def refine_osc(cls, w, coeff_field, a=1):
Cadelta = 1.0
mesh_maxh = w.basis.basis.mesh.hmax()
coeff_min_val, coeff_max_grad = 1e10, 0.0
suppLambda = supp(w.active_indices())
if len(suppLambda) > 0:
try:
# a0_f = coeff_field.mean_func
# print "suppLambda", suppLambda
for m in suppLambda:
coeff, _ = coeff_field[m]
min_val, max_grad = abs(coeff.min_val), abs(coeff.max_grad)
coeff_min_val, coeff_max_grad = min(coeff_min_val, min_val), max(coeff_max_grad, max_grad)
print "DDDD", m, min_val, max_grad, coeff_min_val, coeff_max_grad
logger.debug("\tm: %s %s %s %s %s" % (m, min_val, max_grad, coeff_min_val, coeff_max_grad))
# determine (4.14) c_{a,\delta}
Cadelta = mesh_maxh * coeff_max_grad / coeff_min_val
except:
logger.error("coefficient does not provide max_val and max_grad. OSC refinement not supported for this case...")
# determine maximal mesh size to resolve coefficient oscillations
logger.debug("OSC marking maxh {0} and Cadelta {1} with scaling factor {2}".format(mesh_maxh, Cadelta, a))
print "OSC marking maxh {0} and Cadelta {1} with scaling factor {2}".format(mesh_maxh, Cadelta, a)
maxh = a * mesh_maxh / Cadelta
# create appropriate mesh by refinement and project current solution
new_w = w.refine_maxh(maxh)
return new_w, maxh, Cadelta
else:
logger.info("SKIP OSC refinement since only active mi is deterministic.")
return w, 1.0, Cadelta
def LambdaBoundary(Lambda):
suppLambda = supp(Lambda)
for mu in Lambda:
for m in suppLambda:
mu1 = mu.inc(m)
if mu1 not in Lambda:
yield mu1
mu2 = mu.dec(m)
if mu2 not in Lambda and mu2 is not None:
yield mu2
def evaluateUpperTailBound(cls, w, coeff_field, pde, maxh=1 / 10, add_maxm=10):
"""Estimate upper tail bounds according to Section 3.2."""
@cache
def get_ainfty(m, V):
a0_f = coeff_field.mean_func
if isinstance(a0_f, tuple):
a0_f = a0_f[0]
# determine min \overline{a} on D (approximately)
f = FEniCSVector.from_basis(V, sub_spaces=0)
f.interpolate(a0_f)
min_a0 = f.min_val
am_f, _ = coeff_field[m]
if isinstance(am_f, tuple):
am_f = am_f[0]
# determine ||a_m/\overline{a}||_{L\infty(D)} (approximately)
try:
# use exact bounds if defined
max_am = am_f.max_val
except:
# otherwise interpolate
f.interpolate(am_f)
max_am = f.max_val
ainftym = max_am / min_a0
assert isinstance(ainftym, float)
return ainftym
def prepare_norm_w(energynorm, w):
normw = {}
for mu in w.active_indices():
normw[mu] = energynorm(w[mu]._fefunc)
return normw
def LambdaBoundary(Lambda):
suppLambda = supp(Lambda)
for mu in Lambda:
for m in suppLambda:
mu1 = mu.inc(m)
if mu1 not in Lambda:
yield mu1
mu2 = mu.dec(m)
if mu2 not in Lambda and mu2 is not None:
yield mu2
# evaluate (3.15)
def eval_zeta_bar(mu, suppLambda, coeff_field, normw, V, M):
assert mu in normw.keys()
zz = 0
# print "====zeta bar Z1", mu, M
for m in range(M):
if m in suppLambda:
continue
_, am_rv = coeff_field[m]
beta = am_rv.orth_polys.get_beta(mu[m])
ainfty = get_ainfty(m, V)
zz += (beta[1] * ainfty) ** 2
return normw[mu] * sqrt(zz)
# evaluate (3.11)
def eval_zeta(mu, Lambda, coeff_field, normw, V, M=None, this_m=None):
z = 0
if this_m is None:
for m in range(M):
_, am_rv = coeff_field[m]
beta = am_rv.orth_polys.get_beta(mu[m])
ainfty = get_ainfty(m, V)
mu1 = mu.inc(m)
if mu1 in Lambda:
# print "====zeta Z1", ainfty, beta[1], normw[mu1], " == ", ainfty * beta[1] * normw[mu1]
z += ainfty * beta[1] * normw[mu1]
mu2 = mu.dec(m)
if mu2 in Lambda:
# print "====zeta Z2", ainfty, beta[-1], normw[mu2], " == ", ainfty * beta[-1] * normw[mu2]
z += ainfty * beta[-1] * normw[mu2]
return z
else:
m = this_m
_, am_rv = coeff_field[m]
beta = am_rv.orth_polys.get_beta(mu[m])
ainfty = get_ainfty(m, V)
# print "====zeta Z3", m, ainfty, beta[1], normw[mu], " == ", ainfty * beta[1] * normw[mu]
return ainfty * beta[1] * normw[mu]
# prepare some variables
energynorm = pde.energy_norm
Lambda = w.active_indices()
suppLambda = supp(w.active_indices())
# M = min(w.max_order + add_maxm, len(coeff_field))
M = w.max_order + add_maxm
normw = prepare_norm_w(energynorm, w)
# retrieve (sufficiently fine) function space for maximum norm evaluation
V = w[Multiindex()].basis.refine_maxh(maxh)[0]
# evaluate estimator contributions of (3.16)
from collections import defaultdict
# === (a) zeta ===
zeta = defaultdict(int)
# iterate multiindex extensions
# print "===A1 Lambda", Lambda
for nu in LambdaBoundary(Lambda):
assert nu not in Lambda
# print "===A2 boundary nu", nu
zeta[nu] += eval_zeta(nu, Lambda, coeff_field, normw, V, M)
# === (b) zeta_bar ===
zeta_bar = {}
# iterate over active indices
for mu in Lambda:
zeta_bar[mu] = eval_zeta_bar(mu, suppLambda, coeff_field, normw, V, M)
# evaluate summed estimator (3.16)
global_zeta = sqrt(sum([v ** 2 for v in zeta.values()]) + sum([v ** 2 for v in zeta_bar.values()]))
# also return zeta evaluation for single m (needed for refinement algorithm)
eval_zeta_m = lambda mu, m: eval_zeta(mu=mu, Lambda=Lambda, coeff_field=coeff_field, normw=normw, V=V, M=M, this_m=m)
logger.debug("=== ZETA %s --- %s --- %s", global_zeta, zeta, zeta_bar)
return global_zeta, zeta, zeta_bar, eval_zeta_m
def mark_y(cls, Lambda, zeta_, eval_zeta_m, theta_y, max_new_mi=100, type=1):
"""Carry out Doerfler marking by activation of new indices."""
zeta = zeta_
global_zeta = np.sqrt(sum([z ** 2 for z in zeta_.values()]))
suppLambda = supp(Lambda)
maxm = max(suppLambda)
logger.debug("---- SUPPORT Lambda %s maxm %s Lambda %s ", suppLambda, maxm, Lambda)
# A modified paper marking
# ========================
if type == 0:
new_mi = []
marked_zeta = 0.0
while True:
# break if sufficiently many new mi are selected
if theta_y * global_zeta <= marked_zeta or len(new_mi) >= max_new_mi or len(zeta) == 0:
if len(new_mi) >= max_new_mi:
logger.warn("max new_mi reached (%i) WITHOUT sufficient share of global zeta!" % len(new_mi))
if len(zeta) == 0:
logger.warn("NO MORE MI TO MARK!")
break
sorted_zeta = sorted(zeta.items(), key=itemgetter(1))
logger.debug("SORTED ZETA %s", sorted_zeta)
new_zeta = sorted_zeta[-1]
mu = new_zeta[0]
zeta.pop(mu)
logger.debug("ADDING %s to new_mi %s", mu, new_mi)
assert mu not in Lambda
new_mi.append(mu)
marked_zeta = np.sqrt(marked_zeta ** 2 + new_zeta[1] ** 2)
# extend set of inactive potential indices if necessary (see section 5.7)
# mu2 = mu.dec(maxm)
# NOTE: the following is a slight extension of the algorithm in the paper since it executed the extension on all active multiindices (and not only with the latest activated)
# if mu2 in Lambda:
possible_new_mu = []
minm = min(set(range(1, maxm + 2)).difference(set(suppLambda))) # find min(N\setminus supp\Lambda)
for mu2 in Lambda:
new_mu = mu2.inc(minm)
# assert new_mu not in Lambda
# if new_mu not in Lambda and new_mu not in zeta.keys():
if new_mu not in zeta.keys():
# logger.debug("extending multiindex candidates by %s since %s is at the boundary of Lambda (reachable from %s), minm: %s", new_mu, mu, mu2, minm)
logger.debug("extending multiindex candidates by %s since it is at the boundary of Lambda (reachable from %s), minm: %s", new_mu, mu2, minm)
possible_new_mu += [new_mu]
zeta[new_mu] = eval_zeta_m(mu2, minm)
# update global zeta
global_zeta = np.sqrt(global_zeta ** 2 + zeta[new_mu] ** 2)
logger.debug("new global_zeta is %f", global_zeta)
else:
logger.debug("no further extension of multiindex candidates required")
if len(new_mi) >= max_new_mi:
logger.debug("maximal number new mi reached!")
elif len(zeta) == 0:
logger.debug("no more new indices available!")
logger.info("possible new mu considered %s" % possible_new_mu)
# B minimal y-dimension marking
# =============================
else:
assert type == 1
# === EVALUATE EXTENSION ===
# ==========================
# determine possible new mi
new_y = {}
minm = min(set(range(1, maxm + 2)).difference(set(suppLambda))) # find min(N\setminus supp\Lambda)
for mu2 in Lambda:
new_mu = mu2.inc(minm)
if new_mu not in Lambda and new_mu not in zeta.keys() and new_mu not in new_y.keys():
# if new_mu not in zeta.keys() and new_mu not in new_y.keys():
logger.debug("extending multiindex candidates by %s since it is at the boundary of Lambda (reachable from %s), minm: %s", new_mu, mu2, minm)
new_val = eval_zeta_m(mu2, minm)
# update global zeta
global_zeta = np.sqrt(global_zeta ** 2 + new_val ** 2)
logger.debug("new global_zeta is %f", global_zeta)
# test for new y dimension
if len(set(supp([new_mu])).difference(set(suppLambda))) > 0:
assert new_mu not in new_y.keys()
new_y[new_mu] = new_val
else:
assert new_mu not in zeta.keys()
zeta[new_mu] = new_val
else:
logger.debug("no further extension of multiindex candidates required")
# === DETERMINE NEW Y DIMENSIONS ===
# ==================================
# determine how many new y dimensions are needed
new_mi = []
sorted_new_y = sorted(new_y.items(), key=itemgetter(1))
sum_zeta_val = np.sqrt(sum([z ** 2 for z in zeta.values()]))
# add new dimension y while sum_zeta_val is smaller than required marking value
while sum_zeta_val < theta_y*global_zeta and len(sorted_new_y) > 0:
# add new largest y
new_zeta = sorted_new_y[-1]
mu = new_zeta[0]
sorted_new_y.pop(-1)
logger.debug("ADDING NEW Y %s to new_mi %s while target_zeta is %s", mu, new_mi, theta_y*global_zeta)
assert mu not in Lambda
new_mi.append(mu)
global_zeta = np.sqrt(global_zeta ** 2 - new_zeta[1] ** 2)
if len(sorted_new_y) == 0 and zeta_val < theta_y*global_zeta:
logger.warn("UNABLE to mark sufficiently many NEW MI!")
# === DETERMINE HIGHER ORDER ACTIVE MI EXTENSION ===
# ==================================================
# add mi corresponding to already active y dimensions
sorted_zeta = sorted(zeta.items(), key=itemgetter(1))
logger.debug("SORTED ZETA %s", sorted_zeta)
marked_zeta = 0
while marked_zeta < theta_y*global_zeta and len(sorted_zeta) > 0:
new_zeta = sorted_zeta.pop(-1)
mu = new_zeta[0]
logger.debug("ADDING EXTENSION OF EXISTING MI %s to new_mi %s while marked_zeta is %s", mu, new_mi, marked_zeta)
assert mu not in Lambda
new_mi.append(mu)
marked_zeta = np.sqrt(marked_zeta ** 2 + new_zeta[1] ** 2)
zeta = sorted_zeta
logger.info("possible new mu considered %s and %s" % (new_y.keys(), new_mi) )
if len(zeta) == 0:
if theta_y*global_zeta > marked_zeta:
logger.warning("list of mi candidates is empty and reduction goal NOT REACHED, %f > %f!", theta_y * global_zeta, marked_zeta)
if len(new_mi) > 0:
logger.info("SELECTED NEW MULTIINDICES %s", new_mi)
else:
logger.info("NO NEW MULTIINDICES SELECTED")
return new_mi
if len(sim_stats) > 0:
print sim_stats[0].keys()
for k in sim_stats[0].keys():
# print "DATA", k
if k not in ["CONF", "OPTS"]:
D[k] = [s[k] for s in sim_stats]
# evaluate additional data
D["NUM-MI"] = [len(m) for m in D["MI"]]
try:
D["EFFICIENCY"] = [est / err for est, err in zip(D["ERROR-EST"], D["MC-ERROR-H1A"])]
D["WITH-MC"] = True
except:
D["WITH-MC"] = False
print "WARNING: No MC data found!"
# ...from w_history
D["NUM-Y"] = [len(supp(w.active_indices())) + 1 for w in w_history]
D["MESH-CELLS"] = [w.basis.basis.mesh.num_cells() for w in w_history]
D["MESH-HMIN"] = [w.basis.basis.mesh.hmin() for w in w_history]
D["MESH-HMAX"] = [w.basis.basis.mesh.hmax() for w in w_history]
D["MESH-HMINinv"] = [1 / h ** 2 for h in D["MESH-HMIN"]]
D["MESH-HMAXinv"] = [1 / h ** 2 for h in D["MESH-HMAX"]]
# meshes
if options.withMesh:
if options.meshDofs > 0:
for i, dofs in enumerate(D["DOFS"]):
if dofs >= options.meshDofs or i == len(D["DOFS"]) - 1:
D["MESH"] = w_history[i].basis.basis.mesh
break
else:
D["MESH"] = w_history[-1].basis.basis.mesh
sim_stats = pickle.load(fin)
print "sim_stats has %s iterations" % len(sim_stats)
# prepare data
D = {}
if len(sim_stats) > 0:
print sim_stats[0].keys()
for k in sim_stats[0].keys():
# print "DATA", k
if k not in ["CONF", "OPTS", "PROJ-INACTIVE-ZETA"]:
D[k] = [s[k] for s in sim_stats]
# evaluate additional data
D["NUM-MI"] = [len(m) for m in D["MI"]]
try:
if options.singleP:
D["DIM-Y"] = [len(supp([i[0] for i in ami])) + 1 for ami in D["MI"]]
# WARNING: EGSZ1 writes out the squared estimator!!!
D["EST"] = [sqrt(est) for est in D["EST"]]
D["EFFICIENCY"] = [est / err for est, err in zip(D["EST"], D["MC-H1ERR"])]
else:
D["DIM-Y"] = [len(supp(ami)) + 1 for ami in D["MI"]]
D["EFFICIENCY"] = [est / err for est, err in zip(D["ERROR-EST"], D["MC-ERROR-H1A"])]
D["WITH-MC"] = True
except:
D["WITH-MC"] = False
print "WARNING: No MC data found!"
# store data for plotting
SIM_STATS[P] = D
else:
print "SKIPPING P{0} data since it is empty!".format(P)