"""Generic AMG solver."""
import inspect
from warnings import warn
import scipy as sp
from scipy.linalg import pinv
import scipy.sparse.linalg as sla
from scipy.sparse.linalg import LinearOperator
import numpy as np
from . import krylov
from .util.utils import to_type
from .util.params import set_tol
from .relaxation import smoothing
from .util import upcast
[docs]
class MultilevelSolver:
"""Stores multigrid hierarchy and implements the multigrid cycle.
The class constructs the cycling process and points to the methods for
coarse grid solves. A MultilevelSolver object is typically returned from a
particular AMG method (see ruge_stuben_solver or smoothed_aggregation_solver
for example). A call to MultilevelSolver.solve() is a typical access
point. The class also defines methods for constructing operator, cycle, and
grid complexities.
Attributes
----------
levels : level array
Array of level objects that contain A, R, and P.
coarse_solver : string
String passed to coarse_grid_solver indicating the solve type
Methods
-------
aspreconditioner()
Create a preconditioner using this multigrid cycle
cycle_complexity()
A measure of the cost of a single multigrid cycle.
grid_complexity()
A measure of the rate of coarsening.
operator_complexity()
A measure of the size of the multigrid hierarchy.
solve()
Iteratively solves a linear system for the right hand side.
change_solve_matrix(A)
Change matrix solve/preconditioning matrix.
This also changes the corresponding relaxation routines on the fine
grid. This can be used, for example, to precondition a
quadratic finite element discretization with AMG built from
a linear discretization on quadratic quadrature points.
"""
[docs]
class Level:
"""Stores one level of the multigrid hierarchy.
All level objects will have an 'A' attribute referencing the matrix
of that level. All levels, except for the coarsest level, will
also have 'P' and 'R' attributes referencing the prolongation and
restriction operators that act between each level and the next
coarser level.
Attributes
----------
A : csr_matrix
Problem matrix for Ax=b
R : csr_matrix
Restriction matrix between levels (often R = P.T)
P : csr_matrix
Prolongation or Interpolation matrix.
Notes
-----
The functionality of this class is a struct
"""
[docs]
def __init__(self):
"""Level construct (empty)."""
self.A = None
[docs]
class level(Level): # noqa: N801
"""Deprecated level class."""
[docs]
def __init__(self):
"""Raise deprecation warning on use, not import."""
super().__init__()
warn('level() is deprecated. use Level()',
category=DeprecationWarning, stacklevel=2)
[docs]
def __init__(self, levels, coarse_solver='pinv'):
"""Class constructor to initialize the cycle and ensure list of levels is complete.
Parameters
----------
levels : level array
Array of level objects that contain A, R, and P.
coarse_solver: string, callable, tuple
The solver method is either (1) a string such as 'splu' or 'pinv'
of a callable object which receives only parameters (A, b) and
returns an (approximate or exact) solution to the linear system Ax
= b, or (2) a callable object that takes parameters (A,b) and
returns an (approximate or exact) solution to Ax = b, or (3) a
tuple of the form (string|callable, args), where args is a
dictionary of arguments to be passed to the function denoted by
string or callable.
Sparse direct methods:
* splu : sparse LU solver
Sparse iterative methods:
* any method in scipy.sparse.linalg or pyamg.krylov (e.g. 'cg').
* Methods in pyamg.krylov take precedence.
* relaxation method, such as 'gauss_seidel' or 'jacobi',
Dense methods:
* pinv : pseudoinverse (SVD)
* lu : LU factorization
* cholesky : Cholesky factorization
Notes
-----
If not defined, the R attribute on each level is set to
the transpose of P.
Examples
--------
>>> # manual construction of a two-level AMG hierarchy
>>> from pyamg.gallery import poisson
>>> from pyamg.multilevel import MultilevelSolver
>>> from pyamg.strength import classical_strength_of_connection
>>> from pyamg.classical.interpolate import direct_interpolation
>>> from pyamg.classical.split import RS
>>> # compute necessary operators
>>> A = poisson((100, 100), format='csr')
>>> C = classical_strength_of_connection(A)
>>> splitting = RS(A)
>>> P = direct_interpolation(A, C, splitting)
>>> R = P.T
>>> # store first level data
>>> levels = []
>>> levels.append(MultilevelSolver.Level())
>>> levels.append(MultilevelSolver.Level())
>>> levels[0].A = A
>>> levels[0].C = C
>>> levels[0].splitting = splitting
>>> levels[0].P = P
>>> levels[0].R = R
>>> # store second level data
>>> levels[1].A = R @ A @ P # coarse-level matrix
>>> # create MultilevelSolver
>>> ml = MultilevelSolver(levels, coarse_solver='splu')
>>> print(ml)
MultilevelSolver
Number of Levels: 2
Operator Complexity: 1.891
Grid Complexity: 1.500
Coarse Solver: 'splu'
level unknowns nonzeros
0 10000 49600 [52.88%]
1 5000 44202 [47.12%]
<BLANKLINE>
"""
self.symmetric_smoothing = False # force change_smoothers to set to True
self.levels = levels
self.coarse_solver = coarse_grid_solver(coarse_solver)
for level in levels[:-1]:
if not hasattr(level, 'R'):
level.R = level.P.H
[docs]
def __repr__(self):
"""Print basic statistics about the multigrid hierarchy."""
output = 'MultilevelSolver\n'
output += f'Number of Levels: {len(self.levels)}\n'
output += f'Operator Complexity: {self.operator_complexity():6.3f}\n'
output += f'Grid Complexity: {self.grid_complexity():6.3f}\n'
output += f'Coarse Solver: {self.coarse_solver.name()}\n'
total_nnz = sum(level.A.nnz for level in self.levels)
# 123456712345678901 123456789012 123456789
# 0 10000 49600 [52.88%]
output += ' level unknowns nonzeros\n'
for n, level in enumerate(self.levels):
A = level.A
ratio = 100 * A.nnz / total_nnz
output += f'{n:>6} {A.shape[1]:>11} {A.nnz:>12} [{ratio:2.2f}%]\n'
return output
[docs]
def cycle_complexity(self, cycle='V'):
"""Cycle complexity of V, W, AMLI, and F(1,1) cycle with simple relaxation.
Cycle complexity is an approximate measure of the number of
floating point operations (FLOPs) required to perform a single
multigrid cycle relative to the cost a single smoothing operation.
Parameters
----------
cycle : {'V','W','F','AMLI'}
Type of multigrid cycle to perform in each iteration.
Returns
-------
cc : float
Defined as F_sum / F_0, where
F_sum is the total number of nonzeros in the matrix on all
levels encountered during a cycle and F_0 is the number of
nonzeros in the matrix on the finest level.
Notes
-----
This is only a rough estimate of the true cycle complexity. The
estimate assumes that the cost of pre and post-smoothing are
(each) equal to the number of nonzeros in the matrix on that level.
This assumption holds for smoothers like Jacobi and Gauss-Seidel.
However, the true cycle complexity of cycle using more expensive
methods, like block Gauss-Seidel will be underestimated.
Additionally, if the cycle used in practice isn't a (1,1)-cycle,
then this cost estimate will be off.
"""
cycle = str(cycle).upper()
nnz = [level.A.nnz for level in self.levels]
def V(level):
if len(self.levels) == 1:
return nnz[0]
if level == len(self.levels) - 2:
return 2 * nnz[level] + nnz[level + 1]
return 2 * nnz[level] + V(level + 1)
def W(level):
if len(self.levels) == 1:
return nnz[0]
if level == len(self.levels) - 2:
return 2 * nnz[level] + nnz[level + 1]
return 2 * nnz[level] + 2 * W(level + 1)
def F(level):
if len(self.levels) == 1:
return nnz[0]
if level == len(self.levels) - 2:
return 2 * nnz[level] + nnz[level + 1]
return 2 * nnz[level] + F(level + 1) + V(level + 1)
if cycle == 'V':
flops = V(0)
elif cycle in ('W', 'AMLI'):
flops = W(0)
elif cycle == 'F':
flops = F(0)
else:
raise TypeError(f'Unrecognized cycle type ({cycle})')
return float(flops) / float(nnz[0])
[docs]
def operator_complexity(self):
"""Operator complexity of this multigrid hierarchy.
Defined as:
Number of nonzeros in the matrix on all levels /
Number of nonzeros in the matrix on the finest level
"""
return sum(level.A.nnz for level in self.levels) /\
float(self.levels[0].A.nnz)
[docs]
def grid_complexity(self):
"""Grid complexity of this multigrid hierarchy.
Defined as:
Number of unknowns on all levels /
Number of unknowns on the finest level
"""
return sum(level.A.shape[0] for level in self.levels) /\
float(self.levels[0].A.shape[0])
[docs]
def change_solve_matrix(self, A):
"""Change matrix solve/preconditioning matrix.
Parameters
----------
A : csr_matrix
Target solution matrix
Notes
-----
This also changes the corresponding relaxation routines on the fine
grid. This can be used, for example, to precondition a
quadratic finite element discretization with linears.
"""
self.levels[0].A = A
smoothing.rebuild_smoother(self.levels[0])
[docs]
def psolve(self, b):
"""Legacy solve interface."""
return self.solve(b, maxiter=1)
[docs]
def aspreconditioner(self, cycle='V'):
"""Create a preconditioner using this multigrid cycle.
Parameters
----------
cycle : {'V','W','F','AMLI'}
Type of multigrid cycle to perform in each iteration.
Returns
-------
precond : LinearOperator
Preconditioner suitable for the iterative solvers in defined in
the scipy.sparse.linalg module (e.g. cg, gmres) and any other
solver that uses the LinearOperator interface. Refer to the
LinearOperator documentation in scipy.sparse.linalg
See Also
--------
MultilevelSolver.solve, scipy.sparse.linalg.LinearOperator
Examples
--------
>>> from pyamg.aggregation import smoothed_aggregation_solver
>>> from pyamg.gallery import poisson
>>> from scipy.sparse.linalg import cg
>>> import scipy as sp
>>> A = poisson((100, 100), format='csr') # matrix
>>> b = np.random.rand(A.shape[0]) # random RHS
>>> ml = smoothed_aggregation_solver(A) # AMG solver
>>> M = ml.aspreconditioner(cycle='V') # preconditioner
>>> x, info = cg(A, b, tol=1e-8, maxiter=30, M=M) # solve with CG
"""
shape = self.levels[0].A.shape
dtype = self.levels[0].A.dtype
def matvec(b):
return self.solve(b, maxiter=1, cycle=cycle, tol=1e-12)
return LinearOperator(shape, matvec, dtype=dtype)
[docs]
def solve(self, b, x0=None, tol=1e-5, maxiter=100, cycle='V', accel=None,
callback=None, residuals=None, cycles_per_level=1, return_info=False):
"""Execute multigrid cycling.
Parameters
----------
b : array
Right hand side.
x0 : array
Initial guess.
tol : float
Stopping criteria: relative residual r[k]/||b|| tolerance.
If `accel` is used, the stopping criteria is set by the Krylov method.
maxiter : int
Stopping criteria: maximum number of allowable iterations.
cycle : {'V','W','F','AMLI'}
Type of multigrid cycle to perform in each iteration.
accel : string, function
Defines acceleration method. Can be a string such as 'cg'
or 'gmres' which is the name of an iterative solver in
pyamg.krylov (preferred) or scipy.sparse.linalg.
If accel is not a string, it will be treated like a function
with the same interface provided by the iterative solvers in SciPy.
callback : function
User-defined function called after each iteration. It is
called as callback(xk) where xk is the k-th iterate vector.
residuals : list
List to contain residual norms at each iteration. The residuals
will be the residuals from the Krylov iteration -- see the `accel`
method to see verify whether this ||r|| or ||Mr|| (as in the case of
GMRES).
cycles_per_level: int, default 1
Number of V-cycles on each level of an F-cycle
return_info : bool
If true, will return (x, info)
If false, will return x (default)
Returns
-------
x : array
Approximate solution to Ax=b after k iterations
info : string
Halting status
== =======================================
0 successful exit
>0 convergence to tolerance not achieved,
return iteration count instead.
== =======================================
See Also
--------
aspreconditioner
Examples
--------
>>> from numpy import ones
>>> from pyamg import ruge_stuben_solver
>>> from pyamg.gallery import poisson
>>> A = poisson((100, 100), format='csr')
>>> b = A * ones(A.shape[0])
>>> ml = ruge_stuben_solver(A, max_coarse=10)
>>> residuals = []
>>> x = ml.solve(b, tol=1e-12, residuals=residuals) # standalone solver
"""
if x0 is None:
x = np.zeros_like(b)
else:
x = np.array(x0) # copy
A = self.levels[0].A
cycle = str(cycle).upper()
# AMLI cycles require hermitian matrix
if (cycle == 'AMLI') and hasattr(A, 'symmetry'):
if A.symmetry != 'hermitian':
raise ValueError('AMLI cycles require \
symmetry to be hermitian')
if accel is not None:
# Check for symmetric smoothing scheme when using CG
if (accel == 'cg') and (not self.symmetric_smoothing):
warn('Incompatible non-symmetric multigrid preconditioner '
'detected, due to presmoother/postsmoother combination. '
'CG requires SPD preconditioner, not just SPD matrix.')
# Check for AMLI compatibility
if (accel != 'fgmres') and (cycle == 'AMLI'):
raise ValueError('AMLI cycles require acceleration (accel) '
'to be fgmres, or no acceleration')
# Acceleration is being used
kwargs = {}
if isinstance(accel, str):
kwargs = {}
if hasattr(krylov, accel):
accel = getattr(krylov, accel)
else:
accel = getattr(sla, accel)
M = self.aspreconditioner(cycle=cycle)
try: # try PyAMG style interface which has a residuals parameter
x, info = accel(A, b, x0=x0, tol=tol, maxiter=maxiter, M=M,
callback=callback, residuals=residuals, **kwargs)
if return_info:
return x, info
return x
except TypeError:
# try the scipy.sparse.linalg style interface,
# which requires a callback function if a residual
# history is desired
if residuals is not None:
residuals[:] = [np.linalg.norm(b - A @ x)]
def callback_wrapper(x):
if np.isscalar(x):
residuals.append(x)
else:
residuals.append(np.linalg.norm(b - A @ x))
if callback is not None:
callback(x)
else:
callback_wrapper = callback
# for scipy solvers, see if rtol is available
kwargs['tol'] = tol
kwargs['atol'] = 0
if 'rtol' in inspect.getfullargspec(accel).args:
kwargs['rtol'] = kwargs.pop(tol)
x, info = accel(A, b, x0=x0, maxiter=maxiter, M=M,
callback=callback_wrapper, **kwargs)
if return_info:
return x, info
return x
else:
# Scale tol by normb
# Don't scale tol earlier. The accel routine should also scale tol
normb = np.linalg.norm(b)
if normb == 0.0:
normb = 1.0 # set so that we have an absolute tolerance
# Start cycling (no acceleration)
normr = np.linalg.norm(b - A @ x)
if residuals is not None:
residuals[:] = [normr] # initial residual
# Create uniform types for A, x and b
# Clearly, this logic doesn't handle the case of real A and complex b
tp = upcast(b.dtype, x.dtype, A.dtype)
[b, x] = to_type(tp, [b, x])
b = np.ravel(b)
x = np.ravel(x)
it = 0
while True: # it <= maxiter and normr >= tol:
if len(self.levels) == 1:
# hierarchy has only 1 level
x = self.coarse_solver(A, b)
else:
self.__solve(0, x, b, cycle, cycles_per_level)
it += 1
normr = np.linalg.norm(b - A @ x)
if residuals is not None:
residuals.append(normr)
if callback is not None:
callback(x)
if normr < tol * normb:
if return_info:
return x, 0
return x
if it == maxiter:
if return_info:
return x, it
return x
def __solve(self, lvl, x, b, cycle, cycles_per_level=1):
"""Multigrid cycling.
Parameters
----------
lvl : int
Solve problem on level `lvl`
x : numpy array
Initial guess `x` and return correction
b : numpy array
Right-hand side for Ax=b
cycle : {'V','W','F','AMLI'}
Recursively called cycling function. The
Defines the cycling used:
cycle = 'V', V-cycle
cycle = 'W', W-cycle
cycle = 'F', F-cycle
cycle = 'AMLI', AMLI-cycle
cycles_per_level : int, default 1
Number of V-cycles on each level of an F-cycle
"""
A = self.levels[lvl].A
self.levels[lvl].presmoother(A, x, b)
residual = b - A @ x
coarse_b = self.levels[lvl].R @ residual
coarse_x = np.zeros_like(coarse_b)
if lvl == len(self.levels) - 2:
coarse_x[:] = self.coarse_solver(self.levels[-1].A, coarse_b)
else:
if cycle == 'V':
self.__solve(lvl + 1, coarse_x, coarse_b, 'V')
elif cycle == 'W':
self.__solve(lvl + 1, coarse_x, coarse_b, cycle)
self.__solve(lvl + 1, coarse_x, coarse_b, cycle)
elif cycle == 'F':
self.__solve(lvl + 1, coarse_x, coarse_b, cycle, cycles_per_level)
for _ in range(0, cycles_per_level):
self.__solve(lvl + 1, coarse_x, coarse_b, 'V', 1)
elif cycle == 'AMLI':
# Run nAMLI AMLI cycles, which compute "optimal" corrections by
# orthogonalizing the coarse-grid corrections in the A-norm
nAMLI = 2
Ac = self.levels[lvl + 1].A
p = np.zeros((nAMLI, coarse_b.shape[0]), dtype=coarse_b.dtype)
beta = np.zeros((nAMLI, nAMLI), dtype=coarse_b.dtype)
for k in range(nAMLI):
# New search direction --> M^{-1}*residual
p[k, :] = 1
self.__solve(lvl + 1, p[k, :].reshape(coarse_b.shape),
coarse_b, cycle)
# Orthogonalize new search direction to old directions
for j in range(k): # loops from j = 0...(k-1)
beta[k, j] = np.inner(p[j, :].conj(), Ac * p[k, :]) /\
np.inner(p[j, :].conj(), Ac * p[j, :])
p[k, :] -= beta[k, j] * p[j, :]
# Compute step size
Ap = Ac * p[k, :]
alpha = np.inner(p[k, :].conj(), np.ravel(coarse_b)) /\
np.inner(p[k, :].conj(), Ap)
# Update solution
coarse_x += alpha * p[k, :].reshape(coarse_x.shape)
# Update residual
coarse_b -= alpha * Ap.reshape(coarse_b.shape)
else:
raise TypeError(f'Unrecognized cycle type ({cycle})')
x += self.levels[lvl].P @ coarse_x # coarse grid correction
self.levels[lvl].postsmoother(A, x, b)
[docs]
def coarse_grid_solver(solver):
"""Return a coarse grid solver suitable for MultilevelSolver.
Parameters
----------
solver : string, callable, tuple
The solver method is either (1) a string such as 'splu' or 'pinv' of a
callable object which receives only parameters (A, b) and returns an
(approximate or exact) solution to the linear system Ax = b, or (2) a
callable object that takes parameters (A,b) and returns an (approximate
or exact) solution to Ax = b, or (3) a tuple of the form
(string|callable, args), where args is a dictionary of arguments to
be passed to the function denoted by string or callable.
The set of valid string arguments is:
- Sparse direct methods:
+ splu : sparse LU solver
- Sparse iterative methods:
+ the name of any method in scipy.sparse.linalg or
pyamg.krylov (e.g. 'cg').
Methods in pyamg.krylov take precedence.
+ relaxation method, such as 'gauss_seidel' or 'jacobi',
present in pyamg.relaxation
- Dense methods:
+ pinv : pseudoinverse (SVD)
+ lu : LU factorization
+ cholesky : Cholesky factorization
Returns
-------
ptr : GenericSolver
A class for use as a standalone or coarse grids solver
Examples
--------
>>> import numpy as np
>>> from scipy.sparse import spdiags
>>> from pyamg.gallery import poisson
>>> from pyamg import coarse_grid_solver
>>> A = poisson((10, 10), format='csr')
>>> b = A * np.ones(A.shape[0])
>>> cgs = coarse_grid_solver('lu')
>>> x = cgs(A, b)
"""
def unpack_arg(v):
if isinstance(v, tuple):
return v[0], v[1]
return v, {}
solver, kwargs = unpack_arg(solver)
if solver in ['pinv', 'pinv2']:
def solve(self, A, b):
if not hasattr(self, 'P'):
self.P = pinv(A.toarray(), **kwargs)
return np.dot(self.P, b)
elif solver == 'lu':
def solve(self, A, b):
if not hasattr(self, 'LU'):
self.LU = sp.linalg.lu_factor(A.toarray(), **kwargs)
return sp.linalg.lu_solve(self.LU, b)
elif solver == 'cholesky':
def solve(self, A, b):
if not hasattr(self, 'L'):
self.L = sp.linalg.cho_factor(A.toarray(), **kwargs)
return sp.linalg.cho_solve(self.L, b)
elif solver == 'splu':
def solve(self, A, b):
if not hasattr(self, 'LU'):
# for multiple candidates in B, A will often have a couple zero
# rows/columns that must be removed
Acsc = A.tocsc()
Acsc.eliminate_zeros()
diffptr = Acsc.indptr[:-1] - Acsc.indptr[1:]
nonzero_cols = (diffptr != 0).nonzero()[0]
Map = sp.sparse.eye(Acsc.shape[0], Acsc.shape[1], format='csc')
Map = Map[:, nonzero_cols]
Acsc = Map.T.tocsc() * Acsc * Map
self.LU = sp.sparse.linalg.splu(Acsc, **kwargs)
self.LU_Map = Map
return self.LU_Map * self.LU.solve(np.ravel(self.LU_Map.T * b))
elif solver in ['bicg', 'bicgstab', 'cg', 'cgs', 'gmres', 'qmr', 'minres']:
if hasattr(krylov, solver):
fn = getattr(krylov, solver)
else:
fn = getattr(sla, solver)
def solve(_, A, b):
if 'tol' not in kwargs:
kwargs['tol'] = set_tol(A.dtype)
return fn(A, b, **kwargs)[0]
elif solver in ['gauss_seidel', 'jacobi', 'block_gauss_seidel', 'schwarz',
'block_jacobi', 'richardson', 'sor', 'chebyshev',
'jacobi_ne', 'gauss_seidel_ne', 'gauss_seidel_nr']:
if 'iterations' not in kwargs:
kwargs['iterations'] = 10
def solve(_, A, b):
lvl = MultilevelSolver.Level()
lvl.A = A
fn = getattr(smoothing, 'setup_' + str(solver))
relax = fn(lvl, **kwargs)
x = np.zeros_like(b)
relax(A, x, b)
return x
elif solver is None:
# No coarse grid solve
def solve(_, __, b):
return 0 * b # should this return b instead?
elif callable(solver):
def solve(_, A, b):
return solver(A, b, **kwargs)
else:
raise ValueError(f'unknown solver: {solver}')
class GenericSolver:
"""Generic solver class."""
def __call__(self, A, b):
# make sure x is same dimensions and type as b
b = np.asanyarray(b)
if A.nnz == 0:
# if A.nnz = 0, then we expect no correction
x = np.zeros(b.shape)
else:
x = solve(self, A, b)
if isinstance(b, np.ndarray):
x = np.asarray(x)
elif isinstance(b, np.matrix):
# convert to ndarray
b = np.asarray(b)
x = np.asarray(x)
else:
raise ValueError('unrecognized type')
return x.reshape(b.shape)
def __repr__(self):
return 'coarse_grid_solver(' + repr(solver) + ')'
@classmethod
def name(cls):
"""Return the coarse solver name."""
return repr(solver)
return GenericSolver()
[docs]
class multilevel_solver(MultilevelSolver): # noqa: N801
"""Deprecated level class.
.. deprecated:: 4.2.3
Use :class:`MultilevelSolver` instead.
"""
[docs]
def __init__(self, *args, **kwargs):
"""Raise deprecation warning on use, not import."""
super().__init__(*args, **kwargs)
warn('multilevel_solver is deprecated. use MultilevelSolver()',
category=DeprecationWarning, stacklevel=2)
