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()