def setup_preconditioner(W, which, eps):
'''
This is a block diagonal preconditioner based on
Hdiv x L2 x (H0.5 \cap sqrt(eps)*L2) or ?
'''
from block.algebraic.petsc import LU
from block.algebraic.petsc import LumpedInvDiag
from hsmg import HsNorm
S, V, Q = W
# Hdiv
sigma, tau = TrialFunction(S), TestFunction(S)
b00 = inner(div(sigma), div(tau))*dx + inner(sigma, tau)*dx
# Inverted exactly
B00 = LU(ii_assemble(b00))
# L2
u, v = TrialFunction(V), TestFunction(V)
b11 = inner(u, v)*dx
# Inverted by BoomerAMG
B11 = LumpedInvDiag(ii_assemble(b11))
# The Q norm via spectral
B22 = inverse(HsNorm(Q, s=0.5) + eps*HsNorm(Q, s=0.0))# The norm is inverted exactly
return block_diag_mat([B00, B11, B22])
def setup_preconditioner(W, which, eps):
from block.algebraic.petsc import AMG
from block.algebraic.petsc import LumpedInvDiag, LU
from hsmg import HsNorm
u1, ub, p1, u2, p2, lambda_ = map(TrialFunction, W)
v1, vb, q1, v2, q2, beta_ = map(TestFunction, W)
# Neither is super spectacular
# Completely block diagonal preconditioner with H1 on the bubble space
if which == 0:
b00 = inner(grad(u1), grad(v1)) * dx + inner(u1, v1) * dx
B00 = AMG(ii_assemble(b00))
bbb = inner(grad(ub), grad(vb)) * dx + inner(ub, vb) * dx
Bbb = LumpedInvDiag(ii_assemble(bbb))
b11 = inner(p1, q1) * dx
B11 = AMG(ii_assemble(b11))
b22 = inner(div(u2), div(v2)) * dx + inner(u2, v2) * dx
B22 = LU(ii_assemble(b22))
b33 = inner(p2, q2) * dx
B33 = LumpedInvDiag(ii_assemble(b33))
B44 = inverse(HsNorm(W[-1], s=0.5))
return block_diag_mat([B00, Bbb, B11, B22, B33, B44])
# Monolithic for MINI velocity
b = [[0, 0], [0, 0]]
b[0][0] = inner(grad(u1), grad(v1)) * dx + inner(u1, v1) * dx
b[0][1] = inner(grad(ub), grad(v1)) * dx + inner(ub, v1) * dx
b[1][0] = inner(grad(u1), grad(vb)) * dx + inner(u1, vb) * dx
b[1][1] = inner(grad(ub), grad(vb)) * dx + inner(ub, vb) * dx
# Make into a monolithic matrix
B00 = AMG(ii_convert(ii_assemble(b)))
b11 = inner(p1, q1) * dx
B11 = AMG(ii_assemble(b11))
b22 = inner(div(u2), div(v2)) * dx + inner(u2, v2) * dx
B22 = LU(ii_assemble(b22))
b33 = inner(p2, q2) * dx
B33 = LumpedInvDiag(ii_assemble(b33))
B44 = inverse(HsNorm(W[-1], s=0.5))
# So this is a 5x5 matrix
BB = block_diag_mat([B00, B11, B22, B33, B44])
# But the preconditioner has to be 6x6; reduce 6 to 6 by comcat
# first two, rest stays same
R = ReductionOperator([2, 3, 4, 5, 6], W)
return (R.T) * BB * R
def setup_preconditioner(W, which, eps):
'''
This is a block diagonal preconditioner based on
H1 x H1 x (H-0.5 \cap \sqrt{eps-1}*L2) or ...
'''
# NOTE for eps large the fractional term is expected to dominte
from block.algebraic.petsc import AMG
from block.algebraic.petsc import LumpedInvDiag
from hsmg import HsNorm
V1, V2, Q = W
if which == 0:
# H1
print '\tUsing H1 x H1 x (sqrt(1./%g)*L2 \cap H-0.5) preconditioner' % eps
u1, v1 = TrialFunction(V1), TestFunction(V1)
b00 = inner(grad(u1), grad(v1)) * dx + inner(u1, v1) * dx
# Inverted by BoomerAMG
B00 = AMG(ii_assemble(b00))
u2, v2 = TrialFunction(V2), TestFunction(V2)
b11 = inner(grad(u2), grad(v2)) * dx + inner(u2, v2) * dx
# Inverted by BoomerAMG
B11 = AMG(ii_assemble(b11))
# The Q norm via spectral, the norm is inverted exactly
B22 = inverse((HsNorm(Q, s=-0.5) + (eps**-1) * HsNorm(Q, s=0.0)))
else:
print '\tUsing (H1 \cap H0.5) x (H1 \cap H0.5) x sqrt(%g)*L2 preconditioner' % eps
iface = Q.mesh()
dxGamma = dx(domain=iface)
# H1
u1, v1 = TrialFunction(V1), TestFunction(V1)
Tu1, Tv1 = Trace(u1, iface), Trace(v1, iface)
b00 = inner(grad(u1), grad(v1)) * dx + inner(u1, v1) * dx + inner(
Tu1, Tv1) * dxGamma
# Inverted by BoomerAMG
B00 = AMG(ii_convert(ii_assemble(b00)))
u2, v2 = TrialFunction(V2), TestFunction(V2)
Tu2, Tv2 = Trace(u2, iface), Trace(v2, iface)
b11 = inner(grad(u2), grad(v2)) * dx + inner(u2, v2) * dx + inner(
Tu2, Tv2) * dxGamma
# Inverted by BoomerAMG
B11 = AMG(ii_convert(ii_assemble(b11)))
# The Q norm via spectral
p, q = TrialFunction(Q), TestFunction(Q)
b22 = Constant(1. / eps) * inner(p, q) * dxGamma
B22 = LumpedInvDiag(ii_assemble(b22))
return block_diag_mat([B00, B11, B22])
def setup_preconditioner(W, which, eps):
'''The preconditioner'''
from block.algebraic.petsc import AMG, LumpedInvDiag
from block import block_transpose
from hsmg import HsNorm
u, ub, p, lambda_ = map(TrialFunction, W)
v, vb, q, beta_ = map(TestFunction, W)
# A block diagonal preconditioner
if which == 0:
b00 = inner(grad(u), grad(v)) * dx + inner(u, v) * dx
B00 = AMG(ii_assemble(b00))
b11 = inner(grad(ub), grad(vb)) * dx + inner(ub, vb) * dx
B11 = LumpedInvDiag(ii_assemble(b11))
b22 = inner(p, q) * dx
B22 = AMG(ii_assemble(b22))
M = W[-1]
Mi = M.sub(0).collapse()
B33 = inverse(VectorizedOperator(HsNorm(Mi, s=-0.5), M))
return block_diag_mat([B00, B11, B22, B33])
# Preconditioner monolithic in Stokes velocity
b = [[0, 0], [0, 0]]
b[0][0] = inner(grad(u), grad(v)) * dx + inner(u, v) * dx
b[0][1] = inner(grad(ub), grad(v)) * dx + inner(ub, v) * dx
b[1][0] = inner(grad(u), grad(vb)) * dx + inner(u, vb) * dx
b[1][1] = inner(grad(ub), grad(vb)) * dx + inner(ub, vb) * dx
# Make into a monolithic matrix
B00 = AMG(ii_convert(ii_assemble(b)))
b11 = inner(p, q) * dx
B11 = AMG(ii_assemble(b11))
M = W[-1]
Mi = M.sub(0).collapse()
B22 = inverse(VectorizedOperator(HsNorm(Mi, s=-0.5), M))
# So this is a 3x3 matrix
BB = block_diag_mat([B00, B11, B22])
# But the preconditioner has to be 4x4
R = ReductionOperator([2, 3, 4], W)
return (R.T) * BB * R
def setup_preconditioner(W, p_constant):
'''Just an idea H1 x H1 x (Hs \cap p_constant L2)'''
from block.algebraic.petsc import AMG
from hsmg import HsNorm
u3, u, p = map(TrialFunction, W)
v3, v, q = map(TestFunction, W)
b00 = inner(grad(u3), grad(v3)) * dx + inner(u3, v3) * ds
B00 = assemble(b00)
b11 = inner(grad(u), grad(v)) * dx + inner(u, v) * ds
B11 = assemble(b11)
B22 = inverse(HsNorm(W[-1], s=-0.5) + p_constant * HsNorm(W[-1], s=0.0))
return block_diag_mat([AMG(B00), AMG(B11), B22])
def setup_preconditioner(W, which, eps):
'''
TODO
'''
from block.algebraic.petsc import AMG
from block.algebraic.petsc import LumpedInvDiag
from hsmg import HsNorm
V1, V2, Q = W
gamma_mesh = Q.mesh()
dxGamma = Measure('dx', domain=gamma_mesh)
u1, u2 = map(TrialFunction, (V1, V2))
v1, v2 = map(TestFunction, (V1, V2))
Tu1, Tu2 = map(lambda x: Trace(x, gamma_mesh), (u1, u2))
Tv1, Tv2 = map(lambda x: Trace(x, gamma_mesh), (v1, v2))
if eps > 1:
b00 = Constant(eps) * inner(grad(u1), grad(v1)) * dx + inner(u1,
v1) * dx
B00 = ii_assemble(b00)
else:
b00 = Constant(eps) * inner(grad(u1), grad(v1)) * dx + inner(
u1, v1) * dx + inner(Tu1, Tv1) * dxGamma
B00 = ii_convert(ii_assemble(b00))
B00 = AMG(B00)
b11 = inner(grad(u2), grad(v2)) * dx + inner(u2, v2) * dx
# Inverted by BoomerAMG
B11 = AMG(ii_assemble(b11))
if eps > 1:
B22 = inverse(HsNorm(Q, s=-0.5))
else:
B22 = inverse(HsNorm(Q, s=0.5))
return block_diag_mat([B00, B11, B22])
def setup_preconditioner(W, which, eps=None):
'''
This is a block diagonal preconditioner based on H1 x H^{-0.5}
'''
from xii.linalg.matrix_utils import as_petsc
from numpy import hstack
from petsc4py import PETSc
from hsmg import HsNorm
assert len(eps) == 2
mu_value, lmbda_value = eps
V, Q = W
# H1
u, v = TrialFunction(V), TestFunction(V)
b00 = inner(grad(u), grad(v)) * dx + inner(u, v) * dx
A = as_backend_type(assemble(b00))
# Attach rigid deformations to A
# Functions
Z = [
interpolate(Constant((1, 0)), V),
interpolate(Constant((0, 1)), V),
interpolate(Expression(('x[1]', '-x[0]'), degree=1), V)
]
# The basis
Z = VectorSpaceBasis([z.vector() for z in Z])
Z.orthonormalize()
A.set_nullspace(Z)
A.set_near_nullspace(Z)
A = as_petsc(A)
# Setup the preconditioner in petsc
pc = PETSc.PC().create()
pc.setType(PETSc.PC.Type.HYPRE)
pc.setOperators(A)
# Other options
opts = PETSc.Options()
opts.setValue('pc_hypre_boomeramg_cycle_type', 'V')
opts.setValue('pc_hypre_boomeramg_relax_type_all', 'symmetric-SOR/Jacobi')
opts.setValue('pc_hypre_boomeramg_coarsen_type', 'Falgout')
pc.setFromOptions()
# Wrap for cbc.block
B00 = BlockPC(pc)
# The Q norm via spectral
Qi = Q.sub(0).collapse()
B11 = inverse(VectorizedOperator(HsNorm(Qi, s=-0.5), Q))
return block_diag_mat([B00, B11])
def setup_preconditioner(W, which, eps=None):
'''
This is a block diagonal preconditioner based on Hdiv x H0.5'''
from block.algebraic.petsc import LU
from hsmg import HsNorm
S, Q = W
sigma, tau = TrialFunction(S), TestFunction(S)
# Hdiv
b00 = inner(div(sigma), div(tau)) * dx + inner(sigma, tau) * dx
B00 = LU(ii_assemble(b00)) # Exact
# H0.5
B11 = HsNorm(Q, s=0.5)**-1 # The norm is inverted exactly
return block_diag_mat([B00, B11])
def setup_preconditioner(W, which, eps):
'''The preconditioner'''
from block.algebraic.petsc import AMG, LumpedInvDiag
from hsmg import HsNorm
u, p, lambda_ = map(TrialFunction, W)
v, q, beta_ = map(TestFunction, W)
b00 = inner(grad(u), grad(v)) * dx + inner(u, v) * dx
B00 = AMG(ii_assemble(b00))
b11 = inner(p, q) * dx
B11 = AMG(ii_assemble(b11))
M = W[-1]
Mi = M.sub(0).collapse()
B22 = inverse(VectorizedOperator(HsNorm(Mi, s=-0.5), M))
return block_diag_mat([B00, B11, B22])
def setup_preconditioner(W, which, eps=None):
'''
This is a block diagonal preconditioner based on H1 x L2 x H^{-0.5}
'''
from block.algebraic.petsc import LumpedInvDiag
from xii.linalg.matrix_utils import as_petsc
from numpy import hstack
from petsc4py import PETSc
from hsmg import HsNorm
V, Q, Y = W
# H1
u, v = TrialFunction(V), TestFunction(V)
b00 = inner(grad(u), grad(v)) * dx + inner(u, v) * dx
# NOTE: since interpolation is broken with MINI I don't interpolate
# here the RM basis to attach the vectros to matrix
A = assemble(b00)
A = as_petsc(A)
# Setup the preconditioner in petsc
pc = PETSc.PC().create()
pc.setType(PETSc.PC.Type.HYPRE)
pc.setOperators(A)
# Other options
opts = PETSc.Options()
opts.setValue('pc_hypre_boomeramg_cycle_type', 'V')
opts.setValue('pc_hypre_boomeramg_relax_type_all', 'symmetric-SOR/Jacobi')
opts.setValue('pc_hypre_boomeramg_coarsen_type', 'Falgout')
pc.setFromOptions()
# Wrap for cbc.block
B00 = BlockPC(pc)
p, q = TrialFunction(Q), TestFunction(Q)
B11 = LumpedInvDiag(assemble(inner(p, q) * dx))
# The Y norm via spectral
Yi = Y.sub(0).collapse()
B22 = inverse(VectorizedOperator(HsNorm(Yi, s=-0.5), Y))
return block_diag_mat([B00, B11, B22])
def setup_preconditioner(W, which, eps):
'''The preconditioner'''
from block.algebraic.petsc import AMG
from hsmg import HsNorm
u, p, lambda_ = map(TrialFunction, W)
v, q, beta_ = map(TestFunction, W)
b00 = Constant(eps) * inner(grad(u), grad(v)) * dx + inner(u, v) * dx
B00 = AMG(ii_assemble(b00))
b11a = Constant(1. / eps) * inner(p, q) * dx
b11b = inner(grad(p), grad(q)) * dx
bcs = DirichletBC(W[1], Constant(0), 'on_boundary')
B11b, _ = assemble_system(b11b, inner(Constant(0), q) * dx, bcs)
B11 = AMG(ii_assemble(b11a)) + AMG(B11b)
B22 = inverse((1. / eps) * HsNorm(W[-1], s=-0.5))
return block_diag_mat([B00, B11, B22])
def setup_preconditioner(W, which, eps=None):
'''
This is a block diagonal preconditioner based on
H1 x H-0.5 or (H1 \cap H0.5) x L2
'''
from block.algebraic.petsc import AMG
from block.algebraic.petsc import LumpedInvDiag
from hsmg import HsNorm
V, Q = W
if which == 0:
print 'Using H1 x H-0.5 preconditioner'
# H1
u, v = TrialFunction(V), TestFunction(V)
b00 = inner(grad(u), grad(v)) * dx + inner(u, v) * dx
# Inverted by BoomerAMG
B00 = AMG(ii_assemble(b00))
# The Q norm via spectral
B11 = inverse(HsNorm(Q, s=-0.5)) # The norm is inverted exactly
else:
print 'Using (H1 \cap H0.5) x L2 preconditioner'
bdry = Q.mesh()
dxGamma = dx(domain=bdry)
# Cap space
u, v = TrialFunction(V), TestFunction(V)
Tu, Tv = Trace(u, bdry), Trace(v, bdry)
b00 = inner(grad(u), grad(v)) * dx + inner(u, v) * dx + inner(
Tu, Tv) * dxGamma
# Inverted by BoomrAMG
B00 = AMG(ii_convert(ii_assemble(b00)))
# We don't have to work so hard for multiplier
p, q = TrialFunction(Q), TestFunction(Q)
B11 = LumpedInvDiag(ii_assemble(inner(p, q) * dx))
return block_diag_mat([B00, B11])
def setup_preconditioner(W, which, eps):
from block.algebraic.petsc import AMG
from block.algebraic.petsc import LumpedInvDiag, LU
from hsmg import HsNorm
u1, p1, u2, p2, lambda_ = map(TrialFunction, W)
v1, q1, v2, q2, beta_ = map(TestFunction, W)
# This is not spectacular
b00 = inner(grad(u1), grad(v1)) * dx + inner(u1, v1) * dx
B00 = AMG(ii_assemble(b00))
b11 = inner(p1, q1) * dx
B11 = AMG(ii_assemble(b11))
b22 = inner(div(u2), div(v2)) * dx + inner(u2, v2) * dx
B22 = LU(ii_assemble(b22))
b33 = inner(p2, q2) * dx
B33 = LumpedInvDiag(ii_assemble(b33))
B44 = inverse(HsNorm(W[-1], s=0.5))
return block_diag_mat([B00, B11, B22, B33, B44])
from hsmg import HsNorm
import numpy as np
mesh = UnitSquareMesh(10, 10)
VV = FunctionSpace(mesh,
MixedElement([FiniteElement('Lagrange', triangle, 1)] * 3))
u, v = TrialFunction(VV), TestFunction(VV)
M = assemble(inner(u, v) * dx)
x = M.create_vec()
x.set_local(np.random.rand(x.local_size()))
y = M * x
V = FunctionSpace(mesh, 'CG', 1)
u, v = TrialFunction(V), TestFunction(V)
Mj = assemble(inner(u, v) * dx)
# Should match M
X = VectorizedOperator(Mj, VV)
y0 = X * x
print(x.norm('l2'), (y - y0).norm('linf'))
I = HsNorm(V, s=0.5)
II = VectorizedOperator(I, VV)
foo = II * x
x0 = inverse(II) * foo
print((x - x0).norm('linf'))
def solve_problem(ncells, eps, solver_params):
'''Optim problem on [0, 1]^2'''
# Made up
f = Expression('x[0] + x[1]', degree=1)
mesh = UnitSquareMesh(*(ncells, ) * 2)
bmesh = BoundaryMesh(mesh, 'exterior')
Q = FunctionSpace(mesh, 'DG', 0)
V = FunctionSpace(mesh, 'CG', 1)
B = FunctionSpace(mesh, 'CG', 1)
W = [Q, V, B]
p, u, lmbda = list(map(TrialFunction, W))
q, v, beta = list(map(TestFunction, W))
Tu = Trace(u, bmesh)
Tv = Trace(v, bmesh)
# The line integral
dxGamma = Measure('dx', domain=bmesh)
a = [[0] * len(W) for _ in range(len(W))]
a[0][0] = Constant(eps) * inner(p, q) * dx
a[0][2] = inner(q, lmbda) * dx
# We put zero on the diagonal temporarily and then replace by
# the fractional laplacian
a[1][1] = 0
a[1][2] = inner(grad(v), grad(lmbda)) * dx + inner(v, lmbda) * dx
a[2][0] = inner(p, beta) * dx
a[2][1] = inner(grad(u), grad(beta)) * dx + inner(u, beta) * dx
L = [
inner(Constant(0), q) * dx,
inner(Constant(0), v) * dx, # Same replacement idea here
inner(Constant(0), beta) * dx
]
# All but (1, 1) and 1 are final
AA, bb = list(map(ii_assemble, (a, L)))
# Now I want and operator which corresponds to (Tv, (-Delta^{0.5} T_u))_bdry
TV = FunctionSpace(bmesh, 'CG', 1)
T = PETScMatrix(trace_mat_no_restrict(V, TV))
# The fractional laplacian nodal->dual
fDelta = HsNorm(TV, s=0.5)
# This should be it
A11 = block_transpose(T) * fDelta * T
# Now, while here we can also comute the rhs contribution
# (Tv, (-Delta^{0.5}, f))
f_vec = interpolate(f, V).vector()
b1 = A11 * f_vec
bb[1] = b1
AA[1][1] = A11
wh = ii_Function(W)
# Direct solve
if not solver_params:
AAm, bbm = list(map(ii_convert, (AA, bb)))
LUSolver('umfpack').solve(AAm, wh.vector(), bbm)
return wh, -1
# Magne like precond
if False:
# Preconditioner like in the L2 case but with L2 bdry replaced
b00 = Constant(eps) * inner(p, q) * dx
B00 = LumpedInvDiag(ii_assemble(b00))
H1 = ii_assemble(inner(grad(v), grad(u)) * dx + inner(v, u) * dx)
# From dual to nodal
R = LumpedInvDiag(ii_assemble(inner(u, v) * dx))
# The whole matrix to be inverted is then (second term is H2 approx)
B11 = collapse(A11 + eps * H1 * R * H1)
# And the inverse
B11 = AMG(B11, parameters={'pc_hypre_boomeramg_cycle_type': 'W'})
b22 = Constant(1. / eps) * inner(lmbda, beta) * dx
B22 = LumpedInvDiag(ii_assemble(b22))
# A bit like SZ
else:
b00 = Constant(eps) * inner(p, q) * dx
B00 = LumpedInvDiag(ii_assemble(b00))
# L2 \cap eps H1
H1 = assemble(
Constant(eps) * (inner(grad(v), grad(u)) * dx + inner(v, u) * dx))
B11 = AMG(collapse(A11 + H1))
# (1./eps)*L2 \cap 1./sqrt(eps)H1
a = Constant(1./eps)*inner(beta, lmbda)*dx + \
Constant(1./sqrt(eps))*(inner(grad(beta), grad(lmbda))*dx + inner(beta, lmbda)*dx)
B22 = AMG(assemble(a))
BB = block_diag_mat([B00, B11, B22])
# Want the iterations to start from random (iterative)
wh.block_vec().randomize()
# Default is minres
if '-ksp_type' not in solver_params: solver_params['-ksp_type'] = 'minres'
opts = PETSc.Options()
for key, value in solver_params.items():
opts.setValue(key, None if value == 'none' else value)
ksp = PETSc.KSP().create()
ksp.setOperators(ii_PETScOperator(AA))
ksp.setPC(ii_PETScPreconditioner(BB, ksp))
ksp.setNormType(PETSc.KSP.NormType.NORM_PRECONDITIONED)
ksp.setFromOptions()
ksp.solve(as_petsc_nest(bb), wh.petsc_vec())
niters = ksp.getIterationNumber()
return wh, niters