class Interface:
'''This class defines an interface to the NumPy backend for the
discretization. We use this so we can write higher level methods
such as pseudo-arclength continuation without knowing anything
about the underlying methods such as the solvers that are present
in the backend we are interfacing with.'''
def __init__(self,
parameters,
nx,
ny,
nz,
dim,
dof,
x=None,
y=None,
z=None):
self.nx = nx
self.ny = ny
self.nz = nz
self.dim = dim
self.dof = dof
self.discretization = Discretization(parameters, nx, ny, nz, dim, dof,
x, y, z)
self.parameters = parameters
# Solver caching
self._lu = None
self._prec = None
# Eigenvalue solver caching
self._subspaces = None
def set_parameter(self, name, value):
'''Set a parameter in self.parameters while also letting the
discretization know that we changed a parameter. '''
self.discretization.set_parameter(name, value)
def get_parameter(self, name):
'''Get a parameter from self.parameters through the discretization.'''
return self.discretization.get_parameter(name)
def rhs(self, state):
'''Right-hand side in M * du / dt = F(u).'''
return self.discretization.rhs(state)
def jacobian(self, state):
'''Jacobian J of F in M * du / dt = F(u).'''
return self.discretization.jacobian(state)
def mass_matrix(self):
'''Mass matrix M in M * du / dt = F(u).'''
return self.discretization.mass_matrix()
def solve(self, jac, x):
'''Solve J y = x for y.'''
rhs = x.copy()
# Fix one pressure node
if self.dof > self.dim:
if len(rhs.shape) < 2:
rhs[self.dim] = 0
else:
rhs[self.dim, :] = 0
# First try to use an iterative solver with the previous
# direct solver as preconditioner
if self._prec and jac.dtype == rhs.dtype and jac.dtype == self._prec.dtype and \
self.parameters.get('Use Iterative Solver', False):
out, info = linalg.gmres(jac,
rhs,
restart=5,
maxiter=1,
tol=1e-8,
atol=0,
M=self._prec)
if info == 0:
return out
# Use a direct solver instead
if not jac.lu:
coA = jac.coA
jcoA = jac.jcoA
begA = jac.begA
# Fix one pressure node
if self.dof > self.dim:
coA = numpy.zeros(jac.begA[-1], dtype=jac.coA.dtype)
jcoA = numpy.zeros(jac.begA[-1], dtype=int)
begA = numpy.zeros(len(jac.begA), dtype=int)
idx = 0
for i in range(len(jac.begA) - 1):
if i == self.dim:
coA[idx] = -1.0
jcoA[idx] = i
idx += 1
begA[i + 1] = idx
continue
for j in range(jac.begA[i], jac.begA[i + 1]):
if jac.jcoA[j] != self.dim:
coA[idx] = jac.coA[j]
jcoA[idx] = jac.jcoA[j]
idx += 1
begA[i + 1] = idx
# Convert the matrix to CSC format since splu expects that
A = sparse.csr_matrix((coA, jcoA, begA)).tocsc()
jac.lu = linalg.splu(A)
# Cache the factorization for use in the iterative solver
self._lu = jac.lu
self._prec = linalg.LinearOperator((jac.n, jac.n),
matvec=self._lu.solve,
dtype=jac.dtype)
return jac.solve(rhs)
def eigs(self, state, return_eigenvectors=False):
'''Compute the generalized eigenvalues of beta * J(x) * v = alpha * M * v.'''
from jadapy import jdqz, Target
from fvm.JadaInterface import JadaOp, JadaInterface
jac_op = JadaOp(self.jacobian(state))
mass_op = JadaOp(self.mass_matrix())
jada_interface = JadaInterface(self, jac_op, mass_op, jac_op.shape[0],
numpy.complex128)
parameters = self.parameters.get('Eigenvalue Solver', {})
target = parameters.get('Target', Target.LargestRealPart)
subspace_dimensions = [
parameters.get('Minimum Subspace Dimension', 30),
parameters.get('Maximum Subspace Dimension', 60)
]
tol = parameters.get('Tolerance', 1e-7)
num = parameters.get('Number of Eigenvalues', 5)
result = jdqz.jdqz(jac_op,
mass_op,
num,
tol=tol,
subspace_dimensions=subspace_dimensions,
target=target,
interface=jada_interface,
arithmetic='complex',
prec=jada_interface.shifted_prec,
return_eigenvectors=return_eigenvectors,
return_subspaces=True,
initial_subspaces=self._subspaces)
if return_eigenvectors:
alpha, beta, v, q, z = result
self._subspaces = [q, z]
idx = range(len(alpha))
idx = sorted(idx, key=lambda i: -(alpha[i] / beta[i]).real)
w = v.copy()
eigs = alpha.copy()
for i in range(len(idx)):
w[:, i] = v[:, idx[i]]
eigs[i] = alpha[idx[i]] / beta[idx[i]]
return eigs, w
else:
alpha, beta, q, z = result
self._subspaces = [q, z]
return numpy.array(sorted(alpha / beta, key=lambda x: -x.real))
class Interface:
def __init__(self, parameters, nx, ny, nz, dim, dof):
self.nx = nx
self.ny = ny
self.nz = nz
self.dim = dim
self.dof = dof
self.discretization = Discretization(parameters, nx, ny, nz, dim, dof)
self.parameters = parameters
# Solver caching
self._lu = None
self._prec = None
# Eigenvalue solver caching
self._subspaces = None
def set_parameter(self, name, value):
self.discretization.set_parameter(name, value)
def get_parameter(self, name):
return self.discretization.get_parameter(name)
def rhs(self, state):
return self.discretization.rhs(state)
def jacobian(self, state):
return self.discretization.jacobian(state)
def mass_matrix(self):
return self.discretization.mass_matrix()
# def solve(self, jac, rhs):
# coA = numpy.zeros(jac.begA[-1], dtype=jac.coA.dtype)
# jcoA = numpy.zeros(jac.begA[-1], dtype=int)
# begA = numpy.zeros(len(jac.begA), dtype=int)
#
# idx = 0
# for i in range(len(jac.begA)-1):
# if i == self.dim:
# coA[idx] = -1.0
# jcoA[idx] = i
# idx += 1
# begA[i+1] = idx
# continue
# for j in range(jac.begA[i], jac.begA[i+1]):
# if jac.jcoA[j] != self.dim:
# coA[idx] = jac.coA[j]
# jcoA[idx] = jac.jcoA[j]
# idx += 1
# begA[i+1] = idx
#
# A = sparse.csr_matrix((coA, jcoA, begA))
# if len(rhs.shape) < 2:
# rhs[self.dim] = 0
# x = linalg.spsolve(A, rhs)
# else:
# x = rhs.copy()
# #rhs[self.dim, :] = 0
# for i in range(x.shape[1]):
# x[:, i] = linalg.spsolve(A, rhs[:, i])
# return x
# TODO wei
def solve(self, jac, x):
rhs = x.copy()
# Fix one pressure node
if self.dof > self.dim:
if len(rhs.shape) < 2:
rhs[self.dim] = 0
else:
rhs[self.dim, :] = 0
# Use a direct solver instead
if True:
coA = jac.coA
jcoA = jac.jcoA
begA = jac.begA
# Fix one pressure node
if self.dof > self.dim:
coA = numpy.zeros(jac.begA[-1], dtype=jac.coA.dtype)
jcoA = numpy.zeros(jac.begA[-1], dtype=int)
begA = numpy.zeros(len(jac.begA), dtype=int)
idx = 0
for i in range(len(jac.begA) - 1):
if i == self.dim:
coA[idx] = -1.0
jcoA[idx] = i
idx += 1
begA[i + 1] = idx
continue
for j in range(jac.begA[i], jac.begA[i + 1]):
if jac.jcoA[j] != self.dim:
coA[idx] = jac.coA[j]
jcoA[idx] = jac.jcoA[j]
idx += 1
begA[i + 1] = idx
# Convert the matrix to CSC format since splu expects that
A = sparse.csr_matrix((coA, jcoA, begA)).tocsc()
# jac.lu = linalg.splu(A)
if self.parameters.get('Use Iterative Solver', False):
if self.parameters.get('Use Preconditioner', False):
if self.parameters.get('Use ILU Preconditioner', False):
self._prec = linalg.LinearOperator(
(jac.n, jac.n),
matvec=linalg.spilu(A).solve,
dtype=jac.dtype)
if self._prec and jac.dtype == rhs.dtype and jac.dtype == self._prec.dtype:
out, info = linalg.gmres(A,
rhs,
M=self._prec,
callback=gmres_counter())
if info == 0:
return out
else:
out, info = linalg.gmres(A, rhs, callback=gmres_counter())
if info == 0:
return out
A_lu = linalg.splu(A)
return A_lu.solve(rhs)
def solve_bordered(self, jac, fval, dfval, r_x, r_mu, r):
rhs = fval.copy()
# Fix one pressure node
if self.dof > self.dim:
if len(rhs.shape) < 2:
rhs[self.dim] = 0
else:
rhs[self.dim, :] = 0
# Use a direct solver instead
if True:
coA = jac.coA
jcoA = jac.jcoA
begA = jac.begA
# Fix one pressure node
if self.dof > self.dim:
coA = numpy.zeros(jac.begA[-1], dtype=jac.coA.dtype)
jcoA = numpy.zeros(jac.begA[-1], dtype=int)
begA = numpy.zeros(len(jac.begA), dtype=int)
idx = 0
for i in range(len(jac.begA) - 1):
if i == self.dim:
coA[idx] = -1.0
jcoA[idx] = i
idx += 1
begA[i + 1] = idx
continue
for j in range(jac.begA[i], jac.begA[i + 1]):
if jac.jcoA[j] != self.dim:
coA[idx] = jac.coA[j]
jcoA[idx] = jac.jcoA[j]
idx += 1
begA[i + 1] = idx
# Convert the matrix to CSC format since splu expects that
A = sparse.csr_matrix((coA, jcoA, begA)).tocsc().toarray()
a = numpy.concatenate((A, dfval[:, numpy.newaxis]), axis=1)
b = numpy.append(r_x, r_mu)
A = numpy.concatenate((a, b[:, numpy.newaxis].T), axis=0)
b = numpy.append(-fval, r)
A_sparse = sparse.csc_matrix(A)
A_lu = linalg.splu(A_sparse)
# jac.lu = linalg.splu(A)
if self.parameters.get('Use Iterative Solver', False):
if self.parameters.get('Use Preconditioner', False):
if self.parameters.get('Use LU Preconditioner', False):
self._prec = linalg.LinearOperator(
(jac.n + 1, jac.n + 1),
matvec=linalg.splu(A).solve,
dtype=jac.dtype)
elif self.parameters.get('Use ILU Preconditioner', False):
self._prec = linalg.LinearOperator(
(jac.n + 1, jac.n + 1),
matvec=linalg.spilu(A).solve,
dtype=jac.dtype)
if self._prec and jac.dtype == rhs.dtype and jac.dtype == self._prec.dtype:
out, info = linalg.gmres(A_sparse,
b,
M=self._prec,
callback=gmres_counter())
if info == 0:
return out
else:
out, info = linalg.gmres(A_sparse,
b,
callback=gmres_counter())
if info == 0:
return out
return A_lu.solve(b)
#
# def solve(self, jac, x):
# rhs = x.copy()
#
# # Fix one pressure node
# if self.dof > self.dim:
# if len(rhs.shape) < 2:
# rhs[self.dim] = 0
# else:
# rhs[self.dim, :] = 0
#
# # First try to use an iterative solver with the previous
# # direct solver as preconditioner
# if self._prec and jac.dtype == rhs.dtype and jac.dtype == self._prec.dtype and \
# self.parameters.get('Use Iterative Solver', False):
# out, info = linalg.gmres(jac, rhs, restart=5, maxiter=1, tol=1e-8, atol=0, M=self._prec)
# if info == 0:
# return out
#
# # no preconditioner
# # if self._prec and jac.dtype == rhs.dtype and jac.dtype == self._prec.dtype and \
# # self.parameters.get('Use Iterative Solver', False):
# # out, info = linalg.gmres(jac, rhs, restart=5, maxiter=1, tol=1e-8, atol=0)
# # if info == 0:
# # return out
#
# # Use a direct solver instead
# if not jac.lu:
# coA = jac.coA
# jcoA = jac.jcoA
# begA = jac.begA
#
# # Fix one pressure node
# if self.dof > self.dim:
# coA = numpy.zeros(jac.begA[-1], dtype=jac.coA.dtype)
# jcoA = numpy.zeros(jac.begA[-1], dtype=int)
# begA = numpy.zeros(len(jac.begA), dtype=int)
#
# idx = 0
# for i in range(len(jac.begA) - 1):
# if i == self.dim:
# coA[idx] = -1.0
# jcoA[idx] = i
# idx += 1
# begA[i + 1] = idx
# continue
# for j in range(jac.begA[i], jac.begA[i + 1]):
# if jac.jcoA[j] != self.dim:
# coA[idx] = jac.coA[j]
# jcoA[idx] = jac.jcoA[j]
# idx += 1
# begA[i + 1] = idx
#
# # Convert the matrix to CSC format since splu expects that
# A = sparse.csr_matrix((coA, jcoA, begA)).tocsc()
#
# jac.lu = linalg.splu(A)
#
# # Cache the factorization for use in the iterative solver
# self._lu = jac.lu
# # LU decomposition as the preconditioner
# self._prec = linalg.LinearOperator((jac.n, jac.n), matvec=self._lu.solve, dtype=jac.dtype)
#
# # ILU decomposition as the preconditioner
# # self._prec = linalg.LinearOperator((jac.n, jac.n), matvec=linalg.spilu(A).solve, dtype=jac.dtype)
#
# return jac.solve(rhs)
def eigs(self, state, return_eigenvectors=False):
from jadapy import jdqz, Target
from fvm.JadaInterface import JadaOp, JadaInterface
jac_op = JadaOp(self.jacobian(state))
mass_op = JadaOp(self.mass_matrix())
jada_interface = JadaInterface(self, jac_op, mass_op, jac_op.shape[0],
numpy.complex128)
parameters = self.parameters.get('Eigenvalue Solver', {})
target = parameters.get('Target', Target.LargestRealPart)
subspace_dimensions = [
parameters.get('Minimum Subspace Dimension', 30),
parameters.get('Maximum Subspace Dimension', 60)
]
tol = parameters.get('Tolerance', 1e-7)
num = parameters.get('Number of Eigenvalues', 5)
result = jdqz.jdqz(jac_op,
mass_op,
num,
tol=tol,
subspace_dimensions=subspace_dimensions,
target=target,
interface=jada_interface,
arithmetic='complex',
prec=jada_interface.shifted_prec,
return_eigenvectors=return_eigenvectors,
return_subspaces=True,
initial_subspaces=self._subspaces)
if return_eigenvectors:
alpha, beta, v, q, z = result
self._subspaces = [q, z]
idx = range(len(alpha))
idx = sorted(idx, key=lambda i: -(alpha[i] / beta[i]).real)
w = v.copy()
eigs = alpha.copy()
for i in range(len(idx)):
w[:, i] = v[:, idx[i]]
eigs[i] = alpha[idx[i]] / beta[idx[i]]
return eigs, w
else:
alpha, beta, q, z = result
self._subspaces = [q, z]
return numpy.array(sorted(alpha / beta, key=lambda x: -x.real))