def test_assemblePatchBoundaryMassMatrix2d(self):
NPatch = np.array([1, 2])
CLocGetter = fem.localBoundaryMassMatrixGetter(NPatch)
boundaryMap = np.array([[True, True], [True, True]])
CComputed = fem.assemblePatchBoundaryMatrix(NPatch,
CLocGetter,
boundaryMap=boundaryMap)
CCorrect = 1. / 12 * np.array(
[[6, 2, 1, 0, 0, 0], [2, 6, 0, 1, 0, 0], [1, 0, 4, 0, 1, 0],
[0, 1, 0, 4, 0, 1], [0, 0, 1, 0, 6, 2], [0, 0, 0, 1, 2, 6]]).T
self.assertTrue(np.allclose(CComputed.todense(), CCorrect))
NPatch = np.array([1, 2])
CLocGetter = fem.localBoundaryMassMatrixGetter(NPatch)
boundaryMap = np.array([[True, True], [False, True]])
CComputed = fem.assemblePatchBoundaryMatrix(NPatch,
CLocGetter,
boundaryMap=boundaryMap)
CCorrect = 1. / 12 * np.array(
[[2, 0, 1, 0, 0, 0], [0, 2, 0, 1, 0, 0], [1, 0, 4, 0, 1, 0],
[0, 1, 0, 4, 0, 1], [0, 0, 1, 0, 6, 2], [0, 0, 0, 1, 2, 6]]).T
self.assertTrue(np.allclose(CComputed.todense(), CCorrect))
NPatch = np.array([1, 2])
CLocGetter = fem.localBoundaryMassMatrixGetter(NPatch)
boundaryMap = np.array([[False, False], [True, True]])
CComputed = fem.assemblePatchBoundaryMatrix(NPatch,
CLocGetter,
boundaryMap=boundaryMap)
CCorrect = 1. / 12 * np.array(
[[4, 2, 0, 0, 0, 0], [2, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 2], [0, 0, 0, 0, 2, 4]]).T
self.assertTrue(np.allclose(CComputed.todense(), CCorrect))
def test_assemblePatchBoundaryMatrix1d(self):
NPatch = np.array([4])
CLocGetter = fem.localBoundaryNormalDerivativeMatrixGetter(NPatch)
CComputed = fem.assemblePatchBoundaryMatrix(NPatch, CLocGetter)
CCorrect = np.array([[4, -4, 0, 0, 0], [0, 0, 0, 0,
0], [0, 0, 0, 0, 0],
[0, 0, 0, 0, 0], [0, 0, 0, -4, 4]])
self.assertTrue(np.allclose(CComputed.todense(), CCorrect))
NPatch = np.array([4])
aPatch = np.array([2, 10, 10, 3])
CLocGetter = fem.localBoundaryNormalDerivativeMatrixGetter(NPatch)
CComputed = fem.assemblePatchBoundaryMatrix(NPatch, CLocGetter, aPatch)
CCorrect = np.array([[8, -8, 0, 0, 0], [0, 0, 0, 0, 0],
[0, 0, 0, 0, 0], [0, 0, 0, 0, 0],
[0, 0, 0, -12, 12]])
self.assertTrue(np.allclose(CComputed.todense(), CCorrect))
def test_boundaryNormalDerivativeMatrixProperties(self):
# BoundaryNormalDerivative bilinear form should map constants to 0
NPatch = np.array([3, 4, 5])
CLocGetter = fem.localBoundaryNormalDerivativeMatrixGetter(NPatch)
C = fem.assemblePatchBoundaryMatrix(NPatch, CLocGetter)
constant = np.ones(np.prod(NPatch + 1))
self.assertTrue(np.isclose(np.linalg.norm(C * constant), 0))
# BoundaryNormalDerivative bilinear form (a=constant) should map planes to 0
NPatch = np.array([3, 4, 5, 6])
CLocGetter = fem.localBoundaryNormalDerivativeMatrixGetter(NPatch)
C = fem.assemblePatchBoundaryMatrix(NPatch, CLocGetter)
p = util.pCoordinates(NPatch)
pSum = np.sum(p, axis=1)
ones = np.ones(np.prod(NPatch + 1))
self.assertTrue(np.isclose(np.linalg.norm(np.dot(ones, C * pSum)), 0))
# A function f with df/dx_k = 1 at x_k = 0 and df/dx_k = 0 at
# x_k = 1 should give 1'*C*f = -d
NPatch = np.array([5, 4, 3, 7])
CLocGetter = fem.localBoundaryNormalDerivativeMatrixGetter(NPatch)
C = fem.assemblePatchBoundaryMatrix(NPatch, CLocGetter)
p = util.pCoordinates(NPatch)
p = np.minimum(p, 0.5)
pSum = np.sum(p, axis=1)
ones = np.ones(np.prod(NPatch + 1))
self.assertTrue(np.isclose(np.dot(ones, C * pSum), -4))
# Same test as above, but with a coefficient a
NPatch = np.array([5, 4, 3, 7])
CLocGetter = fem.localBoundaryNormalDerivativeMatrixGetter(NPatch)
p = util.pCoordinates(NPatch)
p0 = p[:, 0]
pElement = util.pCoordinates(NPatch, NPatch=NPatch - 1)
pElement0 = pElement[:, 0]
aPatch = 1. * (pElement0 < 0.5) + 10. * (pElement0 >= 0.5)
C = fem.assemblePatchBoundaryMatrix(NPatch, CLocGetter, aPatch)
ones = np.ones(np.prod(NPatch + 1))
self.assertTrue(np.isclose(np.dot(ones, C * p0), 10 - 1))
def test_assemblePatchBoundaryNormalDerivativeMatrix2d(self):
NPatch = np.array([1, 2])
CLocGetter = fem.localBoundaryNormalDerivativeMatrixGetter(NPatch)
CComputed = fem.assemblePatchBoundaryMatrix(NPatch, CLocGetter)
CCorrect = 1. / 12 * np.array(
[[10, 2, 1, -1, 0, 0], [2, 10, -1, 1, 0, 0],
[-7, -5, 4, -4, -7, -5], [-5, -7, -4, 4, -5, -7],
[0, 0, 1, -1, 10, 2], [0, 0, -1, 1, 2, 10]]).T
self.assertTrue(np.allclose(CComputed.todense(), CCorrect))
NPatch = np.array([1, 2])
aPatch = np.array([1, 10])
CLocGetter = fem.localBoundaryNormalDerivativeMatrixGetter(NPatch)
CComputed = fem.assemblePatchBoundaryMatrix(NPatch, CLocGetter, aPatch)
CCorrect = 1. / 12 * np.array(
[[10, 2, 1, -1, 0, 0], [2, 10, -1, 1, 0, 0],
[-7, -5, 22, -22, -70, -50], [-5, -7, -22, 22, -50, -70],
[0, 0, 10, -10, 100, 20], [0, 0, -10, 10, 20, 100]]).T
self.assertTrue(np.allclose(CComputed.todense(), CCorrect))
def test_2d_flux(self):
# Stripes perpendicular to flow direction gives effective
# permeability as the harmonic mean of the permeabilities of
# the stripes.
NWorldFine = np.array([20, 20])
NpFine = np.prod(NWorldFine + 1)
NtFine = np.prod(NWorldFine)
NWorldCoarse = np.array([2, 2])
NCoarseElement = NWorldFine / NWorldCoarse
NtCoarse = np.prod(NWorldCoarse)
NpCoarse = np.prod(NWorldCoarse + 1)
boundaryConditions = np.array([[0, 0], [1, 1]])
world = World(NWorldCoarse, NCoarseElement, boundaryConditions)
np.random.seed(0)
aBaseSquare = np.exp(5 * np.random.random_sample(NWorldFine[1]))
aBaseCube = np.tile(aBaseSquare[..., np.newaxis], [NWorldFine[0], 1])
aBaseCube = aBaseCube[..., np.newaxis]
aBase = aBaseCube.flatten()
IPatchGenerator = lambda i, N: interp.L2ProjectionPatchMatrix(
i, N, NWorldCoarse, NCoarseElement, boundaryConditions)
aCoef = coef.coefficientFine(NWorldCoarse, NCoarseElement, aBase)
k = 2
printLevel = 0
pglod = pg.PetrovGalerkinLOD(world, k, IPatchGenerator, 0, printLevel)
pglod.updateCorrectors(aCoef)
KmsFull = pglod.assembleMsStiffnessMatrix()
coords = util.pCoordinates(NWorldCoarse)
xC = coords[:, 0]
g = 1 - xC
bFull = -KmsFull * g
boundaryMap = boundaryConditions == 0
fixed = util.boundarypIndexMap(NWorldCoarse, boundaryMap)
free = np.setdiff1d(np.arange(0, NpCoarse), fixed)
KmsFree = KmsFull[free][:, free]
bFree = bFull[free]
xFree = sparse.linalg.spsolve(KmsFree, bFree)
xFull = np.zeros(NpCoarse)
xFull[free] = xFree
MGammaLocGetter = fem.localBoundaryMassMatrixGetter(NWorldCoarse)
MGammaFull = fem.assemblePatchBoundaryMatrix(NWorldCoarse,
MGammaLocGetter,
boundaryMap=boundaryMap)
# Solve (F, w) = a(u0, w) + a(g, w) in space of fixed DoFs only
KmsFixedFull = KmsFull[fixed]
cFixed = KmsFixedFull * (xFull + g)
MGammaFixed = MGammaFull[fixed][:, fixed]
FFixed = sparse.linalg.spsolve(MGammaFixed, cFixed)
FFull = np.zeros(NpCoarse)
FFull[fixed] = FFixed
self.assertTrue(
np.isclose(np.mean(FFixed[FFixed > 0]), stats.hmean(aBaseSquare)))
def helmholtz_nonlinear_adaptive(mapper, fineLvl, coarseLvl, maxit):
fineExp = fineLvl
NFine = np.array([2**fineLvl, 2**fineLvl])
NpFine = np.prod(NFine + 1)
N = 2**coarseLvl
tolList = [2.0, 1.0, 0.5, 0.25, 0.125, 0.0625, 0.]
ell = 2 # localization parameter
k = 15. # wavenumber
maxit_Fine = 200
xt = util.tCoordinates(NFine)
xp = util.pCoordinates(NFine)
# multiscale coefficients on the scale NFine-2
np.random.seed(444)
sizeK = np.size(xt[:, 0])
nFine = NFine[0]
# determine domain D_eps = supp(1-n) = supp(1-A) (all equal for the moment)
indicesIn = (xt[:, 0] > 0.15) & (xt[:, 0] < 0.85) & (xt[:, 1] > 0.15) & (
xt[:, 1] < 0.85)
indicesInEps = (xt[:, 0] > 0.15) & (xt[:, 0] < 0.85) & (
xt[:, 1] > 0.15) & (xt[:, 1] < 0.85)
# coefficients
aFine = np.ones(xt.shape[0])
cn = .05 # lower bound on n
Cn = 1. # upper bound on n
nEpsPro = coeffi(xt[:, 0], xt[:, 1], fineLvl)
k2Fine = k**2 * np.ones(xt.shape[0])
k2Fine[indicesIn] = k**2 * ((Cn - cn) * nEpsPro[indicesIn] + cn)
kFine = k * np.ones(xt.shape[0])
Ceps = 0.3 # upper bound on eps (lower bound is 0)
epsEpsPro = np.ones(sizeK)
epsFine = np.zeros(xt.shape[0])
epsFine[indicesInEps] = Ceps * epsEpsPro[indicesInEps] # 0 OR Ceps
plotC = np.ones(sizeK)
plotC[indicesIn] = nEpsPro[indicesIn]
drawCoefficient(NFine, plotC)
xC = xp[:, 0]
yC = xp[:, 1]
# define right-hand side and boundary condition
def funcF(x, y):
res = 100 * np.ones(x.shape, dtype='complex128')
return res
f = funcF(xC, yC)
# reference solution
uSol = np.zeros(NpFine, dtype='complex128')
# boundary conditions
boundaryConditions = np.array([[1, 1], [1, 1]])
worldFine = World(NFine, np.array([1, 1]), boundaryConditions)
# fine matrices
BdFineFEM = fem.assemblePatchBoundaryMatrix(
NFine, fem.localBoundaryMassMatrixGetter(NFine))
MFineFEM = fem.assemblePatchMatrix(NFine, fem.localMassMatrix(NFine))
KFineFEM = fem.assemblePatchMatrix(
NFine, fem.localStiffnessMatrix(NFine)) # , aFine)
kBdFine = fem.assemblePatchBoundaryMatrix(
NFine, fem.localBoundaryMassMatrixGetter(NFine), kFine)
KFine = fem.assemblePatchMatrix(NFine, fem.localStiffnessMatrix(NFine),
aFine)
print('***computing reference solution***')
uOldFine = np.zeros(NpFine, dtype='complex128')
for it in np.arange(maxit_Fine):
print('-- itFine = %d' % it)
knonlinUpreFine = np.abs(uOldFine)
knonlinUFine = func.evaluateCQ1(NFine, knonlinUpreFine, xt)
k2FineUfine = np.copy(k2Fine)
k2FineUfine[indicesInEps] *= (
1. + epsFine[indicesInEps] * knonlinUFine[indicesInEps]**2
) # full coefficient, including nonlinearity
k2MFine = fem.assemblePatchMatrix(
NFine, fem.localMassMatrix(NFine),
k2FineUfine) # weighted mass matrix, updated in every iteration
nodesFine = np.arange(worldFine.NpFine)
fixFine = util.boundarypIndexMap(NFine, boundaryConditions == 0)
freeFine = np.setdiff1d(nodesFine, fixFine)
# right-hand side
fhQuad = MFineFEM * f
# fine system
lhsh = KFine[freeFine][:, freeFine] - k2MFine[
freeFine][:, freeFine] + 1j * kBdFine[freeFine][:, freeFine]
rhsh = fhQuad[freeFine]
xFreeFine = sparse.linalg.spsolve(lhsh, rhsh)
xFullFine = np.zeros(worldFine.NpFine, dtype='complex128')
xFullFine[freeFine] = xFreeFine
uOldFine = np.copy(xFullFine)
# residual - used as stopping criterion
knonlinU = np.abs(uOldFine)
knonlinUFineIt = func.evaluateCQ1(NFine, knonlinU, xt)
k2FineUfineIt = np.copy(k2Fine)
k2FineUfineIt[indicesInEps] *= (
1. + epsFine[indicesInEps] * knonlinUFineIt[indicesInEps]**2
) # update full coefficient, including nonlinearity
k2MFineIt = fem.assemblePatchMatrix(NFine, fem.localMassMatrix(NFine),
k2FineUfineIt)
Ares = KFine - k2MFineIt + 1j * kBdFine
residual = np.linalg.norm(Ares * xFullFine - fhQuad) / np.linalg.norm(
Ares * xFullFine)
print('---- residual = %.4e' % residual)
if residual < 1e-12:
break # stopping criterion
uSol = xFullFine # final fine reference solution
print('***reference solution computed***\n')
counter = 0 # for figures
print('***computing multiscale approximations***')
relErrEnergy = np.zeros([len(tolList), maxit])
for tol in tolList:
counter += 1
print('H = %.4e, tol = %.4e' % (1. / N, tol))
NWorldCoarse = np.array([N, N])
NCoarseElement = NFine // NWorldCoarse
world = World(NWorldCoarse, NCoarseElement, boundaryConditions)
NpCoarse = np.prod(NWorldCoarse + 1)
uOldUps = np.zeros(NpFine, dtype='complex128')
for it in np.arange(maxit):
print('-- it = %d:' % it)
knonlinUpre = np.abs(uOldUps)
knonlinU = func.evaluateCQ1(NFine, knonlinUpre, xt)
k2FineU = np.copy(k2Fine)
k2FineU[indicesInEps] *= (
1. + epsFine[indicesInEps] * knonlinU[indicesInEps]**2)
print('---- starting computation of correctors')
def computeLocalContribution(TInd):
patch = Patch(world, ell, TInd)
IPatch = lambda: interp.L2ProjectionPatchMatrix(
patch, boundaryConditions)
aPatch = lambda: coef.localizeCoefficient(patch, aFine)
kPatch = lambda: coef.localizeCoefficient(patch, kFine)
k2Patch = lambda: coef.localizeCoefficient(patch, k2FineU)
correctorsList = lod.computeBasisCorrectors_helmholtz(
patch, IPatch, aPatch, kPatch,
k2Patch) # adapted for Helmholtz setting
csi = lod.computeBasisCoarseQuantities_helmholtz(
patch, correctorsList, aPatch, kPatch,
k2Patch) # adapted for Helmholtz setting
return patch, correctorsList, csi.Kmsij, csi.Mmsij, csi.Bdmsij, csi.muTPrime
def computeIndicators(TInd):
k2FineUPatch = lambda: coef.localizeCoefficient(
patchT[TInd], k2FineU)
k2FineUOldPatch = lambda: coef.localizeCoefficient(
patchT[TInd], k2FineUOld)
E_vh = lod.computeErrorIndicatorCoarse_helmholtz(
patchT[TInd], muTPrime[TInd], k2FineUOldPatch,
k2FineUPatch)
return E_vh
def UpdateCorrectors(TInd):
patch = Patch(world, ell, TInd)
IPatch = lambda: interp.L2ProjectionPatchMatrix(
patch, boundaryConditions)
aPatch = lambda: coef.localizeCoefficient(patch, aFine)
kPatch = lambda: coef.localizeCoefficient(patch, kFine)
k2Patch = lambda: coef.localizeCoefficient(patch, k2FineU)
correctorsList = lod.computeBasisCorrectors_helmholtz(
patch, IPatch, aPatch, kPatch, k2Patch)
csi = lod.computeBasisCoarseQuantities_helmholtz(
patch, correctorsList, aPatch, kPatch,
k2Patch) # adapted for Helmholtz setting
return patch, correctorsList, csi.Kmsij, csi.Mmsij, csi.Bdmsij, csi.muTPrime
def UpdateElements(tol, E, Kmsij_old, Mmsij_old, Bdmsij_old,
correctors_old, mu_old):
print('---- apply tolerance')
Elements_to_be_updated = []
for (i, eps) in E.items():
if eps > tol * k**2:
Elements_to_be_updated.append(i)
if len(E) > 0:
print(
'---- percentage of non-zero element correctors to be updated: %.4f'
% (100 * np.size(Elements_to_be_updated) / len(E)),
flush=True)
print(
'---- total percentage of element correctors to be updated: %.4f'
%
(100 * np.size(Elements_to_be_updated) / len(mu_old)),
flush=True)
print('---- update local contributions')
KmsijT_list = list(np.copy(Kmsij_old))
MmsijT_list = list(np.copy(Mmsij_old))
BdmsijT_list = list(np.copy(Bdmsij_old))
muT_list = np.copy(mu_old)
for T in np.setdiff1d(range(world.NtCoarse),
Elements_to_be_updated):
patch = Patch(world, ell, T)
aPatch = lambda: coef.localizeCoefficient(patch, aFine)
kPatch = lambda: coef.localizeCoefficient(patch, kFine)
k2Patch = lambda: coef.localizeCoefficient(patch, k2FineU)
csi = lod.computeBasisCoarseQuantities_helmholtz(
patch, correctors_old[T], aPatch, kPatch, k2Patch)
KmsijT_list[T] = csi.Kmsij
MmsijT_list[T] = csi.Mmsij
BdmsijT_list[T] = csi.Bdmsij
muT_list[T] = csi.muTPrime
if np.size(Elements_to_be_updated) != 0:
#print('---- update correctors')
patchT_irrelevant, correctorsListTNew, KmsijTNew, MmsijTNew, BdmsijTNew, muTPrimeNew = zip(
*mapper(UpdateCorrectors, Elements_to_be_updated))
#print('---- update correctorsList')
correctorsListT_list = list(np.copy(correctors_old))
i = 0
for T in Elements_to_be_updated:
KmsijT_list[T] = KmsijTNew[i]
correctorsListT_list[T] = correctorsListTNew[i]
MmsijT_list[T] = MmsijTNew[i]
BdmsijT_list[T] = BdmsijTNew[i]
muT_list[T] = muTPrimeNew[i]
i += 1
KmsijT = tuple(KmsijT_list)
correctorsListT = tuple(correctorsListT_list)
MmsijT = tuple(MmsijT_list)
BdmsijT = tuple(BdmsijT_list)
muTPrime = tuple(muT_list)
return correctorsListT, KmsijT, MmsijT, BdmsijT, muTPrime
else:
KmsijT = tuple(KmsijT_list)
MmsijT = tuple(MmsijT_list)
BdmsijT = tuple(BdmsijT_list)
muTPrime = tuple(muT_list)
return correctors_old, KmsijT, MmsijT, BdmsijT, muTPrime
if it == 0:
patchT, correctorsListT, KmsijT, MmsijT, BdmsijT, muTPrime = zip(
*mapper(computeLocalContribution, range(world.NtCoarse)))
else:
E_vh = list(mapper(computeIndicators, range(world.NtCoarse)))
print(
'---- maximal value error estimator for basis correctors {}'
.format(np.max(E_vh)))
E = {i: E_vh[i] for i in range(np.size(E_vh)) if E_vh[i] > 0}
# loop over elements with possible recomputation of correctors
correctorsListT, KmsijT, MmsijT, BdmsijT, muTPrime = UpdateElements(
tol, E, KmsijT, MmsijT, BdmsijT, correctorsListT,
muTPrime) # tol scaled by maximal error indicator
print('---- finished computation of correctors')
KLOD = pglod.assembleMsStiffnessMatrix(
world, patchT, KmsijT) # ms stiffness matrix
k2MLOD = pglod.assembleMsStiffnessMatrix(world, patchT,
MmsijT) # ms mass matrix
kBdLOD = pglod.assembleMsStiffnessMatrix(
world, patchT, BdmsijT) # ms boundary matrix
MFEM = fem.assemblePatchMatrix(NWorldCoarse, world.MLocCoarse)
BdFEM = fem.assemblePatchBoundaryMatrix(
NWorldCoarse, fem.localBoundaryMassMatrixGetter(NWorldCoarse))
print('---- coarse matrices assembled')
nodes = np.arange(world.NpCoarse)
fix = util.boundarypIndexMap(NWorldCoarse, boundaryConditions == 0)
free = np.setdiff1d(nodes, fix)
assert (nodes.all() == free.all())
# compute global interpolation matrix
patchGlobal = Patch(world, NFine[0] + 2, 0)
IH = interp.L2ProjectionPatchMatrix(patchGlobal,
boundaryConditions)
assert (IH.shape[0] == NpCoarse)
basis = fem.assembleProlongationMatrix(NWorldCoarse,
NCoarseElement)
fHQuad = basis.T * MFineFEM * f
print('---- solving coarse system')
# coarse system
lhsH = KLOD[free][:, free] - k2MLOD[
free][:, free] + 1j * kBdLOD[free][:, free]
rhsH = fHQuad[free]
xFree = sparse.linalg.spsolve(lhsH, rhsH)
basisCorrectors = pglod.assembleBasisCorrectors(
world, patchT, correctorsListT)
modifiedBasis = basis - basisCorrectors
xFull = np.zeros(world.NpCoarse, dtype='complex128')
xFull[free] = xFree
uLodCoarse = basis * xFull
uLodFine = modifiedBasis * xFull
uOldUps = np.copy(uLodFine)
k2FineUOld = np.copy(k2FineU)
Err = np.sqrt(
np.dot((uSol - uLodFine).conj(), KFineFEM *
(uSol - uLodFine)) + k**2 *
np.dot((uSol - uLodFine).conj(), MFineFEM * (uSol - uLodFine)))
ErrEnergy = Err / np.sqrt(
np.dot((uSol).conj(), KFineFEM *
(uSol)) + k**2 * np.dot((uSol).conj(), MFineFEM *
(uSol)))
print('---- ', np.abs(ErrEnergy),
'\n***********************************************')
# save errors in arrays
relErrEnergy[counter - 1, it] = ErrEnergy
print('\n')
its = np.arange(1, maxit + 1)
plt.figure(1)
plt.title(
'Relative energy errors w.r.t iterations for different tolerances - Ex 3'
)
plt.plot(its, relErrEnergy[0, :], 'x--', color='black', label='tol = 2')
plt.plot(its, relErrEnergy[1, :], 'x-', color='blue', label='tol = 1')
plt.plot(its, relErrEnergy[2, :], 'x-', color='green', label='tol = 0.5')
plt.plot(its, relErrEnergy[3, :], 'x-', color='orange', label='tol = 0.25')
plt.plot(its, relErrEnergy[4, :], 'x-', color='red', label='tol = 0.125')
plt.plot(its,
relErrEnergy[5, :],
'x-',
color='magenta',
label='tol = 0.0625')
plt.plot(its, relErrEnergy[6, :], 'x--', color='black', label='tol = 0')
plt.yscale('log')
plt.legend()
plt.show()
def performTPrimeLoop_helmholtz(patch, lambdasList, correctorsList, aPatch,
kPatch, k2Patch, accumulate):
while callable(aPatch):
aPatch = aPatch()
while callable(kPatch):
kPatch = kPatch()
while callable(k2Patch):
k2Patch = k2Patch()
world = patch.world
NCoarseElement = world.NCoarseElement
NPatchCoarse = patch.NPatchCoarse
NPatchFine = NPatchCoarse * NCoarseElement
NTPrime = np.prod(NPatchCoarse)
NpPatchCoarse = np.prod(NPatchCoarse + 1)
d = np.size(NPatchCoarse)
assert (aPatch.ndim == 1 or aPatch.ndim == 3)
assert (kPatch.ndim == 1)
assert (k2Patch.ndim == 1)
if aPatch.ndim == 1:
ALocFine = world.ALocFine
elif aPatch.ndim == 3:
ALocFine = world.ALocMatrixFine
MLocFine = world.MLocFine
BdLocFine = fem.localBoundaryMassMatrixGetter(NCoarseElement *
world.NWorldCoarse)
lambdas = np.column_stack(lambdasList)
numLambdas = len(lambdasList)
TPrimeCoarsepStartIndices = util.lowerLeftpIndexMap(
NPatchCoarse - 1, NPatchCoarse)
TPrimeCoarsepIndexMap = util.lowerLeftpIndexMap(np.ones_like(NPatchCoarse),
NPatchCoarse)
TPrimeFinetStartIndices = util.pIndexMap(NPatchCoarse - 1, NPatchFine - 1,
NCoarseElement)
TPrimeFinetIndexMap = util.lowerLeftpIndexMap(NCoarseElement - 1,
NPatchFine - 1)
TPrimeFinepStartIndices = util.pIndexMap(NPatchCoarse - 1, NPatchFine,
NCoarseElement)
TPrimeFinepIndexMap = util.lowerLeftpIndexMap(NCoarseElement, NPatchFine)
TInd = util.convertpCoordIndexToLinearIndex(NPatchCoarse - 1,
patch.iElementPatchCoarse)
QPatch = np.column_stack(correctorsList)
# global boundary
bdMapWorld = world.boundaryConditions == 1
for (TPrimeInd,
TPrimeCoarsepStartIndex,
TPrimeFinetStartIndex,
TPrimeFinepStartIndex) \
in zip(np.arange(NTPrime),
TPrimeCoarsepStartIndices,
TPrimeFinetStartIndices,
TPrimeFinepStartIndices):
aTPrime = aPatch[TPrimeFinetStartIndex + TPrimeFinetIndexMap]
kTPrime = kPatch[TPrimeFinetStartIndex + TPrimeFinetIndexMap]
k2TPrime = k2Patch[TPrimeFinetStartIndex + TPrimeFinetIndexMap]
KTPrime = fem.assemblePatchMatrix(NCoarseElement, ALocFine, aTPrime)
MTPrime = fem.assemblePatchMatrix(NCoarseElement, MLocFine, k2TPrime)
L2TPrime = fem.assemblePatchMatrix(NCoarseElement, MLocFine)
# boundary on element
bdMapElement = np.zeros([d, 2], dtype='bool')
iElementTPrime = patch.iPatchWorldCoarse + util.convertpLinearIndexToCoordIndex(
NPatchCoarse - 1, TPrimeInd)
inheritElement0 = iElementTPrime == 0
inheritElement1 = (iElementTPrime + np.ones(d)) == world.NWorldCoarse
bdMapElement[inheritElement0, 0] = bdMapWorld[inheritElement0, 0]
bdMapElement[inheritElement1, 1] = bdMapWorld[inheritElement1, 1]
BdTPrime = fem.assemblePatchBoundaryMatrix(NCoarseElement, BdLocFine,
kTPrime, bdMapElement)
P = lambdas
Q = QPatch[TPrimeFinepStartIndex + TPrimeFinepIndexMap, :]
_KTPrimeij = np.dot(P.T, KTPrime * Q)
TPrimei = TPrimeCoarsepStartIndex + TPrimeCoarsepIndexMap
_MTPrimeij = np.dot(P.T, MTPrime * Q)
_BdTPrimeij = np.dot(P.T, BdTPrime * Q)
CTPrimeij = np.dot(Q.T, L2TPrime * Q)
BTPrimeij = np.dot(P.T, L2TPrime * Q)
accumulate(TPrimeInd, TPrimei, P, Q, KTPrime, _KTPrimeij, MTPrime,
BdTPrime, _MTPrimeij, _BdTPrimeij, L2TPrime, CTPrimeij,
BTPrimeij)
def computeElementCorrector_helmholtz(patch,
IPatch,
aPatch,
kPatch,
k2Patch,
ARhsList=None,
MRhsList=None,
saddleSolver=None):
'''Compute the fine correctors over a patch.
Compute the correctors
B( Q_T_j, vf)_{U_K(T)} = B( ARhs_j, vf)_{T} + (MRhs_j, vf)_{T}
where B is the sesquilinear form associated with the linear Helmholtz eq.
'''
while callable(IPatch):
IPatch = IPatch()
while callable(aPatch):
aPatch = aPatch()
while callable(kPatch):
kPatch = kPatch()
while callable(k2Patch):
k2Patch = k2Patch()
assert (ARhsList is not None or MRhsList is not None)
numRhs = None
if ARhsList is not None:
assert (numRhs is None or numRhs == len(ARhsList))
numRhs = len(ARhsList)
if MRhsList is not None:
assert (numRhs is None or numRhs == len(MRhsList))
numRhs = len(MRhsList)
world = patch.world
NCoarseElement = world.NCoarseElement
NPatchCoarse = patch.NPatchCoarse
d = np.size(NCoarseElement)
NPatchFine = NPatchCoarse * NCoarseElement
NtFine = np.prod(NPatchFine)
NpFineCoarseElement = np.prod(NCoarseElement + 1)
NpCoarse = np.prod(NPatchCoarse + 1)
NpFine = np.prod(NPatchFine + 1)
assert (aPatch.shape[0] == NtFine)
assert (aPatch.ndim == 1 or aPatch.ndim == 3)
assert (kPatch.ndim == 1)
assert (k2Patch.ndim == 1)
if aPatch.ndim == 1:
ALocFine = world.ALocFine
elif aPatch.ndim == 3:
ALocFine = world.ALocMatrixFine
MLocFine = world.MLocFine
BdLocFine = fem.localBoundaryMassMatrixGetter(NCoarseElement *
world.NWorldCoarse)
iElementPatchCoarse = patch.iElementPatchCoarse
elementFinetIndexMap = util.extractElementFine(NPatchCoarse,
NCoarseElement,
iElementPatchCoarse,
extractElements=True)
elementFinepIndexMap = util.extractElementFine(NPatchCoarse,
NCoarseElement,
iElementPatchCoarse,
extractElements=False)
# global boundary?
bdMapWorld = world.boundaryConditions == 1
# on element
bdMapElement = np.zeros([d, 2], dtype='bool')
inheritElement0 = patch.iElementWorldCoarse == 0
inheritElement1 = (patch.iElementWorldCoarse +
np.ones(d)) == world.NWorldCoarse
bdMapElement[inheritElement0, 0] = bdMapWorld[inheritElement0, 0]
bdMapElement[inheritElement1, 1] = bdMapWorld[inheritElement1, 1]
# on patch
inherit0 = patch.iPatchWorldCoarse == 0
inherit1 = (patch.iPatchWorldCoarse + NPatchCoarse) == world.NWorldCoarse
bdMapPatch = np.zeros([d, 2], dtype='bool')
bdMapPatch[inherit0, 0] = bdMapWorld[inherit0, 0]
bdMapPatch[inherit1, 1] = bdMapWorld[inherit1, 1]
if ARhsList is not None:
AElementFull = fem.assemblePatchMatrix(NCoarseElement, ALocFine,
aPatch[elementFinetIndexMap])
k2MElementFull = fem.assemblePatchMatrix(NCoarseElement, MLocFine,
k2Patch[elementFinetIndexMap])
kBdElementFull = fem.assemblePatchBoundaryMatrix(
NCoarseElement, BdLocFine, kPatch[elementFinetIndexMap],
bdMapElement)
if MRhsList is not None:
MElementFull = fem.assemblePatchMatrix(NCoarseElement, MLocFine)
APatchFull = fem.assemblePatchMatrix(NPatchFine, ALocFine, aPatch)
k2MPatchFull = fem.assemblePatchMatrix(NPatchFine, MLocFine, k2Patch)
kBdPatchFull = fem.assemblePatchBoundaryMatrix(NPatchFine, BdLocFine,
kPatch, bdMapPatch)
SPatchFull = APatchFull - k2MPatchFull + 1j * kBdPatchFull
bPatchFullList = []
for rhsIndex in range(numRhs):
bPatchFull = np.zeros(NpFine, dtype='complex128')
if ARhsList is not None:
bPatchFull[elementFinepIndexMap] += (
AElementFull - k2MElementFull +
1j * kBdElementFull) * ARhsList[rhsIndex]
if MRhsList is not None:
bPatchFull[
elementFinepIndexMap] += MElementFull * MRhsList[rhsIndex]
bPatchFullList.append(bPatchFull)
correctorsList = ritzProjectionToFinePatch(patch, SPatchFull,
bPatchFullList, IPatch,
saddleSolver)
return correctorsList
def helmholtz_nonlinear_adaptive(mapper, fineLvl, maxCoarseLvl, maxit):
NFine = np.array([2**fineLvl, 2**fineLvl])
NpFine = np.prod(NFine + 1)
NList = 2**np.arange(1, maxCoarseLvl + 1)
ell = 2 # localization parameter
k = 30. # wavenumber
maxit_Fine = 250
tol = 0.5 # coupled to maximal error indicator
xt = util.tCoordinates(NFine)
xp = util.pCoordinates(NFine)
# multiscale coefficients on the scale NFine-2
np.random.seed(123)
sizeK = np.size(xt[:, 0])
nFine = NFine[0]
# determine domain D_eps = supp(1-n) = supp(1-A) (all equal for this experiment)
indicesIn = (xt[:, 0] > 0.25) & (xt[:, 0] < 0.75) & (xt[:, 1] > 0.25) & (
xt[:, 1] < 0.75)
indicesInEps = (xt[:, 0] > 0.25) & (xt[:, 0] < 0.75) & (
xt[:, 1] > 0.25) & (xt[:, 1] < 0.75)
# coefficients
cA = .2 # lower bound on A
CA = 1. # upper bound on A
aEps = np.random.uniform(0, 1, sizeK // 16)
aEpsPro = np.zeros(sizeK)
for i in range((nFine) // 4):
aEpsPro[4 * i * (nFine):4 * (i + 1) * (nFine)] = np.tile(
np.repeat(aEps[i * (nFine) // 4:(i + 1) * (nFine) // 4], 4), 4)
aFine = np.ones(xt.shape[0])
aFine[indicesIn] = (CA - cA) * aEpsPro[indicesIn] + cA
cn = 1. # lower bound on n
Cn = 1. # upper bound on n
nEps = np.random.uniform(0, 1, sizeK // 16)
nEpsPro = np.zeros(sizeK)
for i in range((nFine) // 4):
nEpsPro[4 * i * (nFine):4 * (i + 1) * (nFine)] = np.tile(
np.repeat(nEps[i * (nFine) // 4:(i + 1) * (nFine) // 4], 4), 4)
k2Fine = k**2 * np.ones(xt.shape[0])
k2Fine[indicesIn] = k**2 * ((Cn - cn) * nEpsPro[indicesIn] + cn)
kFine = k * np.ones(xt.shape[0])
Ceps = .85 # upper bound on eps (lower bound is 0)
lvl = 4
epsEps = np.random.randint(2, size=(sizeK // lvl**2))
epsEpsPro = np.zeros(sizeK)
for i in range((nFine) // lvl):
epsEpsPro[lvl * i * (nFine):lvl * (i + 1) * (nFine)] = np.tile(
np.repeat(epsEps[i * (nFine) // lvl:(i + 1) * (nFine) // lvl],
lvl), lvl)
epsFine = np.zeros(xt.shape[0])
epsFine[indicesInEps] = Ceps * epsEpsPro[indicesInEps] # 0 OR Ceps
drawCoefficient(NFine, epsFine)
xC = xp[:, 0]
yC = xp[:, 1]
fact = 100.
mult = .8
a = .5
b = .25
k2 = 30.
# define right-hand side and boundary condition
def funcF(x, y):
res = mult * (-np.exp(-1.j * k2 * (a * x - b)) *
(2 * a**2 * fact**2 * np.sinh(fact * (a * x - b))**2 /
(np.cosh(fact * (a * x - b)) + 1)**3 -
a**2 * fact**2 * np.cosh(fact * (a * x - b)) /
(np.cosh(fact * (a * x - b)) + 1)**2) +
a**2 * k2**2 * np.exp(-1.j * k2 * (a * x - b)) /
(np.cosh(fact * (a * x - b)) + 1) - 2.j * a**2 * fact *
k2 * np.exp(-1.j * k2 *
(a * x - b)) * np.sinh(fact * (a * x - b)) /
(np.cosh(fact * (a * x - b)) + 1)**2 -
k**2 * np.exp(-1.j * k2 * (a * x - b)) /
(np.cosh(fact * (a * x - b)) + 1))
return res
f = funcF(xC, yC)
g = np.zeros(NpFine, dtype='complex128')
# bottom boundary
g[0:(NFine[0] +
1)] = mult * 1.j * k * 1. / (np.cosh(fact *
(a * xC[0:(NFine[0] + 1)] - b)) +
1) * np.exp(
-1.j * k2 *
(a * xC[0:(NFine[0] + 1)] - b))
# top boundary
g[(NpFine - NFine[0] -
1):] = mult * 1.j * k * 1. / (np.cosh(fact * (a * xC[
(NpFine - NFine[0] - 1):NpFine] - b)) + 1) * np.exp(
-1.j * k2 * (a * xC[(NpFine - NFine[0] - 1):NpFine] - b))
# left boundary
g[0:(NpFine - NFine[0]):(
NFine[0] +
1)] = mult * 1.j * k * np.ones_like(yC[0:(NpFine - NFine[0]):(
NFine[0] + 1)]) / (np.cosh(fact * (a * 0 - b)) + 1) * np.exp(
-1.j * k2 * (a * 0 - b)) + mult * np.ones_like(
yC[0:(NpFine - NFine[0]):(NFine[0] + 1)]) * (
a * 1.j * k2 * np.exp(-1.j * k2 * (a * 0 - b)) /
(np.cosh((a * 0 - b) * fact) + 1) + a * fact * np.sinh(
(a * 0 - b) * fact) * np.exp(-1.j * k2 *
(a * 0 - b)) /
(np.cosh((a * 0 - b) * fact) + 1)**2)
# right boundary
g[NFine[0]:NpFine:(
NFine[0] + 1)] = mult * 1.j * k * np.ones_like(yC[NFine[0]:NpFine:(
NFine[0] + 1)]) / (np.cosh(fact * (a * 1. - b)) + 1) * np.exp(
-1.j * k2 * (a * 1. - b)) - mult * np.ones_like(
yC[NFine[0]:NpFine:(NFine[0] + 1)]) * (
a * 1.j * k2 * np.exp(-1.j * k2 * (a * 1. - b)) /
(np.cosh(
(a * 1. - b) * fact) + 1) + a * fact * np.sinh(
(a * 1. - b) * fact) * np.exp(-1.j * k2 *
(a * 1. - b)) /
(np.cosh((a * 1. - b) * fact) + 1)**2)
# reference solution
uSol = np.zeros(NpFine, dtype='complex128')
# boundary conditions
boundaryConditions = np.array([[1, 1], [1, 1]]) # Robin boundary
worldFine = World(NFine, np.array([1, 1]), boundaryConditions)
# fine matrices
BdFineFEM = fem.assemblePatchBoundaryMatrix(
NFine, fem.localBoundaryMassMatrixGetter(NFine))
MFineFEM = fem.assemblePatchMatrix(NFine, fem.localMassMatrix(NFine))
KFineFEM = fem.assemblePatchMatrix(NFine, fem.localStiffnessMatrix(NFine))
kBdFine = fem.assemblePatchBoundaryMatrix(
NFine, fem.localBoundaryMassMatrixGetter(NFine), kFine)
KFine = fem.assemblePatchMatrix(NFine, fem.localStiffnessMatrix(NFine),
aFine)
# incident beam
uInc = mult / (np.cosh(fact * (a * xC - b)) + 1) * np.exp(-1.j * k2 *
(a * xC - b))
print('***computing reference solution***')
uOldFine = np.zeros(NpFine, dtype='complex128')
for it in np.arange(maxit_Fine):
print('-- itFine = %d' % it)
knonlinUpreFine = np.abs(uOldFine)
knonlinUFine = func.evaluateCQ1(NFine, knonlinUpreFine, xt)
k2FineUfine = np.copy(k2Fine)
k2FineUfine[indicesInEps] *= (
1. + epsFine[indicesInEps] * knonlinUFine[indicesInEps]**2
) # full coefficient, including nonlinearity
k2MFine = fem.assemblePatchMatrix(
NFine, fem.localMassMatrix(NFine),
k2FineUfine) # weighted mass matrix, updated in every iteration
nodesFine = np.arange(worldFine.NpFine)
fixFine = util.boundarypIndexMap(NFine, boundaryConditions == 0)
freeFine = np.setdiff1d(nodesFine, fixFine)
# right-hand side (including boundary condition)
fhQuad = MFineFEM * f + BdFineFEM * g
# fine system
lhsh = KFine[freeFine][:, freeFine] - k2MFine[
freeFine][:, freeFine] + 1j * kBdFine[freeFine][:, freeFine]
rhsh = fhQuad[freeFine]
xFreeFine = sparse.linalg.spsolve(lhsh, rhsh)
xFullFine = np.zeros(worldFine.NpFine, dtype='complex128')
xFullFine[freeFine] = xFreeFine
uOldFine = np.copy(xFullFine)
# residual - used as stopping criterion
knonlinU = np.abs(uOldFine)
knonlinUFineIt = func.evaluateCQ1(NFine, knonlinU, xt)
k2FineUfineIt = np.copy(k2Fine)
k2FineUfineIt[indicesInEps] *= (
1. + epsFine[indicesInEps] * knonlinUFineIt[indicesInEps]**2
) # update full coefficient, including nonlinearity
k2MFineIt = fem.assemblePatchMatrix(NFine, fem.localMassMatrix(NFine),
k2FineUfineIt)
Ares = KFine - k2MFineIt + 1j * kBdFine
residual = np.linalg.norm(Ares * xFullFine - fhQuad) / np.linalg.norm(
Ares * xFullFine)
print('---- residual = %.4e' % residual)
if residual < 1e-12:
break # stopping criterion
uSol = xFullFine # final fine reference solution
print('***reference solution computed***\n')
######################################################################################
print('***computing multiscale approximations***')
relErrEnergy = np.zeros([len(NList), maxit])
counter = 0
for N in NList:
counter += 1
print('H = %.4e' % (1. / N))
NWorldCoarse = np.array([N, N])
NCoarseElement = NFine // NWorldCoarse
world = World(NWorldCoarse, NCoarseElement, boundaryConditions)
NpCoarse = np.prod(NWorldCoarse + 1)
uOldUps = np.zeros(NpFine, dtype='complex128')
for it in np.arange(maxit):
print('-- it = %d:' % it)
knonlinUpre = np.abs(uOldUps)
knonlinU = func.evaluateCQ1(NFine, knonlinUpre, xt)
k2FineU = np.copy(k2Fine)
k2FineU[indicesInEps] *= (
1. + epsFine[indicesInEps] * knonlinU[indicesInEps]**2)
print('---- starting computation of correctors')
def computeLocalContribution(TInd):
patch = Patch(world, ell, TInd)
IPatch = lambda: interp.L2ProjectionPatchMatrix(
patch, boundaryConditions)
aPatch = lambda: coef.localizeCoefficient(patch, aFine)
kPatch = lambda: coef.localizeCoefficient(patch, kFine)
k2Patch = lambda: coef.localizeCoefficient(patch, k2FineU)
correctorsList = lod.computeBasisCorrectors_helmholtz(
patch, IPatch, aPatch, kPatch, k2Patch)
csi = lod.computeBasisCoarseQuantities_helmholtz(
patch, correctorsList, aPatch, kPatch, k2Patch)
return patch, correctorsList, csi.Kmsij, csi.Mmsij, csi.Bdmsij, csi.muTPrime
def computeIndicators(TInd):
k2FineUPatch = lambda: coef.localizeCoefficient(
patchT[TInd], k2FineU)
k2FineUOldPatch = lambda: coef.localizeCoefficient(
patchT[TInd], k2FineUOld)
E_vh = lod.computeErrorIndicatorCoarse_helmholtz(
patchT[TInd], muTPrime[TInd], k2FineUOldPatch,
k2FineUPatch)
return E_vh
def UpdateCorrectors(TInd):
patch = Patch(world, ell, TInd)
IPatch = lambda: interp.L2ProjectionPatchMatrix(
patch, boundaryConditions)
aPatch = lambda: coef.localizeCoefficient(patch, aFine)
kPatch = lambda: coef.localizeCoefficient(patch, kFine)
k2Patch = lambda: coef.localizeCoefficient(patch, k2FineU)
correctorsList = lod.computeBasisCorrectors_helmholtz(
patch, IPatch, aPatch, kPatch, k2Patch)
csi = lod.computeBasisCoarseQuantities_helmholtz(
patch, correctorsList, aPatch, kPatch, k2Patch)
return patch, correctorsList, csi.Kmsij, csi.Mmsij, csi.Bdmsij, csi.muTPrime
def UpdateElements(tol, E, Kmsij_old, Mmsij_old, Bdmsij_old,
correctors_old, mu_old):
print('---- apply tolerance')
Elements_to_be_updated = []
for (i, eps) in E.items():
if eps > tol:
Elements_to_be_updated.append(i)
if len(E) > 0:
print(
'---- total percentage of element correctors to be updated: %.4f'
%
(100 * np.size(Elements_to_be_updated) / len(mu_old)),
flush=True)
print('---- update local contributions')
KmsijT_list = list(np.copy(Kmsij_old))
MmsijT_list = list(np.copy(Mmsij_old))
BdmsijT_list = list(np.copy(Bdmsij_old))
muT_list = np.copy(mu_old)
for T in np.setdiff1d(range(world.NtCoarse),
Elements_to_be_updated):
patch = Patch(world, ell, T)
aPatch = lambda: coef.localizeCoefficient(patch, aFine)
kPatch = lambda: coef.localizeCoefficient(patch, kFine)
k2Patch = lambda: coef.localizeCoefficient(patch, k2FineU)
csi = lod.computeBasisCoarseQuantities_helmholtz(
patch, correctors_old[T], aPatch, kPatch, k2Patch)
KmsijT_list[T] = csi.Kmsij
MmsijT_list[T] = csi.Mmsij
BdmsijT_list[T] = csi.Bdmsij
muT_list[T] = csi.muTPrime
if np.size(Elements_to_be_updated) != 0:
#print('---- update correctors')
patchT_irrelevant, correctorsListTNew, KmsijTNew, MmsijTNew, BdmsijTNew, muTPrimeNew = zip(
*mapper(UpdateCorrectors, Elements_to_be_updated))
#print('---- update correctorsList')
correctorsListT_list = list(np.copy(correctors_old))
i = 0
for T in Elements_to_be_updated:
KmsijT_list[T] = KmsijTNew[i]
correctorsListT_list[T] = correctorsListTNew[i]
MmsijT_list[T] = MmsijTNew[i]
BdmsijT_list[T] = BdmsijTNew[i]
muT_list[T] = muTPrimeNew[i]
i += 1
KmsijT = tuple(KmsijT_list)
correctorsListT = tuple(correctorsListT_list)
MmsijT = tuple(MmsijT_list)
BdmsijT = tuple(BdmsijT_list)
muTPrime = tuple(muT_list)
return correctorsListT, KmsijT, MmsijT, BdmsijT, muTPrime
else:
KmsijT = tuple(KmsijT_list)
MmsijT = tuple(MmsijT_list)
BdmsijT = tuple(BdmsijT_list)
muTPrime = tuple(muT_list)
return correctors_old, KmsijT, MmsijT, BdmsijT, muTPrime
if it == 0:
patchT, correctorsListT, KmsijT, MmsijT, BdmsijT, muTPrime = zip(
*mapper(computeLocalContribution, range(world.NtCoarse)))
else:
E_vh = list(mapper(computeIndicators, range(world.NtCoarse)))
print(
'---- maximal value error estimator for basis correctors {}'
.format(np.max(E_vh)))
E = {i: E_vh[i] for i in range(np.size(E_vh)) if E_vh[i] > 0}
# loop over elements with possible recomputation of correctors
correctorsListT, KmsijT, MmsijT, BdmsijT, muTPrime = UpdateElements(
tol * np.max(E_vh), E, KmsijT, MmsijT, BdmsijT,
correctorsListT,
muTPrime) # tol scaled by maximal error indicator
print('---- finished computation of correctors')
KLOD = pglod.assembleMsStiffnessMatrix(
world, patchT, KmsijT) # ms stiffness matrix
k2MLOD = pglod.assembleMsStiffnessMatrix(world, patchT,
MmsijT) # ms mass matrix
kBdLOD = pglod.assembleMsStiffnessMatrix(
world, patchT, BdmsijT) # ms boundary matrix
MFEM = fem.assemblePatchMatrix(NWorldCoarse, world.MLocCoarse)
BdFEM = fem.assemblePatchBoundaryMatrix(
NWorldCoarse, fem.localBoundaryMassMatrixGetter(NWorldCoarse))
print('---- coarse matrices assembled')
nodes = np.arange(world.NpCoarse)
fix = util.boundarypIndexMap(NWorldCoarse, boundaryConditions == 0)
free = np.setdiff1d(nodes, fix)
assert (nodes.all() == free.all())
# compute global interpolation matrix
patchGlobal = Patch(world, NFine[0] + 2, 0)
IH = interp.L2ProjectionPatchMatrix(patchGlobal,
boundaryConditions)
assert (IH.shape[0] == NpCoarse)
basis = fem.assembleProlongationMatrix(NWorldCoarse,
NCoarseElement)
fHQuad = basis.T * MFineFEM * f + basis.T * BdFineFEM * g
print('---- solving coarse system')
# coarse system
lhsH = KLOD[free][:, free] - k2MLOD[
free][:, free] + 1j * kBdLOD[free][:, free]
rhsH = fHQuad[free]
xFree = sparse.linalg.spsolve(lhsH, rhsH)
basisCorrectors = pglod.assembleBasisCorrectors(
world, patchT, correctorsListT)
modifiedBasis = basis - basisCorrectors
xFull = np.zeros(world.NpCoarse, dtype='complex128')
xFull[free] = xFree
uLodCoarse = basis * xFull
uLodFine = modifiedBasis * xFull
uOldUps = np.copy(uLodFine)
k2FineUOld = np.copy(k2FineU)
# visualization
if it == maxit - 1 and N == 2**4:
grid = uLodFine.reshape(NFine + 1, order='C')
plt.figure(2)
plt.title('LOD_ad, Hlvl=4 - Ex 2')
plt.imshow(grid.real,
extent=(xC.min(), xC.max(), yC.min(), yC.max()),
cmap=plt.cm.hot,
origin='lower',
vmin=-.6,
vmax=.6)
plt.colorbar()
grid2 = uSol.reshape(NFine + 1, order='C')
plt.figure(1)
plt.title('reference solution - Ex 2')
plt.imshow(grid2.real,
extent=(xC.min(), xC.max(), yC.min(), yC.max()),
cmap=plt.cm.hot,
origin='lower',
vmin=-.6,
vmax=.6)
plt.colorbar()
grid3 = uInc.reshape(NFine + 1, order='C')
plt.figure(6)
plt.title('incident beam - Ex 2')
plt.imshow(grid3.real,
extent=(xC.min(), xC.max(), yC.min(), yC.max()),
cmap=plt.cm.hot,
origin='lower',
vmin=-.6,
vmax=.6)
plt.colorbar()
Err = np.sqrt(
np.dot((uSol - uLodFine).conj(), KFineFEM *
(uSol - uLodFine)) + k**2 *
np.dot((uSol - uLodFine).conj(), MFineFEM * (uSol - uLodFine)))
ErrEnergy = Err / np.sqrt(
np.dot((uSol).conj(), KFineFEM *
(uSol)) + k**2 * np.dot((uSol).conj(), MFineFEM *
(uSol)))
print('---- ', np.abs(ErrEnergy),
'\n***********************************************')
# save errors in arrays
relErrEnergy[counter - 1, it] = ErrEnergy
print('\n')
######################################################################################
print(
'***computing multiscale approximations without updates of correctors***'
)
relErrEnergyNoUpdate = np.zeros([len(NList), maxit])
counter = 0
for N in NList:
counter += 1
print('H = %.4e' % (1. / N))
NWorldCoarse = np.array([N, N])
NCoarseElement = NFine // NWorldCoarse
world = World(NWorldCoarse, NCoarseElement, boundaryConditions)
NpCoarse = np.prod(NWorldCoarse + 1)
uOldUps = np.zeros(NpFine, dtype='complex128')
for it in np.arange(maxit):
print('-- it = %d:' % it)
knonlinUpre = np.abs(uOldUps)
knonlinU = func.evaluateCQ1(NFine, knonlinUpre, xt)
k2FineU = np.copy(k2Fine)
k2FineU[indicesInEps] *= (
1. + epsFine[indicesInEps] * knonlinU[indicesInEps]**2)
print('---- starting computation of correctors')
def computeLocalContribution(TInd):
patch = Patch(world, ell, TInd)
IPatch = lambda: interp.L2ProjectionPatchMatrix(
patch, boundaryConditions)
aPatch = lambda: coef.localizeCoefficient(patch, aFine)
kPatch = lambda: coef.localizeCoefficient(patch, kFine)
k2Patch = lambda: coef.localizeCoefficient(patch, k2FineU)
correctorsList = lod.computeBasisCorrectors_helmholtz(
patch, IPatch, aPatch, kPatch,
k2Patch) # adapted for Helmholtz setting
csi = lod.computeBasisCoarseQuantities_helmholtz(
patch, correctorsList, aPatch, kPatch,
k2Patch) # adapted for Helmholtz setting
return patch, correctorsList, csi.Kmsij, csi.Mmsij, csi.Bdmsij, csi.muTPrime
def computeIndicators(TInd):
k2FineUPatch = lambda: coef.localizeCoefficient(
patchT[TInd], k2FineU)
k2FineUOldPatch = lambda: coef.localizeCoefficient(
patchT[TInd], k2FineUOld)
E_vh = lod.computeErrorIndicatorCoarse_helmholtz(
patchT[TInd], muTPrime[TInd], k2FineUOldPatch,
k2FineUPatch)
return E_vh
def UpdateCorrectors(TInd):
patch = Patch(world, ell, TInd)
IPatch = lambda: interp.L2ProjectionPatchMatrix(
patch, boundaryConditions)
aPatch = lambda: coef.localizeCoefficient(patch, aFine)
kPatch = lambda: coef.localizeCoefficient(patch, kFine)
k2Patch = lambda: coef.localizeCoefficient(patch, k2FineU)
correctorsList = lod.computeBasisCorrectors_helmholtz(
patch, IPatch, aPatch, kPatch, k2Patch)
csi = lod.computeBasisCoarseQuantities_helmholtz(
patch, correctorsList, aPatch, kPatch,
k2Patch) # adapted for Helmholtz setting
return patch, correctorsList, csi.Kmsij, csi.Mmsij, csi.Bdmsij, csi.muTPrime
def UpdateElements(tol, E, Kmsij_old, Mmsij_old, Bdmsij_old,
correctors_old, mu_old):
print('---- apply tolerance')
Elements_to_be_updated = []
for (i, eps) in E.items():
if eps > tol:
Elements_to_be_updated.append(i)
if len(E) > 0:
print(
'---- total percentage of element correctors to be updated: %.4f'
%
(100 * np.size(Elements_to_be_updated) / len(mu_old)),
flush=True)
print('---- update local contributions')
KmsijT_list = list(np.copy(Kmsij_old))
MmsijT_list = list(np.copy(Mmsij_old))
BdmsijT_list = list(np.copy(Bdmsij_old))
muT_list = np.copy(mu_old)
for T in np.setdiff1d(range(world.NtCoarse),
Elements_to_be_updated):
patch = Patch(world, ell, T)
aPatch = lambda: coef.localizeCoefficient(patch, aFine)
kPatch = lambda: coef.localizeCoefficient(patch, kFine)
k2Patch = lambda: coef.localizeCoefficient(patch, k2FineU)
csi = lod.computeBasisCoarseQuantities_helmholtz(
patch, correctors_old[T], aPatch, kPatch, k2Patch)
KmsijT_list[T] = csi.Kmsij
MmsijT_list[T] = csi.Mmsij
BdmsijT_list[T] = csi.Bdmsij
muT_list[T] = csi.muTPrime
if np.size(Elements_to_be_updated) != 0:
#print('---- update correctors')
patchT_irrelevant, correctorsListTNew, KmsijTNew, MmsijTNew, BdmsijTNew, muTPrimeNew = zip(
*mapper(UpdateCorrectors, Elements_to_be_updated))
#print('---- update correctorsList')
correctorsListT_list = list(np.copy(correctors_old))
i = 0
for T in Elements_to_be_updated:
KmsijT_list[T] = KmsijTNew[i]
correctorsListT_list[T] = correctorsListTNew[i]
MmsijT_list[T] = MmsijTNew[i]
BdmsijT_list[T] = BdmsijTNew[i]
muT_list[T] = muTPrimeNew[i]
i += 1
KmsijT = tuple(KmsijT_list)
correctorsListT = tuple(correctorsListT_list)
MmsijT = tuple(MmsijT_list)
BdmsijT = tuple(BdmsijT_list)
muTPrime = tuple(muT_list)
return correctorsListT, KmsijT, MmsijT, BdmsijT, muTPrime
else:
KmsijT = tuple(KmsijT_list)
MmsijT = tuple(MmsijT_list)
BdmsijT = tuple(BdmsijT_list)
muTPrime = tuple(muT_list)
return correctors_old, KmsijT, MmsijT, BdmsijT, muTPrime
if it == 0:
patchT, correctorsListT, KmsijT, MmsijT, BdmsijT, muTPrime = zip(
*mapper(computeLocalContribution, range(world.NtCoarse)))
else:
E_vh = list(mapper(computeIndicators, range(world.NtCoarse)))
print(
'---- maximal value error estimator for basis correctors {}'
.format(np.max(E_vh)))
E = {i: E_vh[i] for i in range(np.size(E_vh)) if E_vh[i] > 0}
# loop over elements with possible recomputation of correctors
correctorsListT, KmsijT, MmsijT, BdmsijT, muTPrime = UpdateElements(
2. * np.max(E_vh), E, KmsijT, MmsijT, BdmsijT,
correctorsListT, muTPrime) # no updates
print('---- finished computation of correctors')
KLOD = pglod.assembleMsStiffnessMatrix(
world, patchT, KmsijT) # ms stiffness matrix
k2MLOD = pglod.assembleMsStiffnessMatrix(world, patchT,
MmsijT) # ms mass matrix
kBdLOD = pglod.assembleMsStiffnessMatrix(
world, patchT, BdmsijT) # ms boundary matrix
MFEM = fem.assemblePatchMatrix(NWorldCoarse, world.MLocCoarse)
BdFEM = fem.assemblePatchBoundaryMatrix(
NWorldCoarse, fem.localBoundaryMassMatrixGetter(NWorldCoarse))
print('---- coarse matrices assembled')
nodes = np.arange(world.NpCoarse)
fix = util.boundarypIndexMap(NWorldCoarse, boundaryConditions == 0)
free = np.setdiff1d(nodes, fix)
assert (nodes.all() == free.all())
# compute global interpolation matrix
patchGlobal = Patch(world, NFine[0] + 2, 0)
IH = interp.L2ProjectionPatchMatrix(patchGlobal,
boundaryConditions)
assert (IH.shape[0] == NpCoarse)
basis = fem.assembleProlongationMatrix(NWorldCoarse,
NCoarseElement)
fHQuad = basis.T * MFineFEM * f + basis.T * BdFineFEM * g
print('---- solving coarse system')
# coarse system
lhsH = KLOD[free][:, free] - k2MLOD[
free][:, free] + 1j * kBdLOD[free][:, free]
rhsH = fHQuad[free]
xFree = sparse.linalg.spsolve(lhsH, rhsH)
basisCorrectors = pglod.assembleBasisCorrectors(
world, patchT, correctorsListT)
modifiedBasis = basis - basisCorrectors
xFull = np.zeros(world.NpCoarse, dtype='complex128')
xFull[free] = xFree
uLodCoarse = basis * xFull
uLodFine = modifiedBasis * xFull
uOldUps = np.copy(uLodFine)
k2FineUOld = np.copy(k2FineU)
# visualization
if it == maxit - 1 and N == 2**4:
grid = uLodFine.reshape(NFine + 1, order='C')
plt.figure(3)
plt.title('LOD_inf, Hlvl=4 - Ex 2')
plt.imshow(grid.real,
extent=(xC.min(), xC.max(), yC.min(), yC.max()),
cmap=plt.cm.hot,
origin='lower',
vmin=-.6,
vmax=.6)
plt.colorbar()
Err = np.sqrt(
np.dot((uSol - uLodFine).conj(), KFineFEM *
(uSol - uLodFine)) + k**2 *
np.dot((uSol - uLodFine).conj(), MFineFEM * (uSol - uLodFine)))
ErrEnergy = Err / np.sqrt(
np.dot((uSol).conj(), KFineFEM *
(uSol)) + k**2 * np.dot((uSol).conj(), MFineFEM *
(uSol)))
print('---- ', np.abs(ErrEnergy),
'\n***********************************************')
# save errors in arrays
relErrEnergyNoUpdate[counter - 1, it] = ErrEnergy
print('\n')
######################################################################################
print(
'***computing multiscale approximations where all correctors in the part of the domain with active nonlinearity are recomputed***'
)
relErrEnergyFullUpdate = np.zeros([len(NList), maxit])
counter = 0
for N in NList:
counter += 1
print('H = %.4e' % (1. / N))
NWorldCoarse = np.array([N, N])
NCoarseElement = NFine // NWorldCoarse
world = World(NWorldCoarse, NCoarseElement, boundaryConditions)
NpCoarse = np.prod(NWorldCoarse + 1)
uOldUps = np.zeros(NpFine, dtype='complex128')
for it in np.arange(maxit):
print('-- it = %d:' % it)
knonlinUpre = np.abs(uOldUps)
knonlinU = func.evaluateCQ1(NFine, knonlinUpre, xt)
k2FineU = np.copy(k2Fine)
k2FineU[indicesInEps] *= (
1. + epsFine[indicesInEps] * knonlinU[indicesInEps]**2)
print('---- starting computation of correctors')
def computeLocalContribution(TInd):
patch = Patch(world, ell, TInd)
IPatch = lambda: interp.L2ProjectionPatchMatrix(
patch, boundaryConditions)
aPatch = lambda: coef.localizeCoefficient(patch, aFine)
kPatch = lambda: coef.localizeCoefficient(patch, kFine)
k2Patch = lambda: coef.localizeCoefficient(patch, k2FineU)
correctorsList = lod.computeBasisCorrectors_helmholtz(
patch, IPatch, aPatch, kPatch,
k2Patch) # adapted for Helmholtz setting
csi = lod.computeBasisCoarseQuantities_helmholtz(
patch, correctorsList, aPatch, kPatch,
k2Patch) # adapted for Helmholtz setting
return patch, correctorsList, csi.Kmsij, csi.Mmsij, csi.Bdmsij, csi.muTPrime
def computeIndicators(TInd):
k2FineUPatch = lambda: coef.localizeCoefficient(
patchT[TInd], k2FineU)
k2FineUOldPatch = lambda: coef.localizeCoefficient(
patchT[TInd], k2FineUOld)
E_vh = lod.computeErrorIndicatorCoarse_helmholtz(
patchT[TInd], muTPrime[TInd], k2FineUOldPatch,
k2FineUPatch)
return E_vh
def UpdateCorrectors(TInd):
patch = Patch(world, ell, TInd)
IPatch = lambda: interp.L2ProjectionPatchMatrix(
patch, boundaryConditions)
aPatch = lambda: coef.localizeCoefficient(patch, aFine)
kPatch = lambda: coef.localizeCoefficient(patch, kFine)
k2Patch = lambda: coef.localizeCoefficient(patch, k2FineU)
correctorsList = lod.computeBasisCorrectors_helmholtz(
patch, IPatch, aPatch, kPatch, k2Patch)
csi = lod.computeBasisCoarseQuantities_helmholtz(
patch, correctorsList, aPatch, kPatch,
k2Patch) # adapted for Helmholtz setting
return patch, correctorsList, csi.Kmsij, csi.Mmsij, csi.Bdmsij, csi.muTPrime
def UpdateElements(tol, E, Kmsij_old, Mmsij_old, Bdmsij_old,
correctors_old, mu_old):
print('---- apply tolerance')
Elements_to_be_updated = []
for (i, eps) in E.items():
if eps > tol:
Elements_to_be_updated.append(i)
if len(E) > 0:
print(
'---- total percentage of element correctors to be updated: %.4f'
%
(100 * np.size(Elements_to_be_updated) / len(mu_old)),
flush=True)
print('---- update local contributions')
KmsijT_list = list(np.copy(Kmsij_old))
MmsijT_list = list(np.copy(Mmsij_old))
BdmsijT_list = list(np.copy(Bdmsij_old))
muT_list = np.copy(mu_old)
for T in np.setdiff1d(range(world.NtCoarse),
Elements_to_be_updated):
patch = Patch(world, ell, T)
aPatch = lambda: coef.localizeCoefficient(patch, aFine)
kPatch = lambda: coef.localizeCoefficient(patch, kFine)
k2Patch = lambda: coef.localizeCoefficient(patch, k2FineU)
csi = lod.computeBasisCoarseQuantities_helmholtz(
patch, correctors_old[T], aPatch, kPatch, k2Patch)
KmsijT_list[T] = csi.Kmsij
MmsijT_list[T] = csi.Mmsij
BdmsijT_list[T] = csi.Bdmsij
muT_list[T] = csi.muTPrime
if np.size(Elements_to_be_updated) != 0:
#print('---- update correctors')
patchT_irrelevant, correctorsListTNew, KmsijTNew, MmsijTNew, BdmsijTNew, muTPrimeNew = zip(
*mapper(UpdateCorrectors, Elements_to_be_updated))
#print('---- update correctorsList')
correctorsListT_list = list(np.copy(correctors_old))
i = 0
for T in Elements_to_be_updated:
KmsijT_list[T] = KmsijTNew[i]
correctorsListT_list[T] = correctorsListTNew[i]
MmsijT_list[T] = MmsijTNew[i]
BdmsijT_list[T] = BdmsijTNew[i]
muT_list[T] = muTPrimeNew[i]
i += 1
KmsijT = tuple(KmsijT_list)
correctorsListT = tuple(correctorsListT_list)
MmsijT = tuple(MmsijT_list)
BdmsijT = tuple(BdmsijT_list)
muTPrime = tuple(muT_list)
return correctorsListT, KmsijT, MmsijT, BdmsijT, muTPrime
else:
KmsijT = tuple(KmsijT_list)
MmsijT = tuple(MmsijT_list)
BdmsijT = tuple(BdmsijT_list)
muTPrime = tuple(muT_list)
return correctors_old, KmsijT, MmsijT, BdmsijT, muTPrime
if it == 0:
patchT, correctorsListT, KmsijT, MmsijT, BdmsijT, muTPrime = zip(
*mapper(computeLocalContribution, range(world.NtCoarse)))
else:
E_vh = list(mapper(computeIndicators, range(world.NtCoarse)))
print(
'---- maximal value error estimator for basis correctors {}'
.format(np.max(E_vh)))
E = {i: E_vh[i] for i in range(np.size(E_vh)) if E_vh[i] > 0}
# loop over elements with possible recomputation of correctors
correctorsListT, KmsijT, MmsijT, BdmsijT, muTPrime = UpdateElements(
0., E, KmsijT, MmsijT, BdmsijT, correctorsListT,
muTPrime) # no updates
print('---- finished computation of correctors')
KLOD = pglod.assembleMsStiffnessMatrix(
world, patchT, KmsijT) # ms stiffness matrix
k2MLOD = pglod.assembleMsStiffnessMatrix(world, patchT,
MmsijT) # ms mass matrix
kBdLOD = pglod.assembleMsStiffnessMatrix(
world, patchT, BdmsijT) # ms boundary matrix
MFEM = fem.assemblePatchMatrix(NWorldCoarse, world.MLocCoarse)
BdFEM = fem.assemblePatchBoundaryMatrix(
NWorldCoarse, fem.localBoundaryMassMatrixGetter(NWorldCoarse))
print('---- coarse matrices assembled')
nodes = np.arange(world.NpCoarse)
fix = util.boundarypIndexMap(NWorldCoarse, boundaryConditions == 0)
free = np.setdiff1d(nodes, fix)
assert (nodes.all() == free.all())
# compute global interpolation matrix
patchGlobal = Patch(world, NFine[0] + 2, 0)
IH = interp.L2ProjectionPatchMatrix(patchGlobal,
boundaryConditions)
assert (IH.shape[0] == NpCoarse)
basis = fem.assembleProlongationMatrix(NWorldCoarse,
NCoarseElement)
fHQuad = basis.T * MFineFEM * f + basis.T * BdFineFEM * g
print('---- solving coarse system')
# coarse system
lhsH = KLOD[free][:, free] - k2MLOD[
free][:, free] + 1j * kBdLOD[free][:, free]
rhsH = fHQuad[free]
xFree = sparse.linalg.spsolve(lhsH, rhsH)
basisCorrectors = pglod.assembleBasisCorrectors(
world, patchT, correctorsListT)
modifiedBasis = basis - basisCorrectors
xFull = np.zeros(world.NpCoarse, dtype='complex128')
xFull[free] = xFree
uLodCoarse = basis * xFull
uLodFine = modifiedBasis * xFull
uOldUps = np.copy(uLodFine)
k2FineUOld = np.copy(k2FineU)
# visualization
if it == maxit - 1 and N == 2**4:
grid = uLodFine.reshape(NFine + 1, order='C')
plt.figure(7)
plt.title('LOD_inf, Hlvl=4 - Ex 2')
plt.imshow(grid.real,
extent=(xC.min(), xC.max(), yC.min(), yC.max()),
cmap=plt.cm.hot,
origin='lower',
vmin=-.6,
vmax=.6)
plt.colorbar()
Err = np.sqrt(
np.dot((uSol - uLodFine).conj(), KFineFEM *
(uSol - uLodFine)) + k**2 *
np.dot((uSol - uLodFine).conj(), MFineFEM * (uSol - uLodFine)))
ErrEnergy = Err / np.sqrt(
np.dot((uSol).conj(), KFineFEM *
(uSol)) + k**2 * np.dot((uSol).conj(), MFineFEM *
(uSol)))
print('---- ', np.abs(ErrEnergy),
'\n***********************************************')
# save errors in arrays
relErrEnergyFullUpdate[counter - 1, it] = ErrEnergy
print('\n')
######################################################################################
print('***computing FEM approximations***')
FEMrelErrEnergy = np.zeros([len(NList), maxit])
counter = 0
for N in NList:
counter += 1
print('H = %.4e' % (1. / N))
NWorldCoarse = np.array([N, N])
NCoarseElement = NFine // NWorldCoarse
world = World(NWorldCoarse, NCoarseElement, boundaryConditions)
NpCoarse = np.prod(NWorldCoarse + 1)
xT = util.tCoordinates(NWorldCoarse)
xP = util.pCoordinates(NWorldCoarse)
uOld = np.zeros(NpCoarse, dtype='complex128')
# compute coarse coefficients by averaging
NtC = np.prod(NWorldCoarse)
aCoarse = np.zeros(NtC)
kCoarse = k * np.ones(xT.shape[0])
k2Coarse = np.zeros(NtC)
epsCoarse = np.zeros(NtC)
for Q in range(NtC):
patch = Patch(world, 0, Q)
aPatch = coef.localizeCoefficient(patch, aFine)
epsPatch = coef.localizeCoefficient(patch, epsFine)
k2Patch = coef.localizeCoefficient(patch, k2Fine)
aCoarse[Q] = np.sum(aPatch) / (len(aPatch))
k2Coarse[Q] = np.sum(k2Patch) / (len(k2Patch))
epsCoarse[Q] = np.sum(epsPatch) / (len(epsPatch))
# coarse matrices
KFEM = fem.assemblePatchMatrix(NWorldCoarse,
fem.localStiffnessMatrix(NWorldCoarse),
aCoarse)
kBdFEM = fem.assemblePatchBoundaryMatrix(
NWorldCoarse, fem.localBoundaryMassMatrixGetter(NWorldCoarse),
kCoarse)
MFEM = fem.assemblePatchMatrix(NWorldCoarse, world.MLocCoarse)
BdFEM = fem.assemblePatchBoundaryMatrix(
NWorldCoarse, fem.localBoundaryMassMatrixGetter(NWorldCoarse))
for it in np.arange(maxit):
print('-- it = %d:' % it)
knonlinUpre = np.abs(uOld)
knonlinU = func.evaluateCQ1(NWorldCoarse, knonlinUpre, xT)
k2CoarseU = np.copy(k2Coarse)
k2CoarseU *= (1. + epsCoarse * knonlinU**2)
# update weighted mass matrix
k2MFEM = fem.assemblePatchMatrix(NWorldCoarse,
fem.localMassMatrix(NWorldCoarse),
k2CoarseU)
nodes = np.arange(world.NpCoarse)
fix = util.boundarypIndexMap(NWorldCoarse, boundaryConditions == 0)
free = np.setdiff1d(nodes, fix)
assert (nodes.all() == free.all())
basis = fem.assembleProlongationMatrix(NWorldCoarse,
NCoarseElement)
fHQuad = basis.T * MFineFEM * f + basis.T * BdFineFEM * g
print('---- solving coarse system')
# coarse system
lhsH = KFEM[free][:, free] - k2MFEM[
free][:, free] + 1j * kBdFEM[free][:, free]
rhsH = fHQuad[free]
xFree = sparse.linalg.spsolve(lhsH, rhsH)
xFull = np.zeros(world.NpCoarse, dtype='complex128')
xFull[free] = xFree
uCoarseInt = basis * xFull
uOld = np.copy(xFull)
# visualization
if it == maxit - 1 and N == 2**4:
grid = uCoarseInt.reshape(NFine + 1, order='C')
plt.figure(4)
plt.title('FEM, Hlvl=4 - Ex 2')
plt.imshow(grid.real,
extent=(xC.min(), xC.max(), yC.min(), yC.max()),
cmap=plt.cm.hot,
origin='lower',
vmin=-.6,
vmax=.6)
plt.colorbar()
Err = np.sqrt(
np.dot((uSol -
uCoarseInt).conj(), KFineFEM * (uSol - uCoarseInt)) +
k**2 * np.dot(
(uSol - uCoarseInt).conj(), MFineFEM *
(uSol - uCoarseInt)))
ErrEnergy = Err / np.sqrt(
np.dot((uSol).conj(), KFineFEM *
(uSol)) + k**2 * np.dot((uSol).conj(), MFineFEM *
(uSol)))
print('---- ', np.abs(ErrEnergy),
'\n***********************************************')
# save errors in arrays
FEMrelErrEnergy[counter - 1, it] = ErrEnergy
print('\n')
# error plots
errLOD_2 = np.min(relErrEnergy, 1)
errLOD0_2 = np.min(relErrEnergyNoUpdate, 1)
errLODall_2 = np.min(relErrEnergyFullUpdate, 1)
errFEM_2 = np.min(FEMrelErrEnergy, 1)
Hs = 0.5**np.arange(1, maxCoarseLvl + 1)
plt.figure(5)
plt.title('Relative energy errors w.r.t H - Ex 2')
plt.plot(Hs, errLOD_2, 'x-', color='blue', label='LOD_ad')
plt.plot(Hs, errLOD0_2, 'x-', color='green', label='LOD_inf')
plt.plot(Hs, errLODall_2, 'x-', color='orange', label='LOD_0')
plt.plot(Hs, errFEM_2, 'x-', color='red', label='FEM')
plt.plot([0.5, 0.0078125], [0.75, 0.01171875],
color='black',
linestyle='dashed',
label='order 1')
plt.yscale('log')
plt.xscale('log')
plt.legend()
plt.show()