Source code for pymordemos.burgers_ei

#!/usr/bin/env python
# This file is part of the pyMOR project (
# Copyright 2013-2020 pyMOR developers and contributors. All rights reserved.
# License: BSD 2-Clause License (

import sys
import math
import time

import numpy as np
from typer import Argument, Option, run

from pymor.algorithms.greedy import rb_greedy
from pymor.algorithms.ei import interpolate_operators
from pymor.analyticalproblems.burgers import burgers_problem_2d
from pymor.discretizers.builtin import discretize_instationary_fv, RectGrid, TriaGrid
from pymor.parallel.default import new_parallel_pool
from pymor.reductors.basic import InstationaryRBReductor
from import Choices

[docs]def main( exp_min: float = Argument(..., help='Minimal exponent'), exp_max: float = Argument(..., help='Maximal exponent'), ei_snapshots: int = Argument(..., help='Number of snapshots for empirical interpolation.'), ei_size: int = Argument(..., help='Number of interpolation DOFs.'), snapshots: int = Argument(..., help='Number of snapshots for basis generation.'), rb_size: int = Argument(..., help='Size of the reduced basis.'), cache_region: Choices('none memory disk persistent') = Option( 'disk', help='Name of cache region to use for caching solution snapshots.' ), ei_alg: Choices('ei_greedy deim') = Option('ei_greedy', help='Interpolation algorithm to use.'), grid: int = Option(60, help='Use grid with (2*NI)*NI elements.'), grid_type: Choices('rect tria') = Option('rect', help='Type of grid to use.'), initial_data: Choices('sin bump') = Option('sin', help='Select the initial data (sin, bump).'), ipython_engines: int = Option( 0, help='If positive, the number of IPython cluster engines to use for parallel greedy search. ' 'If zero, no parallelization is performed.'), ipython_profile: str = Option(None, help='IPython profile to use for parallelization.'), lxf_lambda: float = Option(1., help='Parameter lambda in Lax-Friedrichs flux.'), periodic: bool = Option(True, help='If not, solve with dirichlet boundary conditions on left and bottom boundary.'), nt: int = Option(100, help='Number of time steps.'), num_flux: Choices('lax_friedrichs engquist_osher') = Option('engquist_osher', help='Numerical flux to use.'), plot_err: bool = Option(False, help='Plot error.'), plot_ei_err: bool = Option(False, help='Plot empirical interpolation error.'), plot_error_landscape: bool = Option(False, help='Calculate and show plot of reduction error vs. basis sizes.'), plot_error_landscape_M: int = Option(10, help='Number of collateral basis sizes to test.'), plot_error_landscape_N: int = Option(10, help='Number of basis sizes to test.'), plot_solutions: bool = Option(False, help='Plot some example solutions.'), test: int = Option(10, help='Number of snapshots to use for stochastic error estimation.'), vx: float = Option(1., help='Speed in x-direction.'), vy: float = Option(1., help='Speed in y-direction.'), ): """Model order reduction of a two-dimensional Burgers-type equation (see pymor.analyticalproblems.burgers) using the reduced basis method with empirical operator interpolation. """ print('Setup Problem ...') problem = burgers_problem_2d(vx=vx, vy=vy, initial_data_type=initial_data.value, parameter_range=(exp_min, exp_max), torus=periodic) print('Discretize ...') if grid_type == 'rect': grid *= 1. / math.sqrt(2) fom, _ = discretize_instationary_fv( problem, diameter=1. / grid, grid_type=RectGrid if grid_type == 'rect' else TriaGrid, num_flux=num_flux.value, lxf_lambda=lxf_lambda, nt=nt ) if cache_region != 'none': # building a cache_id is only needed for persistent CacheRegions cache_id = (f"pymordemos.burgers_ei {vx} {vy} {initial_data}" f"{periodic} {grid} {grid_type} {num_flux} {lxf_lambda} {nt}") fom.enable_caching(cache_region.value, cache_id) print(fom.operator.grid) print(f'The parameters are {fom.parameters}') if plot_solutions: print('Showing some solutions') Us = () legend = () for mu in problem.parameter_space.sample_uniformly(4): print(f"Solving for exponent = {mu['exponent']} ... ") sys.stdout.flush() Us = Us + (fom.solve(mu),) legend = legend + (f"exponent: {mu['exponent']}",) fom.visualize(Us, legend=legend, title='Detailed Solutions', block=True) pool = new_parallel_pool(ipython_num_engines=ipython_engines, ipython_profile=ipython_profile) eim, ei_data = interpolate_operators(fom, ['operator'], problem.parameter_space.sample_uniformly(ei_snapshots), error_norm=fom.l2_norm, product=fom.l2_product, max_interpolation_dofs=ei_size, alg=ei_alg.value, pool=pool) if plot_ei_err: print('Showing some EI errors') ERRs = () legend = () for mu in problem.parameter_space.sample_randomly(2): print(f"Solving for exponent = \n{mu['exponent']} ... ") sys.stdout.flush() U = fom.solve(mu) U_EI = eim.solve(mu) ERR = U - U_EI ERRs = ERRs + (ERR,) legend = legend + (f"exponent: {mu['exponent']}",) print(f'Error: {np.max(fom.l2_norm(ERR))}') fom.visualize(ERRs, legend=legend, title='EI Errors', separate_colorbars=True) print('Showing interpolation DOFs ...') U = np.zeros(U.dim) dofs = eim.operator.interpolation_dofs U[dofs] = np.arange(1, len(dofs) + 1) U[eim.operator.source_dofs] += int(len(dofs)/2) fom.visualize(fom.solution_space.make_array(U), title='Interpolation DOFs') print('RB generation ...') reductor = InstationaryRBReductor(eim) greedy_data = rb_greedy(fom, reductor, problem.parameter_space.sample_uniformly(snapshots), use_error_estimator=False, error_norm=lambda U: np.max(fom.l2_norm(U)), extension_params={'method': 'pod'}, max_extensions=rb_size, pool=pool) rom = greedy_data['rom'] print('\nSearching for maximum error on random snapshots ...') tic = time.perf_counter() mus = problem.parameter_space.sample_randomly(test) def error_analysis(N, M): print(f'N = {N}, M = {M}: ', end='') rom = reductor.reduce(N) rom = rom.with_(operator=rom.operator.with_cb_dim(M)) l2_err_max = -1 mumax = None for mu in mus: print('.', end='') sys.stdout.flush() u = rom.solve(mu) URB = reductor.reconstruct(u) U = fom.solve(mu) l2_err = np.max(fom.l2_norm(U - URB)) l2_err = np.inf if not np.isfinite(l2_err) else l2_err if l2_err > l2_err_max: l2_err_max = l2_err mumax = mu print() return l2_err_max, mumax error_analysis = np.frompyfunc(error_analysis, 2, 2) real_rb_size = len(reductor.bases['RB']) real_cb_size = len(ei_data['basis']) if plot_error_landscape: N_count = min(real_rb_size - 1, plot_error_landscape_N) M_count = min(real_cb_size - 1, plot_error_landscape_M) Ns = np.linspace(1, real_rb_size, N_count).astype( Ms = np.linspace(1, real_cb_size, M_count).astype( else: Ns = np.array([real_rb_size]) Ms = np.array([real_cb_size]) N_grid, M_grid = np.meshgrid(Ns, Ms) errs, err_mus = error_analysis(N_grid, M_grid) errs = errs.astype(np.float64) l2_err_max = errs[-1, -1] mumax = err_mus[-1, -1] toc = time.perf_counter() t_est = toc - tic print(''' *** RESULTS *** Problem: parameter range: ({exp_min}, {exp_max}) h: sqrt(2)/{grid} grid-type: {grid_type} initial-data: {initial_data} lxf-lambda: {lxf_lambda} nt: {nt} not-periodic: {periodic} num-flux: {num_flux} (vx, vy): ({vx}, {vy}) Greedy basis generation: number of ei-snapshots: {ei_snapshots} prescribed collateral basis size: {ei_size} actual collateral basis size: {real_cb_size} number of snapshots: {snapshots} prescribed basis size: {rb_size} actual basis size: {real_rb_size} elapsed time: {greedy_data[time]} Stochastic error estimation: number of samples: {test} maximal L2-error: {l2_err_max} (mu = {mumax}) elapsed time: {t_est} '''.format(**locals())) sys.stdout.flush() if plot_error_landscape: import matplotlib.pyplot as plt import mpl_toolkits.mplot3d # NOQA fig = plt.figure() ax = fig.add_subplot(111, projection='3d') # rescale the errors since matplotlib does not support logarithmic scales on 3d plots # surf = ax.plot_surface(M_grid, N_grid, np.log(np.minimum(errs, 1)) / np.log(10), rstride=1, cstride=1, cmap='jet') if plot_err: U = fom.solve(mumax) URB = reductor.reconstruct(rom.solve(mumax)) fom.visualize((U, URB, U - URB), legend=('Detailed Solution', 'Reduced Solution', 'Error'), title='Maximum Error Solution', separate_colorbars=True) global test_results test_results = (ei_data, greedy_data)
if __name__ == '__main__': run(main)