#!/usr/bin/env python
# This file is part of the pyMOR project (http://www.pymor.org).
# Copyright 2013-2020 pyMOR developers and contributors. All rights reserved.
# License: BSD 2-Clause License (http://opensource.org/licenses/BSD-2-Clause)
from functools import partial # fix parameters of given function
import numpy as np
from typer import Argument, run
from pymor.basic import * # most common pyMOR functions and classes
from pymor.algorithms.timestepping import ImplicitEulerTimeStepper
from pymor.tools.typer import Choices
# parameters for high-dimensional models
GRID_INTERVALS = 50
FENICS_ORDER = 2
NT = 50
DT = 1. / NT
####################################################################################################
# Main script #
####################################################################################################
[docs]def main(
backend: Choices('pymor fenics') = Argument(..., help='Discretization toolkit to use.'),
alg: Choices('greedy adaptive_greedy pod') = Argument(..., help='The model reduction algorithm to use.'),
snapshots: int = Argument(
...,
help='greedy/pod: number of training set parameters\n\n'
'adaptive_greedy: size of validation set.'
),
rbsize: int = Argument(..., help='Size of the reduced basis.'),
test: int = Argument(..., help='Number of test parameters for reduction error estimation.'),
):
"""Reduced basis approximation of the heat equation."""
# discretize
############
if backend == 'pymor':
fom = discretize_pymor()
elif backend == 'fenics':
fom = discretize_fenics()
else:
raise NotImplementedError
parameter_space=fom.parameters.space(1, 100)
# select reduction algorithm with error estimator
#################################################
coercivity_estimator = ExpressionParameterFunctional('1.', fom.parameters)
reductor = ParabolicRBReductor(fom, product=fom.h1_0_semi_product, coercivity_estimator=coercivity_estimator)
# generate reduced model
########################
if alg == 'greedy':
rom = reduce_greedy(fom, reductor, parameter_space, snapshots, rbsize)
elif alg == 'adaptive_greedy':
rom = reduce_adaptive_greedy(fom, reductor, parameter_space, snapshots, rbsize)
elif alg == 'pod':
rom = reduce_pod(fom, reductor, parameter_space, snapshots, rbsize)
else:
raise NotImplementedError
# evaluate the reduction error
##############################
results = reduction_error_analysis(
rom, fom=fom, reductor=reductor, error_estimator=True,
error_norms=[lambda U: DT * np.sqrt(np.sum(fom.h1_0_semi_norm(U)[1:]**2))],
error_norm_names=['l^2-h^1'],
condition=False, test_mus=parameter_space.sample_randomly(test, seed=999), plot=True
)
# show results
##############
print(results['summary'])
import matplotlib.pyplot as plt
plt.show()
# write results to disk
#######################
from pymor.core.pickle import dump
dump(rom, open('reduced_model.out', 'wb'))
results.pop('figure') # matplotlib figures cannot be serialized
dump(results, open('results.out', 'wb'))
# visualize reduction error for worst-approximated mu
#####################################################
mumax = results['max_error_mus'][0, -1]
U = fom.solve(mumax)
U_RB = reductor.reconstruct(rom.solve(mumax))
if backend == 'fenics': # right now the fenics visualizer does not support time trajectories
U = U[len(U) - 1].copy()
U_RB = U_RB[len(U_RB) - 1].copy()
fom.visualize((U, U_RB, U - U_RB), legend=('Detailed Solution', 'Reduced Solution', 'Error'),
separate_colorbars=True)
return results
####################################################################################################
# High-dimensional models #
####################################################################################################
[docs]def discretize_pymor():
# setup analytical problem
problem = InstationaryProblem(
StationaryProblem(
domain=RectDomain(top='dirichlet', bottom='neumann'),
diffusion=LincombFunction(
[ConstantFunction(1., dim_domain=2),
ExpressionFunction('(x[..., 0] > 0.45) * (x[..., 0] < 0.55) * (x[..., 1] < 0.7) * 1.',
dim_domain=2),
ExpressionFunction('(x[..., 0] > 0.35) * (x[..., 0] < 0.40) * (x[..., 1] > 0.3) * 1. + '
'(x[..., 0] > 0.60) * (x[..., 0] < 0.65) * (x[..., 1] > 0.3) * 1.',
dim_domain=2)],
[1.,
100. - 1.,
ExpressionParameterFunctional('top[0] - 1.', {'top': 1})]
),
rhs=ConstantFunction(value=100., dim_domain=2) * ExpressionParameterFunctional('sin(10*pi*t[0])', {'t': 1}),
dirichlet_data=ConstantFunction(value=0., dim_domain=2),
neumann_data=ExpressionFunction('(x[..., 0] > 0.45) * (x[..., 0] < 0.55) * -1000.',
dim_domain=2),
),
T=1.,
initial_data=ExpressionFunction('(x[..., 0] > 0.45) * (x[..., 0] < 0.55) * (x[..., 1] < 0.7) * 10.',
dim_domain=2)
)
# discretize using continuous finite elements
fom, _ = discretize_instationary_cg(analytical_problem=problem, diameter=1./GRID_INTERVALS, nt=NT)
fom.enable_caching('disk')
return fom
[docs]def discretize_fenics():
from pymor.tools import mpi
if mpi.parallel:
from pymor.models.mpi import mpi_wrap_model
return mpi_wrap_model(_discretize_fenics, use_with=True, pickle_local_spaces=False)
else:
return _discretize_fenics()
def _discretize_fenics():
# assemble system matrices - FEniCS code
########################################
import dolfin as df
# discrete function space
mesh = df.UnitSquareMesh(GRID_INTERVALS, GRID_INTERVALS, 'crossed')
V = df.FunctionSpace(mesh, 'Lagrange', FENICS_ORDER)
u = df.TrialFunction(V)
v = df.TestFunction(V)
# data functions
bottom_diffusion = df.Expression('(x[0] > 0.45) * (x[0] < 0.55) * (x[1] < 0.7) * 1.',
element=df.FunctionSpace(mesh, 'DG', 0).ufl_element())
top_diffusion = df.Expression('(x[0] > 0.35) * (x[0] < 0.40) * (x[1] > 0.3) * 1. +'
'(x[0] > 0.60) * (x[0] < 0.65) * (x[1] > 0.3) * 1.',
element=df.FunctionSpace(mesh, 'DG', 0).ufl_element())
initial_data = df.Expression('(x[0] > 0.45) * (x[0] < 0.55) * (x[1] < 0.7) * 10.',
element=df.FunctionSpace(mesh, 'DG', 0).ufl_element())
neumann_data = df.Expression('(x[0] > 0.45) * (x[0] < 0.55) * 1000.',
element=df.FunctionSpace(mesh, 'DG', 0).ufl_element())
# assemble matrices and vectors
l2_mat = df.assemble(df.inner(u, v) * df.dx)
l2_0_mat = l2_mat.copy()
h1_mat = df.assemble(df.inner(df.nabla_grad(u), df.nabla_grad(v)) * df.dx)
h1_0_mat = h1_mat.copy()
mat0 = h1_mat.copy()
mat0.zero()
bottom_mat = df.assemble(bottom_diffusion * df.inner(df.nabla_grad(u), df.nabla_grad(v)) * df.dx)
top_mat = df.assemble(top_diffusion * df.inner(df.nabla_grad(u), df.nabla_grad(v)) * df.dx)
u0 = df.project(initial_data, V).vector()
f = df.assemble(neumann_data * v * df.ds)
# boundary treatment
def dirichlet_boundary(x, on_boundary):
tol = 1e-14
return on_boundary and (abs(x[0]) < tol or abs(x[0] - 1) < tol or abs(x[1] - 1) < tol)
bc = df.DirichletBC(V, df.Constant(0.), dirichlet_boundary)
bc.apply(l2_0_mat)
bc.apply(h1_0_mat)
bc.apply(mat0)
bc.zero(bottom_mat)
bc.zero(top_mat)
bc.apply(f)
bc.apply(u0)
# wrap everything as a pyMOR model
##################################
from pymor.bindings.fenics import FenicsVectorSpace, FenicsMatrixOperator, FenicsVisualizer
fom = InstationaryModel(
T=1.,
initial_data=FenicsVectorSpace(V).make_array([u0]),
operator=LincombOperator([FenicsMatrixOperator(mat0, V, V),
FenicsMatrixOperator(h1_0_mat, V, V),
FenicsMatrixOperator(bottom_mat, V, V),
FenicsMatrixOperator(top_mat, V, V)],
[1.,
1.,
100. - 1.,
ExpressionParameterFunctional('top[0] - 1.', {'top': 1})]),
rhs=VectorOperator(FenicsVectorSpace(V).make_array([f])),
mass=FenicsMatrixOperator(l2_0_mat, V, V, name='l2'),
products={'l2': FenicsMatrixOperator(l2_mat, V, V, name='l2'),
'l2_0': FenicsMatrixOperator(l2_0_mat, V, V, name='l2_0'),
'h1': FenicsMatrixOperator(h1_mat, V, V, name='h1'),
'h1_0_semi': FenicsMatrixOperator(h1_0_mat, V, V, name='h1_0_semi')},
time_stepper=ImplicitEulerTimeStepper(nt=NT),
visualizer=FenicsVisualizer(FenicsVectorSpace(V))
)
return fom
####################################################################################################
# Reduction algorithms #
####################################################################################################
[docs]def reduce_greedy(fom, reductor, parameter_space, snapshots, basis_size):
training_set = parameter_space.sample_uniformly(snapshots)
pool = new_parallel_pool()
greedy_data = rb_greedy(fom, reductor, training_set, max_extensions=basis_size, pool=pool)
return greedy_data['rom']
[docs]def reduce_adaptive_greedy(fom, reductor, parameter_space, validation_mus, basis_size):
pool = new_parallel_pool()
greedy_data = rb_adaptive_greedy(fom, reductor, parameter_space,
validation_mus=validation_mus,
max_extensions=basis_size, pool=pool)
return greedy_data['rom']
[docs]def reduce_pod(fom, reductor, parameter_space, snapshots, basis_size):
training_set = parameter_space.sample_uniformly(snapshots)
snapshots = fom.operator.source.empty()
for mu in training_set:
snapshots.append(fom.solve(mu))
basis, singular_values = pod(snapshots, modes=basis_size, product=fom.h1_0_semi_product)
reductor.extend_basis(basis, method='trivial')
rom = reductor.reduce()
return rom
if __name__ == '__main__':
run(main)