def error_analysis(N, M):
print('N = {}, M = {}: '.format(N, M), end='')
rd, rc, _ = reduce_to_subbasis(rb_discretization, N, reconstructor)
rd = rd.with_(explicit_operator=rd.explicit_operator.projected_to_subbasis(dim_collateral=M))
l2_err_max = -1
mumax = None
for mu in mus:
print('.', end='')
sys.stdout.flush()
u = rd.solve(mu)
URB = rc.reconstruct(u)
U = discretization.solve(mu)
l2_err = np.max(discretization.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
def burgers_demo(args):
args['--grid'] = int(args['--grid'])
args['--initial-data'] = args['--initial-data'].lower()
assert args['--initial-data'] in ('sin', 'bump')
args['--lxf-lambda'] = float(args['--lxf-lambda'])
args['--nt'] = int(args['--nt'])
args['--not-periodic'] = bool(args['--not-periodic'])
args['--num-flux'] = args['--num-flux'].lower()
assert args['--num-flux'] in ('lax_friedrichs', 'engquist_osher')
args['--plot-error-landscape-N'] = int(args['--plot-error-landscape-N'])
args['--plot-error-landscape-M'] = int(args['--plot-error-landscape-M'])
args['--test'] = int(args['--test'])
args['--vx'] = float(args['--vx'])
args['--vy'] = float(args['--vy'])
args['EXP_MIN'] = int(args['EXP_MIN'])
args['EXP_MAX'] = int(args['EXP_MAX'])
args['DIFF_MIN'] = float(args['DIFF_MIN'])
args['DIFF_MAX'] = float(args['DIFF_MAX'])
args['RBSIZE'] = int(args['RBSIZE'])
args['EXP_SNAPSHOTS'] = int(args['EXP_SNAPSHOTS'])
args['DIFF_SNAPSHOTS'] = int(args['DIFF_SNAPSHOTS'])
args['INIT_EISIZE'] = int(args['INIT_EISIZE'])
args['EXP_INIT_SNAPSHOTS'] = int(args['EXP_INIT_SNAPSHOTS'])
args['DIFF_INIT_SNAPSHOTS'] = int(args['DIFF_INIT_SNAPSHOTS'])
print('Setup Problem ...')
domain_discretizer = partial(discretize_domain_default, grid_type=RectGrid)
problem = ViscousBurgersProblem(vx=args['--vx'], vy=args['--vy'], initial_data=args['--initial-data'],
parameter_range={'exponent': (args['EXP_MIN'], args['EXP_MAX']), 'diffusion': (args['DIFF_MIN'], args['DIFF_MAX'])},
torus=not args['--not-periodic'])
print('Discretize ...')
discretizer = discretize_nonlinear_instationary_advection_diffusion_fv
discretization, _ = discretizer(problem, diameter=m.sqrt(2) / args['--grid'],
num_flux=args['--num-flux'], lxf_lambda=args['--lxf-lambda'],
nt=args['--nt'], domain_discretizer=domain_discretizer)
print(discretization.explicit_operator.grid)
print('The parameter type is {}'.format(discretization.parameter_type))
if args['--plot-solutions']:
print('Showing some solutions')
for mu in discretization.parameter_space.sample_randomly(2):
print('Solving for exponent = \n{} ... '.format(mu['exponent']))
print('Solving for diffusion = \n{} ... '.format(mu['diffusion']))
sys.stdout.flush()
U = discretization.solve(mu)
discretization.visualize(U)
print('CB an RB generation ...')
extension_algorithm = partial(pod_basis_extension)
if args['EXP_MIN'] == args['EXP_MAX']:
init_samples = discretization.parameter_space.sample_uniformly({'exponent': 1, 'diffusion': args['DIFF_INIT_SNAPSHOTS']})
samples = discretization.parameter_space.sample_uniformly({'exponent': 1, 'diffusion': args['DIFF_SNAPSHOTS']})
elif args['DIFF_MIN'] == args['DIFF_MAX']:
init_samples = discretization.parameter_space.sample_uniformly({'exponent': args['EXP_INIT_SNAPSHOTS'], 'diffusion': 1})
samples = discretization.parameter_space.sample_uniformly({'exponent': args['EXP_SNAPSHOTS'], 'diffusion': 1})
else:
init_samples = discretization.parameter_space.sample_uniformly({'exponent': args['EXP_INIT_SNAPSHOTS'], 'diffusion': args['DIFF_INIT_SNAPSHOTS']})
samples = discretization.parameter_space.sample_uniformly({'exponent': args['EXP_SNAPSHOTS'], 'diffusion': args['DIFF_SNAPSHOTS']})
_, ei_initial_data = interpolate_operators(discretization, ['explicit_operator'],
init_samples,
error_norm=discretization.l2_norm, target_error=1e-10,
max_interpolation_dofs=args['INIT_EISIZE'], projection='orthogonal',
product=discretization.l2_product)
rb_initial_data = discretization.initial_data.as_vector()
rb_initial_data *= 1 / rb_initial_data.l2_norm()[0]
ei_greedy_data = ei_rb_greedy(discretization,
operator_names=['explicit_operator'], samples=samples,
error_norm=discretization.l2_norm,
target_error=1e-10, rb_initial_data=rb_initial_data, ei_initial_data=ei_initial_data,
product=discretization.l2_product, max_extensions=args['RBSIZE'],
use_estimator=False, extension_algorithm=extension_algorithm)
rb_discretization, reconstructor = ei_greedy_data['reduced_discretization'], ei_greedy_data['reconstructor']
ei_discretization, ei_data = ei_greedy_data['ei_discretization'], ei_greedy_data['ei_data']
if args['--plot-ei-err']:
print('Showing some EI errors')
for mu in discretization.parameter_space.sample_randomly(2):
print('Solving for exponent = \n{} ... '.format(mu['exponent']))
print('Solving for diffusion = \n{} ... '.format(mu['diffusion']))
sys.stdout.flush()
U = discretization.solve(mu)
U_EI = ei_discretization.solve(mu)
ERR = U - U_EI
print('Error: {}'.format(np.max(discretization.l2_norm(ERR))))
discretization.visualize(ERR)
print('Showing interpolation DOFs ...')
U = np.zeros(U.dim)
dofs = ei_discretization.operator.interpolation_dofs
U[dofs] = np.arange(1, len(dofs) + 1)
U[ei_discretization.operator.source_dofs] += int(len(dofs)/2)
discretization.visualize(NumpyVectorArray(U))
print('\nSearching for maximum error on random snapshots ...')
tic = time.time()
mus = list(discretization.parameter_space.sample_randomly(args['--test']))
#mus.append({'diffusion': np.array([0.0]), 'exponent': np.array(2.0)})
def error_analysis(N, M):
print('N = {}, M = {}: '.format(N, M), end='')
rd, rc, _ = reduce_to_subbasis(rb_discretization, N, reconstructor)
rd = rd.with_(explicit_operator=rd.explicit_operator.projected_to_subbasis(dim_collateral=M))
l2_err_max = -1
mumax = None
for mu in mus:
print('.', end='')
sys.stdout.flush()
u = rd.solve(mu)
URB = rc.reconstruct(u)
U = discretization.solve(mu)
l2_err = np.max(discretization.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(ei_greedy_data['data'])
real_cb_size = len(ei_data['basis'])
if args['--plot-error-landscape']:
N_count = min(real_rb_size - 1, args['--plot-error-landscape-N'])
M_count = min(real_cb_size - 1, args['--plot-error-landscape-M'])
Ns = np.linspace(1, real_rb_size, N_count).astype(np.int)
Ms = np.linspace(1, real_cb_size, M_count).astype(np.int)
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.float)
l2_err_max = errs[-1, -1]
mumax = err_mus[-1, -1]
toc = time.time()
t_est = toc - tic
print('''
*** RESULTS ***
Problem:
parameter range: ({args[EXP_MIN]}, {args[EXP_MAX]}), ({args[DIFF_MIN]}, {args[DIFF_MAX]})
h: sqrt(2)/{args[--grid]}
initial-data: {args[--initial-data]}
lxf-lambda: {args[--lxf-lambda]}
nt: {args[--nt]}
not-periodic: {args[--not-periodic]}
num-flux: {args[--num-flux]}
(vx, vy): ({args[--vx]}, {args[--vy]})
Greedy basis generation:
number of initial snapshots: ({args[EXP_INIT_SNAPSHOTS]}, {args[DIFF_INIT_SNAPSHOTS]})
initial collateral basis size: {args[INIT_EISIZE]}
actual collateral basis size: {real_cb_size}
number of snapshots: ({args[EXP_SNAPSHOTS]}, {args[DIFF_SNAPSHOTS]})
prescribed basis size: {args[RBSIZE]}
actual basis size: {real_rb_size}
elapsed time: {ei_greedy_data[time]}
Stochastic error estimation:
number of samples: {args[--test]}
maximal L2-error: {l2_err_max} (mu = {mumax})
maximum errors: {errs}
N_grid (RB size): {N_grid}
M_grid (CB size): {M_grid}
elapsed time: {t_est}
'''.format(**locals()))
sys.stdout.flush()
if args['--plot-error-landscape']:
import matplotlib.pyplot as plt
import mpl_toolkits.mplot3d
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
# we have to rescale the errors since matplotlib does not support logarithmic scales on 3d plots
# https://github.com/matplotlib/matplotlib/issues/209
surf = ax.plot_surface(M_grid, N_grid, np.log(np.minimum(errs, 1)) / np.log(10),
rstride=1, cstride=1, cmap='jet')
plt.show()
if args['--plot-error-correlation']:
import matplotlib.pyplot as plt
from matplotlib.ticker import FuncFormatter
errs = []
Ns = ei_greedy_data['N_M_correlation'][0]
Ms = ei_greedy_data['N_M_correlation'][1]
corr = []
for i in np.arange(real_rb_size):
N = Ns[i]
M = Ms[i]
if M > real_cb_size:
M = real_cb_size
corr.append(M/N)
print('N = {}, M = {}: '.format(N, M), end='')
rd, rc, _ = reduce_to_subbasis(rb_discretization, N, reconstructor)
rd = rd.with_(explicit_operator=rd.explicit_operator.projected_to_subbasis(dim_collateral=M))
l2_err_max = -1
mumax = None
for mu in mus:
print('.', end='')
sys.stdout.flush()
u = rd.solve(mu)
URB = rc.reconstruct(u)
U = discretization.solve(mu)
l2_err = np.max(discretization.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
errs.append(l2_err_max)
print(mumax)
print()
def tick_function(x, pos):
N = int(x)
if not N == 0:
if N > real_rb_size:
value = (N, Ms[-1])
else:
value = (N, Ms[N-1])
else:
value = (1, Ms[0])
return str(value)
plt.semilogy(Ns, errs)
plt.xlabel('(RB size, CRB size)')
plt.ylabel('max. L2-error')
ax = plt.axes()
ax.xaxis.set_major_formatter( FuncFormatter(tick_function))
plt.show()
plt.plot(Ns, corr)
plt.xlabel('(RB size, CRB size)')
plt.ylabel('M/N correlation')
ax = plt.axes()
ax.xaxis.set_major_formatter( FuncFormatter(tick_function))
plt.show()
if args['--plot-err']:
discretization.visualize(U - URB)
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['--estimator-norm'] = args['--estimator-norm'].lower()
assert args['--estimator-norm'] in {'trivial', 'h1'}
args['--extension-alg'] = args['--extension-alg'].lower()
assert args['--extension-alg'] in {'trivial', 'gram_schmidt', 'h1_gram_schmidt'}
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=m.sqrt(2) / 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 ...')
error_product = discretization.h1_product if args['--estimator-norm'] == 'h1' else None
reductor = partial(reduce_stationary_affine_linear, error_product=error_product)
extension_algorithms = {'trivial': trivial_basis_extension,
'gram_schmidt': gram_schmidt_basis_extension,
'h1_gram_schmidt': partial(gram_schmidt_basis_extension, product=discretization.h1_product)}
extension_algorithm = extension_algorithms[args['--extension-alg']]
greedy_data = greedy(discretization, reductor, discretization.parameter_space.sample_uniformly(args['SNAPSHOTS']),
use_estimator=args['--with-estimator'], error_norm=discretization.h1_norm,
extension_algorithm=extension_algorithm, max_extensions=args['RBSIZE'])
rb_discretization, reconstructor = greedy_data['reduced_discretization'], greedy_data['reconstructor']
print('\nSearching for maximum error on random snapshots ...')
def error_analysis(d, rd, rc, mus):
print('N = {}: '.format(rd.operator.dim_source), end='')
h1_err_max = -1
h1_est_max = -1
cond_max = -1
for mu in mus:
print('.', end='')
u = rd.solve(mu)
URB = rc.reconstruct(u)
U = d.solve(mu)
h1_err = d.h1_norm(U - URB)[0]
h1_est = rd.estimate(u, mu=mu)
cond = np.linalg.cond(rd.operator.assemble(mu)._matrix)
if h1_err > h1_err_max:
h1_err_max = h1_err
mumax = mu
if h1_est > h1_est_max:
h1_est_max = h1_est
mu_est_max = mu
if cond > cond_max:
cond_max = cond
cond_max_mu = mu
print()
return h1_err_max, mumax, h1_est_max, mu_est_max, cond_max, cond_max_mu
tic = time.time()
real_rb_size = len(greedy_data['basis'])
if args['--plot-error-sequence']:
N_count = min(real_rb_size - 1, 25)
Ns = np.linspace(1, real_rb_size, N_count).astype(np.int)
else:
Ns = np.array([real_rb_size])
rd_rcs = [reduce_to_subbasis(rb_discretization, N, reconstructor)[:2] for N in Ns]
mus = list(discretization.parameter_space.sample_randomly(args['--test']))
errs, err_mus, ests, est_mus, conds, cond_mus = zip(*(error_analysis(discretization, rd, rc, mus)
for rd, rc in rd_rcs))
h1_err_max = errs[-1]
mumax = err_mus[-1]
cond_max = conds[-1]
cond_max_mu = cond_mus[-1]
toc = time.time()
t_est = toc - tic
print('''
*** RESULTS ***
Problem:
number of blocks: {args[XBLOCKS]}x{args[YBLOCKS]}
h: sqrt(2)/{args[--grid]}
Greedy basis generation:
number of snapshots: {args[SNAPSHOTS]}^({args[XBLOCKS]}x{args[YBLOCKS]})
used estimator: {args[--with-estimator]}
estimator norm: {args[--estimator-norm]}
extension method: {args[--extension-alg]}
prescribed basis size: {args[RBSIZE]}
actual basis size: {real_rb_size}
elapsed time: {greedy_data[time]}
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-error-sequence']:
plt.semilogy(Ns, errs, Ns, ests)
plt.legend(('error', 'estimator'))
plt.show()
if args['--plot-err']:
U = discretization.solve(mumax)
URB = reconstructor.reconstruct(rb_discretization.solve(mumax))
discretization.visualize((U, URB, U - URB), legend=('Detailed Solution', 'Reduced Solution', 'Error'),
title='Maximum Error Solution', separate_colorbars=True, block=True)