def test_alive(self): NWorldCoarse = np.array([5, 5]) NCoarseElement = np.array([20, 20]) NFine = NWorldCoarse * NCoarseElement NtFine = np.prod(NFine) NpCoarse = np.prod(NWorldCoarse + 1) NpFine = np.prod(NWorldCoarse * NCoarseElement + 1) world = World(NWorldCoarse, NCoarseElement) IPatchGenerator = lambda i, N: interp.L2ProjectionPatchMatrix( i, N, NWorldCoarse, NCoarseElement) k = 5 pglod = pg.PetrovGalerkinLOD(world, k, IPatchGenerator, 0) aBase = np.ones(NtFine) pglod.updateCorrectors(coef.coefficientFine(NWorldCoarse, NCoarseElement, aBase), clearFineQuantities=False) K = pglod.assembleMsStiffnessMatrix() self.assertTrue(np.all(K.shape == NpCoarse)) basisCorrectors = pglod.assembleBasisCorrectors() self.assertTrue(np.all(basisCorrectors.shape == (NpFine, NpCoarse)))
def test_trivial(self): NPatchCoarse = np.array([3,3]) NCoarseElement = np.array([2,2]) NPatchFine = NPatchCoarse*NCoarseElement Nt = np.prod(NPatchFine) Np = np.prod(NPatchFine+1) fixed = util.boundarypIndexMap(NPatchFine) world = World(NPatchCoarse, NCoarseElement) patch = Patch(world, 3, 0) aFlatPatchFine = np.ones(Nt) ALoc = fem.localStiffnessMatrix(NPatchFine) APatchFull = fem.assemblePatchMatrix(NPatchFine, ALoc, aFlatPatchFine) PPatch = fem.assembleProlongationMatrix(NPatchCoarse, NCoarseElement) IPatchNodal = interp.nodalPatchMatrix(patch) #IPatchuncL2 = interp.uncoupledL2ProjectionPatchMatrix(np.array([0, 0]), NPatchCoarse, NPatchCoarse, NCoarseElement) IPatchL2 = interp.L2ProjectionPatchMatrix(patch) for IPatch in [IPatchNodal, IPatchL2]: np.random.seed(0) bPatchFullList = [] self.assertTrue(not lod.ritzProjectionToFinePatch(patch, APatchFull, bPatchFullList, IPatch)) bPatchFullList = [np.zeros(Np)] projections = lod.ritzProjectionToFinePatch(patch, APatchFull, bPatchFullList, IPatch) self.assertEqual(len(projections), 1) self.assertTrue(np.allclose(projections[0], 0*projections[0])) bPatchFull = np.random.rand(Np) bPatchFullList = [bPatchFull] projections = lod.ritzProjectionToFinePatch(patch, APatchFull, bPatchFullList, IPatch) self.assertTrue(np.isclose(np.linalg.norm(IPatch*projections[0]), 0)) self.assertTrue(np.isclose(np.dot(projections[0], APatchFull*projections[0]), np.dot(projections[0], bPatchFullList[0]))) self.assertTrue(np.isclose(np.linalg.norm(projections[0][fixed]), 0)) bPatchFullList = [bPatchFull, -bPatchFull] projections = lod.ritzProjectionToFinePatch(patch, APatchFull, bPatchFullList, IPatch) self.assertTrue(np.allclose(projections[0], -projections[1])) bPatchFullList = [np.random.rand(Np), np.random.rand(Np)] projections = lod.ritzProjectionToFinePatch(patch, APatchFull, bPatchFullList, IPatch) self.assertTrue(np.isclose(np.dot(projections[1], APatchFull*projections[0]), np.dot(projections[1], bPatchFullList[0]))) bPatchFull = np.random.rand(Np) bPatchFullList = [bPatchFull] projectionCheckAgainst = lod.ritzProjectionToFinePatch(patch, APatchFull, bPatchFullList, IPatch)[0] for saddleSolver in [#lod.nullspaceOneLevelHierarchySolver(NPatchCoarse, NCoarseElement), lod.SchurComplementSolver()]: projection = lod.ritzProjectionToFinePatch(patch, APatchFull, bPatchFullList, IPatch, saddleSolver)[0] self.assertTrue(np.isclose(np.max(np.abs(projectionCheckAgainst-projection)), 0))
def computeKmsij(TInd): patch = Patch(world, k, TInd) IPatch = lambda: interp.L2ProjectionPatchMatrix(patch, boundaryConditions) aPatch = lambda: coef.localizeCoefficient(patch, aFine_ref) correctorsList = lod.computeBasisCorrectors(patch, IPatch, aPatch) csi = lod.computeBasisCoarseQuantities(patch, correctorsList, aPatch) return patch, correctorsList, csi.Kmsij, csi
def real_computeKmsij(TInd): print('.', end='', flush=True) patch = Patch(world, k, TInd) IPatch = lambda: interp.L2ProjectionPatchMatrix(patch, boundaryConditions) aPatch = lambda: coef.localizeCoefficient(patch, a_Fine_to_be_approximated) correctorsList = lod.computeBasisCorrectors(patch, IPatch, aPatch) csi = lod.computeBasisCoarseQuantities(patch, correctorsList, aPatch) return patch, correctorsList, csi.Kmsij, csi
def computeRmsi(TInd): patch = Patch(world, k, TInd) IPatch = lambda: interp.L2ProjectionPatchMatrix(patch, boundaryConditions) aPatch = lambda: coef.localizeCoefficient(patch, aFine_ref) MRhsList = [f_ref[util.extractElementFine(world.NWorldCoarse, world.NCoarseElement, patch.iElementWorldCoarse, extractElements=False)]]; correctorRhs = lod.computeElementCorrector(patch, IPatch, aPatch, None, MRhsList)[0] Rmsi = lod.computeRhsCoarseQuantities(patch, correctorRhs, aPatch) return patch, correctorRhs, Rmsi
def computeKmsij(TInd, aPatch, k, boundaryConditions): tic = time.perf_counter() patch = lod_periodic.PatchPeriodic(world, k, TInd) if dim == 1: IPatch = lambda: interp.nodalPatchMatrix(patch) else: IPatch = lambda: interp.L2ProjectionPatchMatrix( patch, boundaryConditions) correctorsList = lod.computeBasisCorrectors(patch, IPatch, aPatch) csi = lod.computeBasisCoarseQuantities(patch, correctorsList, aPatch) toc = time.perf_counter() return patch, correctorsList, csi.Kmsij, csi.muTPrime, toc - tic
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 real_computeRmsi(TInd): print('.', end='', flush=True) patch = Patch(world, k, TInd) IPatch = lambda: interp.L2ProjectionPatchMatrix(patch, boundaryConditions) aPatch = lambda: coef.localizeCoefficient(patch, a_Fine_to_be_approximated) MRhsList = [f_ref[util.extractElementFine(world.NWorldCoarse, world.NCoarseElement, patch.iElementWorldCoarse, extractElements=False)]]; correctorRhs = lod.computeElementCorrector(patch, IPatch, aPatch, None, MRhsList)[0] Rmsi, cetaTPrime = lod.computeRhsCoarseQuantities(patch, correctorRhs, aPatch, True) return patch, correctorRhs, Rmsi, cetaTPrime
def test_stiffessMatrix(self): # Compare stiffness matrix from PG object with the one # computed from correctors and fine stiffness matrix NWorldFine = np.array([10, 10]) 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) world = World(NWorldCoarse, NCoarseElement) np.random.seed(0) aBase = np.random.rand(NtFine) aCoef = coef.coefficientFine(NWorldCoarse, NCoarseElement, aBase) IPatchGenerator = lambda i, N: interp.L2ProjectionPatchMatrix( i, N, NWorldCoarse, NCoarseElement) IWorld = IPatchGenerator(0 * NWorldCoarse, NWorldCoarse) k = 2 printLevel = 0 pglod = pg.PetrovGalerkinLOD(world, k, IPatchGenerator, 0, printLevel) pglod.updateCorrectors(aCoef, clearFineQuantities=False) KmsFull = pglod.assembleMsStiffnessMatrix() KFull = pglod.assembleStiffnessMatrix() basis = fem.assembleProlongationMatrix(NWorldCoarse, NCoarseElement) basisCorrectors = pglod.assembleBasisCorrectors() self.assertTrue( np.isclose(np.linalg.norm(IWorld * basisCorrectors.todense()), 0)) modifiedBasis = basis - basisCorrectors AFine = fem.assemblePatchMatrix(NWorldFine, world.ALocFine, aBase) KmsRef = np.dot(basis.T, AFine * modifiedBasis) KRef = np.dot(basis.T, AFine * basis) self.assertTrue( np.isclose(np.linalg.norm(KFull.todense() - KRef.todense()), 0)) self.assertTrue( np.isclose(np.linalg.norm(KmsFull.todense() - KmsRef.todense()), 0))
def test_testCsi(self): NWorldCoarse = np.array([4, 5, 6]) NCoarseElement = np.array([5, 2, 3]) world = World(NWorldCoarse, NCoarseElement) d = np.size(NWorldCoarse) k = 1 iElementWorldCoarse = np.array([2, 1, 2]) ec = lod.elementCorrector(world, k, iElementWorldCoarse) IPatch = interp.L2ProjectionPatchMatrix(ec.iPatchWorldCoarse, ec.NPatchCoarse, NWorldCoarse, NCoarseElement) NtPatch = np.prod(ec.NPatchCoarse*NCoarseElement) np.random.seed(1) aPatch = np.random.rand(NtPatch) coefficientPatch = coef.coefficientFine(ec.NPatchCoarse, NCoarseElement, aPatch) ec.computeCorrectors(coefficientPatch, IPatch) ec.computeCoarseQuantities() TFinetIndexMap = util.extractElementFine(ec.NPatchCoarse, NCoarseElement, ec.iElementPatchCoarse, extractElements=True) TFinepIndexMap = util.extractElementFine(ec.NPatchCoarse, NCoarseElement, ec.iElementPatchCoarse, extractElements=False) TCoarsepIndexMap = util.extractElementFine(ec.NPatchCoarse, np.ones_like(NCoarseElement), ec.iElementPatchCoarse, extractElements=False) APatchFine = fem.assemblePatchMatrix(ec.NPatchCoarse*NCoarseElement, world.ALocFine, aPatch) AElementFine = fem.assemblePatchMatrix(NCoarseElement, world.ALocFine, aPatch[TFinetIndexMap]) basisPatch = fem.assembleProlongationMatrix(ec.NPatchCoarse, NCoarseElement) correctorsPatch = np.column_stack(ec.fsi.correctorsList) localBasis = world.localBasis KmsijShouldBe = -basisPatch.T*(APatchFine*(correctorsPatch)) KmsijShouldBe[TCoarsepIndexMap,:] += np.dot(localBasis.T, AElementFine*localBasis) self.assertTrue(np.isclose(np.max(np.abs(ec.csi.Kmsij-KmsijShouldBe)), 0))
def test_testCsi_Kmsij(self): NWorldCoarse = np.array([4, 5, 6]) NCoarseElement = np.array([5, 2, 3]) world = World(NWorldCoarse, NCoarseElement) d = np.size(NWorldCoarse) k = 1 iElementWorldCoarse = np.array([2, 1, 2]) TInd = util.convertpCoordIndexToLinearIndex(NWorldCoarse, iElementWorldCoarse) patch = Patch(world, k, TInd) IPatch = interp.L2ProjectionPatchMatrix(patch) NtPatch = patch.NtFine np.random.seed(1) aPatch = np.random.rand(NtPatch) basisCorrectorsList = lod.computeBasisCorrectors(patch, IPatch, aPatch) csi = lod.computeBasisCoarseQuantities(patch, basisCorrectorsList, aPatch) TFinetIndexMap = util.extractElementFine(patch.NPatchCoarse, NCoarseElement, patch.iElementPatchCoarse, extractElements=True) TFinepIndexMap = util.extractElementFine(patch.NPatchCoarse, NCoarseElement, patch.iElementPatchCoarse, extractElements=False) TCoarsepIndexMap = util.extractElementFine(patch.NPatchCoarse, np.ones_like(NCoarseElement), patch.iElementPatchCoarse, extractElements=False) APatchFine = fem.assemblePatchMatrix(patch.NPatchFine, world.ALocFine, aPatch) AElementFine = fem.assemblePatchMatrix(NCoarseElement, world.ALocFine, aPatch[TFinetIndexMap]) basisPatch = fem.assembleProlongationMatrix(patch.NPatchCoarse, NCoarseElement) correctorsPatch = np.column_stack(basisCorrectorsList) localBasis = world.localBasis KmsijShouldBe = -basisPatch.T*(APatchFine*(correctorsPatch)) KmsijShouldBe[TCoarsepIndexMap,:] += np.dot(localBasis.T, AElementFine*localBasis) self.assertTrue(np.isclose(np.max(np.abs(csi.Kmsij-KmsijShouldBe)), 0))
def computeRmsi(TInd): print('.', end='', flush=True) patch = Patch(world, k, TInd) IPatch = lambda: interp.L2ProjectionPatchMatrix(patch, boundaryConditions) aPatch = lambda: coef.localizeCoefficient(patch, aFine_ref) MRhsList = [ f_ref[util.extractElementFine(world.NWorldCoarse, world.NCoarseElement, patch.iElementWorldCoarse, extractElements=False)] ] correctorRhs = lod.computeElementCorrector(patch, IPatch, aPatch, None, MRhsList)[0] Rmsi, cetaTPrime = lod.computeRhsCoarseQuantities(patch, correctorRhs, aPatch, True) eft_patch = Patch(world, 1, TInd) a_eft_Patch = lambda: coef.localizeCoefficient(eft_patch, aFine_ref) etaT = lod.computeSupremumForEf(eft_patch, a_eft_Patch) return patch, correctorRhs, Rmsi, cetaTPrime, etaT
def PGsolver(world, ABase, f,k): NWorldFine = world.NWorldFine NWorldCoarse = world.NWorldCoarse NCoarseElement = world.NCoarseElement boundaryConditions = world.boundaryConditions NpFine = np.prod(NWorldFine+1) NpCoarse = np.prod(NWorldCoarse+1) #interpolant IPatchGenerator = lambda i, N: interp.L2ProjectionPatchMatrix(i, N, NWorldCoarse, NCoarseElement, boundaryConditions) #Coefficient (need flatten form) aCoef = coef.coefficientFine(NWorldCoarse, NCoarseElement, ABase) pglod = pg_pert.PerturbedPetrovGalerkinLOD(aCoef, world, k, IPatchGenerator, 0) pglod.originCorrectors(clearFineQuantities=False) KFull = pglod.assembleMsStiffnessMatrix() MFull = fem.assemblePatchMatrix(NWorldCoarse, world.MLocCoarse) free = util.interiorpIndexMap(NWorldCoarse) bFull = MFull*f KFree = KFull[free][:,free] bFree = bFull[free] xFree = sparse.linalg.spsolve(KFree, bFree) basis = fem.assembleProlongationMatrix(NWorldCoarse, NCoarseElement) basisCorrectors = pglod.assembleBasisCorrectors() modifiedBasis = basis - basisCorrectors xFull = np.zeros(NpCoarse) xFull[free] = xFree uLodCoarse = xFull uLodFine = modifiedBasis*xFull return uLodCoarse, uLodFine
def UpdateCorrectors(self, TInd): # print(" UPDATING {}".format(TInd)) patch = Patch(self.world, self.k, TInd) IPatch = lambda: interp.L2ProjectionPatchMatrix( patch, self.boundaryConditions) rPatch = lambda: coef.localizeCoefficient( patch, self.a_Fine_to_be_approximated) MRhsList = [ self.f_trans[util.extractElementFine(self.world.NWorldCoarse, self.world.NCoarseElement, patch.iElementWorldCoarse, extractElements=False)] ] correctorsList = lod.computeBasisCorrectors(patch, IPatch, rPatch) csi = lod.computeBasisCoarseQuantities(patch, correctorsList, rPatch) correctorRhs = lod.computeElementCorrector(patch, IPatch, rPatch, None, MRhsList)[0] Rmsij = lod.computeRhsCoarseQuantities(patch, correctorRhs, rPatch) return patch, correctorsList, csi.Kmsij, Rmsij, correctorRhs
def test_ritzProjectionToFinePatchBoundaryConditions(self): NPatchCoarse = np.array([4, 4]) NCoarseElement = np.array([10, 10]) world = World(NPatchCoarse, NCoarseElement) patch = Patch(world, 4, 0) NPatchFine = NPatchCoarse*NCoarseElement NpFine = np.prod(NPatchFine + 1) APatchFull = fem.assemblePatchMatrix(NPatchCoarse*NCoarseElement, world.ALocFine) bPatchFullList = [np.ones(NpFine)] fixed = util.boundarypIndexMap(NPatchFine) for IPatch in [interp.L2ProjectionPatchMatrix(patch), interp.nodalPatchMatrix(patch)]: schurComplementSolver = lod.SchurComplementSolver() schurComplementSolution = lod.ritzProjectionToFinePatch(patch, APatchFull, bPatchFullList, IPatch, schurComplementSolver)[0] self.assertTrue(np.isclose(np.max(np.abs(schurComplementSolution[fixed])), 0))
def test_computeSingleT(self): NWorldCoarse = np.array([4, 5, 6]) NCoarseElement = np.array([5, 2, 3]) world = World(NWorldCoarse, NCoarseElement) d = np.size(NWorldCoarse) k = 1 iElementWorldCoarse = np.array([2, 1, 2]) TInd = util.convertpCoordIndexToLinearIndex(NWorldCoarse, iElementWorldCoarse) patch = Patch(world, k, TInd) IPatch = interp.L2ProjectionPatchMatrix(patch) NtPatch = patch.NtFine aPatch = np.ones(NtPatch) basisCorrectorsList = lod.computeBasisCorrectors(patch, IPatch, aPatch) correctorSum = reduce(np.add, basisCorrectorsList) self.assertTrue(np.allclose(correctorSum, 0)) csi = lod.computeBasisCoarseQuantities(patch, basisCorrectorsList, aPatch) # Test that the matrices have the constants in their null space #self.assertTrue(np.allclose(np.sum(ec.csi.LTPrimeij, axis=1), 0)) #self.assertTrue(np.allclose(np.sum(ec.csi.LTPrimeij, axis=2), 0)) self.assertTrue(np.allclose(np.sum(csi.Kij, axis=0), 0)) self.assertTrue(np.allclose(np.sum(csi.Kij, axis=1), 0)) self.assertTrue(np.allclose(np.sum(csi.Kmsij, axis=0), 0)) self.assertTrue(np.allclose(np.sum(csi.Kmsij, axis=1), 0)) # I had difficulties come up with test cases here. This test # verifies that most "energy" is in the element T. elementTIndex = util.convertpCoordIndexToLinearIndex(patch.NPatchCoarse-1, patch.iElementPatchCoarse) self.assertTrue(np.all(csi.muTPrime[elementTIndex] >= csi.muTPrime)) self.assertTrue(not np.all(csi.muTPrime[elementTIndex+1] >= csi.muTPrime))
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 compute_perturbed_MsStiffness(world, aPert, aRef, KmsijRef, muTPrimeRef, k, update_percentage): computePatch = lambda TInd: lod_periodic.PatchPeriodic(world, k, TInd) patchT = list(map(computePatch, range(world.NtCoarse))) dim = np.size(world.NWorldFine) if dim == 2: middle = world.NWorldCoarse[1] // 2 * world.NWorldCoarse[ 0] + world.NWorldCoarse[0] // 2 #2d!!! elif dim == 1: middle = world.NWorldCoarse[0] // 2 patchRef = lod_periodic.PatchPeriodic(world, k, middle) IPatch = lambda: interp.L2ProjectionPatchMatrix(patchRef) def computeIndicator(TInd): aPatch = lambda: lod_periodic.localizeCoefficient( patchT[TInd], aPert, periodic=True) # true coefficient E_vh = lod.computeErrorIndicatorCoarseFromCoefficients( patchT[TInd], muTPrimeRef, aRef, aPatch) return E_vh def UpdateCorrectors(TInd): rPatch = lambda: lod_periodic.localizeCoefficient( patchT[TInd], aPert, periodic=True) correctorsList = lod.computeBasisCorrectors(patchT[TInd], IPatch, rPatch) csi = lod.computeBasisCoarseQuantities(patchT[TInd], correctorsList, rPatch) return patchT[TInd], correctorsList, csi.Kmsij def UpdateElements(tol, E, Kmsij_old): Elements_to_be_updated = [] for (i, eps) in E.items(): if eps > tol: Elements_to_be_updated.append(i) if np.size(Elements_to_be_updated) != 0: patchT_irrelevant, correctorsListT_irrelevant, KmsijTNew = zip( *map(UpdateCorrectors, Elements_to_be_updated)) KmsijT_list = list(np.copy(Kmsij_old)) i = 0 for T in Elements_to_be_updated: KmsijT_list[T] = np.copy(KmsijTNew[i]) i += 1 KmsijT = tuple(KmsijT_list) return KmsijT else: return Kmsij_old E_vh = list(map(computeIndicator, range(world.NtCoarse))) 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 tol_relative = np.quantile(E_vh, 1. - update_percentage, interpolation='higher') KmsijRefList = [KmsijRef for _ in range(world.NtCoarse) ] #tile up the stiffness matrix for one element KmsijT = UpdateElements(tol_relative, E, KmsijRefList) #assembly of matrix KFull = lod_periodic.assembleMsStiffnessMatrix(world, patchT, KmsijT, periodic=True) return KFull, E_vh
def test_2d_exactSolution(self): NWorldFine = np.array([30, 40]) NpFine = np.prod(NWorldFine + 1) NtFine = np.prod(NWorldFine) NWorldCoarse = np.array([3, 4]) 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[0])) aBaseCube = np.tile(aBaseSquare, [NWorldFine[1], 1]) aBaseCube = aBaseCube[..., np.newaxis] aBase = aBaseCube.flatten() IPatchGenerator = lambda i, N: interp.L2ProjectionPatchMatrix( i, N, NWorldCoarse, NCoarseElement, boundaryConditions) rCoarse = np.ones(NtCoarse) aCoef = coef.coefficientCoarseFactor(NWorldCoarse, NCoarseElement, aBase, rCoarse) k = 4 printLevel = 0 epsilonTol = 0.05 pglod = pg.PetrovGalerkinLOD(world, k, IPatchGenerator, epsilonTol, printLevel) pglod.updateCorrectors(aCoef, clearFineQuantities=False) boundaryMap = boundaryConditions == 0 fixed = util.boundarypIndexMap(NWorldCoarse, boundaryMap) free = np.setdiff1d(np.arange(0, NpCoarse), fixed) coords = util.pCoordinates(NWorldCoarse) xC = coords[:, 0] yC = coords[:, 1] g = 1 - xC firstIteration = True # First case is to not modify. Error should be 0 # The other cases modify one, a few or half of the coarse elements to different degrees. rCoarseModPairs = [([], []), ([0], [2.]), ([10], [3.]), ([4, 3, 2], [1.3, 1.5, 1.8])] rCoarseModPairs.append((range(NtCoarse / 2), [2] * NtCoarse)) rCoarseModPairs.append((range(NtCoarse / 2), [0.9] * NtCoarse)) rCoarseModPairs.append((range(NtCoarse / 2), [0.95] * NtCoarse)) for i, rCoarseModPair in zip(count(), rCoarseModPairs): for ind, val in zip(rCoarseModPair[0], rCoarseModPair[1]): rCoarse[ind] *= val aCoef = coef.coefficientCoarseFactor(NWorldCoarse, NCoarseElement, aBase, rCoarse) pglod.updateCorrectors(aCoef, clearFineQuantities=False) KmsFull = pglod.assembleMsStiffnessMatrix() bFull = -KmsFull * g KmsFree = KmsFull[free][:, free] bFree = bFull[free] xFree = sparse.linalg.spsolve(KmsFree, bFree) basis = fem.assembleProlongationMatrix(NWorldCoarse, NCoarseElement) basisCorrectors = pglod.assembleBasisCorrectors() modifiedBasis = basis - basisCorrectors xFull = np.zeros(NpCoarse) xFull[free] = xFree uLodFine = modifiedBasis * (xFull + g) gFine = basis * g uFineFull, AFine, MFine = femsolver.solveFine( world, aCoef.aFine, None, -gFine, boundaryConditions) uFineFull += gFine errorFineA = np.sqrt( np.dot(uFineFull - uLodFine, AFine * (uFineFull - uLodFine))) errorFineM = np.sqrt( np.dot(uFineFull - uLodFine, MFine * (uFineFull - uLodFine))) if firstIteration: self.assertTrue(np.isclose(errorFineA, 0)) self.assertTrue(np.isclose(errorFineM, 0)) # Also compute upscaled solution and compare with uFineFull uLodFineUpscaled = basis * ( xFull + g) - pglod.computeCorrection(ARhsFull=basis * (xFull + g)) self.assertTrue(np.allclose(uLodFineUpscaled, uLodFine)) firstIteration = False else: # For this problem, it seems that # error < 1.1*errorTol self.assertTrue(errorFineA <= 1.1 * epsilonTol)
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 test_1d_toReference(self): NWorldFine = np.array([200]) NWorldCoarse = np.array([10]) NCoarseElement = NWorldFine / NWorldCoarse boundaryConditions = np.array([[0, 0]]) world = World(NWorldCoarse, NCoarseElement, boundaryConditions) NpFine = np.prod(NWorldFine + 1) NtFine = np.prod(NWorldFine) NpCoarse = np.prod(NWorldCoarse + 1) aBase = np.zeros(NtFine) aBase[:90] = 1 aBase[90:] = 2 k = 10 IPatchGenerator = lambda i, N: interp.L2ProjectionPatchMatrix( i, N, NWorldCoarse, NCoarseElement, boundaryConditions) aCoef = coef.coefficientFine(NWorldCoarse, NCoarseElement, aBase) pglod = pg.PetrovGalerkinLOD(world, k, IPatchGenerator, 0) pglod.updateCorrectors(aCoef, clearFineQuantities=False) KmsFull = pglod.assembleMsStiffnessMatrix() KFull = pglod.assembleStiffnessMatrix() MFull = fem.assemblePatchMatrix(NWorldCoarse, world.MLocCoarse) free = util.interiorpIndexMap(NWorldCoarse) coords = util.pCoordinates(NWorldCoarse) g = 1 - coords[:, 0] bFull = -KmsFull * g KmsFree = KmsFull[free][:, free] bFree = bFull[free] xFree = sparse.linalg.spsolve(KmsFree, bFree) basis = fem.assembleProlongationMatrix(NWorldCoarse, NCoarseElement) basisCorrectors = pglod.assembleBasisCorrectors() modifiedBasis = basis - basisCorrectors xFull = np.zeros(NpCoarse) xFull[free] = xFree uLodCoarse = basis * (xFull + g) uLodFine = modifiedBasis * (xFull + g) coordsFine = util.pCoordinates(NWorldFine) gFine = 1 - coordsFine[:, 0] uFineFull, AFine, _ = femsolver.solveFine(world, aBase, None, -gFine, boundaryConditions) uFineFull += gFine errorFine = np.sqrt( np.dot(uFineFull - uLodFine, AFine * (uFineFull - uLodFine))) self.assertTrue(np.isclose(errorFine, 0)) # Also compute upscaled solution and compare with uFineFull uLodFineUpscaled = basis * (xFull + g) - pglod.computeCorrection( ARhsFull=basis * (xFull + g)) self.assertTrue(np.allclose(uLodFineUpscaled, uLodFine))
def test_computeCoarseErrorIndicatorFlux(self): NWorldCoarse = np.array([7, 7], dtype='int64') NCoarseElement = np.array([10, 10], dtype='int64') NWorldFine = NWorldCoarse * NCoarseElement NpWorldFine = np.prod(NWorldFine + 1) NpWorldCoarse = np.prod(NWorldCoarse + 1) NtWorldFine = np.prod(NWorldCoarse * NCoarseElement) NtWorldCoarse = np.prod(NWorldCoarse) np.random.seed(0) world = World(NWorldCoarse, NCoarseElement) d = np.size(NWorldCoarse) aBase = np.exp(np.random.rand(NtWorldFine)) k = np.max(NWorldCoarse) iElementWorldCoarse = np.array([3, 3]) rCoarseFirst = 1 + 3 * np.random.rand(NtWorldCoarse) coefFirst = coef.CoefficientCoarseFactor(NWorldCoarse, NCoarseElement, aBase, rCoarseFirst) IPatchGenerator = lambda i, N: interp.L2ProjectionPatchMatrix( i, N, NWorldCoarse, NCoarseElement) ec = lod_flux.CoarseBasisElementCorrectorFlux(world, k, iElementWorldCoarse, IPatchGenerator) ec.computeCorrectors(coefFirst) ec.computeCoarseQuantities() # If both rCoarseFirst and rCoarseSecond are equal, the error indicator should be zero rCoarseSecond = np.array(rCoarseFirst) self.assertTrue( np.isclose(ec.computeCoarseErrorIndicatorFlux(rCoarseSecond), 0)) coefSecond = coef.CoefficientCoarseFactor(NWorldCoarse, NCoarseElement, aBase, rCoarseSecond) self.assertTrue(np.isclose(ec.computeErrorIndicatorFine(coefSecond), 0)) # If rCoarseSecond is not rCoarseFirst, the error indicator should not be zero rCoarseSecond = 2 * np.array(rCoarseFirst) self.assertTrue( ec.computeCoarseErrorIndicatorFlux(rCoarseSecond) >= 0.1) coefSecond = coef.CoefficientCoarseFactor(NWorldCoarse, NCoarseElement, aBase, rCoarseSecond) self.assertTrue(ec.computeErrorIndicatorFine(coefSecond) >= 0.1) # Fine should be smaller than coarse estimate self.assertTrue( ec.computeErrorIndicatorFine(coefSecond) < ec.computeCoarseErrorIndicatorFlux(rCoarseSecond)) # If rCoarseSecond is different in the element itself, the error # indicator should be large elementCoarseIndex = util.convertpCoordIndexToLinearIndex( NWorldCoarse - 1, iElementWorldCoarse) rCoarseSecond = np.array(rCoarseFirst) rCoarseSecond[elementCoarseIndex] *= 2 saveForNextTest = ec.computeCoarseErrorIndicatorFlux(rCoarseSecond) self.assertTrue(saveForNextTest >= 0.1) coefSecond = coef.CoefficientCoarseFactor(NWorldCoarse, NCoarseElement, aBase, rCoarseSecond) fineResult = ec.computeErrorIndicatorFine(coefSecond) self.assertTrue(fineResult >= 0.1) self.assertTrue( ec.computeErrorIndicatorFine(coefSecond) < ec.computeCoarseErrorIndicatorFlux(rCoarseSecond)) # A difference in the perifery should be smaller than in the center rCoarseSecond = np.array(rCoarseFirst) rCoarseSecond[0] *= 2 self.assertTrue( saveForNextTest > ec.computeCoarseErrorIndicatorFlux(rCoarseSecond) ) # Again, but closer rCoarseSecond = np.array(rCoarseFirst) rCoarseSecond[elementCoarseIndex - 1] *= 2 self.assertTrue( saveForNextTest > ec.computeCoarseErrorIndicatorFlux(rCoarseSecond) )
ii = 0 for p in pList: if p == 0.1: mean_time_true = 0. mean_time_perturbed = 0. mean_time_combined = 0. for N in range(NSamples): aPert = build_coefficient.build_randomcheckerboard( Nepsilon, NFine, alpha, beta, p) #true LOD middle = world.NWorldCoarse[1] // 2 * world.NWorldCoarse[ 0] + world.NWorldCoarse[0] // 2 patchRef = lod_periodic.PatchPeriodic(world, k, middle) IPatch = lambda: interp.L2ProjectionPatchMatrix(patchRef) computeKmsijT = lambda TInd: computeKmsij(TInd, aPert, IPatch) if p == 0.1: tic = time.perf_counter() patchT, correctorsTtrue, KmsijTtrue, _ = zip( *map(computeKmsijT, range(world.NtCoarse))) KFulltrue = lod_periodic.assembleMsStiffnessMatrix(world, patchT, KmsijTtrue, periodic=True) toc = time.perf_counter() mean_time_true += (toc - tic) correctorsTtrue = tuple(correctorsTtrue) modbasistrue = basis - lod_periodic.assembleBasisCorrectors( world, patchT, correctorsTtrue, periodic=True) else:
def test_1d(self): # Example from Peterseim, Variational Multiscale Stabilization and the Exponential Decay of correctors, p. 2 # Two modifications: A with minus and u(here) = 1/4*u(paper). NFine = np.array([3200]) NpFine = np.prod(NFine + 1) NList = [10, 20, 40, 80, 160] epsilon = 1024. / NFine epsilon = 1. / 320 k = 2 pi = np.pi xt = util.tCoordinates(NFine).flatten() xp = util.pCoordinates(NFine).flatten() #aFine = (2 + np.cos(2*pi*xt/epsilon))**(-1) aFine = (2 - np.cos(2 * pi * xt / epsilon))**(-1) uSol = 4 * (xp - xp**2) - 4 * epsilon * ( 1 / (4 * pi) * np.sin(2 * pi * xp / epsilon) - 1 / (2 * pi) * xp * np.sin(2 * pi * xp / epsilon) - epsilon / (4 * pi**2) * np.cos(2 * pi * xp / epsilon) + epsilon / (4 * pi**2)) uSol = uSol / 4 previousErrorCoarse = np.inf previousErrorFine = np.inf for N in NList: NWorldCoarse = np.array([N]) NCoarseElement = NFine / NWorldCoarse boundaryConditions = np.array([[0, 0]]) world = World(NWorldCoarse, NCoarseElement, boundaryConditions) xpCoarse = util.pCoordinates(NWorldCoarse).flatten() NpCoarse = np.prod(NWorldCoarse + 1) IPatchGenerator = lambda i, N: interp.L2ProjectionPatchMatrix( i, N, NWorldCoarse, NCoarseElement, boundaryConditions) aCoef = coef.coefficientFine(NWorldCoarse, NCoarseElement, aFine) pglod = pg.PetrovGalerkinLOD(world, k, IPatchGenerator, 0) pglod.updateCorrectors(aCoef, clearFineQuantities=False) KFull = pglod.assembleMsStiffnessMatrix() MFull = fem.assemblePatchMatrix(NWorldCoarse, world.MLocCoarse) free = util.interiorpIndexMap(NWorldCoarse) f = np.ones(NpCoarse) bFull = MFull * f KFree = KFull[free][:, free] bFree = bFull[free] xFree = sparse.linalg.spsolve(KFree, bFree) basis = fem.assembleProlongationMatrix(NWorldCoarse, NCoarseElement) basisCorrectors = pglod.assembleBasisCorrectors() modifiedBasis = basis - basisCorrectors xFull = np.zeros(NpCoarse) xFull[free] = xFree uLodCoarse = basis * xFull uLodFine = modifiedBasis * xFull AFine = fem.assemblePatchMatrix(NFine, world.ALocFine, aFine) MFine = fem.assemblePatchMatrix(NFine, world.MLocFine) newErrorCoarse = np.sqrt( np.dot(uSol - uLodCoarse, MFine * (uSol - uLodCoarse))) newErrorFine = np.sqrt( np.dot(uSol - uLodFine, AFine * (uSol - uLodFine))) self.assertTrue(newErrorCoarse < previousErrorCoarse) self.assertTrue(newErrorFine < previousErrorFine)
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()
def IPatchGenerator(i, N): return interp.L2ProjectionPatchMatrix(i, N, NWorldCoarse, NCoarseElement, boundaryConditions)
def test_3d(self): return NWorldFine = np.array([60, 220, 50]) NpFine = np.prod(NWorldFine + 1) NtFine = np.prod(NWorldFine) NWorldCoarse = np.array([6, 22, 5]) NCoarseElement = NWorldFine / NWorldCoarse NtCoarse = np.prod(NWorldCoarse) NpCoarse = np.prod(NWorldCoarse + 1) boundaryConditions = np.array([[1, 1], [0, 0], [1, 1]]) world = World(NWorldCoarse, NCoarseElement, boundaryConditions) aBase = np.loadtxt( os.path.dirname(os.path.realpath(__file__)) + '/data/upperness_x.txt') #aBase = aBase[::8] print 'a' coords = util.pCoordinates(NWorldFine) gFine = 1 - coords[:, 1] uFineFull, AFine, _ = femsolver.solveFine(world, aBase, None, -gFine, boundaryConditions) print 'b' rCoarse = np.ones(NtCoarse) self.assertTrue(np.size(aBase) == NtFine) if True: IPatchGenerator = lambda i, N: interp.L2ProjectionPatchMatrix( i, N, NWorldCoarse, NCoarseElement, boundaryConditions) aCoef = coef.coefficientCoarseFactor(NWorldCoarse, NCoarseElement, aBase, rCoarse) k = 2 printLevel = 1 pglod = pg.PetrovGalerkinLOD(world, k, IPatchGenerator, 1e-1, printLevel) pglod.updateCorrectors(aCoef, clearFineQuantities=True) KmsFull = pglod.assembleMsStiffnessMatrix() coords = util.pCoordinates(NWorldCoarse) g = 1 - coords[:, 1] 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) uCoarse = np.zeros(NpCoarse) uCoarse[free] = xFree uCoarse += g uCube = np.reshape(uCoarse, (NWorldCoarse + 1)[::-1]) uCube = np.ascontiguousarray(np.transpose(uCube, axes=[2, 1, 0])) imageToVTK("./image", pointData={"u": uCube}) if False: coord = util.pCoordinates(NWorldCoarse) uCoarse = coord[:, 1].flatten() uCube = np.reshape(uCoarse, (NWorldCoarse + 1)[::-1]) uCube = np.ascontiguousarray(np.transpose(uCube, axes=[2, 1, 0])) imageToVTK("./image", pointData={"u": uCube})
ShapeRestriction = True, ShapeWave = None, Original = True, NewShapeChange = True) # precompute NWorldFine = world.NWorldFine NWorldCoarse = world.NWorldCoarse NCoarseElement = world.NCoarseElement boundaryConditions = world.boundaryConditions NpFine = np.prod(NWorldFine+1) NpCoarse = np.prod(NWorldCoarse+1) #interpolant IPatchGenerator = lambda i, N: interp.L2ProjectionPatchMatrix(i, N, NWorldCoarse, NCoarseElement, boundaryConditions) #old Coefficient (need flatten form) Aold = coef.coefficientFine(NWorldCoarse, NCoarseElement, ABase) for k in range(1,6): print(('<<<<<<<<<<<<<<<< ' + str(k) + ' >>>>>>>>>>>>>>>>')) pglod = pg_pert.PerturbedPetrovGalerkinLOD(Aold, world, k, IPatchGenerator, 1) pglod.originCorrectors(clearFineQuantities=False) #new Coefficient ANew = R2.flatten() Anew = coef.coefficientFine(NWorldCoarse, NCoarseElement, ANew) # tolerance = 0 vis, eps = pglod.updateCorrectors(Anew, 0,clearFineQuantities=False, Computing = False)
def test_00coarseFlux(self): NWorldFine = np.array([20, 20]) NpFine = np.prod(NWorldFine + 1) NtFine = np.prod(NWorldFine) NWorldCoarse = np.array([4, 4]) 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 = 4 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 uFull = xFull + g lodFluxTF = pglod.computeFaceFluxTF(uFull) ## Compute fine solution coords = util.pCoordinates(NWorldFine) xF = coords[:, 0] gFine = 1 - xF uFineFull, AFine, _ = femsolver.solveFine(world, aBase, None, -gFine, boundaryConditions) uFineFull = uFineFull + gFine fineFluxTF = transport.computeHarmonicMeanFaceFlux( NWorldCoarse, NWorldCoarse, NCoarseElement, aBase, uFineFull) self.assertTrue(np.allclose(lodFluxTF, fineFluxTF, rtol=1e-7))