opts.setValue("eps_target_magnitude", None)
opts.setValue("eps_target", 0.)
opts.setValue("eps_tol", 1e-13)
opts.setValue("st_ksp_max_it", 10000)
opts.setValue("st_ksp_rtol", 1e-8)
# opts.setValue("eps_gen_hermitian", None)
# opts.setValue("st_pc_factor_shift_type", "NONZERO")
# opts.setValue("eps_type", "krylovschur")
# opts.setValue("eps_smallest_real", None)
# opts.setValue("eps_tol", 1e-10)
# opts.setValue("eps_ncv", 40)
# Solve for eigenvalues
print('Computing eigenvalues...')
es = SLEPc.EPS().create().create(comm=SLEPc.COMM_WORLD)
es.setDimensions(num_eigenvalues)
es.setOperators(petsc_a, petsc_m)
es.setFromOptions()
print(es.getDimensions())
es.solve()
# Number of converged eigenvalues
nconv = es.getConverged()
eigvecs = []
eigvalues = []
for i in range(nconv):
print(es.getEigenvalue(i).real)
vr, vi = petsc_a.getVecs()
def __init__(self, F_form: typing.List, u: typing.List, lmbda: dolfinx.Function,
A_form=None, B_form=None, prefix=None):
"""SLEPc problem and solver wrapper.
Wrapper for a generalised eigenvalue problem obtained from UFL residual forms.
Parameters
----------
F_form
Residual forms
u
Current solution vectors
lmbda
Eigenvalue function. Residual forms must be linear in lmbda for
linear eigenvalue problems.
A_form, optional
Override automatically derived A
B_form, optional
Override automatically derived B
Note
----
In general, eigenvalue problems have form T(lmbda) * x = 0,
where T(lmbda) is a matrix-valued function.
Linear eigenvalue problems have T(lmbda) = A + lmbda * B, and if B is not identity matrix
then this problem is called generalized (linear) eigenvalue problem.
"""
self.F_form = F_form
self.u = u
self.lmbda = lmbda
self.comm = u[0].function_space.mesh.mpi_comm()
self.ur = []
self.ui = []
for func in u:
self.ur.append(dolfinx.function.Function(func.function_space, name=func.name))
self.ui.append(dolfinx.function.Function(func.function_space, name=func.name))
# Prepare tangent form M0 which has terms involving lambda
self.M0 = [[None for i in range(len(self.u))] for j in range(len(self.u))]
for i in range(len(self.u)):
for j in range(len(self.u)):
self.M0[i][j] = ufl.algorithms.expand_derivatives(ufl.derivative(
F_form[i], self.u[j], ufl.TrialFunction(self.u[j].function_space)))
if B_form is None:
B0 = [[None for i in range(len(self.u))] for j in range(len(self.u))]
for i in range(len(self.u)):
for j in range(len(self.u)):
# Differentiate wrt. lambda and replace all remaining lambda with Zero
B0[i][j] = ufl.algorithms.expand_derivatives(ufl.diff(self.M0[i][j], lmbda))
B0[i][j] = ufl.replace(B0[i][j], {lmbda: ufl.zero()})
if B0[i][j].empty():
B0[i][j] = None
self.B_form = B0
else:
self.B_form = B_form
if A_form is None:
A0 = [[None for i in range(len(self.u))] for j in range(len(self.u))]
for i in range(len(self.u)):
for j in range(len(self.u)):
A0[i][j] = ufl.replace(self.M0[i][j], {lmbda: ufl.zero()})
if A0[i][j].empty():
A0[i][j] = None
continue
self.A_form = A0
else:
self.A_form = A_form
self.eps = SLEPc.EPS().create(self.comm)
self.eps.setOptionsPrefix(prefix)
self.eps.setFromOptions()
self.A = dolfinx.fem.create_matrix_block(self.A_form)
self.B = None
if not self.empty_B():
self.B = dolfinx.fem.create_matrix_block(self.B_form)
for row in df.itertuples():
_, i, j, wcCount = row
wCount = counts[i]
cCount = counts[j]
ppmi = PPMI(wcCount, total, wCount, cCount, alpha=alpha)
data.append(ppmi)
rowIndex.append(i)
columnIndex.append(j)
csr = csr_matrix((data, (rowIndex, columnIndex)), shape=(n, n))
print("Doing SVD")
print("PPMI[99,2]=", csr[99, 2])
A = PETSc.Mat().createAIJ(size=(n, n), csr=(csr.indptr, csr.indices, csr.data))
A.assemble()
svd = SLEPc.SVD()
svd.create()
svd.setOperator(A)
svd.setType(svd.Type.TRLANCZOS)
svd.setDimensions(nsv=50, ncv=100)
svd.setFromOptions()
svd.solve()
iterations = svd.getIterationNumber()
nsv, ncv, mpd = svd.getDimensions()
nconv = svd.getConverged()
v, u = A.createVecs()
U = []
V = []
from petsc4py import PETSc
from slepc4py import SLEPc
from numpy import exp
from math import pi
Print = PETSc.Sys.Print
opts = PETSc.Options()
n = opts.getInt('n', 128)
tau = opts.getReal('tau', 0.001)
a = 20
h = pi/(n+1)
# Setup the solver
nep = SLEPc.NEP().create()
# Create problem matrices
# Identity matrix
Id = PETSc.Mat().create()
Id.setSizes([n, n])
Id.setFromOptions()
Id.setUp()
Id.assemble()
Id.shift(1.0)
Id.setOption(PETSc.Mat.Option.HERMITIAN, True)
# A = 1/h^2*tridiag(1,-2,1) + a*I
A = PETSc.Mat().create()
A.setSizes([n, n])
A.setFromOptions()
A.setUp()
def _init_eigensolver(k=6, which='LM', sigma=None, isherm=True, isgen=False,
EPSType=None, st_opts=None, tol=None, A_is_linop=False,
B_is_linop=False, maxiter=None, ncv=None, l_win=None,
comm=None):
"""Create an advanced eigensystem solver
Parameters
----------
sigma :
Target eigenvalue.
isherm :
Whether problem is hermitian or not.
Returns
-------
SLEPc solver ready to be called.
"""
SLEPc, comm = get_slepc(comm=comm)
eigensolver = SLEPc.EPS().create(comm=comm)
if st_opts is None:
st_opts = {}
if l_win is not None:
EPSType = {'ciss': 'ciss', None: 'ciss'}[EPSType]
which = 'ALL'
rg = eigensolver.getRG()
rg.setType(SLEPc.RG.Type.INTERVAL)
rg.setIntervalEndpoints(*l_win, -0.1, 0.1)
else:
eigensolver.setDimensions(k, ncv)
internal = (sigma is not None) or (l_win is not None)
# pick right backend
if EPSType is None:
if (isgen and B_is_linop) or (internal and A_is_linop):
EPSType = 'gd'
else:
EPSType = 'krylovschur'
# set some preconditioning defaults for 'gd' and 'lobpcg'
if EPSType in ('gd', 'lobpcg', 'blopex', 'primme'):
st_opts.setdefault('STType', 'precond')
st_opts.setdefault('KSPType', 'preonly')
st_opts.setdefault('PCType', 'none')
# set some preconditioning defaults for 'jd'
elif EPSType == 'jd':
st_opts.setdefault('STType', 'precond')
st_opts.setdefault('KSPType', 'bcgs')
st_opts.setdefault('PCType', 'none')
# set the spectral inverter / preconditioner.
if st_opts or internal:
st = _init_spectral_inverter(comm=comm, **st_opts)
eigensolver.setST(st)
# NB: `setTarget` must be called *after* `setST`.
if sigma is not None:
which = "TR"
eigensolver.setTarget(sigma)
if A_is_linop:
st.setMatMode(SLEPc.ST.MatMode.SHELL)
_EPS_PROB_TYPES = {
(False, False): SLEPc.EPS.ProblemType.NHEP,
(True, False): SLEPc.EPS.ProblemType.HEP,
(False, True): SLEPc.EPS.ProblemType.GNHEP,
(True, True): SLEPc.EPS.ProblemType.GHEP,
}
eigensolver.setType(EPSType)
eigensolver.setProblemType(_EPS_PROB_TYPES[(isherm, isgen)])
eigensolver.setWhichEigenpairs(_which_scipy_to_slepc(which))
eigensolver.setConvergenceTest(SLEPc.EPS.Conv.REL)
eigensolver.setTolerances(tol=tol, max_it=maxiter)
eigensolver.setFromOptions()
return eigensolver
def solve_GenEO_eigmax(self, V0s, tauGenEO_eigmax):
"""
Solves the local GenEO eigenvalue problem related to the largest eigenvalue eigmax.
Parameters
==========
V0s : list of local PETSc .vecs
V0s may already contain some local coarse vectors. This routine will possibly add more vectors to the list.
tauGenEO_eigmax: Real.
Threshold for selecting eigenvectors for the coarse space.
"""
if tauGenEO_eigmax > 0:
eps = SLEPc.EPS().create(comm=PETSc.COMM_SELF)
eps.setDimensions(nev=self.nev)
eps.setProblemType(SLEPc.EPS.ProblemType.GHIEP)
eps.setOperators(self.Atildes, self.As)
eps.setWhichEigenpairs(SLEPc.EPS.Which.TARGET_REAL)
eps.setTarget(0.)
if len(V0s) > 0:
eps.setDeflationSpace(V0s)
ST = eps.getST()
ST.setType("sinvert")
ST.setKSP(self.ksp_Atildes)
eps.solve()
if eps.getConverged() < self.nev:
PETSc.Sys.Print(
'WARNING: Only {} eigenvalues converged for GenEO_eigmax in subdomain {} whereas {} were requested'
.format(eps.getConverged(), mpi.COMM_WORLD.rank, self.nev),
comm=self.comm)
if abs(eps.getEigenvalue(eps.getConverged() -
1)) < tauGenEO_eigmax:
PETSc.Sys.Print(
'WARNING: The largest eigenvalue computed for GenEO_eigmax in subdomain {} is {} < the threshold which is {}. Consider setting PCBNN_GenEO_nev to something larger than {}'
.format(mpi.COMM_WORLD.rank,
eps.getEigenvalue(eps.getConverged() - 1),
tauGenEO_eigmax, eps.getConverged()),
comm=self.comm)
for i in range(min(eps.getConverged(), self.maxev)):
if (abs(eps.getEigenvalue(i)) < tauGenEO_eigmax
): #TODO tell slepc that the eigenvalues are real
V0s.append(self.works.duplicate())
eps.getEigenvector(i, V0s[-1])
if self.verbose:
PETSc.Sys.Print(
'GenEO eigenvalue number {} for lambdamax in subdomain {}: {}'
.format(i, mpi.COMM_WORLD.rank,
eps.getEigenvalue(i)),
comm=self.comm)
else:
if self.verbose:
PETSc.Sys.Print(
'GenEO eigenvalue number {} for lambdamax in subdomain {}: {} <-- not selected (> {})'
.format(i, mpi.COMM_WORLD.rank,
eps.getEigenvalue(i), tauGenEO_eigmax),
comm=self.comm)
self.eps_eigmax = eps #TODO FIX the only reason for this line is to make sure self.ksp_Atildes and hence PCBNN.ksp is not destroyed
if self.verbose:
PETSc.Sys.Print(
'Subdomain number {} contributes {} coarse vectors after first GenEO'
.format(mpi.COMM_WORLD.rank, len(V0s)),
comm=self.comm)
def DirectModeSLEPc(L, B, shift, nev):
"""
Computes generalized eigenvectors and eigenvalues for the problem
Lq = lambda Bq
using SLEPc
inputs : B,L, PETSc Mats
shift, scalar (same on all procs). Shift parameter for
the shift-invert method
nev, integer. Number of requested eigenvalues
outputs: omega, complex array(nconv). Conputed eigenvalues
modes, complex array(nconv,ndofs). Computed eigenvectors
residual, real array(nconv). Actual residuals for each mode
ALL OUTPUT ARE ONLY ON PROC 0 (=None on other procs)
TO DO: compute left eigenvectors (adjoint problem)
"""
comm = MPI.COMM_WORLD
rank = comm.Get_rank()
ev = L.getVecRight()
Lq = ev.duplicate()
Bq = ev.duplicate()
ndof = ev.getSize()
# Setup EPS
Print(" - Setting up the EPS and the ST")
SI = SLEPc.ST().create()
SI.setType(SLEPc.ST.Type.SINVERT)
SI.setOperators(L, B)
SI.setShift(shift)
SI.setFromOptions()
S = SLEPc.EPS()
S.create(comm)
S.setTarget(shift)
S.setWhichEigenpairs(SLEPc.EPS.Which.TARGET_MAGNITUDE)
S.setST(SI)
S.setDimensions(nev=nev, ncv=60)
S.setTolerances(tol=tol_ev, max_it=100)
S.setFromOptions()
# Solve the EVP
Print(" - Solving the EPS")
S.solve()
its = S.getIterationNumber()
nconv = S.getConverged()
if rank == 0:
residual = zeros(nconv)
omega = zeros(nconv, 'complex')
modes = zeros([ndof, nconv], 'complex')
else:
residual = None
omega = None
modes = None
for i in range(nconv):
eigval = S.getEigenpair(i, ev)
L.mult(ev, Lq)
B.mult(ev, Bq)
Bq.aypx(-eigval, Lq)
res = Bq.norm() / ev.norm()
v = Vec2DOF(ev)
if rank == 0:
omega[i] = eigval / (-1j)
modes[:, i] = v
residual[i] = res
return omega, modes, residual
v = TestFunction(V)
# Following Ritz (1909)
nu = Constant(0.225)
a = (inner(u.dx(0).dx(0),
v.dx(0).dx(0)) + inner(u.dx(1).dx(1),
v.dx(1).dx(1)) + nu *
(inner(u.dx(1).dx(1),
v.dx(0).dx(0)) + inner(u.dx(0).dx(0),
v.dx(1).dx(1))) + 2 *
(1 - nu) * inner(u.dx(0).dx(1),
v.dx(0).dx(1))) * dx
mss = inner(u, v) * dx
# This prefix allows us to control factorisation package from options.
A = assemble(a, options_prefix="st_").M.handle
M = assemble(mss, options_prefix="st_").M.handle
E = SLEPc.EPS().create(comm=msh.comm)
E.setOperators(A, M)
E.setProblemType(SLEPc.EPS.ProblemType.GHEP)
# Defaults -eps_nev 5 -eps_tol 1e-11 -eps_target 10 -st_type sinvert
E.setTolerances(tol=1e-11)
E.setTarget(10)
E.setDimensions(nev=10)
E.st.setType(SLEPc.ST.Type.SINVERT)
E.setFromOptions()
E.solve()
Print = PETSc.Sys.Print
Print()
Print("******************************")
Print("*** SLEPc Solution Results ***")
Print("******************************")
Print()
def test_curl_curl_eigenvalue(family, order):
"""curl curl eigenvalue problem.
Solved using H(curl)-conforming finite element method.
See https://www-users.cse.umn.edu/~arnold/papers/icm2002.pdf for details.
"""
slepc4py = pytest.importorskip("slepc4py") # noqa: F841
from slepc4py import SLEPc
mesh = create_rectangle(
MPI.COMM_WORLD,
[np.array([0.0, 0.0]), np.array([np.pi, np.pi])], [24, 24],
CellType.triangle)
element = ufl.FiniteElement(family, ufl.triangle, order)
V = FunctionSpace(mesh, element)
u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)
a = inner(ufl.curl(u), ufl.curl(v)) * dx
b = inner(u, v) * dx
boundary_facets = locate_entities_boundary(
mesh, mesh.topology.dim - 1,
lambda x: np.full(x.shape[1], True, dtype=bool))
boundary_dofs = locate_dofs_topological(V, mesh.topology.dim - 1,
boundary_facets)
zero_u = Function(V)
zero_u.x.array[:] = 0.0
bcs = [dirichletbc(zero_u, boundary_dofs)]
a, b = form(a), form(b)
A = assemble_matrix(a, bcs=bcs)
A.assemble()
B = assemble_matrix(b, bcs=bcs, diagonal=0.01)
B.assemble()
eps = SLEPc.EPS().create()
eps.setOperators(A, B)
PETSc.Options()["eps_type"] = "krylovschur"
PETSc.Options()["eps_gen_hermitian"] = ""
PETSc.Options()["eps_target_magnitude"] = ""
PETSc.Options()["eps_target"] = 5.0
PETSc.Options()["eps_view"] = ""
PETSc.Options()["eps_nev"] = 12
eps.setFromOptions()
eps.solve()
num_converged = eps.getConverged()
eigenvalues_unsorted = np.zeros(num_converged, dtype=np.complex128)
for i in range(0, num_converged):
eigenvalues_unsorted[i] = eps.getEigenvalue(i)
assert np.isclose(np.imag(eigenvalues_unsorted), 0.0).all()
eigenvalues_sorted = np.sort(np.real(eigenvalues_unsorted))[:-1]
eigenvalues_sorted = eigenvalues_sorted[np.logical_not(
eigenvalues_sorted < 1E-8)]
eigenvalues_exact = np.array([1.0, 1.0, 2.0, 4.0, 4.0, 5.0, 5.0, 8.0, 9.0])
assert np.isclose(eigenvalues_sorted[0:eigenvalues_exact.shape[0]],
eigenvalues_exact,
rtol=1E-2).all()
a = dot(grad(u), grad(v))*dx
m = u*v*dx
print V.dim()
# Assemble the stiffness matrix and the mass matrix.
A = assemble(a)
M = assemble(m)
A = as_backend_type(A).mat()
M = as_backend_type(M).mat()
from slepc4py import SLEPc
from petsc4py import PETSc
# Setup the eigensolver
solver = SLEPc.EPS().create()
solver.setOperators(A, M)
solver.setType(solver.Type.KRYLOVSCHUR)
solver.setDimensions(3, PETSc.DECIDE)
solver.setWhichEigenpairs(solver.Which.SMALLEST_REAL)
solver.setTolerances(1E-8, 1000)
solver.solve()
nconv = solver.getConverged()
niters = solver.getIterationNumber()
if MPI.rank(mesh.mpi_comm()) == 0:
print "Number of eigenvalues successfully computed: ", nconv
print "Iterations used", niters
eigenvalues = []
def solve_GenEO_eigmin(self, rbm_vecs, tauGenEO_eigmin):
"""
Solves the local GenEO eigenvalue problem related to the smallest eigenvalue eigmin.
Parameters
==========
rbm_vecs : list of local PETSc .vecs
rbm_vecs may already contain some local coarse vectors. This routine will possibly add more vectors to the list.
tauGenEO_eigmin: Real.
Threshold for selecting eigenvectors for the coarse space.
"""
if tauGenEO_eigmin > 0:
#to compute the smallest eigenvalues of the preconditioned matrix, A_scaled must be factorized
tempksp = PETSc.KSP().create(comm=PETSc.COMM_SELF)
tempksp.setOperators(self.A_scaled)
tempksp.setType('preonly')
temppc = tempksp.getPC()
temppc.setType('cholesky')
temppc.setFactorSolverType('mumps')
temppc.setFactorSetUpSolverType()
tempF = temppc.getFactorMatrix()
tempF.setMumpsIcntl(7, 2)
tempF.setMumpsIcntl(24, 1)
tempF.setMumpsCntl(3, self.mumpsCntl3)
temppc.setUp()
tempnrb = tempF.getMumpsInfog(28)
for i in range(tempnrb):
tempF.setMumpsIcntl(25, i+1)
self.workl.set(0.)
tempksp.solve(self.workl, self.workl)
rbm_vecs.append(self.workl.copy())
tempF.setMumpsIcntl(25, 0)
if self.verbose:
PETSc.Sys.Print('Subdomain number {} contributes {} coarse vectors as zero energy modes of the scaled local operator (in GenEO for eigmin)'.format(mpi.COMM_WORLD.rank, tempnrb), comm=self.comm)
#Eigenvalue Problem for smallest eigenvalues
eps = SLEPc.EPS().create(comm=PETSc.COMM_SELF)
eps.setDimensions(nev=self.nev)
eps.setProblemType(SLEPc.EPS.ProblemType.GHIEP)
eps.setOperators(self.A_scaled,self.Alocal)
eps.setWhichEigenpairs(SLEPc.EPS.Which.TARGET_REAL)
eps.setTarget(0.)
ST = eps.getST()
ST.setType("sinvert")
ST.setKSP(tempksp)
if len(rbm_vecs) > 0 :
eps.setDeflationSpace(rbm_vecs)
eps.solve()
if eps.getConverged() < self.nev:
PETSc.Sys.Print('WARNING: Only {} eigenvalues converged for GenEO_eigmin in subdomain {} whereas {} were requested'.format(eps.getConverged(), mpi.COMM_WORLD.rank, self.nev), comm=self.comm)
for i in range(min(eps.getConverged(),self.maxev)):
if(abs(eps.getEigenvalue(i))<tauGenEO_eigmin): #TODO tell slepc that the eigenvalues are real
rbm_vecs.append(self.workl.duplicate())
eps.getEigenvector(i,rbm_vecs[-1])
if self.verbose:
PETSc.Sys.Print('GenEO eigenvalue number {} for lambdamin in subdomain {}: {}'.format(i, mpi.COMM_WORLD.rank, eps.getEigenvalue(i)), comm=self.comm)
else:
if self.verbose:
PETSc.Sys.Print('GenEO eigenvalue number {} for lambdamin in subdomain {}: {} <-- not selected (> {})'.format(i, mpi.COMM_WORLD.rank, eps.getEigenvalue(i), tauGenEO_eigmin), comm=self.comm)
self.eps_eigmin=eps #the only reason for this line is to make sure self.ksp and hence PCBNN.ksp is not destroyed
def getEigenvalues(func, args, N=64, EigvType="All"):
import sys, slepc4py
slepc4py.init(sys.argv)
from petsc4py import PETSc
from slepc4py import SLEPc
import numpy as np
# Equation
class A(object):
def __init__(self):
pass
def mult(self, A, x, y):
y[...] = func(x[...], args)
context = A()
A = PETSc.Mat().createPython((N, N), context)
A.setUp()
"""
def setupBMatrix():
B = PETSc.Mat(dtype=complex); B.create()
B.setSizes([N,N])
B.setFromOptions()
for n in range(N):
for m in range(N):
if m == n : B[n,m] = - eps * 2. * omega[n]
else : B[n,m] = 0.
B.assemble()
return B
B = setupBMatrix()
"""
S = SLEPc.EPS().create()
S.setOperators(A)
#S.setOperators(A,B)
S.setFromOptions()
A.setFromOptions()
S.setProblemType(SLEPc.EPS.ProblemType.NHEP)
S.setBalance()
S.setDimensions(N)
#F1 = PETSc.Vec().createSeq(Nv)
#F1.setValues(range(Nv), getInitialF1(ky))
#S.setInitialSpace(F1)
S.setTolerances(tol=1.e-15, max_it=500000)
S.solve()
its = S.getIterationNumber()
sol_type = S.getType()
nev, ncv, mpd = S.getDimensions()
tol, maxit = S.getTolerances()
nconv = S.getConverged()
print("Number of converged eigenpairs: %d" % nconv)
eigval = []
eigvec = []
if nconv > 0:
### (1) Get All eigenvalues ###
xr, tmp = A.getVecs()
xi, tmp = A.getVecs()
for i in range(nconv):
k = S.getEigenpair(i, xr, xi)
error = S.computeRelativeError(i)
#print(" %9f%+9f j %12g" % (k.real, k.imag, error))
eigval.append(k.real + k.imag * 1.j)
eigvec.append(xr[...] + 1.j * xi[...])
return np.array(eigval), np.array(eigvec)
def eigendecompose(space,
A_action,
*,
B_action=None,
N_eigenvalues=None,
solver_type=None,
problem_type=None,
which=None,
tolerance=1.0e-12,
configure=None):
# First written 2018-03-01
"""
Matrix-free interface with SLEPc via slepc4py, loosely following
the slepc4py 3.6.0 demo demo/ex3.py, for use in the calculation of a
Hessian eigendecomposition with a real control space.
Arguments:
space Eigenvector space.
A_action Callable accepting a function and returning a function or
NumPy array, defining the complex conjugate of the action
of the left-hand-side matrix, e.g. as returned by
Hessian.action_fn.
B_action (Optional) Callable accepting a function and returning a
function or NumPy array, defining the complex conjugate of
the action of the right-hand-side matrix.
N_eigenvalues (Optional) Number of eigenvalues to attempt to find.
Defaults to a full spectrum.
solver_type (Optional) The solver type.
problem_type (Optional) The problem type. If not supplied
slepc4py.SLEPc.EPS.ProblemType.NHEP or
slepc4py.SLEPc.EPS.ProblemType.GNHEP are used.
which (Optional) Which eigenvalues to find. Defaults to
slepc4py.SLEPc.EPS.Which.LARGEST_MAGNITUDE.
tolerance (Optional) Tolerance, using slepc4py.SLEPc.EPS.Conv.REL
convergence criterion.
configure (Optional) Function handle accepting the EPS. Can be used
for manual configuration.
Returns:
A tuple (lam, V), where lam is an array of eigenvalues. For Hermitian
problems or with complex PETSc V is a tuple of functions containing
corresponding eigenvectors. Otherwise V is a tuple (V_r, V_i) where V_r and
V_i are each tuples of functions containing the real and imaginary parts of
corresponding eigenvectors.
"""
import petsc4py.PETSc as PETSc
import slepc4py.SLEPc as SLEPc
A_action = wrapped_action(space, A_action)
if B_action is not None:
B_action = wrapped_action(space, B_action)
if problem_type is None:
if B_action is None:
problem_type = SLEPc.EPS.ProblemType.NHEP
else:
problem_type = SLEPc.EPS.ProblemType.GNHEP
if which is None:
which = SLEPc.EPS.Which.LARGEST_MAGNITUDE
X = space_new(space)
n, N = function_local_size(X), function_global_size(X)
del X
N_ev = N if N_eigenvalues is None else N_eigenvalues
comm = space_comm(space) # .Dup()
A_matrix = PETSc.Mat().createPython(((n, N), (n, N)),
PythonMatrix(A_action),
comm=comm)
A_matrix.setUp()
if B_action is None:
B_matrix = None
else:
B_matrix = PETSc.Mat().createPython(((n, N), (n, N)),
PythonMatrix(B_action),
comm=comm)
B_matrix.setUp()
esolver = SLEPc.EPS().create(comm=comm)
if solver_type is not None:
esolver.setType(solver_type)
esolver.setProblemType(problem_type)
if B_matrix is None:
esolver.setOperators(A_matrix)
else:
esolver.setOperators(A_matrix, B_matrix)
esolver.setWhichEigenpairs(which)
esolver.setDimensions(nev=N_ev, ncv=SLEPc.DECIDE, mpd=SLEPc.DECIDE)
esolver.setConvergenceTest(SLEPc.EPS.Conv.REL)
esolver.setTolerances(tol=tolerance, max_it=SLEPc.DECIDE)
if configure is not None:
configure(esolver)
esolver.setUp()
assert not _flagged_error[0]
esolver.solve()
if _flagged_error[0]:
raise EigendecompositionException("Error encountered in "
"SLEPc.EPS.solve")
if esolver.getConverged() < N_ev:
raise EigendecompositionException("Not all requested eigenpairs "
"converged")
lam = np.full(
N_ev,
np.NAN,
dtype=PETSc.RealType if esolver.isHermitian() else PETSc.ComplexType)
v_r = A_matrix.getVecRight()
V_r = tuple(space_new(space) for n in range(N_ev))
if issubclass(PETSc.ScalarType, (complex, np.complexfloating)):
v_i = None
V_i = None
else:
v_i = A_matrix.getVecRight()
if esolver.isHermitian():
V_i = None
else:
V_i = tuple(space_new(space) for n in range(N_ev))
for i in range(lam.shape[0]):
lam_i = esolver.getEigenpair(i, v_r, v_i)
if esolver.isHermitian():
assert lam_i.imag == 0.0
lam[i] = lam_i.real
if v_i is not None:
with v_i as v_i_a:
assert abs(v_i_a).max() == 0.0
# else:
# # Complex note: If v_i is None then v_r may be non-real
# pass
with v_r as v_r_a:
function_set_values(V_r[i], v_r_a)
else:
lam[i] = lam_i
with v_r as v_r_a:
function_set_values(V_r[i], v_r_a)
if v_i is not None:
with v_i as v_i_a:
function_set_values(V_i[i], v_i_a)
# comm.Free()
if V_i is None:
return lam, V_r
else:
return lam, (V_r, V_i)
def solve_eig(self):
if self.solver == 'PETSc':
import slepc4py
import sys
from petsc4py import PETSc
from slepc4py import SLEPc
opts = PETSc.Options()
n = opts.getInt('n', self.N)
A = PETSc.Mat().create()
A.setSizes([n, n])
A.setFromOptions()
A.setUp()
A = PETSc.Mat().createDense(size=self.A.shape, array=self.A)
A.assemble()
B = PETSc.Mat().create()
B.setSizes([n, n])
B.setFromOptions()
B.setUp()
B = PETSc.Mat().createDense(size=self.B.shape, array=self.B)
B.assemble()
E = SLEPc.EPS()
E.create()
E.setOperators(A, B)
E.setProblemType(SLEPc.EPS.ProblemType.PGNHEP)
E.setFromOptions()
E.solve()
Print = PETSc.Sys.Print
Print()
Print("******************************")
Print("*** SLEPc Solution Results ***")
Print("******************************")
Print()
its = E.getIterationNumber()
Print("Number of iterations of the method: %d" % its)
eps_type = E.getType()
Print("Solution method: %s" % eps_type)
nev, ncv, mpd = E.getDimensions()
Print("Number of requested eigenvalues: %d" % nev)
tol, maxit = E.getTolerances()
Print("Stopping condition: tol=%.4g, maxit=%d" % (tol, maxit))
nconv = E.getConverged()
Print("Number of converged eigenpairs %d" % nconv)
vec_k = np.zeros(n) * (0 + 1j)
vec_vr = np.zeros([n, n]) * (0 + 1j)
vec_vi = np.zeros([n, n]) * (0 + 1j)
if nconv > 0:
# Create the results vectors
vr, wr = A.getVecs()
vi, wi = A.getVecs()
#
Print()
Print(" k ||Ax-kx||/||kx|| ")
Print("----------------- ------------------")
for i in range(nconv):
k = E.getEigenpair(i, vr, vi)
er = PETSc.Vec.getArray(vr)
ei = PETSc.Vec.getArray(vi)
vec_k[i] = k
vec_vr[i] = er
vec_vi[i] = ei
error = E.computeError(i)
if k.imag != 0.0:
Print(" %9f%+9f j %12g" % (k.real, k.imag, error))
else:
Print(" %12f %12g" % (k.real, error))
Print()
self.eigv = vec_k
self.eigf = vec_vr + vec_vi
""" solve the eigenvalues problem with the LINPACK subrutines"""
self.eigv, self.eigf = lin.eig(self.A, self.B)
#self.eigv, self.eigf = lins.eigs(self.A, k=10, M=self.B, which='LI')
# doesent work ARPACK problems with Arnoldi factorization
# remove the infinite and nan eigenvectors, and their eigenfunctions
selector = np.isfinite(self.eigv)
self.eigv = self.eigv[selector]
self.eigf = self.eigf[:, selector]
def get_null_space(A, npairs, comm=PETSc.COMM_WORLD):
"""
Returns 'npairs' eigenpairs with eigenvalues close to 0. Uses shift-and-invert.
Parameters:
-----------
A: scipy CSR matrix, or numpy array or PETSc mat
npairs : integer
Number of eigenvalues/vectors to return
comm: MPI communicator
Returns:
--------
evals: 1D array of npairs, eigenvalues from largest to smallest
evecs: 2D array of (n, npairs) where n is size of A matrix
"""
matrixtype = str(type(A))
if 'scipy' in matrixtype:
n = A.shape[0]
A = get_petsc_matrix(A, comm)
elif 'petsc' in matrixtype:
n = A.getSize()
elif 'numpy' in matrixtype:
A = csr_matrix(A)
n = A.shape[0]
A = get_petsc_matrix(A, comm)
else:
Print(
'Matrix type {} is not compatible. Use scipy CSR or PETSc AIJ type.'
.format(type(A)))
mpisize = comm.size
Print('Matrix size: {}'.format(n))
eps = SLEPc.EPS()
eps.create()
eps.setOperators(A)
eps.setProblemType(SLEPc.EPS.ProblemType.HEP)
eps.setWhichEigenpairs(SLEPc.EPS.Which.TARGET_REAL)
eps.setTarget(0.)
eps.setDimensions(nev=npairs, ncv=SLEPc.DECIDE, mpd=SLEPc.DECIDE)
st = eps.getST()
ksp = st.getKSP()
pc = ksp.getPC()
pc.setType(PETSc.PC.Type.CHOLESKY)
pc.setFactorShift(shift_type=A.FactorShiftType.POSITIVE_DEFINITE)
pc.setFactorSolverPackage('mumps')
st.setType(SLEPc.ST.Type.SINVERT)
eps.setTolerances(tol=1.e-14)
eps.setFromOptions() #Use command line options
eps.setUp()
eps.solve()
nconv = eps.getConverged()
vr = A.getVecLeft()
vi = A.getVecLeft()
if nconv < npairs:
Print("{} eigenvalues required, {} converged.".format(npairs, nconv))
npairs = nconv
evals = np.zeros(npairs)
evecs = np.zeros((n, npairs))
for i in range(npairs):
k = eps.getEigenpair(i, vr, vi)
if abs(k.imag) > 1.e-10:
Print("Imaginary eigenvalue: {} + {}j".format(k.real, k.imag))
Print("Error: {}".format(eps.computeError(i)))
if mpisize > 1:
evecs[:, i] = get_numpy_array(vr)
else:
evecs[:, i] = vr.getArray()
evals[i] = k.real
return evals, evecs
zero_tol = 1e-8
smallest_singular_values = smallest_singular_values[
smallest_singular_values > zero_tol]
condition_number = largest_singular_values.max(
) / smallest_singular_values.min()
time_scipy = time.time() - time_begin_scipy
print(f'Is symmetric? {petsc_mat.isSymmetric(tol=1e-8)}')
print(f'nnz: {nnz}')
print(f'DoFs: {W.dim()}')
print(f'Condition Number (scipy): {condition_number}')
time_begin_slepc = time.time()
S = SLEPc.SVD()
S.create()
S.setOperator(petsc_mat)
S.setType(SLEPc.SVD.Type.LAPACK)
S.setWhichSingularTriplets(which=S.Which.LARGEST)
S.setFromOptions()
S.solve()
# Create the results vectors
vr, vi = petsc_mat.getVecs()
nconv = S.getConverged()
singular_values_list = list()
num_of_extreme_singular_values = 5
if nconv > 0:
for i in range(num_of_extreme_singular_values):
singular_value_low = S.getSingularTriplet(i, vr, vi)
def get_eigenpairs(A, npairs=8, largest=True, comm=PETSc.COMM_WORLD):
"""
Parameters:
-----------
A: scipy CSR matrix, or numpy array or PETSc mat
npairs: int, number of eigenpairs required
largest: boolean, True (False) if largest (smallest) magnitude eigenvalues are requried
comm: MPI communicator
Returns:
--------
evals: 1D array of npairs, eigenvalues from largest to smallest
evecs: 2D array of (n, npairs) where n is size of A matrix
"""
matrixtype = str(type(A))
if 'scipy' in matrixtype:
n = A.shape[0]
A = get_petsc_matrix(A, comm)
elif 'petsc' in matrixtype:
n = A.getSize()
elif 'numpy' in matrixtype:
A = csr_matrix(A)
n = A.shape[0]
A = get_petsc_matrix(A, comm)
else:
Print(
'Matrix type {} is not compatible. Use scipy CSR or PETSc AIJ type.'
.format(type(A)))
mpisize = comm.size
Print('Matrix size: {}'.format(n))
eps = SLEPc.EPS()
eps.create()
eps.setOperators(A)
eps.setProblemType(SLEPc.EPS.ProblemType.HEP)
if largest:
eps.setWhichEigenpairs(SLEPc.EPS.Which.LARGEST_REAL)
else:
eps.setWhichEigenpairs(SLEPc.EPS.Which.SMALLEST_REAL)
eps.setDimensions(nev=npairs, ncv=SLEPc.DECIDE, mpd=SLEPc.DECIDE)
eps.setTolerances(tol=1.e-14)
eps.setFromOptions() #Use command line options
eps.setUp()
eps.solve()
nconv = eps.getConverged()
vr = A.getVecLeft()
vi = A.getVecLeft()
if nconv < npairs:
Print("{} eigenvalues required, {} converged.".format(npairs, nconv))
npairs = nconv
evals = np.zeros(npairs)
evecs = np.zeros((n, npairs))
for i in range(npairs):
k = eps.getEigenpair(i, vr, vi)
if abs(k.imag) > 1.e-10:
Print("Imaginary eigenvalue: {} + {}j".format(k.real, k.imag))
Print("Error: {}".format(eps.computeError(i)))
if mpisize > 1:
evecs[:, i] = get_numpy_array(vr)
else:
evecs[:, i] = vr.getArray()
evals[i] = k.real
return evals, evecs
def test_moore_spence():
try:
from slepc4py import SLEPc
except ImportError:
pytest.skip(msg="SLEPc unavailable, skipping eigenvalue test")
msh = IntervalMesh(1000, 1)
V = FunctionSpace(msh, "CG", 1)
R = FunctionSpace(msh, "R", 0)
# elastica residual
def residual(theta, lmbda, ttheta):
return inner(grad(theta),
grad(ttheta)) * dx - lmbda**2 * sin(theta) * ttheta * dx
th = Function(V)
x = SpatialCoordinate(msh)[0]
tth = TestFunction(V)
lm = Constant(3.142)
# Using guess for parameter lm, solve for state theta (th)
A = residual(th, lm, tth)
bcs = [DirichletBC(V, 0.0, "on_boundary")]
solve(A == 0, th, bcs=bcs)
# Now solve eigenvalue problem for $F_u(u, \lambda)\phi = r\phi$
# Want eigenmode phi with minimal eigenvalue r
B = derivative(residual(th, lm, TestFunction(V)), th, TrialFunction(V))
petsc_M = assemble(inner(TestFunction(V), TrialFunction(V)) * dx,
bcs=bcs).M.handle
petsc_B = assemble(B, bcs=bcs).M.handle
num_eigenvalues = 1
opts = PETSc.Options()
opts.setValue("eps_target_magnitude", None)
opts.setValue("eps_target", 0)
opts.setValue("st_type", "sinvert")
es = SLEPc.EPS().create(comm=COMM_WORLD)
es.setDimensions(num_eigenvalues)
es.setOperators(petsc_B, petsc_M)
es.setProblemType(SLEPc.EPS.ProblemType.GHEP)
es.setFromOptions()
es.solve()
ev_re, ev_im = petsc_B.getVecs()
es.getEigenpair(0, ev_re, ev_im)
eigenmode = Function(V)
eigenmode.vector().set_local(ev_re)
Z = MixedFunctionSpace([V, R, V])
# Set initial guesses for state, parameter, null eigenmode
z = Function(Z)
z.split()[0].assign(th)
z.split()[1].assign(lm)
z.split()[2].assign(eigenmode)
# Write Moore-Spence system of equations
theta, lmbda, phi = split(z)
ttheta, tlmbda, tphi = TestFunctions(Z)
F1 = residual(theta, lmbda, ttheta)
F2 = derivative(residual(theta, lmbda, tphi), z, as_vector([phi, 0, 0]))
F3 = (inner(phi, phi) - 1) * tlmbda * dx
F = F1 + F2 + F3
bcs = [
DirichletBC(Z.sub(0), 0.0, "on_boundary"),
DirichletBC(Z.sub(2), 0.0, "on_boundary")
]
# Need to fieldsplit onto the real variable as assembly doesn't work with R
sp = {
"mat_type": "matfree",
"snes_type": "newtonls",
"snes_monitor": None,
"snes_converged_reason": None,
"snes_linesearch_type": "basic",
"ksp_type": "fgmres",
"ksp_monitor_true_residual": None,
"ksp_max_it": 10,
"pc_type": "fieldsplit",
"pc_fieldsplit_type": "schur",
"pc_fieldsplit_schur_fact_type": "full",
"pc_fieldsplit_0_fields": "0,2",
"pc_fieldsplit_1_fields": "1",
"fieldsplit_0_ksp_type": "preonly",
"fieldsplit_0_pc_type": "python",
"fieldsplit_0_pc_python_type": "firedrake.AssembledPC",
"fieldsplit_0_assembled_pc_type": "lu",
"fieldsplit_0_assembled_pc_factor_mat_solver_type": "mumps",
"fieldsplit_0_assembled_mat_mumps_icntl_14": 200,
"mat_mumps_icntl_14": 200,
"fieldsplit_1_ksp_type": "gmres",
"fieldsplit_1_ksp_monitor_true_residual": None,
"fieldsplit_1_ksp_max_it": 1,
"fieldsplit_1_ksp_convergence_test": "skip",
"fieldsplit_1_pc_type": "none",
}
solve(F == 0, z, bcs=bcs, solver_parameters=sp)
with z.sub(1).dat.vec_ro as x:
param = x.norm()
assert abs(param - pi) < 1.0e-4
row = index
cols = [index - xsize, index - 1, index, index + 1, index + xsize]
data = [-1, -1, diag, -1, -1]
# ovo se moze i sa MatZeroRowsColumns (vjerojatno je efikasnije)
if y == 0 or x == 0 or x == (xsize - 1) or y == (ysize - 1):
cols = [index]
data = [diag]
A.setValues(row, cols, data)
A.assemblyBegin()
A.assemblyEnd()
# postavi eigenvalue solver
E = SLEPc.EPS()
E.create()
E.setOperators(A)
# matrica nije hermitska zbog rubnih uvjeta
E.setProblemType(SLEPc.EPS.ProblemType.NHEP)
# vrati 3 svojstvene vrijednosti
E.setDimensions(3, PETSc.DECIDE, PETSc.DECIDE)
# prvo najmanja pa onda veće
E.setWhichEigenpairs(SLEPc.EPS.Which.SMALLEST_MAGNITUDE)
E.setFromOptions()
E.solve()
# stvori vektore za rješenje
xr, xi = A.createVecs()
def mfn_multiply_slepc(mat, vec,
fntype='exp',
MFNType='AUTO',
comm=None,
isherm=False):
"""Compute the action of ``func(mat) @ vec``.
Parameters
----------
mat : matrix-like
Matrix to compute function action of.
vec : vector-like
Vector to compute matrix function action on.
func : {'exp', 'sqrt', 'log'}, optional
Function to use.
MFNType : {'krylov', 'expokit'}, optional
Method of computing the matrix function action, 'expokit' is only
available for func='exp'.
comm : mpi4py.MPI.Comm instance, optional
The mpi communicator.
isherm : bool, optional
If `mat` is known to be hermitian, this might speed things up in
some circumstances.
Returns
-------
fvec : np.matrix
The vector output of ``func(mat) @ vec``.
"""
SLEPc, comm = get_slepc(comm=comm)
mat = convert_mat_to_petsc(mat, comm=comm)
if isherm:
mat.setOption(mat.Option.HERMITIAN, True)
vec = convert_vec_to_petsc(vec, comm=comm)
out = new_petsc_vec(vec.size, comm=comm)
if MFNType.upper() == 'AUTO':
if (fntype == 'exp') and (vec.size <= 2**16):
MFNType = 'EXPOKIT'
else:
MFNType = 'KRYLOV'
# set up the matrix function options and objects
mfn = SLEPc.MFN().create(comm=comm)
mfn.setType(getattr(SLEPc.MFN.Type, MFNType.upper()))
mfn_fn = mfn.getFN()
mfn_fn.setType(getattr(SLEPc.FN.Type, fntype.upper()))
mfn_fn.setScale(1.0, 1.0)
mfn.setFromOptions()
mfn.setOperator(mat)
# 'solve' / perform the matrix function
mfn.solve(vec, out)
# --> gather the (distributed) petsc vector to a numpy matrix on master
all_out = gather_petsc_array(
out, comm=comm, out_shape=(-1, 1), matrix=True)
comm.Barrier()
mfn.destroy()
return all_out
def sorted_krylov_schur(
P: Union[spmatrix, np.ndarray], k: int, z: str = "LM", tol: float = 1e-16
) -> Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray]:
r"""
Calculate an orthonormal basis of the subspace associated with the `k`
dominant eigenvalues of `P` using the Krylov-Schur method as implemented in SLEPc.
This functions requires :mod:`petsc4py` and :mod:`slepc4py`.
Parameters
----------
%(P)s
%(k)s
%(z)s
tol
Convergence criterion used by SLEPc internally. If you are dealing
with ill-conditioned matrices, consider decreasing this value to
get accurate results.
Returns
-------
Tuple of the following:
R
%(R_sort)s
Q
%(Q_sort)s
eigenvalues
%(eigenvalues_k)s
eigenvalues_error
Array of shape `(k,)` containing the error, based on the residual
norm, of the `i`th eigenpair at index `i`.
""" # noqa: D205, D400
# We like to thank A. Sikorski and M. Weber for pointing us to SLEPc for partial Schur decompositions of
# sparse matrices.
# Further parts of sorted_krylov_schur were developed based on the function krylov_schur
# https://github.com/zib-cmd/cmdtools/blob/1c6b6d8e1c35bb487fcf247c5c1c622b4b665b0a/src/cmdtools/analysis/pcca.py#L64,
# written by Alexander Sikorski.
from petsc4py import PETSc
from slepc4py import SLEPc
M = PETSc.Mat().create()
_initialize_matrix(M, P)
# Creates EPS object.
E = SLEPc.EPS()
E.create()
# Set the matrix associated with the eigenvalue problem.
E.setOperators(M)
# Select the particular solver to be used in the EPS object: Krylov-Schur
E.setType(SLEPc.EPS.Type.KRYLOVSCHUR)
# Set the number of eigenvalues to compute and the dimension of the subspace.
E.setDimensions(nev=k)
# set the tolerance used in the convergence criterion
E.setTolerances(tol=tol)
# Specify which portion of the spectrum is to be sought.
# LARGEST_MAGNITUDE: Largest magnitude (default).
# LARGEST_REAL: Largest real parts.
# All possible Options can be found here:
# (see: https://slepc.upv.es/slepc4py-current/docs/apiref/slepc4py.SLEPc.EPS.Which-class.html)
if z == "LM":
E.setWhichEigenpairs(E.Which.LARGEST_MAGNITUDE)
elif z == "LR":
E.setWhichEigenpairs(E.Which.LARGEST_REAL)
else:
raise ValueError(f"Invalid spectrum sorting options `{z}`. Valid options are: 'LM', 'LR'.")
# Solve the eigensystem.
E.solve()
# getInvariantSubspace() gets an orthonormal basis of the computed invariant subspace.
# It returns a list of vectors.
# The returned real vectors span an invariant subspace associated with the computed eigenvalues.
# We take the sequence of 1-D arrays and stack them as columns to make a single 2-D array.
Q = np.column_stack([x.array for x in E.getInvariantSubspace()])
# Get the schur form
R = E.getDS().getMat(SLEPc.DS.MatType.A)
R.view()
R = R.getDenseArray().astype(np.float64)
# Gets the number of converged eigenpairs.
nconv = E.getConverged()
# Warn, if nconv smaller than k.
if nconv < k:
warnings.warn(f"The number of converged eigenpairs is `{nconv}`, but `{k}` were requested.")
# Collect the k dominant eigenvalues.
eigenvalues = []
eigenvalues_error = []
for i in range(nconv):
# Get the i-th eigenvalue as computed by solve().
eigenval = E.getEigenvalue(i)
eigenvalues.append(eigenval)
# Computes the error (based on the residual norm) associated with the i-th computed eigenpair.
eigenval_error = E.computeError(i)
eigenvalues_error.append(eigenval_error)
# convert lists with eigenvalues and errors to arrays (while keeping excess eigenvalues and errors)
eigenvalues = np.asarray(eigenvalues)
eigenvalues_error = np.asarray(eigenvalues_error)
return R, Q, eigenvalues, eigenvalues_error
