args['N'] = int(args['N'])
norm = args['NORM']
norm = float(norm) if not norm.lower() in ('h1', 'l2') else norm.lower()
rhss = [ExpressionFunction('ones(x.shape[:-1]) * 10', 2, ()),
LincombFunction(
[ExpressionFunction('ones(x.shape[:-1]) * 10', 2, ()), ConstantFunction(1.,2)],
[ProjectionParameterFunctional('mu'), 0.1])]
dirichlets = [ExpressionFunction('zeros(x.shape[:-1])', 2, ()),
LincombFunction(
[ExpressionFunction('2 * x[..., 0]', 2, ()), ConstantFunction(1.,2)],
[ProjectionParameterFunctional('mu'), 0.5])]
neumanns = [None,
LincombFunction(
[ExpressionFunction('1 - x[..., 1]', 2, ()), ConstantFunction(1.,2)],
[ProjectionParameterFunctional('mu'), 0.5**2])]
robins = [None,
(LincombFunction(
[ExpressionFunction('x[..., 1]', 2, ()), ConstantFunction(1.,2)],
[ProjectionParameterFunctional('mu'), 1]),
ConstantFunction(1.,2))]
domains = [RectDomain(),
RectDomain(right='neumann', top='dirichlet', bottom='robin')]
rhs = rhss[args['PROBLEM-NUMBER']]
dirichlet = dirichlets[args['PROBLEM-NUMBER']]
neumann = neumanns[args['PROBLEM-NUMBER']]
domain = domains[args['PROBLEM-NUMBER']]
robin = robins[args['PROBLEM-NUMBER']]
problem = StationaryProblem(
domain=domain,
rhs=rhs,
diffusion=LincombFunction(
[ExpressionFunction('1 - x[..., 0]', 2, ()), ExpressionFunction('x[..., 0]', 2, ())],
[ProjectionParameterFunctional('mu'), 1]
),
dirichlet_data=dirichlet,
neumann_data=neumann,
robin_data=robin,
parameter_ranges=(0.1, 1),
name='2DProblem'
)
if isinstance(norm, float) and not args['--fv']:
# use a random parameter to construct an energy product
mu_bar = problem.parameters.parse(norm)
else:
mu_bar = None
print('Discretize ...')
if args['--fv']:
m, data = discretize_stationary_fv(problem, diameter=1. / args['N'])
else:
m, data = discretize_stationary_cg(problem, diameter=1. / args['N'], mu_energy_product=mu_bar)
print(data['grid'])
print()
print('Solve ...')
U = m.solution_space.empty()
for mu in problem.parameter_space.sample_uniformly(10):
U.append(m.solve(mu))
if mu_bar is not None:
# use the given energy product
norm_squared = U[-1].norm(m.products['energy'])[0]
print('Energy norm of the last snapshot: ', np.sqrt(norm_squared))
if not args['--fv']:
if args['NORM'] == 'h1':
norm_squared = U[-1].norm(m.products['h1_0_semi'])[0]
print('H^1_0 semi norm of the last snapshot: ', np.sqrt(norm_squared))
if args['NORM'] == 'l2':
norm_squared = U[-1].norm(m.products['l2_0'])[0]
print('L^2_0 norm of the last snapshot: ', np.sqrt(norm_squared))
m.visualize(U, title='Solution for mu in [0.1, 1]')
def text_problem(text='pyMOR', font_name=None):
import numpy as np
from PIL import Image, ImageDraw, ImageFont
from tempfile import NamedTemporaryFile
font_list = [font_name] if font_name else [
'DejaVuSansMono.ttf', 'VeraMono.ttf', 'UbuntuMono-R.ttf', 'Arial.ttf'
]
font = None
for filename in font_list:
try:
font = ImageFont.truetype(
filename, 64) # load some font from file of given size
except (OSError, IOError):
pass
if font is None:
raise ValueError('Could not load TrueType font')
size = font.getsize(text) # compute width and height of rendered text
size = (size[0] + 20, size[1] + 20
) # add a border of 10 pixels around the text
def make_bitmap_function(
char_num
): # we need to genereate a BitmapFunction for each character
img = Image.new('L',
size) # create new Image object of given dimensions
d = ImageDraw.Draw(img) # create ImageDraw object for the given Image
# in order to position the character correctly, we first draw all characters from the first
# up to the wanted character
d.text((10, 10), text[:char_num + 1], font=font, fill=255)
# next we erase all previous character by drawing a black rectangle
if char_num > 0:
d.rectangle(
((0, 0), (font.getsize(text[:char_num])[0] + 10, size[1])),
fill=0,
outline=0)
# open a new temporary file
with NamedTemporaryFile(
suffix='.png'
) as f: # after leaving this 'with' block, the temporary
# file is automatically deleted
img.save(f, format='png')
return BitmapFunction(f.name,
bounding_box=[(0, 0), size],
range=[0., 1.])
# create BitmapFunctions for each character
dfs = [make_bitmap_function(n) for n in range(len(text))]
# create an indicator function for the background
background = ConstantFunction(1., 2) - LincombFunction(
dfs, np.ones(len(dfs)))
# form the linear combination
dfs = [background] + dfs
coefficients = [1] + [
ProjectionParameterFunctional('diffusion', len(text), i)
for i in range(len(text))
]
diffusion = LincombFunction(dfs, coefficients)
return StationaryProblem(domain=RectDomain(dfs[1].bounding_box,
bottom='neumann'),
neumann_data=ConstantFunction(-1., 2),
diffusion=diffusion,
parameter_ranges=(0.1, 1.))
def main(
problem_number: int = Argument(..., min=0, max=1, help='Selects the problem to solve [0 or 1].'),
n: int = Argument(..., help='Triangle count per direction'),
norm: str = Argument(
...,
help="h1: compute the h1-norm of the last snapshot.\n\n"
"l2: compute the l2-norm of the last snapshot.\n\n"
"k: compute the energy norm of the last snapshot, where the energy-product"
"is constructed with a parameter {'mu': k}."
),
fv: bool = Option(False, help='Use finite volume discretization instead of finite elements.'),
):
"""Solves the Poisson equation in 2D using pyMOR's builtin discreization toolkit."""
norm = float(norm) if not norm.lower() in ('h1', 'l2') else norm.lower()
rhss = [ExpressionFunction('ones(x.shape[:-1]) * 10', 2, ()),
LincombFunction(
[ExpressionFunction('ones(x.shape[:-1]) * 10', 2, ()), ConstantFunction(1., 2)],
[ProjectionParameterFunctional('mu'), 0.1])]
dirichlets = [ExpressionFunction('zeros(x.shape[:-1])', 2, ()),
LincombFunction(
[ExpressionFunction('2 * x[..., 0]', 2, ()), ConstantFunction(1., 2)],
[ProjectionParameterFunctional('mu'), 0.5])]
neumanns = [None,
LincombFunction(
[ExpressionFunction('1 - x[..., 1]', 2, ()), ConstantFunction(1., 2)],
[ProjectionParameterFunctional('mu'), 0.5**2])]
robins = [None,
(LincombFunction(
[ExpressionFunction('x[..., 1]', 2, ()), ConstantFunction(1., 2)],
[ProjectionParameterFunctional('mu'), 1]), ConstantFunction(1., 2))]
domains = [RectDomain(),
RectDomain(right='neumann', top='dirichlet', bottom='robin')]
rhs = rhss[problem_number]
dirichlet = dirichlets[problem_number]
neumann = neumanns[problem_number]
domain = domains[problem_number]
robin = robins[problem_number]
problem = StationaryProblem(
domain=domain,
rhs=rhs,
diffusion=LincombFunction(
[ExpressionFunction('1 - x[..., 0]', 2, ()), ExpressionFunction('x[..., 0]', 2, ())],
[ProjectionParameterFunctional('mu'), 1]
),
dirichlet_data=dirichlet,
neumann_data=neumann,
robin_data=robin,
parameter_ranges=(0.1, 1),
name='2DProblem'
)
if isinstance(norm, float) and not fv:
# use a random parameter to construct an energy product
mu_bar = problem.parameters.parse(norm)
else:
mu_bar = None
print('Discretize ...')
if fv:
m, data = discretize_stationary_fv(problem, diameter=1. / n)
else:
m, data = discretize_stationary_cg(problem, diameter=1. / n, mu_energy_product=mu_bar)
print(data['grid'])
print()
print('Solve ...')
U = m.solution_space.empty()
for mu in problem.parameter_space.sample_uniformly(10):
U.append(m.solve(mu))
if mu_bar is not None:
# use the given energy product
norm_squared = U[-1].norm(m.products['energy'])[0]
print('Energy norm of the last snapshot: ', np.sqrt(norm_squared))
if not fv:
if norm == 'h1':
norm_squared = U[-1].norm(m.products['h1_0_semi'])[0]
print('H^1_0 semi norm of the last snapshot: ', np.sqrt(norm_squared))
if norm == 'l2':
norm_squared = U[-1].norm(m.products['l2_0'])[0]
print('L^2_0 norm of the last snapshot: ', np.sqrt(norm_squared))
m.visualize(U, title='Solution for mu in [0.1, 1]')
def main(
diameter: float = Argument(
0.01, help='Diameter option for the domain discretizer.'),
r: int = Argument(5, help='Order of the ROMs.'),
):
"""Parametric 1D heat equation example."""
set_log_levels({'pymor.algorithms.gram_schmidt.gram_schmidt': 'WARNING'})
# Model
p = InstationaryProblem(StationaryProblem(
domain=LineDomain([0., 1.], left='robin', right='robin'),
diffusion=LincombFunction([
ExpressionFunction('(x[...,0] <= 0.5) * 1.', 1),
ExpressionFunction('(0.5 < x[...,0]) * 1.', 1)
], [1, ProjectionParameterFunctional('diffusion')]),
robin_data=(ConstantFunction(1., 1),
ExpressionFunction('(x[...,0] < 1e-10) * 1.', 1)),
outputs=(('l2_boundary',
ExpressionFunction('(x[...,0] > (1 - 1e-10)) * 1.', 1)), ),
),
ConstantFunction(0., 1),
T=3.)
fom, _ = discretize_instationary_cg(p, diameter=diameter, nt=100)
fom.visualize(fom.solve(mu=0.1))
fom.visualize(fom.solve(mu=1))
fom.visualize(fom.solve(mu=10))
lti = fom.to_lti()
print(f'order of the model = {lti.order}')
print(f'number of inputs = {lti.dim_input}')
print(f'number of outputs = {lti.dim_output}')
mu_list = [0.1, 1, 10]
w_list = np.logspace(-1, 3, 100)
# System poles
fig, ax = plt.subplots()
for mu in mu_list:
poles = lti.poles(mu=mu)
ax.plot(poles.real, poles.imag, '.', label=fr'$\mu = {mu}$')
ax.set_title('System poles')
ax.legend()
plt.show()
# Magnitude plots
fig, ax = plt.subplots()
for mu in mu_list:
lti.mag_plot(w_list, ax=ax, mu=mu, label=fr'$\mu = {mu}$')
ax.set_title('Magnitude plot of the full model')
ax.legend()
plt.show()
# Hankel singular values
fig, ax = plt.subplots()
for mu in mu_list:
hsv = lti.hsv(mu=mu)
ax.semilogy(range(1, len(hsv) + 1), hsv, label=fr'$\mu = {mu}$')
ax.set_title('Hankel singular values')
ax.legend()
plt.show()
# System norms
for mu in mu_list:
print(f'mu = {mu}:')
print(f' H_2-norm of the full model: {lti.h2_norm(mu=mu):e}')
if config.HAVE_SLYCOT:
print(
f' H_inf-norm of the full model: {lti.hinf_norm(mu=mu):e}')
print(f' Hankel-norm of the full model: {lti.hankel_norm(mu=mu):e}')
# Model order reduction
run_mor_method_param(lti, r, w_list, mu_list, BTReductor, 'BT')
run_mor_method_param(lti, r, w_list, mu_list, LQGBTReductor, 'LQGBT')
run_mor_method_param(lti, r, w_list, mu_list, BRBTReductor, 'BRBT')
run_mor_method_param(lti, r, w_list, mu_list, IRKAReductor, 'IRKA')
run_mor_method_param(lti, r, w_list, mu_list, TSIAReductor, 'TSIA')
run_mor_method_param(lti,
r,
w_list,
mu_list,
OneSidedIRKAReductor,
'OS-IRKA',
version='V')