def evolution_strength_of_connection(A,
B=None,
epsilon=4.0,
k=2,
proj_type="l2",
weighting='diagonal',
symmetrize_measure=True,
cost=[0]):
"""
Construct strength of connection matrix using an Evolution-based measure
Parameters
----------
A : {csr_matrix, bsr_matrix}
Sparse NxN matrix
B : {string, array}
If B=None, 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
weighting : {string}
'block', 'diagonal' or 'local' construction of the D-inverse
used to precondition A before "evolving" delta-functions. The
local option is the cheapest.
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 as np
>>> from pyamg.gallery import stencil_grid
>>> from pyamg.strength import evolution_strength_of_connection
>>> n=3
>>> stencil = np.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, np.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 is None:
Bmat = np.mat(np.ones((A.shape[0], 1), dtype=A.dtype))
else:
Bmat = np.mat(B)
# Is matrix A CSR?
if (not sparse.isspmatrix_csr(A)):
numPDEs = A.blocksize[0]
csrflag = False
else:
numPDEs = 1
csrflag = True
# Pre-process A. We need A in CSR, to be devoid of explicit 0's, have
# sorted indices and be scaled by D-inverse
if weighting == 'block':
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()
cost[0] += 1
elif weighting == 'diagonal':
D = A.diagonal()
Dinv = get_diagonal(A, norm_eq=False, inv=True)
Dinv[D == 0] = 1.0
Dinv_A = scale_rows(A, Dinv, copy=True)
cost[0] += 1
elif weighting == 'local':
D = np.abs(A) * np.ones((A.shape[0], 1), dtype=A.dtype)
Dinv = np.zeros_like(D)
Dinv[D != 0] = 1.0 / np.abs(D[D != 0])
Dinv[D == 0] = 1.0
Dinv_A = scale_rows(A, Dinv, copy=True)
cost[0] += 1
else:
raise ValueError('Unrecognized weighting for Evolution measure')
A = A.tocsr()
A.eliminate_zeros()
A.sort_indices()
# Handle preliminaries for the algorithm
dimen = A.shape[1]
NullDim = Bmat.shape[1]
if weighting == 'diagonal' or weighting == 'block':
# Get spectral radius of Dinv*A, scales the time step size for the ODE
rho_DinvA = approximate_spectral_radius(Dinv_A)
cost[0] += 15 # 15 lanczos iterations to approximate spectral radius
else:
# Using local weighting, no need for spectral radius
rho_DinvA = 1.0
# Calculate D_A for later use in the minimization problem
if proj_type == "D_A":
D = A.diagonal()
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
# We transpose the product, so that we can efficiently access
# the columns in CSR format. We want the columns (not rows) because
# strength is based on the columns of (I - delta_t Dinv A)^k, i.e.,
# relaxed delta functions
# 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()
cost[0] += 1
# 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(np.arange(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.")
JacobiStep = csr_matrix(Atilde, copy=True)
# Calculate (Atilde^nsquare)^T = (Atilde^T)^nsquare
for i in range(nsquare):
cost[0] += mat_mat_complexity(Atilde, Atilde)
Atilde = Atilde * Atilde
for i in range(ninc):
cost[0] += mat_mat_complexity(Atilde, JacobiStep)
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()
cost[0] += Atilde.nnz / float(A.nnz)
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):
cost[0] += mat_mat_complexity(Atilde, Atilde)
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)
cost[0] += mat_mat_complexity(Atilde, mask, incomplete=True) / float(
A.nnz)
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
cost[0] += Atilde.shape[0] / float(A.nnz)
# 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))
cost[0] += 2.0 * Atilde.nnz / float(A.nnz)
# 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)
cost[0] += Atilde.nnz / float(A.nnz)
if Atilde.dtype is 'complex':
cost[0] += Atilde.nnz / float(A.nnz)
# Calculate Approximation ratio
Atilde.data = Atilde.data / data
cost[0] += Atilde.nnz / float(A.nnz)
# 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)
cost[0] += Atilde.nnz / float(A.nnz)
# 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(np.arange(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
cost[0] += B.shape[0] / float(A.nnz)
# 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.
# This function is doing similar to NullDim=1 with more bad guys.
# Complexity accounts for computing the block inverse, and
# hat{z_i} = B_i*x, hat{z_i} .* hat{z_i},
# hat{z_i} = hat{z_i} / z_i, and abs(1.0 - hat{z_i}).
cost[0] += (Atilde.nnz * (3 + NullDim) +
(NullDim**3) * dimen) / float(A.nnz)
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 epsilon != np.inf:
cost[0] += Atilde.nnz / float(A.nnz)
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
cost[0] += Atilde.shape[0] / float(A.nnz)
# 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=(int(Atilde.shape[0] / numPDEs),
int(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
cost[0] += Atilde.nnz / float(A.nnz)
# Scale C by the largest magnitude entry in each row
Atilde = scale_rows_by_largest_entry(Atilde)
cost[0] += Atilde.nnz / float(A.nnz)
# Symmetrize
if symmetrize_measure:
Atilde = 0.5 * (Atilde + Atilde.T)
cost[0] += Atilde.nnz / float(A.nnz)
return Atilde
def gmres_prolongation_smoothing(A,
T,
B,
BtBinv,
Sparsity_Pattern,
maxiter,
tol,
weighting='local',
Cpt_params=None,
cost=[0.0]):
'''
Helper function for energy_prolongation_smoother(...).
Use GMRES to smooth T by solving A T = 0, subject to nullspace
and sparsity constraints.
Parameters
----------
A : {csr_matrix, bsr_matrix}
SPD sparse NxN matrix
Should be at least nonsymmetric or indefinite
T : {bsr_matrix}
Tentative prolongator, a NxM sparse matrix (M < N).
This is initial guess for the equation A T = 0.
Assumed that T B_c = B_f
B : {array}
Near-nullspace modes for coarse grid, i.e., B_c.
Has shape (M,k) where k is the number of coarse candidate vectors.
BtBinv : {array}
3 dimensional array such that,
BtBinv[i] = pinv(B_i.H Bi), and B_i is B restricted
to the neighborhood (in the matrix graph) of dof of i.
Sparsity_Pattern : {csr_matrix, bsr_matrix}
Sparse NxM matrix
This is the sparsity pattern constraint to enforce on the
eventual prolongator
maxiter : int
maximum number of iterations
tol : float
residual tolerance for A T = 0
weighting : {string}
'block', 'diagonal' or 'local' construction of the diagonal
preconditioning
Cpt_params : {tuple}
Tuple of the form (bool, dict). If the Cpt_params[0] = False, then
the standard SA prolongation smoothing is carried out. If True, then
dict must be a dictionary of parameters containing, (1) P_I: P_I.T is
the injection matrix for the Cpts, (2) I_F: an identity matrix
for only the F-points (i.e. I, but with zero rows and columns for
C-points) and I_C: the C-point analogue to I_F.
Returns
-------
T : {bsr_matrix}
Smoothed prolongator using GMRES to solve A T = 0,
subject to the constraints, T B_c = B_f, and T has no nonzero
outside of the sparsity pattern in Sparsity_Pattern.
See Also
--------
The principal calling routine,
pyamg.aggregation.smooth.energy_prolongation_smoother
'''
# For non-SPD system, apply GMRES with Diagonal Preconditioning
# Preallocate space for new search directions
uones = np.zeros(Sparsity_Pattern.data.shape, dtype=T.dtype)
AV = sparse.bsr_matrix(
(uones, Sparsity_Pattern.indices, Sparsity_Pattern.indptr),
shape=(Sparsity_Pattern.shape))
# Preallocate for Givens Rotations, Hessenberg matrix and Krylov Space
xtype = sparse.sputils.upcast(A.dtype, T.dtype, B.dtype)
Q = [] # Givens Rotations
V = [] # Krylov Space
# vs = [] # vs store the pointers to each column of V for speed
# Upper Hessenberg matrix, converted to upper tri with Givens Rots
H = np.zeros((maxiter + 1, maxiter + 1), dtype=xtype)
# GMRES will be run with diagonal preconditioning
if weighting == 'diagonal':
Dinv = get_diagonal(A, norm_eq=False, inv=True)
elif weighting == 'block':
Dinv = get_block_diag(A, blocksize=A.blocksize[0], inv_flag=True)
Dinv = sparse.bsr_matrix(
(Dinv, np.arange(Dinv.shape[0]), np.arange(Dinv.shape[0] + 1)),
shape=A.shape)
elif weighting == 'local':
# Based on Gershgorin estimate
D = np.abs(A) * np.ones((A.shape[0], 1), dtype=A.dtype)
Dinv = np.zeros_like(D)
Dinv[D != 0] = 1.0 / np.abs(D[D != 0])
cost[0] += 1.0
else:
raise ValueError('weighting value is invalid')
# Calculate initial residual
# Equivalent to R = -A*T; R = R.multiply(Sparsity_Pattern)
# with the added constraint that R has an explicit 0 wherever
# R is 0 and Sparsity_Pattern is not
uones = np.zeros(Sparsity_Pattern.data.shape, dtype=T.dtype)
R = sparse.bsr_matrix(
(uones, Sparsity_Pattern.indices, Sparsity_Pattern.indptr),
shape=(Sparsity_Pattern.shape))
pyamg.amg_core.incomplete_mat_mult_bsr(
A.indptr, A.indices, np.ravel(A.data), T.indptr, T.indices,
np.ravel(T.data), R.indptr, R.indices, np.ravel(R.data),
int(T.shape[0] / T.blocksize[0]), int(T.shape[1] / T.blocksize[1]),
A.blocksize[0], A.blocksize[1], T.blocksize[1])
R.data *= -1.0
# T is block diagonal, using sparsity pattern of R with
# incomplete=True significantly overestimates complexity.
# More accurate to use full mat-mat with block diagonal T.
cost[0] += mat_mat_complexity(A, T, incomplete=False) / float(A.nnz)
# Apply diagonal preconditioner
if weighting == 'local' or weighting == 'diagonal':
R = scale_rows(R, Dinv)
else:
R = Dinv * R
cost[0] += R.nnz / float(A.nnz)
# Enforce R*B = 0
temp_cost = [0.0]
Satisfy_Constraints(R, B, BtBinv, cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
if R.nnz == 0:
print("Error in sa_energy_min(..). Initial R no nonzeros on a level. \
Returning tentative prolongator\n")
return T
# This is the RHS vector for the problem in the Krylov Space
normr = np.sqrt((R.data.conjugate() * R.data).sum())
g = np.zeros((maxiter + 1, ), dtype=xtype)
g[0] = normr
# First Krylov vector
# V[0] = r/normr
if normr > 0.0:
V.append((1.0 / normr) * R)
i = -1
while i < maxiter - 1 and normr > tol:
i = i + 1
# Calculate new search direction
# Equivalent to: AV = A*V; AV = AV.multiply(Sparsity_Pattern)
# with the added constraint that explicit zeros are in AP wherever
# AP = 0 and Sparsity_Pattern does not
AV.data[:] = 0.0
pyamg.amg_core.incomplete_mat_mult_bsr(
A.indptr, A.indices, np.ravel(A.data), V[i].indptr, V[i].indices,
np.ravel(V[i].data), AV.indptr, AV.indices, np.ravel(AV.data),
int(T.shape[0] / T.blocksize[0]), int(T.shape[1] / T.blocksize[1]),
A.blocksize[0], A.blocksize[1], T.blocksize[1])
cost[0] += mat_mat_complexity(A, AV, incomplete=True) / float(A.nnz)
if weighting == 'local' or weighting == 'diagonal':
AV = scale_rows(AV, Dinv)
else:
AV = Dinv * AV
cost[0] += AV.nnz / float(A.nnz)
# Enforce AV*B = 0
temp_cost = [0.0]
Satisfy_Constraints(AV, B, BtBinv, cost=temp_cost)
V.append(AV.copy())
cost[0] += temp_cost[0] / float(A.nnz)
# Modified Gram-Schmidt
for j in range(i + 1):
# Frobenius inner-product
H[j, i] = (V[j].conjugate().multiply(V[i + 1])).sum()
V[i + 1] = V[i + 1] - H[j, i] * V[j]
cost[0] += 2.0 * max(V[i + 1].nnz, V[j].nnz) / float(A.nnz)
# Frobenius Norm
H[i + 1, i] = np.sqrt(
(V[i + 1].data.conjugate() * V[i + 1].data).sum())
cost[0] += V[i + 1].nnz / float(A.nnz)
# Check for breakdown
if H[i + 1, i] != 0.0:
V[i + 1] = (1.0 / H[i + 1, i]) * V[i + 1]
cost[0] += V[i + 1].nnz / float(A.nnz)
# Apply previous Givens rotations to H
if i > 0:
apply_givens(Q, H[:, i], i)
# Calculate and apply next complex-valued Givens Rotation
if H[i + 1, i] != 0:
h1 = H[i, i]
h2 = H[i + 1, i]
h1_mag = np.abs(h1)
h2_mag = np.abs(h2)
if h1_mag < h2_mag:
mu = h1 / h2
tau = np.conjugate(mu) / np.abs(mu)
else:
mu = h2 / h1
tau = mu / np.abs(mu)
denom = np.sqrt(h1_mag**2 + h2_mag**2)
c = h1_mag / denom
s = h2_mag * tau / denom
Qblock = np.array([[c, np.conjugate(s)], [-s, c]], dtype=xtype)
Q.append(Qblock)
# Apply Givens Rotation to g,
# the RHS for the linear system in the Krylov Subspace.
g[i:i + 2] = sp.dot(Qblock, g[i:i + 2])
# Apply effect of Givens Rotation to H
H[i, i] = sp.dot(Qblock[0, :], H[i:i + 2, i])
H[i + 1, i] = 0.0
normr = np.abs(g[i + 1])
# End while loop
# Find best update to x in Krylov Space, V. Solve (i x i) system.
if i != -1:
y = la.solve(H[0:i + 1, 0:i + 1], g[0:i + 1])
for j in range(i + 1):
T = T + y[j] * V[j]
cost[0] += max(T.nnz, V[j].nnz) / float(A.nnz)
# Ensure identity at C-pts
if Cpt_params[0]:
T = Cpt_params[1]['I_F'] * T + Cpt_params[1]['P_I']
return T
def energy_prolongation_smoother(A,
T,
Atilde,
B,
Bf,
Cpt_params,
krylov='cg',
maxiter=4,
tol=1e-8,
degree=1,
weighting='local',
prefilter={},
postfilter={},
cost=[0.0]):
"""Minimize the energy of the coarse basis functions (columns of T). Both
root-node and non-root-node style prolongation smoothing is available, see
Cpt_params description below.
Parameters
----------
A : {csr_matrix, bsr_matrix}
Sparse NxN matrix
T : {bsr_matrix}
Tentative prolongator, a NxM sparse matrix (M < N)
Atilde : {csr_matrix}
Strength of connection matrix
B : {array}
Near-nullspace modes for coarse grid. Has shape (M,k) where
k is the number of coarse candidate vectors.
Bf : {array}
Near-nullspace modes for fine grid. Has shape (N,k) where
k is the number of coarse candidate vectors.
Cpt_params : {tuple}
Tuple of the form (bool, dict). If the Cpt_params[0] = False, then the
standard SA prolongation smoothing is carried out. If True, then
root-node style prolongation smoothing is carried out. The dict must
be a dictionary of parameters containing, (1) P_I: P_I.T is the
injection matrix for the Cpts, (2) I_F: an identity matrix for only the
F-points (i.e. I, but with zero rows and columns for C-points) and I_C:
the C-point analogue to I_F. See Notes below for more information.
krylov : {string}
'cg' : for SPD systems. Solve A T = 0 in a constraint space with CG
'cgnr' : for nonsymmetric and/or indefinite systems.
Solve A T = 0 in a constraint space with CGNR
'gmres' : for nonsymmetric and/or indefinite systems.
Solve A T = 0 in a constraint space with GMRES
maxiter : integer
Number of energy minimization steps to apply to the prolongator
tol : {scalar}
Minimization tolerance
degree : {int}
Generate sparsity pattern for P based on (Atilde^degree T)
weighting : {string}
'block', 'diagonal' or 'local' construction of the
diagonal preconditioning
'local': Uses a local row-wise weight based on the Gershgorin estimate.
Avoids any potential under-damping due to inaccurate spectral
radius estimates.
'block': If A is a BSR matrix, use a block diagonal inverse of A
'diagonal': Use inverse of the diagonal of A
prefilter : {dictionary} : Default {}
Filters elements by row in sparsity pattern for P to reduce operator and
setup complexity. If None or empty dictionary, no dropping in P is done.
If postfilter has key 'k', then the largest 'k' entries are kept in each
row. If postfilter has key 'theta', all entries such that
P[i,j] < kwargs['theta']*max(abs(P[i,:]))
are dropped. If postfilter['k'] and postfiler['theta'] are present, then
they are used in conjunction, with the union of their patterns used.
postfilter : {dictionary} : Default {}
Filters elements by row in smoothed P to reduce operator complexity.
Only supported if using the rootnode_solver. If None or empty dictionary,
no dropping in P is done. If postfilter has key 'k', then the largest 'k'
entries are kept in each row. If postfilter has key 'theta', all entries
such that
P[i,j] < kwargs['theta']*max(abs(P[i,:]))
are dropped. If postfilter['k'] and postfiler['theta'] are present, then
they are used in conjunction, with the union of their patterns used.
Returns
-------
T : {bsr_matrix}
Smoothed prolongator
Notes
-----
Only 'diagonal' weighting is supported for the CGNR method, because
we are working with A^* A and not A.
When Cpt_params[0] == True, root-node style prolongation smoothing
is used to minimize the energy of columns of T. Essentially, an
identity block is maintained in T, corresponding to injection from
the coarse-grid to the fine-grid root-nodes. See [2] for more details,
and see util.utils.get_Cpt_params for the helper function to generate
Cpt_params.
If Cpt_params[0] == False, the energy of columns of T are still
minimized, but without maintaining the identity block.
Examples
--------
>>> from pyamg.aggregation import energy_prolongation_smoother
>>> from pyamg.gallery import poisson
>>> from scipy.sparse import coo_matrix
>>> import numpy as np
>>> data = np.ones((6,))
>>> row = np.arange(0,6)
>>> col = np.kron([0,1],np.ones((3,)))
>>> T = coo_matrix((data,(row,col)),shape=(6,2)).tocsr()
>>> print T.todense()
[[ 1. 0.]
[ 1. 0.]
[ 1. 0.]
[ 0. 1.]
[ 0. 1.]
[ 0. 1.]]
>>> A = poisson((6,),format='csr')
>>> B = np.ones((2,1),dtype=float)
>>> P = energy_prolongation_smoother(A,T,A,B, None, (False,{}))
>>> print P.todense()
[[ 1. 0. ]
[ 1. 0. ]
[ 0.66666667 0.33333333]
[ 0.33333333 0.66666667]
[ 0. 1. ]
[ 0. 1. ]]
References
----------
.. [1] Jan Mandel, Marian Brezina, and Petr Vanek
"Energy Optimization of Algebraic Multigrid Bases"
Computing 62, 205-228, 1999
http://dx.doi.org/10.1007/s006070050022
.. [2] Olson, L. and Schroder, J. and Tuminaro, R.,
"A general interpolation strategy for algebraic
multigrid using energy minimization", SIAM Journal
on Scientific Computing (SISC), vol. 33, pp.
966--991, 2011.
"""
# Test Inputs
if maxiter < 0:
raise ValueError('maxiter must be > 0')
if tol > 1:
raise ValueError('tol must be <= 1')
if sparse.isspmatrix_csr(A):
A = A.tobsr(blocksize=(1, 1), copy=False)
elif sparse.isspmatrix_bsr(A):
pass
else:
raise TypeError("A must be csr_matrix or bsr_matrix")
if sparse.isspmatrix_csr(T):
T = T.tobsr(blocksize=(1, 1), copy=False)
elif sparse.isspmatrix_bsr(T):
pass
else:
raise TypeError("T must be csr_matrix or bsr_matrix")
if T.blocksize[0] != A.blocksize[0]:
raise ValueError("T row-blocksize should be the same as A blocksize")
if B.shape[0] != T.shape[1]:
raise ValueError("B is the candidates for the coarse grid. \
num_rows(b) = num_cols(T)")
if min(T.nnz, A.nnz) == 0:
return T
if not sparse.isspmatrix_csr(Atilde):
raise TypeError("Atilde must be csr_matrix")
if ('theta' in prefilter) and (prefilter['theta'] == 0):
prefilter.pop('theta', None)
if ('theta' in postfilter) and (postfilter['theta'] == 0):
postfilter.pop('theta', None)
# Prepocess Atilde, the strength matrix
if Atilde is None:
Atilde = sparse.csr_matrix(
(np.ones(len(A.indices)), A.indices.copy(), A.indptr.copy()),
shape=(A.shape[0] / A.blocksize[0], A.shape[1] / A.blocksize[1]))
# If Atilde has no nonzeros, then return T
if min(T.nnz, A.nnz) == 0:
return T
# Expand allowed sparsity pattern for P through multiplication by Atilde
if degree > 0:
# Construct Sparsity_Pattern by multiplying with Atilde
T.sort_indices()
shape = (int(T.shape[0] / T.blocksize[0]),
int(T.shape[1] / T.blocksize[1]))
Sparsity_Pattern = sparse.csr_matrix(
(np.ones(T.indices.shape), T.indices, T.indptr), shape=shape)
AtildeCopy = Atilde.copy()
for i in range(degree):
cost[0] += mat_mat_complexity(AtildeCopy,
Sparsity_Pattern) / float(A.nnz)
Sparsity_Pattern = AtildeCopy * Sparsity_Pattern
# Optional filtering of sparsity pattern before smoothing
# - Complexity: two passes through T for theta-filter, a sort on
# each row for k-filter, adding matrices if both.
if 'theta' in prefilter and 'k' in prefilter:
temp_cost = [0.0]
Sparsity_theta = filter_matrix_rows(Sparsity_Pattern,
prefilter['theta'],
cost=temp_cost)
Sparsity_Pattern = truncate_rows(Sparsity_Pattern,
prefilter['k'],
cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
# Union two sparsity patterns
Sparsity_Pattern += Sparsity_theta
cost[0] += Sparsity_Pattern.nnz / float(A.nnz)
elif 'k' in prefilter:
temp_cost = [0.0]
Sparsity_Pattern = truncate_rows(Sparsity_Pattern,
prefilter['k'],
cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
elif 'theta' in prefilter:
temp_cost = [0.0]
Sparsity_Pattern = filter_matrix_rows(Sparsity_Pattern,
prefilter['theta'],
cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
elif len(prefilter) > 0:
raise ValueError("Unrecognized prefilter option")
# UnAmal returns a BSR matrix with 1's in the nonzero locations
Sparsity_Pattern = UnAmal(Sparsity_Pattern, T.blocksize[0],
T.blocksize[1])
Sparsity_Pattern.sort_indices()
else:
# If degree is 0, just copy T for the sparsity pattern
Sparsity_Pattern = T.copy()
if 'theta' in prefilter and 'k' in prefilter:
temp_cost = [0.0]
Sparsity_theta = filter_matrix_rows(Sparsity_Pattern,
prefilter['theta'],
cost=temp_cost)
Sparsity_Pattern = truncate_rows(Sparsity_Pattern,
prefilter['k'],
cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
# Union two sparsity patterns
Sparsity_Pattern += Sparsity_theta
cost[0] += Sparsity_Pattern.nnz / float(A.nnz)
elif 'k' in prefilter:
temp_cost = [0.0]
Sparsity_Pattern = truncate_rows(Sparsity_Pattern,
prefilter['k'],
cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
elif 'theta' in prefilter:
temp_cost = [0.0]
Sparsity_Pattern = filter_matrix_rows(Sparsity_Pattern,
prefilter['theta'],
cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
elif len(prefilter) > 0:
raise ValueError("Unrecognized prefilter option")
Sparsity_Pattern.data[:] = 1.0
Sparsity_Pattern.sort_indices()
# If using root nodes, enforce identity at C-points
if Cpt_params[0]:
Sparsity_Pattern = Cpt_params[1]['I_F'] * Sparsity_Pattern
Sparsity_Pattern = Cpt_params[1]['P_I'] + Sparsity_Pattern
cost[0] += Sparsity_Pattern.nnz / float(A.nnz)
# Construct array of inv(Bi'Bi), where Bi is B restricted to row i's
# sparsity pattern in Sparsity Pattern. This array is used multiple times
# in Satisfy_Constraints(...).
temp_cost = [0.0]
BtBinv = compute_BtBinv(B, Sparsity_Pattern, cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
# If using root nodes and B has more columns that A's blocksize, then
# T must be updated so that T*B = Bfine. Note, if this is a 'secondpass'
# after dropping entries in P, then we must re-enforce the constraints
if (Cpt_params[0] and
(B.shape[1] > A.blocksize[0])) or ('secondpass' in postfilter):
temp_cost = [0.0]
T = filter_operator(T, Sparsity_Pattern, B, Bf, BtBinv, cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
# Ensure identity at C-pts
if Cpt_params[0]:
T = Cpt_params[1]['I_F'] * T + Cpt_params[1]['P_I']
# Iteratively minimize the energy of T subject to the constraints of
# Sparsity_Pattern and maintaining T's effect on B, i.e. T*B =
# (T+Update)*B, i.e. Update*B = 0
if krylov == 'cg':
T = cg_prolongation_smoothing(A, T, B, BtBinv, Sparsity_Pattern,
maxiter, tol, weighting, Cpt_params,
cost)
elif krylov == 'cgnr':
T = cgnr_prolongation_smoothing(A, T, B, BtBinv, Sparsity_Pattern,
maxiter, tol, weighting, Cpt_params,
cost)
elif krylov == 'gmres':
T = gmres_prolongation_smoothing(A, T, B, BtBinv, Sparsity_Pattern,
maxiter, tol, weighting, Cpt_params,
cost)
T.eliminate_zeros()
# Filter entries in P, only in the rootnode case, i.e., Cpt_params[0] == True
if (len(postfilter) == 0) or ('secondpass'
in postfilter) or (Cpt_params[0] is False):
return T
else:
if 'theta' in postfilter and 'k' in postfilter:
temp_cost = [0.0]
T_theta = filter_matrix_rows(T,
postfilter['theta'],
cost=temp_cost)
T_k = truncate_rows(T, postfilter['k'], cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
# Union two sparsity patterns
T_theta.data[:] = 1.0
T_k.data[:] = 1.0
T_filter = T_theta + T_k
T_filter.data[:] = 1.0
T_filter = T.multiply(T_filter)
elif 'k' in postfilter:
temp_cost = [0.0]
T_filter = truncate_rows(T, postfilter['k'], cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
elif 'theta' in postfilter:
temp_cost = [0.0]
T_filter = filter_matrix_rows(T,
postfilter['theta'],
cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
else:
raise ValueError("Unrecognized postfilter option")
# Re-smooth T_filter and re-fit the modes B into the span.
# Note, we set 'secondpass', because this is the second
# filtering pass
T = energy_prolongation_smoother(A,
T_filter,
Atilde,
B,
Bf,
Cpt_params,
krylov=krylov,
maxiter=1,
tol=1e-8,
degree=0,
weighting=weighting,
prefilter={},
postfilter={'secondpass': True},
cost=cost)
return T
def cg_prolongation_smoothing(A,
T,
B,
BtBinv,
Sparsity_Pattern,
maxiter,
tol,
weighting='local',
Cpt_params=None,
cost=[0.0]):
'''
Helper function for energy_prolongation_smoother(...)
Use CG to smooth T by solving A T = 0, subject to nullspace
and sparsity constraints.
Parameters
----------
A : {csr_matrix, bsr_matrix}
SPD sparse NxN matrix
T : {bsr_matrix}
Tentative prolongator, a NxM sparse matrix (M < N).
This is initial guess for the equation A T = 0.
Assumed that T B_c = B_f
B : {array}
Near-nullspace modes for coarse grid, i.e., B_c.
Has shape (M,k) where k is the number of coarse candidate vectors.
BtBinv : {array}
3 dimensional array such that,
BtBinv[i] = pinv(B_i.H Bi), and B_i is B restricted
to the neighborhood (in the matrix graph) of dof of i.
Sparsity_Pattern : {csr_matrix, bsr_matrix}
Sparse NxM matrix
This is the sparsity pattern constraint to enforce on the
eventual prolongator
maxiter : int
maximum number of iterations
tol : float
residual tolerance for A T = 0
weighting : {string}
'block', 'diagonal' or 'local' construction of the diagonal
preconditioning
Cpt_params : {tuple}
Tuple of the form (bool, dict). If the Cpt_params[0] = False, then
the standard SA prolongation smoothing is carried out. If True, then
dict must be a dictionary of parameters containing, (1) P_I: P_I.T is
the injection matrix for the Cpts, (2) I_F: an identity matrix
for only the F-points (i.e. I, but with zero rows and columns for
C-points) and I_C: the C-point analogue to I_F.
Returns
-------
T : {bsr_matrix}
Smoothed prolongator using conjugate gradients to solve A T = 0,
subject to the constraints, T B_c = B_f, and T has no nonzero
outside of the sparsity pattern in Sparsity_Pattern.
See Also
--------
The principal calling routine,
pyamg.aggregation.smooth.energy_prolongation_smoother
'''
# Preallocate
AP = sparse.bsr_matrix(
(np.zeros(Sparsity_Pattern.data.shape, dtype=T.dtype),
Sparsity_Pattern.indices, Sparsity_Pattern.indptr),
shape=(Sparsity_Pattern.shape))
# CG will be run with diagonal preconditioning
if weighting == 'diagonal':
Dinv = get_diagonal(A, norm_eq=False, inv=True)
elif weighting == 'block':
Dinv = get_block_diag(A, blocksize=A.blocksize[0], inv_flag=True)
Dinv = sparse.bsr_matrix(
(Dinv, np.arange(Dinv.shape[0]), np.arange(Dinv.shape[0] + 1)),
shape=A.shape)
elif weighting == 'local':
# Based on Gershgorin estimate
D = np.abs(A) * np.ones((A.shape[0], 1), dtype=A.dtype)
Dinv = np.zeros_like(D)
Dinv[D != 0] = 1.0 / np.abs(D[D != 0])
cost[0] += 1
else:
raise ValueError('weighting value is invalid')
# Calculate initial residual
# Equivalent to R = -A*T; R = R.multiply(Sparsity_Pattern)
# with the added constraint that R has an explicit 0 wherever
# R is 0 and Sparsity_Pattern is not
uones = np.zeros(Sparsity_Pattern.data.shape, dtype=T.dtype)
R = sparse.bsr_matrix(
(uones, Sparsity_Pattern.indices, Sparsity_Pattern.indptr),
shape=(Sparsity_Pattern.shape))
pyamg.amg_core.incomplete_mat_mult_bsr(
A.indptr, A.indices, np.ravel(A.data), T.indptr, T.indices,
np.ravel(T.data), R.indptr, R.indices, np.ravel(R.data),
int(T.shape[0] / T.blocksize[0]), int(T.shape[1] / T.blocksize[1]),
A.blocksize[0], A.blocksize[1], T.blocksize[1])
R.data *= -1.0
# T is block diagonal, using sparsity pattern of R with
# incomplete=True significantly overestimates complexity.
# More accurate to use full mat-mat with block diagonal T.
cost[0] += mat_mat_complexity(A, T, incomplete=False) / float(A.nnz)
# Enforce R*B = 0
temp_cost = [0.0]
Satisfy_Constraints(R, B, BtBinv, cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
if R.nnz == 0:
print("Error in sa_energy_min(..). Initial R no nonzeros on a level. \
Returning tentative prolongator\n")
return T
# Calculate Frobenius norm of the residual
resid = R.nnz # np.sqrt((R.data.conjugate()*R.data).sum())
i = 0
while i < maxiter and resid > tol:
# Apply diagonal preconditioner
if weighting == 'local' or weighting == 'diagonal':
Z = scale_rows(R, Dinv)
else:
Z = Dinv * R
cost[0] += R.nnz / float(A.nnz)
# Frobenius inner-product of (R,Z) = sum( np.conjugate(rk).*zk)
newsum = (R.conjugate().multiply(Z)).sum()
cost[0] += Z.nnz / float(A.nnz)
if newsum < tol:
# met tolerance, so halt
break
# P is the search direction, not the prolongator, which is T.
if (i == 0):
P = Z
oldsum = newsum
else:
beta = newsum / oldsum
P = Z + beta * P
cost[0] += max(Z.nnz, P.nnz) / float(A.nnz)
oldsum = newsum
# Calculate new direction and enforce constraints
# Equivalent to: AP = A*P; AP = AP.multiply(Sparsity_Pattern)
# with the added constraint that explicit zeros are in AP wherever
# AP = 0 and Sparsity_Pattern does not !!!!
AP.data[:] = 0.0
pyamg.amg_core.incomplete_mat_mult_bsr(
A.indptr, A.indices, np.ravel(A.data), P.indptr, P.indices,
np.ravel(P.data), AP.indptr, AP.indices, np.ravel(AP.data),
int(T.shape[0] / T.blocksize[0]), int(T.shape[1] / T.blocksize[1]),
A.blocksize[0], A.blocksize[1], P.blocksize[1])
cost[0] += mat_mat_complexity(A, AP, incomplete=True) / float(A.nnz)
# Enforce AP*B = 0
temp_cost = [0.0]
Satisfy_Constraints(AP, B, BtBinv, cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
# Frobenius inner-product of (P, AP)
alpha = newsum / (P.conjugate().multiply(AP)).sum()
cost[0] += max(P.nnz, AP.nnz) / float(A.nnz)
# Update the prolongator, T
T = T + alpha * P
cost[0] += max(P.nnz, T.nnz) / float(A.nnz)
# Ensure identity at C-pts
if Cpt_params[0]:
T = Cpt_params[1]['I_F'] * T + Cpt_params[1]['P_I']
# Update residual
R = R - alpha * AP
cost[0] += max(R.nnz, AP.nnz) / float(A.nnz)
# Calculate Frobenius norm of the residual
resid = R.nnz # np.sqrt((R.data.conjugate()*R.data).sum())
i += 1
return T
def cgnr_prolongation_smoothing(A,
T,
B,
BtBinv,
Sparsity_Pattern,
maxiter,
tol,
weighting='local',
Cpt_params=None,
cost=[0.0]):
'''
Helper function for energy_prolongation_smoother(...)
Use CGNR to smooth T by solving A T = 0, subject to nullspace
and sparsity constraints.
Parameters
----------
A : {csr_matrix, bsr_matrix}
SPD sparse NxN matrix
Should be at least nonsymmetric or indefinite
T : {bsr_matrix}
Tentative prolongator, a NxM sparse matrix (M < N).
This is initial guess for the equation A T = 0.
Assumed that T B_c = B_f
B : {array}
Near-nullspace modes for coarse grid, i.e., B_c.
Has shape (M,k) where k is the number of coarse candidate vectors.
BtBinv : {array}
3 dimensional array such that,
BtBinv[i] = pinv(B_i.H Bi), and B_i is B restricted
to the neighborhood (in the matrix graph) of dof of i.
Sparsity_Pattern : {csr_matrix, bsr_matrix}
Sparse NxM matrix
This is the sparsity pattern constraint to enforce on the
eventual prolongator
maxiter : int
maximum number of iterations
tol : float
residual tolerance for A T = 0
weighting : {string}
'block', 'diagonal' or 'local' construction of the diagonal
preconditioning
IGNORED here, only 'diagonal' preconditioning is used.
Cpt_params : {tuple}
Tuple of the form (bool, dict). If the Cpt_params[0] = False, then
the standard SA prolongation smoothing is carried out. If True, then
dict must be a dictionary of parameters containing, (1) P_I: P_I.T is
the injection matrix for the Cpts, (2) I_F: an identity matrix
for only the F-points (i.e. I, but with zero rows and columns for
C-points) and I_C: the C-point analogue to I_F.
Returns
-------
T : {bsr_matrix}
Smoothed prolongator using CGNR to solve A T = 0,
subject to the constraints, T B_c = B_f, and T has no nonzero
outside of the sparsity pattern in Sparsity_Pattern.
See Also
--------
The principal calling routine,
pyamg.aggregation.smooth.energy_prolongation_smoother
'''
# For non-SPD system, apply CG on Normal Equations with Diagonal
# Preconditioning (requires transpose)
Ah = A.H
Ah.sort_indices()
# Preallocate
uones = np.zeros(Sparsity_Pattern.data.shape, dtype=T.dtype)
AP = sparse.bsr_matrix(
(uones, Sparsity_Pattern.indices, Sparsity_Pattern.indptr),
shape=(Sparsity_Pattern.shape))
# D for A.H*A
Dinv = get_diagonal(A, norm_eq=1, inv=True)
# Calculate initial residual
# Equivalent to R = -Ah*(A*T); R = R.multiply(Sparsity_Pattern)
# with the added constraint that R has an explicit 0 wherever
# R is 0 and Sparsity_Pattern is not
uones = np.zeros(Sparsity_Pattern.data.shape, dtype=T.dtype)
R = sparse.bsr_matrix(
(uones, Sparsity_Pattern.indices, Sparsity_Pattern.indptr),
shape=(Sparsity_Pattern.shape))
AT = -1.0 * A * T
cost[0] += T.nnz / float(T.shape[0])
R.data[:] = 0.0
pyamg.amg_core.incomplete_mat_mult_bsr(Ah.indptr, Ah.indices,
np.ravel(Ah.data),
AT.indptr, AT.indices,
np.ravel(AT.data), R.indptr,
R.indices, np.ravel(R.data),
int(T.shape[0] / T.blocksize[0]),
int(T.shape[1] / T.blocksize[1]),
Ah.blocksize[0], Ah.blocksize[1],
T.blocksize[1])
# T is block diagonal, sparsity of AT should be well contained
# in R. incomplete=True significantly overestimates complexity
# with R. More accurate to use full mat-mat with block diagonal T.
cost[0] += mat_mat_complexity(Ah, AT, incomplete=False) / float(A.nnz)
# Enforce R*B = 0
temp_cost = [0.0]
Satisfy_Constraints(R, B, BtBinv, cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
if R.nnz == 0:
print("Error in sa_energy_min(..). Initial R no nonzeros on a level. \
Returning tentative prolongator\n")
return T
# Calculate Frobenius norm of the residual
resid = R.nnz # np.sqrt((R.data.conjugate()*R.data).sum())
i = 0
while i < maxiter and resid > tol:
# Apply diagonal preconditioner
Z = scale_rows(R, Dinv)
cost[0] += R.nnz / float(A.nnz)
# Frobenius innerproduct of (R,Z) = sum(rk.*zk)
newsum = (R.conjugate().multiply(Z)).sum()
cost[0] += R.nnz / float(A.nnz)
if newsum < tol:
# met tolerance, so halt
break
# P is the search direction, not the prolongator, which is T.
if (i == 0):
P = Z
oldsum = newsum
else:
beta = newsum / oldsum
P = Z + beta * P
cost[0] += max(Z.nnz, P.nnz) / float(A.nnz)
oldsum = newsum
# Calculate new direction
# Equivalent to: AP = Ah*(A*P); AP = AP.multiply(Sparsity_Pattern)
# with the added constraint that explicit zeros are in AP wherever
# AP = 0 and Sparsity_Pattern does not
AP_temp = A * P
cost[0] += P.nnz / float(P.shape[0])
AP.data[:] = 0.0
pyamg.amg_core.incomplete_mat_mult_bsr(
Ah.indptr, Ah.indices, np.ravel(Ah.data), AP_temp.indptr,
AP_temp.indices, np.ravel(AP_temp.data), AP.indptr, AP.indices,
np.ravel(AP.data), int(T.shape[0] / T.blocksize[0]),
int(T.shape[1] / T.blocksize[1]), Ah.blocksize[0], Ah.blocksize[1],
T.blocksize[1])
cost[0] += mat_mat_complexity(A, AP, incomplete=True) / float(A.nnz)
del AP_temp
# Enforce AP*B = 0
temp_cost = [0.0]
Satisfy_Constraints(AP, B, BtBinv, cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
# Frobenius inner-product of (P, AP)
alpha = newsum / (P.conjugate().multiply(AP)).sum()
cost[0] += max(P.nnz, AP.nnz) / float(A.nnz)
# Update the prolongator, T
T = T + alpha * P
cost[0] += max(T.nnz, P.nnz) / float(A.nnz)
# Ensure identity at C-pts
if Cpt_params[0]:
T = Cpt_params[1]['I_F'] * T + Cpt_params[1]['P_I']
# Update residual
R = R - alpha * AP
cost[0] += max(R.nnz, AP.nnz) / float(A.nnz)
# Calculate Frobenius norm of the residual
resid = R.nnz # np.sqrt((R.data.conjugate()*R.data).sum())
i += 1
return T
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 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, CF, interp, restrict, filter_operator,
coarse_grid_P, coarse_grid_R, keep):
""" helper function for local methods """
# Filter operator. Need to keep original matrix on fineest level for
# computing residuals
if (filter_operator is not None) and (filter_operator[1] != 0):
if len(levels) == 1:
A = deepcopy(levels[-1].A)
else:
A = levels[-1].A
filter_matrix_rows(A, filter_operator[1], diagonal=True, lump=filter_operator[0])
else:
A = levels[-1].A
# Check if matrix was filtered to be diagonal --> coarsest grid
if A.nnz == A.shape[0]:
return 1
# Zero initial complexities for strength, splitting and interpolation
levels[-1].complexity['CF'] = 0.0
levels[-1].complexity['strength'] = 0.0
levels[-1].complexity['interpolate'] = 0.0
# Compute the strength-of-connection matrix C, where larger
# C[i,j] denote stronger couplings between i and j.
fn, kwargs = unpack_arg(strength)
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'):
C = evolution_strength_of_connection(A, **kwargs)
elif fn == 'energy_based':
C = energy_based_strength_of_connection(A, **kwargs)
elif fn == 'algebraic_distance':
C = algebraic_distance(A, **kwargs)
elif fn == 'affinity':
C = affinity_distance(A, **kwargs)
elif fn is None:
C = A
else:
raise ValueError('unrecognized strength of connection method: %s' %
str(fn))
levels[-1].complexity['strength'] += kwargs['cost'][0] * A.nnz / float(A.nnz)
# Generate the C/F splitting
fn, kwargs = unpack_arg(CF)
if fn == 'RS':
splitting = RS(C, **kwargs)
elif fn == 'PMIS':
splitting = PMIS(C, **kwargs)
elif fn == 'PMISc':
splitting = PMISc(C, **kwargs)
elif fn == 'CLJP':
splitting = CLJP(C, **kwargs)
elif fn == 'CLJPc':
splitting = CLJPc(C, **kwargs)
elif fn == 'CR':
splitting = CR(C, **kwargs)
elif fn == 'weighted_matching':
splitting, soc = weighted_matching(C, **kwargs)
if soc is not None:
C = soc
else:
raise ValueError('unknown C/F splitting method (%s)' % CF)
levels[-1].complexity['CF'] += kwargs['cost'][0] * C.nnz / float(A.nnz)
temp = np.sum(splitting)
if (temp == len(splitting)) or (temp == 0):
return 1
# Generate the interpolation matrix that maps from the coarse-grid to the
# fine-grid
r_flag = False
fn, kwargs = unpack_arg(interp)
if fn == 'standard':
P = standard_interpolation(A, C, splitting, **kwargs)
elif fn == 'distance_two':
P = distance_two_interpolation(A, C, splitting, **kwargs)
elif fn == 'direct':
P = direct_interpolation(A, C, splitting, **kwargs)
elif fn == 'one_point':
P = one_point_interpolation(A, C, splitting, **kwargs)
elif fn == 'inject':
P = injection_interpolation(A, splitting, **kwargs)
elif fn == 'neumann':
P = neumann_ideal_interpolation(A, splitting, **kwargs)
elif fn == 'scaledAfc':
P = scaled_Afc_interpolation(A, splitting, **kwargs)
elif fn == 'air':
if isspmatrix_bsr(A):
temp_A = bsr_matrix(A.T)
P = local_AIR(temp_A, splitting, **kwargs)
P = bsr_matrix(P.T)
else:
temp_A = csr_matrix(A.T)
P = local_AIR(temp_A, splitting, **kwargs)
P = csr_matrix(P.T)
elif fn == 'restrict':
r_flag = True
else:
raise ValueError('unknown interpolation method (%s)' % interp)
levels[-1].complexity['interpolate'] += kwargs['cost'][0] * A.nnz / float(A.nnz)
# Build restriction operator
fn, kwargs = unpack_arg(restrict)
if fn is None:
R = P.T
elif fn == 'air':
R = local_AIR(A, splitting, **kwargs)
elif fn == 'neumann':
R = neumann_AIR(A, splitting, **kwargs)
elif fn == 'one_point': # Don't need A^T here
temp_C = C.T.tocsr()
R = one_point_interpolation(A, temp_C, splitting, **kwargs)
if isspmatrix_bsr(A):
R = R.T.tobsr()
else:
R = R.T.tocsr()
elif fn == 'inject': # Don't need A^T or C^T here
R = injection_interpolation(A, splitting, **kwargs)
if isspmatrix_bsr(A):
R = R.T.tobsr()
else:
R = R.T.tocsr()
elif fn == 'standard':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R = standard_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = standard_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'distance_two':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R = distance_two_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = distance_two_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'direct':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
else:
raise ValueError('unknown restriction method (%s)' % restrict)
# If set P = R^T
if r_flag:
P = R.T
# Optional different interpolation for RAP
fn, kwargs = unpack_arg(coarse_grid_P)
if fn == 'standard':
P_temp = standard_interpolation(A, C, splitting, **kwargs)
elif fn == 'distance_two':
P_temp = distance_two_interpolation(A, C, splitting, **kwargs)
elif fn == 'direct':
P_temp = direct_interpolation(A, C, splitting, **kwargs)
elif fn == 'one_point':
P_temp = one_point_interpolation(A, C, splitting, **kwargs)
elif fn == 'inject':
P_temp = injection_interpolation(A, splitting, **kwargs)
elif fn == 'neumann':
P_temp = neumann_ideal_interpolation(A, splitting, **kwargs)
elif fn == 'air':
if isspmatrix_bsr(A):
temp_A = bsr_matrix(A.T)
P_temp = local_AIR(temp_A, splitting, **kwargs)
P_temp = bsr_matrix(P_temp.T)
else:
temp_A = csr_matrix(A.T)
P_temp = local_AIR(temp_A, splitting, **kwargs)
P_temp = csr_matrix(P_temp.T)
else:
P_temp = P
# Optional different restriction for RAP
fn, kwargs = unpack_arg(coarse_grid_R)
if fn == 'air':
R_temp = local_AIR(A, splitting, **kwargs)
elif fn == 'neumann':
R_temp = neumann_AIR(A, splitting, **kwargs)
elif fn == 'one_point': # Don't need A^T here
temp_C = C.T.tocsr()
R_temp = one_point_interpolation(A, temp_C, splitting, **kwargs)
if isspmatrix_bsr(A):
R_temp = R_temp.T.tobsr()
else:
R_temp = R_temp.T.tocsr()
elif fn == 'inject': # Don't need A^T or C^T here
R_temp = injection_interpolation(A, splitting, **kwargs)
if isspmatrix_bsr(A):
R_temp = R_temp.T.tobsr()
else:
R_temp = R_temp.T.tocsr()
elif fn == 'standard':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R_temp = standard_interpolation(temp_A, temp_C, splitting, **kwargs)
R_temp = R_temp.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R_temp = standard_interpolation(temp_A, temp_C, splitting, **kwargs)
R_temp = R_temp.T.tocsr()
elif fn == 'distance_two':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R_temp = distance_two_interpolation(temp_A, temp_C, splitting, **kwargs)
R_temp = R_temp.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R_temp = distance_two_interpolation(temp_A, temp_C, splitting, **kwargs)
R_temp = R_temp.T.tocsr()
elif fn == 'direct':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R_temp = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R_temp = R_temp.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R_temp = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R_temp = R_temp.T.tocsr()
else:
R_temp = R
# Store relevant information for this level
if keep:
levels[-1].C = C # strength of connection matrix
levels[-1].P = P # prolongation operator
levels[-1].R = R # restriction operator
levels[-1].splitting = splitting # C/F splitting
# Form coarse grid operator, get complexity
#levels[-1].complexity['RAP'] = mat_mat_complexity(R_temp,A) / float(A.nnz)
#RA = R_temp * A
#levels[-1].complexity['RAP'] += mat_mat_complexity(RA,P_temp) / float(A.nnz)
#A = RA * P_temp
# RL: RAP = R*(A*P)
levels[-1].complexity['RAP'] = mat_mat_complexity(A, P_temp) / float(A.nnz)
AP = A * P_temp
levels[-1].complexity['RAP'] += mat_mat_complexity(R_temp, AP) / float(A.nnz)
A = R_temp * AP
# Make sure coarse-grid operator is in correct sparse format
if (isspmatrix_csr(P) and (not isspmatrix_csr(A))):
A = A.tocsr()
elif (isspmatrix_bsr(P) and (not isspmatrix_bsr(A))):
A = A.tobsr()
A.eliminate_zeros()
levels.append(multilevel_solver.level())
levels[-1].A = A
return 0
def extend_hierarchy(levels, strength, CF, interp, restrict, filter_operator,
coarse_grid_P, coarse_grid_R, keep):
""" helper function for local methods """
# Filter operator. Need to keep original matrix on fineest level for
# computing residuals
if (filter_operator is not None) and (filter_operator[1] != 0):
if len(levels) == 1:
A = deepcopy(levels[-1].A)
else:
A = levels[-1].A
filter_matrix_rows(A,
filter_operator[1],
diagonal=True,
lump=filter_operator[0])
else:
A = levels[-1].A
# Check if matrix was filtered to be diagonal --> coarsest grid
if A.nnz == A.shape[0]:
return 1
# Zero initial complexities for strength, splitting and interpolation
levels[-1].complexity['CF'] = 0.0
levels[-1].complexity['strength'] = 0.0
levels[-1].complexity['interpolate'] = 0.0
# Compute the strength-of-connection matrix C, where larger
# C[i,j] denote stronger couplings between i and j.
fn, kwargs = unpack_arg(strength)
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'):
C = evolution_strength_of_connection(A, **kwargs)
elif fn == 'energy_based':
C = energy_based_strength_of_connection(A, **kwargs)
elif fn == 'algebraic_distance':
C = algebraic_distance(A, **kwargs)
elif fn == 'affinity':
C = affinity_distance(A, **kwargs)
elif fn is None:
C = A
else:
raise ValueError('unrecognized strength of connection method: %s' %
str(fn))
levels[-1].complexity['strength'] += kwargs['cost'][0] * A.nnz / float(
A.nnz)
# Generate the C/F splitting
fn, kwargs = unpack_arg(CF)
if fn == 'RS':
splitting = RS(C, **kwargs)
elif fn == 'PMIS':
splitting = PMIS(C, **kwargs)
elif fn == 'PMISc':
splitting = PMISc(C, **kwargs)
elif fn == 'CLJP':
splitting = CLJP(C, **kwargs)
elif fn == 'CLJPc':
splitting = CLJPc(C, **kwargs)
elif fn == 'CR':
splitting = CR(C, **kwargs)
elif fn == 'weighted_matching':
splitting, soc = weighted_matching(C, **kwargs)
if soc is not None:
C = soc
else:
raise ValueError('unknown C/F splitting method (%s)' % CF)
levels[-1].complexity['CF'] += kwargs['cost'][0] * C.nnz / float(A.nnz)
temp = np.sum(splitting)
if (temp == len(splitting)) or (temp == 0):
return 1
# Generate the interpolation matrix that maps from the coarse-grid to the
# fine-grid
r_flag = False
fn, kwargs = unpack_arg(interp)
if fn == 'standard':
P = standard_interpolation(A, C, splitting, **kwargs)
elif fn == 'distance_two':
P = distance_two_interpolation(A, C, splitting, **kwargs)
elif fn == 'direct':
P = direct_interpolation(A, C, splitting, **kwargs)
elif fn == 'one_point':
P = one_point_interpolation(A, C, splitting, **kwargs)
elif fn == 'inject':
P = injection_interpolation(A, splitting, **kwargs)
elif fn == 'neumann':
P = neumann_ideal_interpolation(A, splitting, **kwargs)
elif fn == 'scaledAfc':
P = scaled_Afc_interpolation(A, splitting, **kwargs)
elif fn == 'air':
if isspmatrix_bsr(A):
temp_A = bsr_matrix(A.T)
P = local_AIR(temp_A, splitting, **kwargs)
P = bsr_matrix(P.T)
else:
temp_A = csr_matrix(A.T)
P = local_AIR(temp_A, splitting, **kwargs)
P = csr_matrix(P.T)
elif fn == 'restrict':
r_flag = True
else:
raise ValueError('unknown interpolation method (%s)' % interp)
levels[-1].complexity['interpolate'] += kwargs['cost'][0] * A.nnz / float(
A.nnz)
# Build restriction operator
fn, kwargs = unpack_arg(restrict)
if fn is None:
R = P.T
elif fn == 'air':
R = local_AIR(A, splitting, **kwargs)
elif fn == 'neumann':
R = neumann_AIR(A, splitting, **kwargs)
elif fn == 'one_point': # Don't need A^T here
temp_C = C.T.tocsr()
R = one_point_interpolation(A, temp_C, splitting, **kwargs)
if isspmatrix_bsr(A):
R = R.T.tobsr()
else:
R = R.T.tocsr()
elif fn == 'inject': # Don't need A^T or C^T here
R = injection_interpolation(A, splitting, **kwargs)
if isspmatrix_bsr(A):
R = R.T.tobsr()
else:
R = R.T.tocsr()
elif fn == 'standard':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R = standard_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = standard_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'distance_two':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R = distance_two_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = distance_two_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'direct':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
else:
raise ValueError('unknown restriction method (%s)' % restrict)
# If set P = R^T
if r_flag:
P = R.T
# Optional different interpolation for RAP
fn, kwargs = unpack_arg(coarse_grid_P)
if fn == 'standard':
P_temp = standard_interpolation(A, C, splitting, **kwargs)
elif fn == 'distance_two':
P_temp = distance_two_interpolation(A, C, splitting, **kwargs)
elif fn == 'direct':
P_temp = direct_interpolation(A, C, splitting, **kwargs)
elif fn == 'one_point':
P_temp = one_point_interpolation(A, C, splitting, **kwargs)
elif fn == 'inject':
P_temp = injection_interpolation(A, splitting, **kwargs)
elif fn == 'neumann':
P_temp = neumann_ideal_interpolation(A, splitting, **kwargs)
elif fn == 'air':
if isspmatrix_bsr(A):
temp_A = bsr_matrix(A.T)
P_temp = local_AIR(temp_A, splitting, **kwargs)
P_temp = bsr_matrix(P_temp.T)
else:
temp_A = csr_matrix(A.T)
P_temp = local_AIR(temp_A, splitting, **kwargs)
P_temp = csr_matrix(P_temp.T)
else:
P_temp = P
# Optional different restriction for RAP
fn, kwargs = unpack_arg(coarse_grid_R)
if fn == 'air':
R_temp = local_AIR(A, splitting, **kwargs)
elif fn == 'neumann':
R_temp = neumann_AIR(A, splitting, **kwargs)
elif fn == 'one_point': # Don't need A^T here
temp_C = C.T.tocsr()
R_temp = one_point_interpolation(A, temp_C, splitting, **kwargs)
if isspmatrix_bsr(A):
R_temp = R_temp.T.tobsr()
else:
R_temp = R_temp.T.tocsr()
elif fn == 'inject': # Don't need A^T or C^T here
R_temp = injection_interpolation(A, splitting, **kwargs)
if isspmatrix_bsr(A):
R_temp = R_temp.T.tobsr()
else:
R_temp = R_temp.T.tocsr()
elif fn == 'standard':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R_temp = standard_interpolation(temp_A, temp_C, splitting,
**kwargs)
R_temp = R_temp.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R_temp = standard_interpolation(temp_A, temp_C, splitting,
**kwargs)
R_temp = R_temp.T.tocsr()
elif fn == 'distance_two':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R_temp = distance_two_interpolation(temp_A, temp_C, splitting,
**kwargs)
R_temp = R_temp.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R_temp = distance_two_interpolation(temp_A, temp_C, splitting,
**kwargs)
R_temp = R_temp.T.tocsr()
elif fn == 'direct':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R_temp = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R_temp = R_temp.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R_temp = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R_temp = R_temp.T.tocsr()
else:
R_temp = R
# Store relevant information for this level
if keep:
levels[-1].C = C # strength of connection matrix
levels[-1].P = P # prolongation operator
levels[-1].R = R # restriction operator
levels[-1].splitting = splitting # C/F splitting
# Form coarse grid operator, get complexity
#levels[-1].complexity['RAP'] = mat_mat_complexity(R_temp,A) / float(A.nnz)
#RA = R_temp * A
#levels[-1].complexity['RAP'] += mat_mat_complexity(RA,P_temp) / float(A.nnz)
#A = RA * P_temp
# RL: RAP = R*(A*P)
levels[-1].complexity['RAP'] = mat_mat_complexity(A, P_temp) / float(A.nnz)
AP = A * P_temp
levels[-1].complexity['RAP'] += mat_mat_complexity(R_temp, AP) / float(
A.nnz)
A = R_temp * AP
# Make sure coarse-grid operator is in correct sparse format
if (isspmatrix_csr(P) and (not isspmatrix_csr(A))):
A = A.tocsr()
elif (isspmatrix_bsr(P) and (not isspmatrix_bsr(A))):
A = A.tobsr()
A.eliminate_zeros()
levels.append(multilevel_solver.level())
levels[-1].A = A
return 0
def extend_hierarchy(levels, strength, CF, interpolation, restriction, keep):
""" helper function for local methods """
A = levels[-1].A
# Compute the strength-of-connection matrix C, where larger
# C[i,j] denote stronger couplings between i and j.
fn, kwargs = unpack_arg(strength)
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'):
C = evolution_strength_of_connection(A, **kwargs)
elif fn == 'energy_based':
C = energy_based_strength_of_connection(A, **kwargs)
elif fn == 'algebraic_distance':
C = algebraic_distance(A, **kwargs)
elif fn == 'affinity':
C = affinity_distance(A, **kwargs)
elif fn is None:
C = A
else:
raise ValueError('unrecognized strength of connection method: %s' %
str(fn))
levels[-1].complexity['strength'] = kwargs['cost'][0]
# Generate the C/F splitting
fn, kwargs = unpack_arg(CF)
if fn == 'RS':
splitting = split.RS(C, **kwargs)
elif fn == 'PMIS':
splitting = split.PMIS(C, **kwargs)
elif fn == 'PMISc':
splitting = split.PMISc(C, **kwargs)
elif fn == 'CLJP':
splitting = split.CLJP(C, **kwargs)
elif fn == 'CLJPc':
splitting = split.CLJPc(C, **kwargs)
elif fn == 'CR':
splitting = CR(C, **kwargs)
else:
raise ValueError('unknown C/F splitting method (%s)' % CF)
levels[-1].complexity['CF'] = kwargs['cost'][0]
# Generate the interpolation matrix that maps from the coarse-grid to the
# fine-grid
fn, kwargs = unpack_arg(interpolation)
if fn == 'standard':
P = standard_interpolation(A, C, splitting, **kwargs)
elif fn == 'distance_two':
P = distance_two_interpolation(A, C, splitting, **kwargs)
elif fn == 'direct':
P = direct_interpolation(A, C, splitting, **kwargs)
elif fn == 'one_point':
P = one_point_interpolation(A, C, splitting, **kwargs)
elif fn == 'injection':
P = injection_interpolation(A, splitting, **kwargs)
else:
raise ValueError('unknown interpolation method (%s)' % interpolation)
levels[-1].complexity['interpolate'] = kwargs['cost'][0]
# Generate the restriction matrix that maps from the fine-grid to the
# coarse-grid. Must make sure transpose matrices remain in CSR or BSR
fn, kwargs = unpack_arg(restriction)
if isspmatrix_csr(A):
if restriction == 'galerkin':
R = P.T.tocsr()
elif fn == 'standard':
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = standard_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'distance_two':
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = distance_two_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'direct':
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'one_point': # Don't need A^T here
temp_C = C.T.tocsr()
R = one_point_interpolation(A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'injection': # Don't need A^T or C^T here
R = injection_interpolation(A, splitting, **kwargs)
R = R.T.tocsr()
else:
raise ValueError('unknown interpolation method (%s)' % interpolation)
else:
if restriction == 'galerkin':
R = P.T.tobsr()
elif fn == 'standard':
temp_A = A.T.tobsr()
temp_C = C.T.tocsr()
R = standard_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
elif fn == 'distance_two':
temp_A = A.T.tobsr()
temp_C = C.T.tocsr()
R = distance_two_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
elif fn == 'direct':
temp_A = A.T.tobsr()
temp_C = C.T.tocsr()
R = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
elif fn == 'one_point': # Don't need A^T here
temp_C = C.T.tocsr()
R = one_point_interpolation(A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
elif fn == 'injection': # Don't need A^T or C^T here
R = injection_interpolation(A, splitting, **kwargs)
R = R.T.tobsr()
else:
raise ValueError('unknown interpolation method (%s)' % interpolation)
levels[-1].complexity['restriction'] = kwargs['cost'][0]
# Store relevant information for this level
if keep:
levels[-1].C = C # strength of connection matrix
levels[-1].P = P # prolongation operator
levels[-1].R = R # restriction operator
levels[-1].splitting = splitting # C/F splitting
# 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
# Make sure coarse-grid operator is in correct sparse format
if (isspmatrix_csr(P) and (not isspmatrix_csr(A))):
A = A.tocsr()
elif (isspmatrix_bsr(P) and (not isspmatrix_bsr(A))):
A = A.tobsr()
# Form next level through Galerkin product
levels.append(multilevel_solver.level())
levels[-1].A = A
def extend_hierarchy(levels, strength, CF, interpolation, restriction, keep):
""" helper function for local methods """
A = levels[-1].A
# Compute the strength-of-connection matrix C, where larger
# C[i,j] denote stronger couplings between i and j.
fn, kwargs = unpack_arg(strength)
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'):
C = evolution_strength_of_connection(A, **kwargs)
elif fn == 'energy_based':
C = energy_based_strength_of_connection(A, **kwargs)
elif fn == 'algebraic_distance':
C = algebraic_distance(A, **kwargs)
elif fn == 'affinity':
C = affinity_distance(A, **kwargs)
elif fn is None:
C = A
else:
raise ValueError('unrecognized strength of connection method: %s' %
str(fn))
levels[-1].complexity['strength'] = kwargs['cost'][0]
# Generate the C/F splitting
fn, kwargs = unpack_arg(CF)
if fn == 'RS':
splitting = split.RS(C, **kwargs)
elif fn == 'PMIS':
splitting = split.PMIS(C, **kwargs)
elif fn == 'PMISc':
splitting = split.PMISc(C, **kwargs)
elif fn == 'CLJP':
splitting = split.CLJP(C, **kwargs)
elif fn == 'CLJPc':
splitting = split.CLJPc(C, **kwargs)
elif fn == 'CR':
splitting = CR(C, **kwargs)
else:
raise ValueError('unknown C/F splitting method (%s)' % CF)
levels[-1].complexity['CF'] = kwargs['cost'][0]
# Generate the interpolation matrix that maps from the coarse-grid to the
# fine-grid
fn, kwargs = unpack_arg(interpolation)
if fn == 'standard':
P = standard_interpolation(A, C, splitting, **kwargs)
elif fn == 'distance_two':
P = distance_two_interpolation(A, C, splitting, **kwargs)
elif fn == 'direct':
P = direct_interpolation(A, C, splitting, **kwargs)
elif fn == 'one_point':
P = one_point_interpolation(A, C, splitting, **kwargs)
elif fn == 'injection':
P = injection_interpolation(A, splitting, **kwargs)
else:
raise ValueError('unknown interpolation method (%s)' % interpolation)
levels[-1].complexity['interpolate'] = kwargs['cost'][0]
# Generate the restriction matrix that maps from the fine-grid to the
# coarse-grid. Must make sure transpose matrices remain in CSR or BSR
fn, kwargs = unpack_arg(restriction)
if isspmatrix_csr(A):
if restriction == 'galerkin':
R = P.T.tocsr()
elif fn == 'standard':
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = standard_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'distance_two':
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = distance_two_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'direct':
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'one_point': # Don't need A^T here
temp_C = C.T.tocsr()
R = one_point_interpolation(A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'injection': # Don't need A^T or C^T here
R = injection_interpolation(A, splitting, **kwargs)
R = R.T.tocsr()
else:
raise ValueError('unknown interpolation method (%s)' %
interpolation)
else:
if restriction == 'galerkin':
R = P.T.tobsr()
elif fn == 'standard':
temp_A = A.T.tobsr()
temp_C = C.T.tocsr()
R = standard_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
elif fn == 'distance_two':
temp_A = A.T.tobsr()
temp_C = C.T.tocsr()
R = distance_two_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
elif fn == 'direct':
temp_A = A.T.tobsr()
temp_C = C.T.tocsr()
R = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
elif fn == 'one_point': # Don't need A^T here
temp_C = C.T.tocsr()
R = one_point_interpolation(A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
elif fn == 'injection': # Don't need A^T or C^T here
R = injection_interpolation(A, splitting, **kwargs)
R = R.T.tobsr()
else:
raise ValueError('unknown interpolation method (%s)' %
interpolation)
levels[-1].complexity['restriction'] = kwargs['cost'][0]
# Store relevant information for this level
if keep:
levels[-1].C = C # strength of connection matrix
levels[-1].P = P # prolongation operator
levels[-1].R = R # restriction operator
levels[-1].splitting = splitting # C/F splitting
# 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
# Make sure coarse-grid operator is in correct sparse format
if (isspmatrix_csr(P) and (not isspmatrix_csr(A))):
A = A.tocsr()
elif (isspmatrix_bsr(P) and (not isspmatrix_bsr(A))):
A = A.tobsr()
# Form next level through Galerkin product
levels.append(multilevel_solver.level())
levels[-1].A = A
def evolution_strength_of_connection(A, B=None, epsilon=4.0, k=2,
proj_type="l2", weighting='diagonal',
symmetrize_measure=True, cost=[0]):
"""
Construct strength of connection matrix using an Evolution-based measure
Parameters
----------
A : {csr_matrix, bsr_matrix}
Sparse NxN matrix
B : {string, array}
If B=None, 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
weighting : {string}
'block', 'diagonal' or 'local' construction of the D-inverse
used to precondition A before "evolving" delta-functions. The
local option is the cheapest.
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 as np
>>> from pyamg.gallery import stencil_grid
>>> from pyamg.strength import evolution_strength_of_connection
>>> n=3
>>> stencil = np.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, np.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 is None:
Bmat = np.mat(np.ones((A.shape[0], 1), dtype=A.dtype))
else:
Bmat = np.mat(B)
# Is matrix A CSR?
if (not sparse.isspmatrix_csr(A)):
numPDEs = A.blocksize[0]
csrflag = False
else:
numPDEs = 1
csrflag = True
# Pre-process A. We need A in CSR, to be devoid of explicit 0's, have
# sorted indices and be scaled by D-inverse
if weighting == 'block':
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()
cost[0] += 1
elif weighting == 'diagonal':
D = A.diagonal()
Dinv = get_diagonal(A, norm_eq=False, inv=True)
Dinv[D == 0] = 1.0
Dinv_A = scale_rows(A, Dinv, copy=True)
cost[0] += 1
elif weighting == 'local':
D = np.abs(A)*np.ones((A.shape[0], 1), dtype=A.dtype)
Dinv = np.zeros_like(D)
Dinv[D != 0] = 1.0 / np.abs(D[D != 0])
Dinv[D == 0] = 1.0
Dinv_A = scale_rows(A, Dinv, copy=True)
cost[0] += 1
else:
raise ValueError('Unrecognized weighting for Evolution measure')
A = A.tocsr()
A.eliminate_zeros()
A.sort_indices()
# Handle preliminaries for the algorithm
dimen = A.shape[1]
NullDim = Bmat.shape[1]
if weighting == 'diagonal' or weighting == 'block':
# Get spectral radius of Dinv*A, scales the time step size for the ODE
rho_DinvA = approximate_spectral_radius(Dinv_A)
cost[0] += 15 # 15 lanczos iterations to approximate spectral radius
else:
# Using local weighting, no need for spectral radius
rho_DinvA = 1.0
# Calculate D_A for later use in the minimization problem
if proj_type == "D_A":
D = A.diagonal()
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
# We transpose the product, so that we can efficiently access
# the columns in CSR format. We want the columns (not rows) because
# strength is based on the columns of (I - delta_t Dinv A)^k, i.e.,
# relaxed delta functions
# 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()
cost[0] += 1
# 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(np.arange(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.")
JacobiStep = csr_matrix(Atilde, copy=True)
# Calculate (Atilde^nsquare)^T = (Atilde^T)^nsquare
for i in range(nsquare):
cost[0] += mat_mat_complexity(Atilde,Atilde)
Atilde = Atilde*Atilde
for i in range(ninc):
cost[0] += mat_mat_complexity(Atilde,JacobiStep)
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()
cost[0] += Atilde.nnz / float(A.nnz)
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):
cost[0] += mat_mat_complexity(Atilde,Atilde)
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)
cost[0] += mat_mat_complexity(Atilde,mask,incomplete=True) / float(A.nnz)
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
cost[0] += Atilde.shape[0] / float(A.nnz)
# 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))
cost[0] += 2.0 * Atilde.nnz / float(A.nnz)
# 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)
cost[0] += Atilde.nnz / float(A.nnz)
if Atilde.dtype is 'complex':
cost[0] += Atilde.nnz / float(A.nnz)
# Calculate Approximation ratio
Atilde.data = Atilde.data/data
cost[0] += Atilde.nnz / float(A.nnz)
# 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)
cost[0] += Atilde.nnz / float(A.nnz)
# 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(np.arange(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
cost[0] += B.shape[0] / float(A.nnz)
# 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.
# This function is doing similar to NullDim=1 with more bad guys.
# Complexity accounts for computing the block inverse, and
# hat{z_i} = B_i*x, hat{z_i} .* hat{z_i},
# hat{z_i} = hat{z_i} / z_i, and abs(1.0 - hat{z_i}).
cost[0] += ( Atilde.nnz*(3+NullDim) + (NullDim**3)*dimen ) / float(A.nnz)
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 epsilon != np.inf:
cost[0] += Atilde.nnz / float(A.nnz)
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
cost[0] += Atilde.shape[0] / float(A.nnz)
# 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=(int(Atilde.shape[0] / numPDEs),
int(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
cost[0] += Atilde.nnz / float(A.nnz)
# Scale C by the largest magnitude entry in each row
Atilde = scale_rows_by_largest_entry(Atilde)
cost[0] += Atilde.nnz / float(A.nnz)
# Symmetrize
if symmetrize_measure:
Atilde = 0.5*(Atilde + Atilde.T)
cost[0] += Atilde.nnz / float(A.nnz)
return Atilde