def elliptic_demo(args):
args['PROBLEM-NUMBER'] = int(args['PROBLEM-NUMBER'])
assert 0 <= args['PROBLEM-NUMBER'] <= 1, ValueError('Invalid problem number')
args['DIRICHLET-NUMBER'] = int(args['DIRICHLET-NUMBER'])
assert 0 <= args['DIRICHLET-NUMBER'] <= 2, ValueError('Invalid Dirichlet boundary number.')
args['NEUMANN-NUMBER'] = int(args['NEUMANN-NUMBER'])
assert 0 <= args['NEUMANN-NUMBER'] <= 2, ValueError('Invalid Neumann boundary number.')
args['NEUMANN-COUNT'] = int(args['NEUMANN-COUNT'])
assert 0 <= args['NEUMANN-COUNT'] <= 3, ValueError('Invalid Neumann boundary count.')
rhss = [ExpressionFunction('ones(x.shape[:-1]) * 10', 2, ()),
ExpressionFunction('(x[..., 0] - 0.5) ** 2 * 1000', 2, ())]
dirichlets = [ExpressionFunction('zeros(x.shape[:-1])', 2, ()),
ExpressionFunction('ones(x.shape[:-1])', 2, ()),
ExpressionFunction('x[..., 0]', 2, ())]
neumanns = [None,
ConstantFunction(3., dim_domain=2),
ExpressionFunction('50*(0.1 <= x[..., 1]) * (x[..., 1] <= 0.2)'
'+50*(0.8 <= x[..., 1]) * (x[..., 1] <= 0.9)', 2, ())]
domains = [RectDomain(),
RectDomain(right='neumann'),
RectDomain(right='neumann', top='neumann'),
RectDomain(right='neumann', top='neumann', bottom='neumann')]
rhs = rhss[args['PROBLEM-NUMBER']]
dirichlet = dirichlets[args['DIRICHLET-NUMBER']]
neumann = neumanns[args['NEUMANN-NUMBER']]
domain = domains[args['NEUMANN-COUNT']]
problem = StationaryProblem(
domain=domain,
diffusion=ConstantFunction(1, dim_domain=2),
rhs=rhs,
dirichlet_data=dirichlet,
neumann_data=neumann
)
for n in [32, 128]:
print('Discretize ...')
discretizer = discretize_stationary_fv if args['--fv'] else discretize_stationary_cg
m, data = discretizer(
analytical_problem=problem,
grid_type=RectGrid if args['--rect'] else TriaGrid,
diameter=np.sqrt(2) / n if args['--rect'] else 1. / n
)
grid = data['grid']
print(grid)
print()
print('Solve ...')
U = m.solve()
m.visualize(U, title=repr(grid))
print()
def burgers_problem_2d(vx=1.,
vy=1.,
torus=True,
initial_data_type='sin',
parameter_range=(1., 2.)):
"""Two-dimensional Burgers-type problem.
The problem is to solve ::
∂_t u(x, t, μ) + ∇ ⋅ (v * u(x, t, μ)^μ) = 0
u(x, 0, μ) = u_0(x)
for u with t in [0, 0.3], x in [0, 2] x [0, 1].
Parameters
----------
vx
The x component of the velocity vector v.
vy
The y component of the velocity vector v.
torus
If `True`, impose periodic boundary conditions. Otherwise,
Dirichlet left and bottom, outflow top and right.
initial_data_type
Type of initial data (`'sin'` or `'bump'`).
parameter_range
The interval in which μ is allowed to vary.
"""
assert initial_data_type in ('sin', 'bump')
if initial_data_type == 'sin':
initial_data = ExpressionFunction(
"0.5 * (sin(2 * pi * x[..., 0]) * sin(2 * pi * x[..., 1]) + 1.)",
2, ())
dirichlet_data = ConstantFunction(dim_domain=2, value=0.5)
else:
initial_data = ExpressionFunction(
"(x[..., 0] >= 0.5) * (x[..., 0] <= 1) * 1", 2, ())
dirichlet_data = ConstantFunction(dim_domain=2, value=0.)
return InstationaryProblem(
StationaryProblem(
domain=TorusDomain([[0, 0], [2, 1]])
if torus else RectDomain([[0, 0], [2, 1]], right=None, top=None),
dirichlet_data=dirichlet_data,
rhs=None,
nonlinear_advection=ExpressionFunction("abs(x)**exponent * v", 1,
(2, ), {'exponent': 1},
{'v': np.array([vx, vy])}),
nonlinear_advection_derivative=ExpressionFunction(
"exponent * abs(x)**(exponent-1) * sign(x) * v", 1, (2, ),
{'exponent': 1}, {'v': np.array([vx, vy])}),
),
initial_data=initial_data,
T=0.3,
parameter_ranges=parameter_range,
name=f"burgers_problem_2d({vx}, {vy}, {torus}, '{initial_data_type}')")
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 test_visualize_patch(backend_gridtype):
backend, gridtype = backend_gridtype
domain = LineDomain() if gridtype is OnedGrid else RectDomain()
dim = 1 if gridtype is OnedGrid else 2
rhs = GenericFunction(lambda X: np.ones(X.shape[:-1]) * 10, dim) # NOQA
dirichlet = GenericFunction(lambda X: np.zeros(X.shape[:-1]), dim) # NOQA
diffusion = GenericFunction(lambda X: np.ones(X.shape[:-1]), dim) # NOQA
problem = StationaryProblem(domain=domain,
rhs=rhs,
dirichlet_data=dirichlet,
diffusion=diffusion)
grid, bi = discretize_domain_default(problem.domain, grid_type=gridtype)
m, data = discretize_stationary_cg(analytical_problem=problem,
grid=grid,
boundary_info=bi)
U = m.solve()
try:
visualize_patch(data['grid'], U=U, backend=backend)
except QtMissing as ie:
pytest.xfail("Qt missing")
finally:
stop_gui_processes()
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]')
thermalblock_problems = picklable_thermalblock_problems + non_picklable_thermalblock_problems
burgers_problems = \
[burgers_problem(),
burgers_problem(v=0.2, circle=False),
burgers_problem(v=0.4, initial_data_type='bump'),
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
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(
problem_number: int = Argument(
..., min=0, max=1, help='Selects the problem to solve [0 or 1].'),
dirichlet_number: int = Argument(
...,
min=0,
max=2,
help='Selects the Dirichlet data function [0 to 2].'),
neumann_number: int = Argument(
..., min=0, max=2, help='Selects the Neumann data function.'),
neumann_count: int = Argument(
...,
min=0,
max=3,
help='0: no neumann boundary\n\n'
'1: right edge is neumann boundary\n\n'
'2: right+top edges are neumann boundary\n\n'
'3: right+top+bottom edges are neumann boundary\n\n'),
fv: bool = Option(
False,
help='Use finite volume discretization instead of finite elements.'
),
rect: bool = Option(False, help='Use RectGrid instead of TriaGrid.'),
):
"""Solves the Poisson equation in 2D using pyMOR's builtin discreization toolkit."""
rhss = [
ExpressionFunction('ones(x.shape[:-1]) * 10', 2, ()),
ExpressionFunction('(x[..., 0] - 0.5) ** 2 * 1000', 2, ())
]
dirichlets = [
ExpressionFunction('zeros(x.shape[:-1])', 2, ()),
ExpressionFunction('ones(x.shape[:-1])', 2, ()),
ExpressionFunction('x[..., 0]', 2, ())
]
neumanns = [
None,
ConstantFunction(3., dim_domain=2),
ExpressionFunction(
'50*(0.1 <= x[..., 1]) * (x[..., 1] <= 0.2)'
'+50*(0.8 <= x[..., 1]) * (x[..., 1] <= 0.9)', 2, ())
]
domains = [
RectDomain(),
RectDomain(right='neumann'),
RectDomain(right='neumann', top='neumann'),
RectDomain(right='neumann', top='neumann', bottom='neumann')
]
rhs = rhss[problem_number]
dirichlet = dirichlets[dirichlet_number]
neumann = neumanns[neumann_number]
domain = domains[neumann_count]
problem = StationaryProblem(domain=domain,
diffusion=ConstantFunction(1, dim_domain=2),
rhs=rhs,
dirichlet_data=dirichlet,
neumann_data=neumann)
for n in [32, 128]:
print('Discretize ...')
discretizer = discretize_stationary_fv if fv else discretize_stationary_cg
m, data = discretizer(analytical_problem=problem,
grid_type=RectGrid if rect else TriaGrid,
diameter=np.sqrt(2) / n if rect else 1. / n)
grid = data['grid']
print(grid)
print()
print('Solve ...')
U = m.solve()
m.visualize(U, title=repr(grid))
print()
def main(
diameter: float = Argument(
0.1, help='Diameter option for the domain discretizer.'),
r: int = Argument(5, help='Order of the ROMs.'),
):
r"""2D heat equation demo.
Discretization of the PDE:
.. math::
:nowrap:
\begin{align*}
\partial_t z(x, y, t) &= \Delta z(x, y, t), & 0 < x, y < 1,\ t > 0 \\
-\nabla z(0, y, t) \cdot n &= z(0, y, t) - u(t), & 0 < y < 1, t > 0 \\
-\nabla z(1, y, t) \cdot n &= z(1, y, t), & 0 < y < 1, t > 0 \\
-\nabla z(0, x, t) \cdot n &= z(0, x, t), & 0 < x < 1, t > 0 \\
-\nabla z(1, x, t) \cdot n &= z(1, x, t), & 0 < x < 1, t > 0 \\
z(x, y, 0) &= 0 & 0 < x, y < 1 \\
y(t) &= \int_0^1 z(1, y, t) dy, & t > 0 \\
\end{align*}
where :math:`u(t)` is the input and :math:`y(t)` is the output.
"""
set_log_levels({'pymor.algorithms.gram_schmidt.gram_schmidt': 'WARNING'})
p = InstationaryProblem(StationaryProblem(
domain=RectDomain([[0., 0.], [1., 1.]],
left='robin',
right='robin',
top='robin',
bottom='robin'),
diffusion=ConstantFunction(1., 2),
robin_data=(ConstantFunction(1., 2),
ExpressionFunction('(x[...,0] < 1e-10) * 1.', 2)),
outputs=[('l2_boundary',
ExpressionFunction('(x[...,0] > (1 - 1e-10)) * 1.', 2))]),
ConstantFunction(0., 2),
T=1.)
fom, _ = discretize_instationary_cg(p, diameter=diameter, nt=100)
fom.visualize(fom.solve())
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}')
# System poles
poles = lti.poles()
fig, ax = plt.subplots()
ax.plot(poles.real, poles.imag, '.')
ax.set_title('System poles')
plt.show()
# Magnitude plot of the full model
w = np.logspace(-1, 3, 100)
fig, ax = plt.subplots()
lti.mag_plot(w, ax=ax)
ax.set_title('Magnitude plot of the full model')
plt.show()
# Hankel singular values
hsv = lti.hsv()
fig, ax = plt.subplots()
ax.semilogy(range(1, len(hsv) + 1), hsv, '.-')
ax.set_title('Hankel singular values')
plt.show()
# Norms of the system
print(f'FOM H_2-norm: {lti.h2_norm():e}')
if config.HAVE_SLYCOT:
print(f'FOM H_inf-norm: {lti.hinf_norm():e}')
else:
print('Skipped H_inf-norm calculation due to missing slycot.')
print(f'FOM Hankel-norm: {lti.hankel_norm():e}')
# Model order reduction
run_mor_method(lti, w, BTReductor(lti), 'BT', r, tol=1e-5)
run_mor_method(lti, w, LQGBTReductor(lti), 'LQGBT', r, tol=1e-5)
run_mor_method(lti, w, BRBTReductor(lti), 'BRBT', r, tol=1e-5)
run_mor_method(lti, w, IRKAReductor(lti), 'IRKA', r)
run_mor_method(lti, w, TSIAReductor(lti), 'TSIA', r)
run_mor_method(lti, w, OneSidedIRKAReductor(lti, 'V'), 'OS-IRKA', r)