Fpts = [i for i in range(0, n) if i not in Cpts] num_Fpts = len(Fpts) num_Cpts = len(Cpts) num_bad_guys = 1 cf_ratio = float(num_Cpts) / num_Fpts # Permutation matrix to sort rows permute = identity(n, format='csr') permute.indices = np.concatenate((Fpts, Cpts)) permute = permute.T # Smooth bad guys b = np.ones((A.shape[0], 1), dtype=A.dtype) smooth_fn = ('gauss_seidel', {'sweep': 'symmetric', 'iterations': 4}) B = relaxation_as_linear_operator((smooth_fn), A, b) * B # Compute eigenvalues / eigenvectors [eval_A, evec_A] = linalg.eigsh(A, k=n / 4, which='SM') [norm_A, dum] = linalg.eigsh(A, k=1, which='LM') norm_A = norm_A[0] del dum # B = np.array(evec_A[:,0:2]) # Let bad guy be smoothest eigenvector(s) # B = np.array(evec_A[:,0:1]) # Let bad guy be smoothest eigenvector(s) # B = np.array((evec_A[:,0],evec_A[:,0])).T # B = B / np.mean(B) # Form operators Afc = -A[Fpts, :][:, Cpts] Acf = Afc.transpose()
def smoothed_aggregation_helmholtz_solver(A, planewaves, use_constant=(True, {'last_level':0}), symmetry='symmetric', strength='symmetric', aggregate='standard', smooth=('energy', {'krylov': 'gmres'}), presmoother=('gauss_seidel_nr',{'sweep':'symmetric'}), postsmoother=('gauss_seidel_nr',{'sweep':'symmetric'}), improve_candidates='default', max_levels = 10, max_coarse = 100, **kwargs): """ Create a multilevel solver using Smoothed Aggregation (SA) for a 2D Helmholtz operator Parameters ---------- A : {csr_matrix, bsr_matrix} Sparse NxN matrix in CSR or BSR format planewaves : { list } [pw_0, pw_1, ..., pw_n], where the k-th tuple pw_k is of the form (fn, args). fn is a callable and args is a dictionary of arguments for fn. This k-th tuple is used to define any new planewaves (i.e., new coarse grid basis functions) to be appended to the existing B_k at that level. The function fn must return functions defined on the finest level, i.e., a collection of vector(s) of length A.shape[0]. These vectors are then restricted to the appropriate level, where they enrich the coarse space. Instead of a tuple, None can be used to stipulate no introduction of planewaves at that level. If len(planewaves) < max_levels, the last entry is used to define coarser level planewaves. use_constant : {tuple} Tuple of the form (bool, {'last_level':int}). The boolean denotes whether to introduce the constant in B at level 0. 'last_level' denotes the final level to use the constant in B. That is, if 'last_level' is 1, then the vector in B corresponding to the constant on level 0 is dropped from B at level 2. This is important, because using constant based interpolation beyond the Nyquist rate will result in poor solver performance. symmetry : {string} 'symmetric' refers to both real and complex symmetric 'hermitian' refers to both complex Hermitian and real Hermitian 'nonsymmetric' i.e. nonsymmetric in a hermitian sense Note that for the strictly real case, symmetric and hermitian are the same Note that this flag does not denote definiteness of the operator. strength : ['symmetric', 'classical', 'evolution', ('predefined', {'C' : csr_matrix}), None] Method used to determine the strength of connection between unknowns of the linear system. Method-specific parameters may be passed in using a tuple, e.g. strength=('symmetric',{'theta' : 0.25 }). If strength=None, all nonzero entries of the matrix are considered strong. See notes below for varying this parameter on a per level basis. Also, see notes below for using a predefined strength matrix on each level. aggregate : ['standard', 'lloyd', 'naive', ('predefined', {'AggOp' : csr_matrix})] Method used to aggregate nodes. See notes below for varying this parameter on a per level basis. Also, see notes below for using a predefined aggregation on each level. smooth : ['jacobi', 'richardson', 'energy', None] Method used to smooth the tentative prolongator. Method-specific parameters may be passed in using a tuple, e.g. smooth= ('jacobi',{'filter' : True }). See notes below for varying this parameter on a per level basis. presmoother : {tuple, string, list} : default ('block_gauss_seidel', {'sweep':'symmetric'}) Defines the presmoother for the multilevel cycling. The default block Gauss-Seidel option defaults to point-wise Gauss-Seidel, if the matrix is CSR or is a BSR matrix with blocksize of 1. See notes below for varying this parameter on a per level basis. postsmoother : {tuple, string, list} Same as presmoother, except defines the postsmoother. improve_candidates : {list} : default [('block_gauss_seidel', {'sweep':'symmetric'}), None] The ith entry defines the method used to improve the candidates B on level i. If the list is shorter than max_levels, then the last entry will define the method for all levels lower. The list elements are relaxation descriptors of the form used for presmoother and postsmoother. A value of None implies no action on B. max_levels : {integer} : default 10 Maximum number of levels to be used in the multilevel solver. max_coarse : {integer} : default 500 Maximum number of variables permitted on the coarse grid. Other Parameters ---------------- coarse_solver : ['splu','lu', ... ] Solver used at the coarsest level of the MG hierarchy Returns ------- ml : multilevel_solver Multigrid hierarchy of matrices and prolongation operators See Also -------- multilevel_solver, smoothed_aggregation_solver Notes ----- - The additional parameters are passed through as arguments to multilevel_solver. Refer to pyamg.multilevel_solver for additional documentation. - The parameters smooth, strength, aggregate, presmoother, postsmoother can be varied on a per level basis. For different methods on different levels, use a list as input so that the ith entry defines the method at the ith level. If there are more levels in the hierarchy than list entries, the last entry will define the method for all levels lower. Examples are: smooth=[('jacobi', {'omega':1.0}), None, 'jacobi'] presmoother=[('block_gauss_seidel', {'sweep':symmetric}), 'sor'] aggregate=['standard', 'naive'] strength=[('symmetric', {'theta':0.25}), ('symmetric',{'theta':0.08})] - Predefined strength of connection and aggregation schemes can be specified. These options are best used together, but aggregation can be predefined while strength of connection is not. For predefined strength of connection, use a list consisting of tuples of the form ('predefined', {'C' : C0}), where C0 is a csr_matrix and each degree-of-freedom in C0 represents a supernode. For instance to predefine a three-level hierarchy, use [('predefined', {'C' : C0}), ('predefined', {'C' : C1}) ]. Similarly for predefined aggregation, use a list of tuples. For instance to predefine a three-level hierarchy, use [('predefined', {'AggOp' : Agg0}), ('predefined', {'AggOp' : Agg1}) ], where the dimensions of A, Agg0 and Agg1 are compatible, i.e. Agg0.shape[1] == A.shape[0] and Agg1.shape[1] == Agg0.shape[0]. Each AggOp is a csr_matrix. Examples -------- >>> from pyamg import smoothed_aggregation_helmholtz_solver, poisson >>> from scipy.sparse.linalg import cg >>> from scipy import rand >>> A = poisson((100,100), format='csr') # matrix >>> b = rand(A.shape[0]) # random RHS >>> ml = smoothed_aggregation_solver(A) # AMG solver >>> M = ml.aspreconditioner(cycle='V') # preconditioner >>> x,info = cg(A, b, tol=1e-8, maxiter=30, M=M) # solve with CG References ---------- .. [1] L. N. Olson and J. B. Schroder. Smoothed Aggregation for Helmholtz Problems. Numerical Linear Algebra with Applications. pp. 361--386. 17 (2010). """ if not (isspmatrix_csr(A) or isspmatrix_bsr(A)): raise TypeError('argument A must have type csr_matrix or bsr_matrix') A = A.asfptype() if (symmetry != 'symmetric') and (symmetry != 'hermitian') and (symmetry != 'nonsymmetric'): raise ValueError('expected \'symmetric\', \'nonsymmetric\' or \'hermitian\' for the symmetry parameter ') A.symmetry = symmetry if A.shape[0] != A.shape[1]: raise ValueError('expected square matrix') ## # Preprocess and extend planewaves to length max_levels planewaves = preprocess_planewaves(planewaves, max_levels) # Check that the user has defined functions for B at each level use_const, args = unpack_arg(use_constant) first_planewave_level = -1 for pw in planewaves: first_planewave_level += 1 if pw is not None: break ## if (use_const == False) and (planewaves[0] == None): raise ValueError('No functions defined for B on the finest level, ' + \ 'either use_constant must be true, or planewaves must be defined for level 0') elif (use_const == True) and (args['last_level'] < first_planewave_level-1): raise ValueError('Some levels have no function(s) defined for B. ' + \ 'Change use_constant and/or planewave arguments.') ## # Levelize the user parameters, so that they become lists describing the # desired user option on each level. max_levels, max_coarse, strength =\ levelize_strength_or_aggregation(strength, max_levels, max_coarse) max_levels, max_coarse, aggregate =\ levelize_strength_or_aggregation(aggregate, max_levels, max_coarse) improve_candidates = levelize_smooth_or_improve_candidates(improve_candidates, max_levels) smooth = levelize_smooth_or_improve_candidates(smooth, max_levels) ## # Start first level levels = [] levels.append( multilevel_solver.level() ) levels[-1].A = A # matrix levels[-1].B = numpy.zeros((A.shape[0],0)) # place-holder for near-nullspace candidates zeros_0 = numpy.zeros((levels[0].A.shape[0],), dtype=A.dtype) while len(levels) < max_levels and levels[-1].A.shape[0] > max_coarse: A = levels[0].A A_l = levels[-1].A zeros_l = numpy.zeros((levels[-1].A.shape[0],), dtype=A.dtype) ## # Generate additions to n-th level candidates if planewaves[len(levels)-1] != None: fn, args = unpack_arg(planewaves[len(levels)-1]) Bcoarse2 = numpy.array(fn(**args)) ## # As in alpha-SA, relax the candidates before restriction if improve_candidates[0] is not None: Bcoarse2 = relaxation_as_linear_operator(improve_candidates[0], A, zeros_0)*Bcoarse2 ## # Restrict Bcoarse2 to current level for i in range(len(levels)-1): Bcoarse2 = levels[i].R*Bcoarse2 # relax after restriction if improve_candidates[len(levels)-1] is not None: Bcoarse2 =relaxation_as_linear_operator(improve_candidates[len(levels)-1],A_l,zeros_l)*Bcoarse2 else: Bcoarse2 = numpy.zeros((A_l.shape[0],0),dtype=A.dtype) ## # Deal with the use of constant in interpolation use_const, args = unpack_arg(use_constant) if use_const and len(levels) == 1: # If level 0, and the constant is to be used in interpolation levels[0].B = numpy.hstack( (numpy.ones((A.shape[0],1), dtype=A.dtype), Bcoarse2) ) elif use_const and args['last_level'] == len(levels)-2: # If the previous level was the last level to use the constant, then remove the # coarse grid function based on the constant from B levels[-1].B = numpy.hstack( (levels[-1].B[:,1:], Bcoarse2) ) else: levels[-1].B = numpy.hstack((levels[-1].B, Bcoarse2)) ## # Create and Append new level extend_hierarchy(levels, strength, aggregate, smooth, [None for i in range(max_levels)] ,keep=True) ml = multilevel_solver(levels, **kwargs) change_smoothers(ml, presmoother, postsmoother) return ml
def smoothed_aggregation_helmholtz_solver(A, planewaves, use_constant=(True, { 'last_level': 0 }), symmetry='symmetric', strength='symmetric', aggregate='standard', smooth=('energy', { 'krylov': 'gmres' }), presmoother=('gauss_seidel_nr', { 'sweep': 'symmetric' }), postsmoother=('gauss_seidel_nr', { 'sweep': 'symmetric' }), improve_candidates='default', max_levels=10, max_coarse=100, **kwargs): """ Create a multilevel solver using Smoothed Aggregation (SA) for a 2D Helmholtz operator Parameters ---------- A : {csr_matrix, bsr_matrix} Sparse NxN matrix in CSR or BSR format planewaves : { list } [pw_0, pw_1, ..., pw_n], where the k-th tuple pw_k is of the form (fn, args). fn is a callable and args is a dictionary of arguments for fn. This k-th tuple is used to define any new planewaves (i.e., new coarse grid basis functions) to be appended to the existing B_k at that level. The function fn must return functions defined on the finest level, i.e., a collection of vector(s) of length A.shape[0]. These vectors are then restricted to the appropriate level, where they enrich the coarse space. Instead of a tuple, None can be used to stipulate no introduction of planewaves at that level. If len(planewaves) < max_levels, the last entry is used to define coarser level planewaves. use_constant : {tuple} Tuple of the form (bool, {'last_level':int}). The boolean denotes whether to introduce the constant in B at level 0. 'last_level' denotes the final level to use the constant in B. That is, if 'last_level' is 1, then the vector in B corresponding to the constant on level 0 is dropped from B at level 2. This is important, because using constant based interpolation beyond the Nyquist rate will result in poor solver performance. symmetry : {string} 'symmetric' refers to both real and complex symmetric 'hermitian' refers to both complex Hermitian and real Hermitian 'nonsymmetric' i.e. nonsymmetric in a hermitian sense Note that for the strictly real case, symmetric and hermitian are the same Note that this flag does not denote definiteness of the operator. strength : ['symmetric', 'classical', 'evolution', ('predefined', {'C' : csr_matrix}), None] Method used to determine the strength of connection between unknowns of the linear system. Method-specific parameters may be passed in using a tuple, e.g. strength=('symmetric',{'theta' : 0.25 }). If strength=None, all nonzero entries of the matrix are considered strong. See notes below for varying this parameter on a per level basis. Also, see notes below for using a predefined strength matrix on each level. aggregate : ['standard', 'lloyd', 'naive', ('predefined', {'AggOp' : csr_matrix})] Method used to aggregate nodes. See notes below for varying this parameter on a per level basis. Also, see notes below for using a predefined aggregation on each level. smooth : ['jacobi', 'richardson', 'energy', None] Method used to smooth the tentative prolongator. Method-specific parameters may be passed in using a tuple, e.g. smooth= ('jacobi',{'filter' : True }). See notes below for varying this parameter on a per level basis. presmoother : {tuple, string, list} : default ('block_gauss_seidel', {'sweep':'symmetric'}) Defines the presmoother for the multilevel cycling. The default block Gauss-Seidel option defaults to point-wise Gauss-Seidel, if the matrix is CSR or is a BSR matrix with blocksize of 1. See notes below for varying this parameter on a per level basis. postsmoother : {tuple, string, list} Same as presmoother, except defines the postsmoother. improve_candidates : {list} : default [('block_gauss_seidel', {'sweep':'symmetric'}), None] The ith entry defines the method used to improve the candidates B on level i. If the list is shorter than max_levels, then the last entry will define the method for all levels lower. The list elements are relaxation descriptors of the form used for presmoother and postsmoother. A value of None implies no action on B. max_levels : {integer} : default 10 Maximum number of levels to be used in the multilevel solver. max_coarse : {integer} : default 500 Maximum number of variables permitted on the coarse grid. Other Parameters ---------------- coarse_solver : ['splu','lu', ... ] Solver used at the coarsest level of the MG hierarchy Returns ------- ml : multilevel_solver Multigrid hierarchy of matrices and prolongation operators See Also -------- multilevel_solver, smoothed_aggregation_solver Notes ----- - The additional parameters are passed through as arguments to multilevel_solver. Refer to pyamg.multilevel_solver for additional documentation. - The parameters smooth, strength, aggregate, presmoother, postsmoother can be varied on a per level basis. For different methods on different levels, use a list as input so that the ith entry defines the method at the ith level. If there are more levels in the hierarchy than list entries, the last entry will define the method for all levels lower. Examples are: smooth=[('jacobi', {'omega':1.0}), None, 'jacobi'] presmoother=[('block_gauss_seidel', {'sweep':symmetric}), 'sor'] aggregate=['standard', 'naive'] strength=[('symmetric', {'theta':0.25}), ('symmetric',{'theta':0.08})] - Predefined strength of connection and aggregation schemes can be specified. These options are best used together, but aggregation can be predefined while strength of connection is not. For predefined strength of connection, use a list consisting of tuples of the form ('predefined', {'C' : C0}), where C0 is a csr_matrix and each degree-of-freedom in C0 represents a supernode. For instance to predefine a three-level hierarchy, use [('predefined', {'C' : C0}), ('predefined', {'C' : C1}) ]. Similarly for predefined aggregation, use a list of tuples. For instance to predefine a three-level hierarchy, use [('predefined', {'AggOp' : Agg0}), ('predefined', {'AggOp' : Agg1}) ], where the dimensions of A, Agg0 and Agg1 are compatible, i.e. Agg0.shape[1] == A.shape[0] and Agg1.shape[1] == Agg0.shape[0]. Each AggOp is a csr_matrix. Examples -------- >>> from pyamg import smoothed_aggregation_helmholtz_solver, poisson >>> from scipy.sparse.linalg import cg >>> from scipy import rand >>> A = poisson((100,100), format='csr') # matrix >>> b = rand(A.shape[0]) # random RHS >>> ml = smoothed_aggregation_solver(A) # AMG solver >>> M = ml.aspreconditioner(cycle='V') # preconditioner >>> x,info = cg(A, b, tol=1e-8, maxiter=30, M=M) # solve with CG References ---------- .. [1] L. N. Olson and J. B. Schroder. Smoothed Aggregation for Helmholtz Problems. Numerical Linear Algebra with Applications. pp. 361--386. 17 (2010). """ if not (isspmatrix_csr(A) or isspmatrix_bsr(A)): raise TypeError('argument A must have type csr_matrix or bsr_matrix') A = A.asfptype() if (symmetry != 'symmetric') and (symmetry != 'hermitian') and ( symmetry != 'nonsymmetric'): raise ValueError( 'expected \'symmetric\', \'nonsymmetric\' or \'hermitian\' for the symmetry parameter ' ) A.symmetry = symmetry if A.shape[0] != A.shape[1]: raise ValueError('expected square matrix') ## # Preprocess and extend planewaves to length max_levels planewaves = preprocess_planewaves(planewaves, max_levels) # Check that the user has defined functions for B at each level use_const, args = unpack_arg(use_constant) first_planewave_level = -1 for pw in planewaves: first_planewave_level += 1 if pw is not None: break ## if (use_const == False) and (planewaves[0] == None): raise ValueError('No functions defined for B on the finest level, ' + \ 'either use_constant must be true, or planewaves must be defined for level 0') elif (use_const == True) and (args['last_level'] < first_planewave_level - 1): raise ValueError('Some levels have no function(s) defined for B. ' + \ 'Change use_constant and/or planewave arguments.') ## # Levelize the user parameters, so that they become lists describing the # desired user option on each level. max_levels, max_coarse, strength =\ levelize_strength_or_aggregation(strength, max_levels, max_coarse) max_levels, max_coarse, aggregate =\ levelize_strength_or_aggregation(aggregate, max_levels, max_coarse) improve_candidates = levelize_smooth_or_improve_candidates( improve_candidates, max_levels) smooth = levelize_smooth_or_improve_candidates(smooth, max_levels) ## # Start first level levels = [] levels.append(multilevel_solver.level()) levels[-1].A = A # matrix levels[-1].B = numpy.zeros( (A.shape[0], 0)) # place-holder for near-nullspace candidates zeros_0 = numpy.zeros((levels[0].A.shape[0], ), dtype=A.dtype) while len(levels) < max_levels and levels[-1].A.shape[0] > max_coarse: A = levels[0].A A_l = levels[-1].A zeros_l = numpy.zeros((levels[-1].A.shape[0], ), dtype=A.dtype) ## # Generate additions to n-th level candidates if planewaves[len(levels) - 1] != None: fn, args = unpack_arg(planewaves[len(levels) - 1]) Bcoarse2 = numpy.array(fn(**args)) ## # As in alpha-SA, relax the candidates before restriction if improve_candidates[0] is not None: Bcoarse2 = relaxation_as_linear_operator( improve_candidates[0], A, zeros_0) * Bcoarse2 ## # Restrict Bcoarse2 to current level for i in range(len(levels) - 1): Bcoarse2 = levels[i].R * Bcoarse2 # relax after restriction if improve_candidates[len(levels) - 1] is not None: Bcoarse2 = relaxation_as_linear_operator( improve_candidates[len(levels) - 1], A_l, zeros_l) * Bcoarse2 else: Bcoarse2 = numpy.zeros((A_l.shape[0], 0), dtype=A.dtype) ## # Deal with the use of constant in interpolation use_const, args = unpack_arg(use_constant) if use_const and len(levels) == 1: # If level 0, and the constant is to be used in interpolation levels[0].B = numpy.hstack((numpy.ones((A.shape[0], 1), dtype=A.dtype), Bcoarse2)) elif use_const and args['last_level'] == len(levels) - 2: # If the previous level was the last level to use the constant, then remove the # coarse grid function based on the constant from B levels[-1].B = numpy.hstack((levels[-1].B[:, 1:], Bcoarse2)) else: levels[-1].B = numpy.hstack((levels[-1].B, Bcoarse2)) ## # Create and Append new level extend_hierarchy(levels, strength, aggregate, smooth, [None for i in range(max_levels)], keep=True) ml = multilevel_solver(levels, **kwargs) change_smoothers(ml, presmoother, postsmoother) return ml
def extend_hierarchy(levels, strength, aggregate, smooth, improve_candidates, diagonal_dominance=False, keep=True): """Service routine to implement the strength of connection, aggregation, tentative prolongation construction, and prolongation smoothing. Called by smoothed_aggregation_solver. """ A = levels[-1].A B = levels[-1].B if A.symmetry == "nonsymmetric": AH = A.H.asformat(A.format) BH = levels[-1].BH # Compute the strength-of-connection matrix C, where larger # C[i,j] denote stronger couplings between i and j. fn, kwargs = unpack_arg(strength[len(levels)-1]) if fn == 'symmetric': C = symmetric_strength_of_connection(A, **kwargs) elif fn == 'classical': C = classical_strength_of_connection(A, **kwargs) elif fn == 'distance': C = distance_strength_of_connection(A, **kwargs) elif (fn == 'ode') or (fn == 'evolution'): if 'B' in kwargs: C = evolution_strength_of_connection(A, **kwargs) else: C = evolution_strength_of_connection(A, B, **kwargs) elif fn == 'energy_based': C = energy_based_strength_of_connection(A, **kwargs) elif fn == 'predefined': C = kwargs['C'].tocsr() elif fn == 'algebraic_distance': C = algebraic_distance(A, **kwargs) elif fn == 'affinity': C = affinity_distance(A, **kwargs) elif fn is None: C = A.tocsr() else: raise ValueError('unrecognized strength of connection method: %s' % str(fn)) levels[-1].complexity['strength'] = kwargs['cost'][0] # Avoid coarsening diagonally dominant rows flag, kwargs = unpack_arg(diagonal_dominance) if flag: C = eliminate_diag_dom_nodes(A, C, **kwargs) levels[-1].complexity['diag_dom'] = kwargs['cost'][0] # Compute the aggregation matrix AggOp (i.e., the nodal coarsening of A). # AggOp is a boolean matrix, where the sparsity pattern for the k-th column # denotes the fine-grid nodes agglomerated into k-th coarse-grid node. fn, kwargs = unpack_arg(aggregate[len(levels)-1]) if fn == 'standard': AggOp = standard_aggregation(C, **kwargs)[0] elif fn == 'naive': AggOp = naive_aggregation(C, **kwargs)[0] elif fn == 'lloyd': AggOp = lloyd_aggregation(C, **kwargs)[0] elif fn == 'predefined': AggOp = kwargs['AggOp'].tocsr() else: raise ValueError('unrecognized aggregation method %s' % str(fn)) levels[-1].complexity['aggregation'] = kwargs['cost'][0] * (float(C.nnz)/A.nnz) # Improve near nullspace candidates by relaxing on A B = 0 temp_cost = [0.0] fn, kwargs = unpack_arg(improve_candidates[len(levels)-1], cost=False) if fn is not None: b = np.zeros((A.shape[0], 1), dtype=A.dtype) B = relaxation_as_linear_operator((fn, kwargs), A, b, temp_cost) * B levels[-1].B = B if A.symmetry == "nonsymmetric": BH = relaxation_as_linear_operator((fn, kwargs), AH, b, temp_cost) * BH levels[-1].BH = BH levels[-1].complexity['candidates'] = temp_cost[0] * B.shape[1] # Compute the tentative prolongator, T, which is a tentative interpolation # matrix from the coarse-grid to the fine-grid. T exactly interpolates # B_fine = T B_coarse. Orthogonalization complexity ~ 2nk^2, k=B.shape[1]. temp_cost=[0.0] T, B = fit_candidates(AggOp, B, cost=temp_cost) if A.symmetry == "nonsymmetric": TH, BH = fit_candidates(AggOp, BH, cost=temp_cost) levels[-1].complexity['tentative'] = temp_cost[0]/A.nnz # Smooth the tentative prolongator, so that it's accuracy is greatly # improved for algebraically smooth error. fn, kwargs = unpack_arg(smooth[len(levels)-1]) if fn == 'jacobi': P = jacobi_prolongation_smoother(A, T, C, B, **kwargs) elif fn == 'richardson': P = richardson_prolongation_smoother(A, T, **kwargs) elif fn == 'energy': P = energy_prolongation_smoother(A, T, C, B, None, (False, {}), **kwargs) elif fn is None: P = T else: raise ValueError('unrecognized prolongation smoother method %s' % str(fn)) levels[-1].complexity['smooth_P'] = kwargs['cost'][0] # Compute the restriction matrix, R, which interpolates from the fine-grid # to the coarse-grid. If A is nonsymmetric, then R must be constructed # based on A.H. Otherwise R = P.H or P.T. symmetry = A.symmetry if symmetry == 'hermitian': R = P.H elif symmetry == 'symmetric': R = P.T elif symmetry == 'nonsymmetric': fn, kwargs = unpack_arg(smooth[len(levels)-1]) if fn == 'jacobi': R = jacobi_prolongation_smoother(AH, TH, C, BH, **kwargs).H elif fn == 'richardson': R = richardson_prolongation_smoother(AH, TH, **kwargs).H elif fn == 'energy': R = energy_prolongation_smoother(AH, TH, C, BH, None, (False, {}), **kwargs) R = R.H elif fn is None: R = T.H else: raise ValueError('unrecognized prolongation smoother method %s' % str(fn)) levels[-1].complexity['smooth_R'] = kwargs['cost'][0] if keep: levels[-1].C = C # strength of connection matrix levels[-1].AggOp = AggOp # aggregation operator levels[-1].T = T # tentative prolongator levels[-1].P = P # smoothed prolongator levels[-1].R = R # restriction operator # Form coarse grid operator, get complexity levels[-1].complexity['RAP'] = mat_mat_complexity(R,A) / float(A.nnz) RA = R * A levels[-1].complexity['RAP'] += mat_mat_complexity(RA,P) / float(A.nnz) A = RA * P # Galerkin operator, Ac = RAP A.symmetry = symmetry levels.append(multilevel_solver.level()) levels[-1].A = A levels[-1].B = B # right near nullspace candidates if A.symmetry == "nonsymmetric": levels[-1].BH = BH # left near nullspace candidates
def extend_hierarchy(levels, strength, aggregate, smooth, improve_candidates, diagonal_dominance=False, keep=True): """Service routine to implement the strength of connection, aggregation, tentative prolongation construction, and prolongation smoothing. Called by smoothed_aggregation_solver. """ def unpack_arg(v): if isinstance(v, tuple): return v[0], v[1] else: return v, {} A = levels[-1].A B = levels[-1].B if A.symmetry == "nonsymmetric": AH = A.H.asformat(A.format) BH = levels[-1].BH # Compute the strength-of-connection matrix C, where larger # C[i, j] denote stronger couplings between i and j. fn, kwargs = unpack_arg(strength[len(levels) - 1]) if fn == "symmetric": C = symmetric_strength_of_connection(A, **kwargs) elif fn == "classical": C = classical_strength_of_connection(A, **kwargs) elif fn == "distance": C = distance_strength_of_connection(A, **kwargs) elif (fn == "ode") or (fn == "evolution"): if "B" in kwargs: C = evolution_strength_of_connection(A, **kwargs) else: C = evolution_strength_of_connection(A, B, **kwargs) elif fn == "energy_based": C = energy_based_strength_of_connection(A, **kwargs) elif fn == "predefined": C = kwargs["C"].tocsr() elif fn == "algebraic_distance": C = algebraic_distance(A, **kwargs) elif fn is None: C = A.tocsr() else: raise ValueError("unrecognized strength of connection method: %s" % str(fn)) # Avoid coarsening diagonally dominant rows flag, kwargs = unpack_arg(diagonal_dominance) if flag: C = eliminate_diag_dom_nodes(A, C, **kwargs) # Compute the aggregation matrix AggOp (i.e., the nodal coarsening of A). # AggOp is a boolean matrix, where the sparsity pattern for the k-th column # denotes the fine-grid nodes agglomerated into k-th coarse-grid node. fn, kwargs = unpack_arg(aggregate[len(levels) - 1]) if fn == "standard": AggOp, Cnodes = standard_aggregation(C, **kwargs) elif fn == "naive": AggOp, Cnodes = naive_aggregation(C, **kwargs) elif fn == "lloyd": AggOp, Cnodes = lloyd_aggregation(C, **kwargs) elif fn == "predefined": AggOp = kwargs["AggOp"].tocsr() Cnodes = kwargs["Cnodes"] else: raise ValueError("unrecognized aggregation method %s" % str(fn)) # Improve near nullspace candidates by relaxing on A B = 0 fn, kwargs = unpack_arg(improve_candidates[len(levels) - 1]) if fn is not None: b = np.zeros((A.shape[0], 1), dtype=A.dtype) B = relaxation_as_linear_operator((fn, kwargs), A, b) * B levels[-1].B = B if A.symmetry == "nonsymmetric": BH = relaxation_as_linear_operator((fn, kwargs), AH, b) * BH levels[-1].BH = BH # Compute the tentative prolongator, T, which is a tentative interpolation # matrix from the coarse-grid to the fine-grid. T exactly interpolates # B_fine[:, 0:blocksize(A)] = T B_coarse[:, 0:blocksize(A)]. T, dummy = fit_candidates(AggOp, B[:, 0 : blocksize(A)]) del dummy if A.symmetry == "nonsymmetric": TH, dummyH = fit_candidates(AggOp, BH[:, 0 : blocksize(A)]) del dummyH # Create necessary root node matrices Cpt_params = (True, get_Cpt_params(A, Cnodes, AggOp, T)) T = scale_T(T, Cpt_params[1]["P_I"], Cpt_params[1]["I_F"]) if A.symmetry == "nonsymmetric": TH = scale_T(TH, Cpt_params[1]["P_I"], Cpt_params[1]["I_F"]) # Set coarse grid near nullspace modes as injected fine grid near # null-space modes B = Cpt_params[1]["P_I"].T * levels[-1].B if A.symmetry == "nonsymmetric": BH = Cpt_params[1]["P_I"].T * levels[-1].BH # Smooth the tentative prolongator, so that it's accuracy is greatly # improved for algebraically smooth error. fn, kwargs = unpack_arg(smooth[len(levels) - 1]) if fn == "energy": P = energy_prolongation_smoother(A, T, C, B, levels[-1].B, Cpt_params=Cpt_params, **kwargs) elif fn is None: P = T else: raise ValueError( "unrecognized prolongation smoother \ method %s" % str(fn) ) # Compute the restriction matrix R, which interpolates from the fine-grid # to the coarse-grid. If A is nonsymmetric, then R must be constructed # based on A.H. Otherwise R = P.H or P.T. symmetry = A.symmetry if symmetry == "hermitian": R = P.H elif symmetry == "symmetric": R = P.T elif symmetry == "nonsymmetric": fn, kwargs = unpack_arg(smooth[len(levels) - 1]) if fn == "energy": R = energy_prolongation_smoother(AH, TH, C, BH, levels[-1].BH, Cpt_params=Cpt_params, **kwargs) R = R.H elif fn is None: R = T.H else: raise ValueError( "unrecognized prolongation smoother \ method %s" % str(fn) ) if keep: levels[-1].C = C # strength of connection matrix levels[-1].AggOp = AggOp # aggregation operator levels[-1].T = T # tentative prolongator levels[-1].Fpts = Cpt_params[1]["Fpts"] # Fpts levels[-1].P_I = Cpt_params[1]["P_I"] # Injection operator levels[-1].I_F = Cpt_params[1]["I_F"] # Identity on F-pts levels[-1].I_C = Cpt_params[1]["I_C"] # Identity on C-pts levels[-1].P = P # smoothed prolongator levels[-1].R = R # restriction operator levels[-1].Cpts = Cpt_params[1]["Cpts"] # Cpts (i.e., rootnodes) levels.append(multilevel_solver.level()) A = R * A * P # Galerkin operator A.symmetry = symmetry levels[-1].A = A levels[-1].B = B # right near nullspace candidates if A.symmetry == "nonsymmetric": levels[-1].BH = BH # left near nullspace candidates
def extend_hierarchy(levels, strength, aggregate, smooth, improve_candidates, diagonal_dominance=False, keep=True): """Service routine to implement the strength of connection, aggregation, tentative prolongation construction, and prolongation smoothing. Called by smoothed_aggregation_solver. """ A = levels[-1].A B = levels[-1].B if A.symmetry == "nonsymmetric": AH = A.H.asformat(A.format) BH = levels[-1].BH # Compute the strength-of-connection matrix C, where larger # C[i, j] denote stronger couplings between i and j. fn, kwargs = unpack_arg(strength[len(levels)-1]) if fn == 'symmetric': C = symmetric_strength_of_connection(A, **kwargs) elif fn == 'classical': C = classical_strength_of_connection(A, **kwargs) elif fn == 'distance': C = distance_strength_of_connection(A, **kwargs) elif (fn == 'ode') or (fn == 'evolution'): if 'B' in kwargs: C = evolution_strength_of_connection(A, **kwargs) else: C = evolution_strength_of_connection(A, B, **kwargs) elif fn == 'energy_based': C = energy_based_strength_of_connection(A, **kwargs) elif fn == 'predefined': C = kwargs['C'].tocsr() elif fn == 'algebraic_distance': C = algebraic_distance(A, **kwargs) elif fn == 'affinity': C = affinity_distance(A, **kwargs) elif fn is None: C = A.tocsr() else: raise ValueError('unrecognized strength of connection method: %s' % str(fn)) levels[-1].complexity['strength'] = kwargs['cost'][0] # Avoid coarsening diagonally dominant rows flag, kwargs = unpack_arg(diagonal_dominance) if flag: C = eliminate_diag_dom_nodes(A, C, **kwargs) levels[-1].complexity['diag_dom'] = kwargs['cost'][0] # Compute the aggregation matrix AggOp (i.e., the nodal coarsening of A). # AggOp is a boolean matrix, where the sparsity pattern for the k-th column # denotes the fine-grid nodes agglomerated into k-th coarse-grid node. fn, kwargs = unpack_arg(aggregate[len(levels)-1]) if fn == 'standard': AggOp, Cnodes = standard_aggregation(C, **kwargs) elif fn == 'naive': AggOp, Cnodes = naive_aggregation(C, **kwargs) elif fn == 'lloyd': AggOp, Cnodes = lloyd_aggregation(C, **kwargs) elif fn == 'predefined': AggOp = kwargs['AggOp'].tocsr() Cnodes = kwargs['Cnodes'] else: raise ValueError('unrecognized aggregation method %s' % str(fn)) levels[-1].complexity['aggregation'] = kwargs['cost'][0] * (float(C.nnz)/A.nnz) # Improve near nullspace candidates by relaxing on A B = 0 temp_cost = [0.0] fn, kwargs = unpack_arg(improve_candidates[len(levels)-1],cost=False) if fn is not None: b = np.zeros((A.shape[0], 1), dtype=A.dtype) B = relaxation_as_linear_operator((fn, kwargs), A, b, temp_cost) * B levels[-1].B = B if A.symmetry == "nonsymmetric": BH = relaxation_as_linear_operator((fn, kwargs), AH, b, temp_cost) * BH levels[-1].BH = BH levels[-1].complexity['candidates'] = temp_cost[0] * B.shape[1] # Compute the tentative prolongator, T, which is a tentative interpolation # matrix from the coarse-grid to the fine-grid. T exactly interpolates # B_fine[:, 0:blocksize(A)] = T B_coarse[:, 0:blocksize(A)]. # Orthogonalization complexity ~ 2nk^2, k = blocksize(A). temp_cost=[0.0] T, dummy = fit_candidates(AggOp, B[:, 0:blocksize(A)], cost=temp_cost) del dummy if A.symmetry == "nonsymmetric": TH, dummyH = fit_candidates(AggOp, BH[:, 0:blocksize(A)], cost=temp_cost) del dummyH levels[-1].complexity['tentative'] = temp_cost[0]/A.nnz # Create necessary root node matrices Cpt_params = (True, get_Cpt_params(A, Cnodes, AggOp, T)) T = scale_T(T, Cpt_params[1]['P_I'], Cpt_params[1]['I_F']) levels[-1].complexity['tentative'] += T.nnz / float(A.nnz) if A.symmetry == "nonsymmetric": TH = scale_T(TH, Cpt_params[1]['P_I'], Cpt_params[1]['I_F']) levels[-1].complexity['tentative'] += TH.nnz / float(A.nnz) # Set coarse grid near nullspace modes as injected fine grid near # null-space modes B = Cpt_params[1]['P_I'].T*levels[-1].B if A.symmetry == "nonsymmetric": BH = Cpt_params[1]['P_I'].T*levels[-1].BH # Smooth the tentative prolongator, so that it's accuracy is greatly # improved for algebraically smooth error. fn, kwargs = unpack_arg(smooth[len(levels)-1]) if fn == 'energy': P = energy_prolongation_smoother(A, T, C, B, levels[-1].B, Cpt_params=Cpt_params, **kwargs) elif fn is None: P = T else: raise ValueError('unrecognized prolongation smoother \ method %s' % str(fn)) levels[-1].complexity['smooth_P'] = kwargs['cost'][0] # Compute the restriction matrix R, which interpolates from the fine-grid # to the coarse-grid. If A is nonsymmetric, then R must be constructed # based on A.H. Otherwise R = P.H or P.T. symmetry = A.symmetry if symmetry == 'hermitian': R = P.H elif symmetry == 'symmetric': R = P.T elif symmetry == 'nonsymmetric': fn, kwargs = unpack_arg(smooth[len(levels)-1]) if fn == 'energy': R = energy_prolongation_smoother(AH, TH, C, BH, levels[-1].BH, Cpt_params=Cpt_params, **kwargs) R = R.H levels[-1].complexity['smooth_R'] = kwargs['cost'][0] elif fn is None: R = T.H else: raise ValueError('unrecognized prolongation smoother \ method %s' % str(fn)) if keep: levels[-1].C = C # strength of connection matrix levels[-1].AggOp = AggOp # aggregation operator levels[-1].T = T # tentative prolongator levels[-1].Fpts = Cpt_params[1]['Fpts'] # Fpts levels[-1].P_I = Cpt_params[1]['P_I'] # Injection operator levels[-1].I_F = Cpt_params[1]['I_F'] # Identity on F-pts levels[-1].I_C = Cpt_params[1]['I_C'] # Identity on C-pts levels[-1].P = P # smoothed prolongator levels[-1].R = R # restriction operator levels[-1].Cpts = Cpt_params[1]['Cpts'] # Cpts (i.e., rootnodes) # Form coarse grid operator, get complexity levels[-1].complexity['RAP'] = mat_mat_complexity(R,A) / float(A.nnz) RA = R * A levels[-1].complexity['RAP'] += mat_mat_complexity(RA,P) / float(A.nnz) A = RA * P # Galerkin operator, Ac = RAP A.symmetry = symmetry levels.append(multilevel_solver.level()) levels[-1].A = A levels[-1].B = B # right near nullspace candidates if A.symmetry == "nonsymmetric": levels[-1].BH = BH # left near nullspace candidates
def adaptive_pairwise_solver(A, initial_targets=None, symmetry='hermitian', desired_convergence=0.5, test_iterations=10, test_cycle='V', test_accel=None, strength=None, smooth=None, aggregate=('drake', { 'levels': 2 }), presmoother=('block_gauss_seidel', { 'sweep': 'symmetric' }), postsmoother=('block_gauss_seidel', { 'sweep': 'symmetric' }), max_levels=30, max_coarse=100, diagonal_dominance=False, coarse_solver='pinv', keep=False, additive=False, reconstruct=False, max_hierarchies=10, use_ritz=False, improve_candidates=[('block_gauss_seidel', { 'sweep': 'symmetric', 'iterations': 4 })], **kwargs): def unpack_arg(v): if isinstance(v, tuple): return v[0], v[1] elif v is None: return None else: return v, {} if isspmatrix_bsr(A): warn("Only currently implemented for CSR matrices.") if not (isspmatrix_csr(A) or isspmatrix_bsr(A)): try: A = csr_matrix(A) warn("Implicit conversion of A to CSR", SparseEfficiencyWarning) except: raise TypeError('Argument A must have type csr_matrix or\ bsr_matrix, or be convertible to csr_matrix') if (symmetry != 'symmetric') and (symmetry != 'hermitian') and\ (symmetry != 'nonsymmetric'): raise ValueError('expected \'symmetric\', \'nonsymmetric\' or\ \'hermitian\' for the symmetry parameter ') if A.shape[0] != A.shape[1]: raise ValueError('expected square matrix') A = A.asfptype() A.symmetry = symmetry n = A.shape[0] test_rhs = np.zeros((n, 1)) # SHOULD I START WITH CONSTANT VECTOR OR SMOOTHED RANDOM VECTOR? # Right near nullspace candidates if initial_targets is None: initial_targets = np.kron( np.ones((A.shape[0] / blocksize(A), 1), dtype=A.dtype), np.eye(blocksize(A))) else: initial_targets = np.asarray(initial_targets, dtype=A.dtype) if len(initial_targets.shape) == 1: initial_targets = initial_targets.reshape(-1, 1) if initial_targets.shape[0] != A.shape[0]: raise ValueError( 'The near null-space modes initial_targets have incorrect \ dimensions for matrix A') if initial_targets.shape[1] < blocksize(A): raise ValueError( 'initial_targets.shape[1] must be >= the blocksize of A') # Improve near nullspace candidates by relaxing on A B = 0 if improve_candidates is not None: fn, temp_args = unpack_arg(improve_candidates[0]) else: fn = None if fn is not None: b = np.zeros((A.shape[0], 1), dtype=A.dtype) initial_targets = relaxation_as_linear_operator( (fn, temp_args), A, b) * initial_targets if A.symmetry == "nonsymmetric": AH = A.H.asformat(A.format) BH = relaxation_as_linear_operator((fn, temp_args), AH, b) * BH # Empty set of solver hierarchies solvers = multilevel_solver_set() target = initial_targets B = initial_targets cf = 1.0 # Aggregation process on the finest level is the same each iteration. # To prevent repeating processes, we compute it here and provide it to the # sovler construction. AggOp = get_aggregate(A, strength=strength, aggregate=aggregate, diagonal_dominance=diagonal_dominance, B=initial_targets) if isinstance(aggregate, tuple): aggregate = [('predefined', {'AggOp': AggOp}), aggregate] elif isinstance(aggregate, list): aggregate.insert(0, ('predefined', {'AggOp': AggOp})) else: raise TypeError("Aggregate variable must be list or tuple.") # Continue adding hierarchies until desired convergence factor achieved, # or maximum number of hierarchies constructed it = 0 while (cf > desired_convergence) and (it < max_hierarchies): # pdb.set_trace() # Make target vector orthogonal and energy orthonormal and reconstruct hierarchy if use_ritz and it > 0: B = global_ritz_process(A, B, weak_tol=100) reconstruct_hierarchy(solver_set=solvers, A=A, new_B=B, symmetry=symmetry, aggregate=aggregate, presmoother=presmoother, postsmoother=postsmoother, smooth=smooth, strength=strength, max_levels=max_levels, max_coarse=max_coarse, coarse_solver=coarse_solver, diagonal_dominance=diagonal_dominance, keep=keep, **kwargs) print "Hierarchy reconstructed." # Otherwise just add new hierarchy to solver set. else: solvers.add_hierarchy( smoothed_aggregation_solver( A, B=B[:, -1], symmetry=symmetry, aggregate=aggregate, presmoother=presmoother, postsmoother=postsmoother, smooth=smooth, strength=strength, max_levels=max_levels, max_coarse=max_coarse, diagonal_dominance=diagonal_dominance, coarse_solver=coarse_solver, improve_candidates=improve_candidates, keep=keep, **kwargs)) # Test for convergence factor using new hierarchy. x0 = np.random.rand(n, 1) residuals = [] target = solvers.solve(test_rhs, x0=x0, tol=1e-12, maxiter=test_iterations, cycle=test_cycle, accel=test_accel, residuals=residuals, additive=additive) cf = residuals[-1] / residuals[-2] B = np.hstack((B, target)) it += 1 print "Added new hierarchy, convergence factor = ", cf B = B[:, :-1] # B2 = global_ritz_process(A, B, weak_tol=1.0) angles = test_targets(A, B) # angles = test_targets(A, B2) # -------------------------------------------------------------------------------------- # # -------------------------------------------------------------------------------------- # # -------------------------------------------------------------------------------------- # # b = np.zeros((n,1)) # asa_residuals = [] # sol = solvers.solve(b, x0, tol=1e-8, residuals=asa_residuals, accel=None) # asa_conv_factors = np.zeros((len(asa_residuals)-1,1)) # for i in range(0,len(asa_residuals)-1): # asa_conv_factors[i] = asa_residuals[i]/asa_residuals[i-1] # print "Original adaptive SA/AMG - ", np.mean(asa_conv_factors[1:]) # if reconstruct: # reconstruct_hierarchy(solver_set=solvers, A=A, new_B=B2, symmetry=symmetry, # aggregate=aggregate, presmoother=presmoother, # postsmoother=postsmoother, smooth=smooth, # strength=strength, max_levels=max_levels, # max_coarse=max_coarse, coarse_solver=coarse_solver, # diagonal_dominance=diagonal_dominance, # keep=keep, **kwargs) # print "Hierarchy reconstructed." # asa_residuals2 = [] # sol = solvers.solve(b, x0, tol=1e-8, residuals=asa_residuals2, accel=None) # asa_conv_factors2 = np.zeros((len(asa_residuals2)-1,1)) # for i in range(0,len(asa_residuals2)-1): # asa_conv_factors2[i] = asa_residuals2[i]/asa_residuals2[i-1] # print "Ritz adaptive SA/AMG - ", np.mean(asa_conv_factors2[1:]) # if reconstruct: # reconstruct_hierarchy(solver_set=solvers, A=A, new_B=B[:,:-1], symmetry=symmetry, # aggregate=aggregate, presmoother=presmoother, # postsmoother=postsmoother, smooth=smooth, # strength=strength, max_levels=max_levels, # max_coarse=max_coarse, coarse_solver=coarse_solver, # diagonal_dominance=diagonal_dominance, # keep=keep, **kwargs) # print "Hierarchy reconstructed." # asa_residuals2 = [] # sol = solvers.solve(b, x0, tol=1e-8, residuals=asa_residuals2, accel=None) # asa_conv_factors2 = np.zeros((len(asa_residuals2)-1,1)) # for i in range(0,len(asa_residuals2)-1): # asa_conv_factors2[i] = asa_residuals2[i]/asa_residuals2[i-1] # print "Original(-1) SA/AMG - ", np.mean(asa_conv_factors2[1:]) # if reconstruct: # reconstruct_hierarchy(solver_set=solvers, A=A, new_B=B2[:,:-1], symmetry=symmetry, # aggregate=aggregate, presmoother=presmoother, # postsmoother=postsmoother, smooth=smooth, # strength=strength, max_levels=max_levels, # max_coarse=max_coarse, coarse_solver=coarse_solver, # diagonal_dominance=diagonal_dominance, # keep=keep, **kwargs) # print "Hierarchy reconstructed." # asa_residuals2 = [] # sol = solvers.solve(b, x0, tol=1e-8, residuals=asa_residuals2, accel=None) # asa_conv_factors2 = np.zeros((len(asa_residuals2)-1,1)) # for i in range(0,len(asa_residuals2)-1): # asa_conv_factors2[i] = asa_residuals2[i]/asa_residuals2[i-1] # print "Ritz(-1) SA/AMG - ", np.mean(asa_conv_factors2[1:]) # pdb.set_trace() # -------------------------------------------------------------------------------------- # # -------------------------------------------------------------------------------------- # # -------------------------------------------------------------------------------------- # return solvers
def pairwise_aggregation(A, B, Bh=None, symmetry='hermitian', algorithm='drake_C', matchings=1, weights=None, improve_candidates=None, strength=None, **kwargs): """ Pairwise aggregation of nodes. Parameters ---------- A : csr_matrix or bsr_matrix matrix for linear system. B : array_like Right near-nullspace candidates stored in the columns of an NxK array. BH : array_like : default None Left near-nullspace candidates stored in the columns of an NxK array. BH is only used if symmetry is 'nonsymmetric'. The default value B=None is equivalent to BH=B.copy() algorithm : string : default 'drake' Algorithm to perform pairwise matching. Current options are 'drake', 'preis', 'notay', referring to the Drake (2003), Preis (1999), and Notay (2010), respectively. matchings : int : default 1 Number of pairwise matchings to do. k matchings will lead to a coarsening factor of under 2^k. weights : function handle : default None Optional function handle to compute weights used in the matching, e.g. a strength of connection routine. Additional arguments for this routine should be provided in **kwargs. improve_candidates : {tuple, string, list} : default None The list elements are relaxation descriptors of the form used for presmoother and postsmoother. A value of None implies no action on B. strength : {(int, string) or None} : default None If a strength of connection matrix should be returned along with aggregation. If None, no SOC matrix returned. To return a SOC matrix, pass in a tuple (a,b), for int a > matchings telling how many matchings to use to construct a SOC matrix, and string b with data type. E.g. strength = (4,'bool'). THINGS TO NOTE -------------- - Not implemented for non-symmetric and/or block systems + Need to set up pairwise aggregation to be applicable for nonsymmetric matrices (think it actually is...) + Need to define how a matching is done nodally. + Also must consider what targets are used to form coarse grid' in nodal approach... - Once we are done w/ Python implementations of matching, we can remove the deepcopy of A to W --> Don't need it, waste of time/memory. """ def unpack_arg(v): if isinstance(v, tuple): return v[0], v[1] else: return v, {} if not isinstance(matchings, int): raise TypeError("Number of matchings must be an integer.") if matchings < 1: raise ValueError("Number of matchings must be > 0.") if (algorithm is not 'drake') and (algorithm is not 'preis') and \ (algorithm is not 'notay') and (algorithm is not 'drake_C'): raise ValueError("Only drake, notay and preis algorithms implemeted.") if (symmetry != 'symmetric') and (symmetry != 'hermitian') and \ (symmetry != 'nonsymmetric'): raise ValueError('expected \'symmetric\', \'nonsymmetric\' or\ \'hermitian\' for the symmetry parameter ') if strength is not None: if strength[0] < matchings: warn("Expect number of matchings for SOC >= matchings for aggregation.") diff = 0 else: diff = strength[0] - matchings # How many more matchings to do for SOC else: diff = 0 # Compute weights if function provided, otherwise let W = A if weights is not None: W = weights(A, **kwargs) else: W = deepcopy(A) if not isspmatrix_csr(W): warn("Requires CSR matrix - trying to convert.", SparseEfficiencyWarning) try: W = W.tocsr() except: raise TypeError("Could not convert to csr matrix.") n = A.shape[0] if (symmetry == 'nonsymmetric') and (Bh == None): print "Warning - no left near null-space vector provided for nonsymmetric matrix.\n\ Copying right near null-space vector." Bh = deepcopy(B[0:n,0:1]) # Dictionary of function names for matching algorithms get_matching = { 'drake': drake_matching, 'preis': preis_matching_1999, 'notay': notay_matching_2010 } # Get initial matching [M,S] = get_matching[algorithm](W, order='backward', **kwargs) num_pairs = M.shape[0] num_sing = S.shape[0] Nc = num_pairs+num_sing # Pick C-points and save in list Cpts = np.zeros((Nc,),dtype=int) Cpts[0:num_pairs] = M[:,0] Cpts[num_pairs:Nc] = S # Form sparse P from pairwise aggregation row_inds = np.empty(n) row_inds[0:(2*num_pairs)] = M.flatten() row_inds[(2*num_pairs):n] = S col_inds = np.empty(n) col_inds[0:(2*num_pairs)] = ( np.array( ( np.arange(0,num_pairs),np.arange(0,num_pairs) ) ).T).flatten() col_inds[(2*num_pairs):n] = np.arange(num_pairs,Nc) AggOp = csr_matrix( (np.ones((n,), dtype=bool), (row_inds,col_inds)), shape=(n,Nc) ) # Predefine SOC matrix is only one pairwise pass is done for aggregation if (matchings == 1) and (diff > 0): AggOp2 = csr_matrix(AggOp, dtype=strength[1]) # If performing multiple pairwise matchings, form coarse grid operator # and repeat process if (matchings+diff) > 1: P = csr_matrix( (B[0:n,0], (row_inds,col_inds)), shape=(n,Nc) ) Bc = np.ones((Nc,1)) if symmetry == 'hermitian': R = P.H Ac = R*A*P elif symmetry == 'symmetric': R = P.T Ac = R*A*P elif symmetry == 'nonsymmetric': R = csr_matrix( (Bh[0:n,0], (col_inds,row_inds)), shape=(Nc,n) ) Ac = R*A*P AcH = Ac.H.asformat(Ac.format) Bhc = np.ones((Nc,1)) # Loop over the number of pairwise matchings to be done for i in range(1,(matchings+diff)): if weights is not None: W = weights(Ac, **kwargs) else: W = Ac # Get matching [M,S] = get_matching[algorithm](W, order='forward', **kwargs) n = Ac.shape[0] num_pairs = M.shape[0] num_sing = S.shape[0] Nc = num_pairs+num_sing # Pick C-points and save in list temp = np.zeros((Nc,),dtype=int) temp[0:num_pairs] = M[:,0] temp[num_pairs:Nc] = S Cpts = Cpts[temp] # Improve near nullspace candidates by relaxing on A B = 0 fn, kwargs = unpack_arg(improve_candidates) if fn is not None: b = np.zeros((n, 1), dtype=Ac.dtype) Bc = relaxation_as_linear_operator((fn, kwargs), Ac, b) * Bc if symmetry == "nonsymmetric": Bhc = relaxation_as_linear_operator((fn, kwargs), AcH, b) * Bhc # Form sparse P from pairwise aggregation row_inds = np.empty(n) row_inds[0:(2*num_pairs)] = M.flatten() row_inds[(2*num_pairs):n] = S col_inds = np.empty(n) col_inds[0:(2*num_pairs)] = ( np.array( ( np.arange(0,num_pairs),np.arange(0,num_pairs) ) ).T).flatten() col_inds[(2*num_pairs):n] = np.arange(num_pairs,Nc) # Form coarse grid operator and update aggregation matrix if i < (matchings-1): P = csr_matrix( (Bc[0:n,0], (row_inds,col_inds)), shape=(n,Nc) ) if symmetry == 'hermitian': R = P.H Ac = R*Ac*P elif symmetry == 'symmetric': R = P.T Ac = R*Ac*P elif symmetry == 'nonsymmetric': R = csr_matrix( (Bhc[0:n,0], (col_inds,row_inds)), shape=(Nc,n) ) Ac = R*Ac*P AcH = Ac.H.asformat(Ac.format) Bhc = np.ones((Nc,1)) AggOp = csr_matrix(AggOp * P, dtype=bool) Bc = np.ones((Nc,1)) # Construct final aggregation matrix elif i == (matchings-1): P = csr_matrix( (np.ones((n,), dtype=bool), (row_inds,col_inds)), shape=(n,Nc) ) AggOp = csr_matrix(AggOp * P, dtype=bool) # Construct coarse grids and additional aggregation matrix # only if doing more matchings for SOC. if diff > 0: if symmetry == 'hermitian': R = P.H Ac = R*Ac*P elif symmetry == 'symmetric': R = P.T Ac = R*Ac*P elif symmetry == 'nonsymmetric': R = csr_matrix( (Bhc[0:n,0], (col_inds,row_inds)), shape=(Nc,n) ) Ac = R*Ac*P AcH = Ac.H.asformat(Ac.format) Bhc = np.ones((Nc,1)) Bc = np.ones((Nc,1)) AggOp2 = csr_matrix(AggOp, dtype=strength[1]) # Pairwise iterations for SOC elif i < (matchings+diff-1): P = csr_matrix( (Bc[0:n,0], (row_inds,col_inds)), shape=(n,Nc) ) if symmetry == 'hermitian': R = P.H Ac = R*Ac*P elif symmetry == 'symmetric': R = P.T Ac = R*Ac*P elif symmetry == 'nonsymmetric': R = csr_matrix( (Bhc[0:n,0], (col_inds,row_inds)), shape=(Nc,n) ) Ac = R*Ac*P AcH = Ac.H.asformat(Ac.format) Bhc = np.ones((Nc,1)) AggOp2 = csr_matrix(AggOp2 * P, dtype=bool) Bc = np.ones((Nc,1)) # Final matching for SOC matrix. Construct SOC as AggOp*AggOp^T. elif i == (matchings+diff-1): P = csr_matrix( (np.ones((n,), dtype=bool), (row_inds,col_inds)), shape=(n,Nc) ) AggOp2 = csr_matrix(AggOp2 * P, dtype=bool) AggOp2 = csr_matrix(AggOp2*AggOp2.T, dtype=strength[1]) if strength is None: return AggOp, Cpts else: return AggOp, Cpts, AggOp2
def extend_hierarchy(levels, strength, aggregate, smooth, improve_candidates, diagonal_dominance=False, keep=True, test_ind=0): """Service routine to implement the strength of connection, aggregation, tentative prolongation construction, and prolongation smoothing. Called by smoothed_aggregation_solver. """ def unpack_arg(v): if isinstance(v, tuple): return v[0], v[1] else: return v, {} A = levels[-1].A B = levels[-1].B if A.symmetry == "nonsymmetric": AH = A.H.asformat(A.format) BH = levels[-1].BH # Improve near nullspace candidates by relaxing on A B = 0 fn, kwargs = unpack_arg(improve_candidates[len(levels)-1]) if fn is not None: b = np.zeros((A.shape[0], 1), dtype=A.dtype) B = relaxation_as_linear_operator((fn, kwargs), A, b) * B levels[-1].B = B if A.symmetry == "nonsymmetric": BH = relaxation_as_linear_operator((fn, kwargs), AH, b) * BH levels[-1].BH = BH # Compute the strength-of-connection matrix C, where larger # C[i, j] denote stronger couplings between i and j. fn, kwargs = unpack_arg(strength[len(levels)-1]) if fn == 'symmetric': C = symmetric_strength_of_connection(A, **kwargs) elif fn == 'classical': C = classical_strength_of_connection(A, **kwargs) elif fn == 'distance': C = distance_strength_of_connection(A, **kwargs) elif (fn == 'ode') or (fn == 'evolution'): if 'B' in kwargs: C = evolution_strength_of_connection(A, **kwargs) else: C = evolution_strength_of_connection(A, B, **kwargs) elif fn == 'energy_based': C = energy_based_strength_of_connection(A, **kwargs) elif fn == 'predefined': C = kwargs['C'].tocsr() elif fn == 'algebraic_distance': C = algebraic_distance(A, **kwargs) elif fn is None: C = A.tocsr() else: raise ValueError('unrecognized strength of connection method: %s' % str(fn)) # Avoid coarsening diagonally dominant rows flag, kwargs = unpack_arg(diagonal_dominance) if flag: C = eliminate_diag_dom_nodes(A, C, **kwargs) # Compute the aggregation matrix AggOp (i.e., the nodal coarsening of A). # AggOp is a boolean matrix, where the sparsity pattern for the k-th column # denotes the fine-grid nodes agglomerated into k-th coarse-grid node. fn, kwargs = unpack_arg(aggregate[len(levels)-1]) if fn == 'standard': AggOp, Cnodes = standard_aggregation(C, **kwargs) elif fn == 'naive': AggOp, Cnodes = naive_aggregation(C, **kwargs) elif fn == 'lloyd': AggOp, Cnodes = lloyd_aggregation(C, **kwargs) elif fn == 'pairwise': AggOp, Cnodes = pairwise_aggregation(A, B, **kwargs) elif fn == 'predefined': AggOp = kwargs['AggOp'].tocsr() Cnodes = kwargs['Cnodes'] else: raise ValueError('unrecognized aggregation method %s' % str(fn)) # ----------------------------------------------------------------------------- # # ------------------- New ideal interpolation constructed -------------------- # # ----------------------------------------------------------------------------- # # pdb.set_trace() # splitting = CR(A) # Cpts = [i for i in range(0,AggOp.shape[0]) if splitting[i]==1] # Compute prolongation operator. if test_ind==0: T = new_ideal_interpolation(A=A, AggOp=AggOp, Cnodes=Cnodes, B=B[:, 0:blocksize(A)], SOC=C) else: T = py_ideal_interpolation(A=A, AggOp=AggOp, Cnodes=Cnodes, B=B[:, 0:blocksize(A)], SOC=C) print "\nSize of sparsity pattern - ", T.nnz # Smooth the tentative prolongator, so that it's accuracy is greatly # improved for algebraically smooth error. # fn, kwargs = unpack_arg(smooth[len(levels)-1]) # if fn == 'jacobi': # P = jacobi_prolongation_smoother(A, T, C, B, **kwargs) # elif fn == 'richardson': # P = richardson_prolongation_smoother(A, T, **kwargs) # elif fn == 'energy': # P = energy_prolongation_smoother(A, T, C, B, None, (False, {}), # **kwargs) # elif fn is None: # P = T # else: # raise ValueError('unrecognized prolongation smoother method %s' % # str(fn)) P = T # ----------------------------------------------------------------------------- # # ----------------------------------------------------------------------------- # # Compute the restriction matrix R, which interpolates from the fine-grid # to the coarse-grid. If A is nonsymmetric, then R must be constructed # based on A.H. Otherwise R = P.H or P.T. symmetry = A.symmetry if symmetry == 'hermitian': # symmetrically scale out the diagonal, include scaling in P, R A = P.H * A * P [dum, Dinv, dum] = symmetric_rescaling(A,copy=False) P = bsr_matrix(P * diags(Dinv,offsets=0,format='csr'), blocksize=A.blocksize) del dum R = P.H elif symmetry == 'symmetric': # symmetrically scale out the diagonal, include scaling in P, R A = P.T * A * P [dum, Dinv, dum] = symmetric_rescaling(A,copy=False) P = bsr_matrix(P * diags(Dinv,offsets=0,format='csr'), blocksize=A.blocksize) del dum R = P.T elif symmetry == 'nonsymmetric': raise TypeError('New ideal interpolation not implemented for non-symmetric matrix.') if keep: levels[-1].C = C # strength of connection matrix levels[-1].AggOp = AggOp # aggregation operator levels[-1].Fpts = [i for i in range(0,AggOp.shape[0]) if i not in Cnodes] levels[-1].P = P # smoothed prolongator levels[-1].R = R # restriction operator levels[-1].Cpts = Cnodes # Cpts (i.e., rootnodes) levels.append(multilevel_solver.level()) A.symmetry = symmetry levels[-1].A = A levels[-1].B = R*B # right near nullspace candidates test = A.tocsr() print "\nSize of coarse operator - ", test.nnz if A.symmetry == "nonsymmetric": levels[-1].BH = BH # left near nullspace candidates