Source code for pyamg.strength

"""
Strength of Connection functions

Requirements for the strength matrix C are:
    1) Nonzero diagonal whenever A has a nonzero diagonal
    2) Non-negative entries (float or bool) in [0,1]
    3) Large entries denoting stronger connections
    4) C denotes nodal connections, i.e., if A is an nxn BSR matrix with
       row block size of m, then C is (n/m) x (n/m)

"""

__docformat__ = "restructuredtext en"

from warnings import warn

import numpy as np
import pyamg
from pyamg.util.utils import scale_rows_by_largest_entry, amalgamate
from scipy import sparse
from pyamg import amg_core

__all__ = ['classical_strength_of_connection',
           'symmetric_strength_of_connection',
           'evolution_strength_of_connection',
           'distance_strength_of_connection',
           'algebraic_distance',
           # deprecated:
           'ode_strength_of_connection']


[docs]def distance_strength_of_connection(A, V, theta=2.0, relative_drop=True): """ Distance based strength-of-connection Parameters ---------- A : csr_matrix or bsr_matrix Square, sparse matrix in CSR or BSR format V : array Coordinates of the vertices of the graph of A relative_drop : bool If false, then a connection must be within a distance of theta from a point to be strongly connected. If true, then the closest connection is always strong, and other points must be within theta times the smallest distance to be strong Returns ------- C : csr_matrix C(i,j) = distance(point_i, point_j) Strength of connection matrix where strength values are distances, i.e. the smaller the value, the stronger the connection. Sparsity pattern of C is copied from A. Notes ----- - theta is a drop tolerance that is applied row-wise - If a BSR matrix given, then the return matrix is still CSR. The strength is given between super nodes based on the BSR block size. Examples -------- >>> from pyamg.gallery import load_example >>> from pyamg.strength import distance_strength_of_connection >>> data = load_example('airfoil') >>> A = data['A'].tocsr() >>> S = distance_strength_of_connection(data['A'], data['vertices']) """ # Amalgamate for the supernode case if sparse.isspmatrix_bsr(A): sn = A.shape[0]/A.blocksize[0] u = np.ones((A.data.shape[0],)) A = sparse.csr_matrix((u, A.indices, A.indptr), shape=(sn, sn)) if not sparse.isspmatrix_csr(A): warn("Implicit conversion of A to csr", sparse.SparseEfficiencyWarning) A = sparse.csr_matrix(A) dim = V.shape[1] # Create two arrays for differencing the different coordinates such # that C(i,j) = distance(point_i, point_j) cols = A.indices rows = np.repeat(np.arange(A.shape[0]), A.indptr[1:] - A.indptr[0:-1]) # Insert difference for each coordinate into C C = (V[rows, 0] - V[cols, 0])**2 for d in range(1, dim): C += (V[rows, d] - V[cols, d])**2 C = np.sqrt(C) C[C < 1e-6] = 1e-6 C = sparse.csr_matrix((C, A.indices.copy(), A.indptr.copy()), shape=A.shape) # Apply drop tolerance if relative_drop is True: if theta != np.inf: amg_core.apply_distance_filter(C.shape[0], theta, C.indptr, C.indices, C.data) else: amg_core.apply_absolute_distance_filter(C.shape[0], theta, C.indptr, C.indices, C.data) C.eliminate_zeros() C = C + sparse.eye(C.shape[0], C.shape[1], format='csr') # Standardized strength values require small values be weak and large # values be strong. So, we invert the distances. C.data = 1.0/C.data # Scale C by the largest magnitude entry in each row C = scale_rows_by_largest_entry(C) return C
[docs]def classical_strength_of_connection(A, theta=0.0): """ Return a strength of connection matrix using the classical AMG measure An off-diagonal entry A[i,j] is a strong connection iff:: | A[i,j] | >= theta * max(| A[i,k] |), where k != i Parameters ---------- A : csr_matrix or bsr_matrix Square, sparse matrix in CSR or BSR format theta : float Threshold parameter in [0,1]. Returns ------- S : csr_matrix Matrix graph defining strong connections. S[i,j]=1 if vertex i is strongly influenced by vertex j. See Also -------- symmetric_strength_of_connection : symmetric measure used in SA evolution_strength_of_connection : relaxation based strength measure Notes ----- - A symmetric A does not necessarily yield a symmetric strength matrix S - Calls C++ function classical_strength_of_connection - The version as implemented is designed form M-matrices. Trottenberg et al. use max A[i,k] over all negative entries, which is the same. A positive edge weight never indicates a strong connection. References ---------- .. [1] Briggs, W. L., Henson, V. E., McCormick, S. F., "A multigrid tutorial", Second edition. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. xii+193 pp. ISBN: 0-89871-462-1 .. [2] Trottenberg, U., Oosterlee, C. W., Schuller, A., "Multigrid", Academic Press, Inc., San Diego, CA, 2001. xvi+631 pp. ISBN: 0-12-701070-X Examples -------- >>> import numpy >>> from pyamg.gallery import stencil_grid >>> from pyamg.strength import classical_strength_of_connection >>> n=3 >>> stencil = numpy.array([[-1.0,-1.0,-1.0], ... [-1.0, 8.0,-1.0], ... [-1.0,-1.0,-1.0]]) >>> A = stencil_grid(stencil, (n,n), format='csr') >>> S = classical_strength_of_connection(A, 0.0) """ if sparse.isspmatrix_bsr(A): blocksize = A.blocksize[0] else: blocksize = 1 if not sparse.isspmatrix_csr(A): warn("Implicit conversion of A to csr", sparse.SparseEfficiencyWarning) A = sparse.csr_matrix(A) if (theta < 0 or theta > 1): raise ValueError('expected theta in [0,1]') Sp = np.empty_like(A.indptr) Sj = np.empty_like(A.indices) Sx = np.empty_like(A.data) fn = amg_core.classical_strength_of_connection fn(A.shape[0], theta, A.indptr, A.indices, A.data, Sp, Sj, Sx) S = sparse.csr_matrix((Sx, Sj, Sp), shape=A.shape) if blocksize > 1: S = amalgamate(S, blocksize) # Strength represents "distance", so take the magnitude S.data = np.abs(S.data) # Scale S by the largest magnitude entry in each row S = scale_rows_by_largest_entry(S) return S
[docs]def symmetric_strength_of_connection(A, theta=0): """ Compute strength of connection matrix using the standard symmetric measure An off-diagonal connection A[i,j] is strong iff:: abs(A[i,j]) >= theta * sqrt( abs(A[i,i]) * abs(A[j,j]) ) Parameters ---------- A : csr_matrix Matrix graph defined in sparse format. Entry A[i,j] describes the strength of edge [i,j] theta : float Threshold parameter (positive). Returns ------- S : csr_matrix Matrix graph defining strong connections. S[i,j]=1 if vertex i is strongly influenced by vertex j. See Also -------- symmetric_strength_of_connection : symmetric measure used in SA evolution_strength_of_connection : relaxation based strength measure Notes ----- - For vector problems, standard strength measures may produce undesirable aggregates. A "block approach" from Vanek et al. is used to replace vertex comparisons with block-type comparisons. A connection between nodes i and j in the block case is strong if:: ||AB[i,j]|| >= theta * sqrt( ||AB[i,i]||*||AB[j,j]|| ) where AB[k,l] is the matrix block (degrees of freedom) associated with nodes k and l and ||.|| is a matrix norm, such a Frobenius. References ---------- .. [1] Vanek, P. and Mandel, J. and Brezina, M., "Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems", Computing, vol. 56, no. 3, pp. 179--196, 1996. http://citeseer.ist.psu.edu/vanek96algebraic.html Examples -------- >>> import numpy >>> from pyamg.gallery import stencil_grid >>> from pyamg.strength import symmetric_strength_of_connection >>> n=3 >>> stencil = numpy.array([[-1.0,-1.0,-1.0], ... [-1.0, 8.0,-1.0], ... [-1.0,-1.0,-1.0]]) >>> A = stencil_grid(stencil, (n,n), format='csr') >>> S = symmetric_strength_of_connection(A, 0.0) """ if theta < 0: raise ValueError('expected a positive theta') if sparse.isspmatrix_csr(A): # if theta == 0: # return A Sp = np.empty_like(A.indptr) Sj = np.empty_like(A.indices) Sx = np.empty_like(A.data) fn = amg_core.symmetric_strength_of_connection fn(A.shape[0], theta, A.indptr, A.indices, A.data, Sp, Sj, Sx) S = sparse.csr_matrix((Sx, Sj, Sp), shape=A.shape) elif sparse.isspmatrix_bsr(A): M, N = A.shape R, C = A.blocksize if R != C: raise ValueError('matrix must have square blocks') if theta == 0: data = np.ones(len(A.indices), dtype=A.dtype) S = sparse.csr_matrix((data, A.indices.copy(), A.indptr.copy()), shape=(M / R, N / C)) else: # the strength of connection matrix is based on the # Frobenius norms of the blocks data = (np.conjugate(A.data) * A.data).reshape(-1, R*C).sum(axis=1) A = sparse.csr_matrix((data, A.indices, A.indptr), shape=(M / R, N / C)) return symmetric_strength_of_connection(A, theta) else: raise TypeError('expected csr_matrix or bsr_matrix') # Strength represents "distance", so take the magnitude S.data = np.abs(S.data) # Scale S by the largest magnitude entry in each row S = scale_rows_by_largest_entry(S) return S
def energy_based_strength_of_connection(A, theta=0.0, k=2): """ Compute a strength of connection matrix using an energy-based measure. Parameters ---------- A : {sparse-matrix} matrix from which to generate strength of connection information theta : {float} Threshold parameter in [0,1] k : {int} Number of relaxation steps used to generate strength information Returns ------- S : {csr_matrix} Matrix graph defining strong connections. The sparsity pattern of S matches that of A. For BSR matrices, S is a reduced strength of connection matrix that describes connections between supernodes. Notes ----- This method relaxes with weighted-Jacobi in order to approximate the matrix inverse. A normalized change of energy is then used to define point-wise strength of connection values. Specifically, let v be the approximation to the i-th column of the inverse, then (S_ij)^2 = <v_j, v_j>_A / <v, v>_A, where v_j = v, such that entry j in v has been zeroed out. As is common, larger values imply a stronger connection. Current implementation is a very slow pure-python implementation for experimental purposes, only. References ---------- .. [1] Brannick, Brezina, MacLachlan, Manteuffel, McCormick. "An Energy-Based AMG Coarsening Strategy", Numerical Linear Algebra with Applications, vol. 13, pp. 133-148, 2006. Examples -------- >>> import numpy >>> from pyamg.gallery import stencil_grid >>> from pyamg.strength import energy_based_strength_of_connection >>> n=3 >>> stencil = numpy.array([[-1.0,-1.0,-1.0], ... [-1.0, 8.0,-1.0], ... [-1.0,-1.0,-1.0]]) >>> A = stencil_grid(stencil, (n,n), format='csr') >>> S = energy_based_strength_of_connection(A, 0.0) """ if (theta < 0): raise ValueError('expected a positive theta') if not sparse.isspmatrix(A): raise ValueError('expected sparse matrix') if (k < 0): raise ValueError('expected positive number of steps') if not isinstance(k, int): raise ValueError('expected integer') if sparse.isspmatrix_bsr(A): bsr_flag = True numPDEs = A.blocksize[0] if A.blocksize[0] != A.blocksize[1]: raise ValueError('expected square blocks in BSR matrix A') else: bsr_flag = False # Convert A to csc and Atilde to csr if sparse.isspmatrix_csr(A): Atilde = A.copy() A = A.tocsc() else: A = A.tocsc() Atilde = A.copy() Atilde = Atilde.tocsr() # Calculate the weighted-Jacobi parameter from pyamg.util.linalg import approximate_spectral_radius D = A.diagonal() Dinv = 1.0 / D Dinv[D == 0] = 0.0 Dinv = sparse.csc_matrix((Dinv, (np.arange(A.shape[0]), np.arange(A.shape[1]))), shape=A.shape) DinvA = Dinv*A omega = 1.0/approximate_spectral_radius(DinvA) del DinvA # Approximate A-inverse with k steps of w-Jacobi and a zero initial guess S = sparse.csc_matrix(A.shape, dtype=A.dtype) # empty matrix I = sparse.eye(A.shape[0], A.shape[1], format='csc') for i in range(k+1): S = S + omega*(Dinv*(I - A * S)) # Calculate the strength entries in S column-wise, but only strength # values at the sparsity pattern of A for i in range(Atilde.shape[0]): v = np.mat(S[:, i].todense()) Av = np.mat(A * v) denom = np.sqrt(np.conjugate(v).T * Av) # replace entries in row i with strength values for j in range(Atilde.indptr[i], Atilde.indptr[i+1]): col = Atilde.indices[j] vj = v[col].copy() v[col] = 0.0 # = (||v_j||_A - ||v||_A) / ||v||_A val = np.sqrt(np.conjugate(v).T * A * v)/denom - 1.0 # Negative values generally imply a weak connection if val > -0.01: Atilde.data[j] = abs(val) else: Atilde.data[j] = 0.0 v[col] = vj # Apply drop tolerance Atilde = classical_strength_of_connection(Atilde, theta=theta) Atilde.eliminate_zeros() # Put ones on the diagonal Atilde = Atilde + I.tocsr() Atilde.sort_indices() # Amalgamate Atilde for the BSR case, using ones for all strong connections if bsr_flag: Atilde = Atilde.tobsr(blocksize=(numPDEs, numPDEs)) nblocks = Atilde.indices.shape[0] uone = np.ones((nblocks,)) Atilde = sparse.csr_matrix((uone, Atilde.indices, Atilde.indptr), shape=( Atilde.shape[0] / numPDEs, Atilde.shape[1] / numPDEs)) # Scale C by the largest magnitude entry in each row Atilde = scale_rows_by_largest_entry(Atilde) return Atilde @np.deprecate
[docs]def ode_strength_of_connection(A, B='ones', epsilon=4.0, k=2, proj_type="l2", block_flag=False, symmetrize_measure=True): """Use evolution_strength_of_connection instead""" return evolution_strength_of_connection(A, B, epsilon, k, proj_type, block_flag, symmetrize_measure)
[docs]def evolution_strength_of_connection(A, B='ones', epsilon=4.0, k=2, proj_type="l2", block_flag=False, symmetrize_measure=True): """ Construct strength of connection matrix using an Evolution-based measure Parameters ---------- A : {csr_matrix, bsr_matrix} Sparse NxN matrix B : {string, array} If B='ones', then the near nullspace vector used is all ones. If B is an (NxK) array, then B is taken to be the near nullspace vectors. epsilon : scalar Drop tolerance k : integer ODE num time steps, step size is assumed to be 1/rho(DinvA) proj_type : {'l2','D_A'} Define norm for constrained min prob, i.e. define projection block_flag : {boolean} If True, use a block D inverse as preconditioner for A during weighted-Jacobi Returns ------- Atilde : {csr_matrix} Sparse matrix of strength values References ---------- .. [1] Olson, L. N., Schroder, J., Tuminaro, R. S., "A New Perspective on Strength Measures in Algebraic Multigrid", submitted, June, 2008. Examples -------- >>> import numpy >>> from pyamg.gallery import stencil_grid >>> from pyamg.strength import evolution_strength_of_connection >>> n=3 >>> stencil = numpy.array([[-1.0,-1.0,-1.0], ... [-1.0, 8.0,-1.0], ... [-1.0,-1.0,-1.0]]) >>> A = stencil_grid(stencil, (n,n), format='csr') >>> S = evolution_strength_of_connection(A, numpy.ones((A.shape[0],1))) """ # local imports for evolution_strength_of_connection from pyamg.util.utils import scale_rows, get_block_diag, scale_columns from pyamg.util.linalg import approximate_spectral_radius # ==================================================================== # Check inputs if epsilon < 1.0: raise ValueError("expected epsilon > 1.0") if k <= 0: raise ValueError("number of time steps must be > 0") if proj_type not in ['l2', 'D_A']: raise ValueError("proj_type must be 'l2' or 'D_A'") if (not sparse.isspmatrix_csr(A)) and (not sparse.isspmatrix_bsr(A)): raise TypeError("expected csr_matrix or bsr_matrix") # ==================================================================== # Format A and B correctly. # B must be in mat format, this isn't a deep copy if B == 'ones': Bmat = np.mat(np.ones((A.shape[0], 1), dtype=A.dtype)) else: Bmat = np.mat(B) # Pre-process A. We need A in CSR, to be devoid of explicit 0's and have # sorted indices if (not sparse.isspmatrix_csr(A)): csrflag = False numPDEs = A.blocksize[0] D = A.diagonal() # Calculate Dinv*A if block_flag: Dinv = get_block_diag(A, blocksize=numPDEs, inv_flag=True) Dinv = sparse.bsr_matrix((Dinv, np.arange(Dinv.shape[0]), np.arange(Dinv.shape[0] + 1)), shape=A.shape) Dinv_A = (Dinv * A).tocsr() else: Dinv = np.zeros_like(D) mask = (D != 0.0) Dinv[mask] = 1.0 / D[mask] Dinv[D == 0] = 1.0 Dinv_A = scale_rows(A, Dinv, copy=True) A = A.tocsr() else: csrflag = True numPDEs = 1 D = A.diagonal() Dinv = np.zeros_like(D) mask = (D != 0.0) Dinv[mask] = 1.0 / D[mask] Dinv[D == 0] = 1.0 Dinv_A = scale_rows(A, Dinv, copy=True) A.eliminate_zeros() A.sort_indices() # Handle preliminaries for the algorithm dimen = A.shape[1] NullDim = Bmat.shape[1] # Get spectral radius of Dinv*A, this will be used to scale the time step # size for the ODE rho_DinvA = approximate_spectral_radius(Dinv_A) # Calculate D_A for later use in the minimization problem if proj_type == "D_A": D_A = sparse.spdiags([D], [0], dimen, dimen, format='csr') else: D_A = sparse.eye(dimen, dimen, format="csr", dtype=A.dtype) # Calculate (I - delta_t Dinv A)^k # In order to later access columns, we calculate the transpose in # CSR format so that columns will be accessed efficiently # Calculate the number of time steps that can be done by squaring, and # the number of time steps that must be done incrementally nsquare = int(np.log2(k)) ninc = k - 2**nsquare # Calculate one time step I = sparse.eye(dimen, dimen, format="csr", dtype=A.dtype) Atilde = (I - (1.0/rho_DinvA)*Dinv_A) Atilde = Atilde.T.tocsr() # Construct a sparsity mask for Atilde that will restrict Atilde^T to the # nonzero pattern of A, with the added constraint that row i of Atilde^T # retains only the nonzeros that are also in the same PDE as i. mask = A.copy() # Restrict to same PDE if numPDEs > 1: row_length = np.diff(mask.indptr) my_pde = np.mod(range(dimen), numPDEs) my_pde = np.repeat(my_pde, row_length) mask.data[np.mod(mask.indices, numPDEs) != my_pde] = 0.0 del row_length, my_pde mask.eliminate_zeros() # If the total number of time steps is a power of two, then there is # a very efficient computational short-cut. Otherwise, we support # other numbers of time steps, through an inefficient algorithm. if ninc > 0: warn("The most efficient time stepping for the Evolution Strength\ Method is done in powers of two.\nYou have chosen " + str(k) + " time steps.") # Calculate (Atilde^nsquare)^T = (Atilde^T)^nsquare for i in range(nsquare): Atilde = Atilde*Atilde JacobiStep = (I - (1.0/rho_DinvA)*Dinv_A).T.tocsr() for i in range(ninc): Atilde = Atilde*JacobiStep del JacobiStep # Apply mask to Atilde, zeros in mask have already been eliminated at # start of routine. mask.data[:] = 1.0 Atilde = Atilde.multiply(mask) Atilde.eliminate_zeros() Atilde.sort_indices() elif nsquare == 0: if numPDEs > 1: # Apply mask to Atilde, zeros in mask have already been eliminated # at start of routine. mask.data[:] = 1.0 Atilde = Atilde.multiply(mask) Atilde.eliminate_zeros() Atilde.sort_indices() else: # Use computational short-cut for case (ninc == 0) and (nsquare > 0) # Calculate Atilde^k only at the sparsity pattern of mask. for i in range(nsquare-1): Atilde = Atilde*Atilde # Call incomplete mat-mat mult AtildeCSC = Atilde.tocsc() AtildeCSC.sort_indices() mask.sort_indices() Atilde.sort_indices() amg_core.incomplete_mat_mult_csr(Atilde.indptr, Atilde.indices, Atilde.data, AtildeCSC.indptr, AtildeCSC.indices, AtildeCSC.data, mask.indptr, mask.indices, mask.data, dimen) del AtildeCSC, Atilde Atilde = mask Atilde.eliminate_zeros() Atilde.sort_indices() del Dinv, Dinv_A, mask # Calculate strength based on constrained min problem of # min( z - B*x ), such that # (B*x)|_i = z|_i, i.e. they are equal at point i # z = (I - (t/k) Dinv A)^k delta_i # # Strength is defined as the relative point-wise approx. error between # B*x and z. We don't use the full z in this problem, only that part of # z that is in the sparsity pattern of A. # # Can use either the D-norm, and inner product, or l2-norm and inner-prod # to solve the constrained min problem. Using D gives scale invariance. # # This is a quadratic minimization problem with a linear constraint, so # we can build a linear system and solve it to find the critical point, # i.e. minimum. # # We exploit a known shortcut for the case of NullDim = 1. The shortcut is # mathematically equivalent to the longer constrained min. problem if NullDim == 1: # Use shortcut to solve constrained min problem if B is only a vector # Strength(i,j) = | 1 - (z(i)/b(j))/(z(j)/b(i)) | # These ratios can be calculated by diagonal row and column scalings # Create necessary vectors for scaling Atilde # Its not clear what to do where B == 0. This is an # an easy programming solution, that may make sense. Bmat_forscaling = np.ravel(Bmat) Bmat_forscaling[Bmat_forscaling == 0] = 1.0 DAtilde = Atilde.diagonal() DAtildeDivB = np.ravel(DAtilde) / Bmat_forscaling # Calculate best approximation, z_tilde, in span(B) # Importantly, scale_rows and scale_columns leave zero entries # in the matrix. For previous implementations this was useful # because we assume data and Atilde.data are the same length below data = Atilde.data.copy() Atilde.data[:] = 1.0 Atilde = scale_rows(Atilde, DAtildeDivB) Atilde = scale_columns(Atilde, np.ravel(Bmat_forscaling)) # If angle in the complex plane between z and z_tilde is # greater than 90 degrees, then weak. We can just look at the # dot product to determine if angle is greater than 90 degrees. angle = np.real(Atilde.data) * np.real(data) +\ np.imag(Atilde.data) * np.imag(data) angle = angle < 0.0 angle = np.array(angle, dtype=bool) # Calculate Approximation ratio Atilde.data = Atilde.data/data # If approximation ratio is less than tol, then weak connection weak_ratio = (np.abs(Atilde.data) < 1e-4) # Calculate Approximation error Atilde.data = abs(1.0 - Atilde.data) # Set small ratios and large angles to weak Atilde.data[weak_ratio] = 0.0 Atilde.data[angle] = 0.0 # Set near perfect connections to 1e-4 Atilde.eliminate_zeros() Atilde.data[Atilde.data < np.sqrt(np.finfo(float).eps)] = 1e-4 del data, weak_ratio, angle else: # For use in computing local B_i^H*B, precompute the element-wise # multiply of each column of B with each other column. We also scale # by 2.0 to account for BDB's eventual use in a constrained # minimization problem BDBCols = int(np.sum(range(NullDim + 1))) BDB = np.zeros((dimen, BDBCols), dtype=A.dtype) counter = 0 for i in range(NullDim): for j in range(i, NullDim): BDB[:, counter] = 2.0 *\ (np.conjugate(np.ravel(np.asarray(B[:, i]))) * np.ravel(np.asarray(D_A * B[:, j]))) counter = counter + 1 # Choose tolerance for dropping "numerically zero" values later t = Atilde.dtype.char eps = np.finfo(np.float).eps feps = np.finfo(np.single).eps geps = np.finfo(np.longfloat).eps _array_precision = {'f': 0, 'd': 1, 'g': 2, 'F': 0, 'D': 1, 'G': 2} tol = {0: feps*1e3, 1: eps*1e6, 2: geps*1e6}[_array_precision[t]] # Use constrained min problem to define strength amg_core.evolution_strength_helper(Atilde.data, Atilde.indptr, Atilde.indices, Atilde.shape[0], np.ravel(np.asarray(B)), np.ravel(np.asarray( (D_A * np.conjugate(B)).T)), np.ravel(np.asarray(BDB)), BDBCols, NullDim, tol) Atilde.eliminate_zeros() # All of the strength values are real by this point, so ditch the complex # part Atilde.data = np.array(np.real(Atilde.data), dtype=float) # Apply drop tolerance if symmetrize_measure: Atilde = 0.5*(Atilde + Atilde.T) if epsilon != np.inf: amg_core.apply_distance_filter(dimen, epsilon, Atilde.indptr, Atilde.indices, Atilde.data) Atilde.eliminate_zeros() # Set diagonal to 1.0, as each point is strongly connected to itself. I = sparse.eye(dimen, dimen, format="csr") I.data -= Atilde.diagonal() Atilde = Atilde + I # If converted BSR to CSR, convert back and return amalgamated matrix, # i.e. the sparsity structure of the blocks of Atilde if not csrflag: Atilde = Atilde.tobsr(blocksize=(numPDEs, numPDEs)) n_blocks = Atilde.indices.shape[0] blocksize = Atilde.blocksize[0]*Atilde.blocksize[1] CSRdata = np.zeros((n_blocks,)) amg_core.min_blocks(n_blocks, blocksize, np.ravel(np.asarray(Atilde.data)), CSRdata) # Atilde = sparse.csr_matrix((data, row, col), shape=(*,*)) Atilde = sparse.csr_matrix((CSRdata, Atilde.indices, Atilde.indptr), shape=(Atilde.shape[0] / numPDEs, Atilde.shape[1] / numPDEs)) # Standardized strength values require small values be weak and large # values be strong. So, we invert the algebraic distances computed here Atilde.data = 1.0/Atilde.data # Scale C by the largest magnitude entry in each row Atilde = scale_rows_by_largest_entry(Atilde) return Atilde
[docs]def algebraic_distance(A, alpha=0.5, R=5, k=20, theta=0.1, p=2): """Construct an AMG strength of connection matrix using an algebraic distance measure. Parameters ---------- A : {csr_matrix} Sparse NxN matrix alpha : scalar Weight for Jacobi R : integer Number of random vectors k : integer Number of relaxation passes theta : scalar Drop values larger than theta p : scalar or inf p-norm of the measure Returns ------- C : {csr_matrix} Sparse matrix of strength values References ---------- .. [1] "Advanced Coarsening Schemes for Graph Partitioning" by Ilya Safro, Peter Sanders, and Christian Schulz Notes ----- No unit testing yet. Does not handle BSR matrices yet. """ print A.format if not sparse.isspmatrix_csr(A): warn("Implicit conversion of A to csr", sparse.SparseEfficiencyWarning) A = sparse.csr_matrix(A) if alpha < 0: raise ValueError('expected alpha>0') if R <= 0 or not isinstance(R, int): raise ValueError('expected integer R>0') if k <= 0 or not isinstance(k, int): raise ValueError('expected integer k>0') if theta < 0: raise ValueError('expected theta>0.0') if p < 1: raise ValueError('expected p>1 or equal to numpy.inf') # random n x R block in column ordering n = A.shape[0] x = np.random.rand(n * R) - 0.5 x = np.reshape(x, (n, R), order='F') b = np.zeros((n, 1)) # relax k times for r in range(0, R): pyamg.relaxation.jacobi(A, x[:, r], b, iterations=k, omega=alpha) # get distance measure d C = A.tocoo() I = C.row J = C.col # TODO : this is not right vvvvvvv if p != np.inf: d = np.sum((x[I] - x[J])**p, axis=1)**(1.0/p) else: d = np.abs(x[I] - x[J]).max(axis=1) del d # drop weak connections larger than theta weak = np.where(C.data > theta)[0] C.data[weak] = 0 C = C.tocsr() C.eliminate_zeros() # Strength represents "distance", so take the magnitude C.data = np.abs(C.data) # Standardized strength values require small values be weak and large # values be strong. So, we invert the distances. C.data = 1.0/C.data # Put an identity on the diagonal C = C + sparse.eye(C.shape[0], C.shape[1], format='csr') # Scale C by the largest magnitude entry in each row C = scale_rows_by_largest_entry(C) return C