def test_krylov_solver_norm_type():
"""Check setting of norm type used in testing for convergence by
PETScKrylovSolver
"""
norm_type = (PETScKrylovSolver.norm_type_default_norm,
PETScKrylovSolver.norm_type_natural,
PETScKrylovSolver.norm_type_preconditioned,
PETScKrylovSolver.norm_type_none,
PETScKrylovSolver.norm_type_unpreconditioned)
for norm in norm_type:
# Solve a system of equations
mesh = UnitSquareMesh(4, 4)
V = FunctionSpace(mesh, "Lagrange", 1)
u, v = TrialFunction(V), TestFunction(V)
a = u*v*dx
L = Constant(1.0)*v*dx
A, b = assemble(a), assemble(L)
solver = PETScKrylovSolver("cg")
solver.parameters["maximum_iterations"] = 2
solver.parameters["error_on_nonconvergence"] = False
solver.set_norm_type(norm)
solver.set_operator(A)
solver.solve(b.copy(), b)
solver.get_norm_type()
if norm is not PETScKrylovSolver.norm_type_default_norm:
assert solver.get_norm_type() == norm
def test_krylov_solver_options_prefix(pushpop_parameters):
"Test set/get PETScKrylov solver prefix option"
# Set backend
parameters["linear_algebra_backend"] = "PETSc"
# Prefix
prefix = "test_foo_"
# Create solver and set prefix
solver = PETScKrylovSolver()
solver.set_options_prefix(prefix)
# Check prefix (pre solve)
assert solver.get_options_prefix() == prefix
# Solve a system of equations
mesh = UnitSquareMesh(4, 4)
V = FunctionSpace(mesh, "Lagrange", 1)
u, v = TrialFunction(V), TestFunction(V)
a, L = u*v*dx, Constant(1.0)*v*dx
A, b = assemble(a), assemble(L)
solver.set_operator(A)
solver.solve(b.copy(), b)
# Check prefix (post solve)
assert solver.get_options_prefix() == prefix
def test_krylov_solver_options_prefix(pushpop_parameters):
"Test set/get PETScKrylov solver prefix option"
# Set backend
parameters["linear_algebra_backend"] = "PETSc"
# Prefix
prefix = "test_foo_"
# Create solver and set prefix
solver = PETScKrylovSolver()
solver.set_options_prefix(prefix)
# Check prefix (pre solve)
assert solver.get_options_prefix() == prefix
# Solve a system of equations
mesh = UnitSquareMesh(4, 4)
V = FunctionSpace(mesh, "Lagrange", 1)
u, v = TrialFunction(V), TestFunction(V)
a, L = u*v*dx, Constant(1.0)*v*dx
A, b = assemble(a), assemble(L)
solver.set_operator(A)
solver.solve(b.copy(), b)
# Check prefix (post solve)
assert solver.get_options_prefix() == prefix
def test_krylov_solver_norm_type():
"""Check setting of norm type used in testing for convergence by
PETScKrylovSolver
"""
norm_type = (PETScKrylovSolver.norm_type.default_norm,
PETScKrylovSolver.norm_type.natural,
PETScKrylovSolver.norm_type.preconditioned,
PETScKrylovSolver.norm_type.none,
PETScKrylovSolver.norm_type.unpreconditioned)
for norm in norm_type:
# Solve a system of equations
mesh = UnitSquareMesh(4, 4)
V = FunctionSpace(mesh, "Lagrange", 1)
u, v = TrialFunction(V), TestFunction(V)
a = u*v*dx
L = Constant(1.0)*v*dx
A, b = assemble(a), assemble(L)
solver = PETScKrylovSolver("cg")
solver.parameters["maximum_iterations"] = 2
solver.parameters["error_on_nonconvergence"] = False
solver.set_norm_type(norm)
solver.set_operator(A)
solver.solve(b.copy(), b)
solver.get_norm_type()
if norm is not PETScKrylovSolver.norm_type.default_norm:
assert solver.get_norm_type() == norm
def getprecond(self):
Prec = PETScKrylovSolver("richardson", "amg")
Prec.parameters["maximum_iterations"] = 1
Prec.parameters["error_on_nonconvergence"] = False
Prec.parameters["nonzero_initial_guess"] = False
Prec.set_operator(self.Regul.get_precond())
return Prec
def getprecond(self):
""" precondition by TV + small fraction of mass matrix """
solver = PETScKrylovSolver('cg', self.amgprecond)
solver.parameters["maximum_iterations"] = 2000
solver.parameters["relative_tolerance"] = 1e-24
solver.parameters["absolute_tolerance"] = 1e-24
solver.parameters["error_on_nonconvergence"] = True
solver.parameters["nonzero_initial_guess"] = False
solver.set_operator(self.H + self.sMass)
return solver
def getprecond(self):
""" precondition by TV + small fraction of mass matrix """
#TODO: does not appear to be a great way to apply preconditioner (DIVERGED_ITS)
solver = PETScKrylovSolver('cg', self.amgprecond)
solver.parameters["maximum_iterations"] = 3000
solver.parameters["relative_tolerance"] = 1e-24
solver.parameters["absolute_tolerance"] = 1e-24
solver.parameters["error_on_nonconvergence"] = True
solver.parameters["nonzero_initial_guess"] = False
self.precond = self.H + self.sMass
solver.set_operator(self.precond)
return solver
def getprecond(self):
# if self.coeff_cg + self.coeff_ncg > 0.0:
# solver = PETScKrylovSolver('gmres', self.amgprecond)
# solver.parameters["maximum_iterations"] = 10000
# else:
solver = PETScKrylovSolver('cg', self.amgprecond)
solver.parameters["maximum_iterations"] = 2000
solver.parameters["absolute_tolerance"] = 1e-24
solver.parameters["relative_tolerance"] = 1e-24
solver.parameters["error_on_nonconvergence"] = True
solver.parameters["nonzero_initial_guess"] = False
solver.set_operator(self.precond)
return solver
def getprecond(self):
"""
solver = PETScKrylovSolver("richardson", "amg")
solver.parameters["maximum_iterations"] = 1
solver.parameters["error_on_nonconvergence"] = False
solver.parameters["nonzero_initial_guess"] = False
"""
solver = PETScKrylovSolver('cg', self.amgsolver)
solver.parameters["maximum_iterations"] = 1000
solver.parameters["relative_tolerance"] = 1e-24
solver.parameters["absolute_tolerance"] = 1e-24
solver.parameters["error_on_nonconvergence"] = True
solver.parameters["nonzero_initial_guess"] = False
solver.set_operator(self.precond)
return solver
class LaplacianPrior(GaussianPrior):
"""Gaussian prior
Parameters must be a dictionary containing:
gamma = multiplicative factor applied to <Grad u, Grad v> term
beta = multiplicative factor applied to <u,v> term (default=0.0)
m0 = mean (or reference parameter when used as regularization)
Vm = function space for parameter
cost = 1/2 * (m-m0)^T.R.(m-m0)"""
def _assemble(self):
# Get input:
self.gamma = self.Parameters['gamma']
if self.Parameters.has_key('beta'): self.beta = self.Parameters['beta']
else: self.beta = 0.0
self.Vm = self.Parameters['Vm']
self.m0 = Function(self.Vm)
if self.Parameters.has_key('m0'):
setfct(self.m0, self.Parameters['m0'])
self.mtrial = TrialFunction(self.Vm)
self.mtest = TestFunction(self.Vm)
self.mysample = Function(self.Vm)
self.draw = Function(self.Vm)
# Assemble:
self.R = assemble(inner(nabla_grad(self.mtrial), \
nabla_grad(self.mtest))*dx)
self.M = assemble(inner(self.mtrial, self.mtest) * dx)
self.Msolver = PETScKrylovSolver('cg', 'jacobi')
self.Msolver.parameters["maximum_iterations"] = 2000
self.Msolver.parameters["relative_tolerance"] = 1e-24
self.Msolver.parameters["absolute_tolerance"] = 1e-24
self.Msolver.parameters["error_on_nonconvergence"] = True
self.Msolver.parameters["nonzero_initial_guess"] = False
self.Msolver.set_operator(self.M)
# preconditioner is Gamma^{-1}:
if self.beta > 1e-10:
self.precond = self.gamma * self.R + self.beta * self.M
else:
self.precond = self.gamma * self.R + (1e-10) * self.M
# Minvprior is M.A^2 (if you use M inner-product):
self.Minvprior = self.gamma * self.R + self.beta * self.M
# L is used to sample
def Minvpriordot(self, vect):
return self.Minvprior * vect
def init_vector(self, u, dim):
self.R.init_vector(u, dim)
def getprecond(self):
""" Precondition by inverting the TV Hessian """
solver = PETScKrylovSolver('cg', self.amgprecond)
solver.parameters["maximum_iterations"] = 2000
solver.parameters["relative_tolerance"] = 1e-24
solver.parameters["absolute_tolerance"] = 1e-24
solver.parameters["error_on_nonconvergence"] = True
solver.parameters["nonzero_initial_guess"] = False
# used to compare iterative application of preconditioner
# with exact application of preconditioner:
#solver = PETScLUSolver("petsc")
#solver.parameters['symmetric'] = True
#solver.parameters['reuse_factorization'] = True
solver.set_operator(self.precond)
return solver
class TV():
"""
Define Total Variation regularization
"""
def __init__(self, parameters=[]):
"""
TV regularization in primal form:
|f|_TV = int k(x) sqrt{|grad f|^2 + eps} dx
Input parameters:
* k = regularization parameter
* eps = regularization constant (see above)
* GNhessian = use GN format (aka, lagged diffusivity) (bool)
* PCGN = use GN Hessian to precondition (bool); only used if 'GNhessian = False'
* print (bool)
"""
self.parameters = {}
self.parameters['k'] = 1.0
self.parameters['eps'] = 1e-2
self.parameters['GNhessian'] = False
self.parameters['PCGN'] = False
self.parameters['print'] = False
self.parameters['correctcost'] = True
self.parameters['amg'] = 'default'
assert parameters.has_key('Vm')
self.parameters.update(parameters)
GN = self.parameters['GNhessian']
self.Vm = self.parameters['Vm']
eps = self.parameters['eps']
k = self.parameters['k']
isprint = self.parameters['print']
amg = self.parameters['amg']
self.m = Function(self.Vm)
test, trial = TestFunction(self.Vm), TrialFunction(self.Vm)
factM = 1e-2 * k
M = assemble(inner(test, trial) * dx)
self.sMass = M * factM
self.Msolver = PETScKrylovSolver('cg', 'jacobi')
self.Msolver.parameters["maximum_iterations"] = 2000
self.Msolver.parameters["relative_tolerance"] = 1e-24
self.Msolver.parameters["absolute_tolerance"] = 1e-24
self.Msolver.parameters["error_on_nonconvergence"] = True
self.Msolver.parameters["nonzero_initial_guess"] = False
self.Msolver.set_operator(M)
self.fTV = inner(nabla_grad(self.m), nabla_grad(
self.m)) + Constant(eps)
self.kovsq = Constant(k) / sqrt(self.fTV)
if self.parameters['correctcost']:
meshtmp = UnitSquareMesh(self.Vm.mesh().mpi_comm(), 10, 10)
Vtmp = FunctionSpace(meshtmp, 'CG', 1)
x = SpatialCoordinate(meshtmp)
correctioncost = 1. / assemble(sqrt(4.0 * x[0] * x[0]) * dx)
else:
correctioncost = 1.0
self.wkformcost = Constant(k * correctioncost) * sqrt(self.fTV) * dx
self.wkformgrad = self.kovsq * inner(nabla_grad(self.m),
nabla_grad(test)) * dx
self.wkformGNhess = self.kovsq * inner(nabla_grad(trial),
nabla_grad(test)) * dx
self.wkformFhess = self.kovsq*( \
inner(nabla_grad(trial), nabla_grad(test)) - \
inner(nabla_grad(self.m), nabla_grad(test))*\
inner(nabla_grad(trial), nabla_grad(self.m))/self.fTV )*dx
if GN:
self.wkformhess = self.wkformGNhess
else:
self.wkformhess = self.wkformFhess
if amg == 'default': self.amgprecond = amg_solver()
else: self.amgprecond = amg
if isprint:
print '[TV] TV regularization',
if GN:
print ' -- GN Hessian',
else:
print ' -- full Hessian',
if self.parameters['PCGN']:
print ' -- PCGN',
print ' -- k={}, eps={}'.format(k, eps)
print '[TV] preconditioner = {}'.format(self.amgprecond)
print '[TV] correction cost with factor={}'.format(correctioncost)
def isTV(self):
return True
def isPD(self):
return False
def cost(self, m_in):
""" returns the cost functional for self.m=m_in """
setfct(self.m, m_in)
return assemble(self.wkformcost)
def costvect(self, m_in):
return self.cost(m_in)
def grad(self, m_in):
""" returns the gradient (in vector format) evaluated at self.m = m_in """
setfct(self.m, m_in)
return assemble(self.wkformgrad)
def gradvect(self, m_in):
return self.grad(m_in)
def assemble_hessian(self, m_in):
""" Assemble the Hessian of TV at m_in """
setfct(self.m, m_in)
self.H = assemble(self.wkformhess)
PCGN = self.parameters['PCGN']
if PCGN:
HGN = assemble(self.wkformGNhess)
self.precond = HGN + self.sMass
else:
self.precond = self.H + self.sMass
def assemble_GNhessian(self, m_in):
""" Assemble the Gauss-Newton Hessian at m_in
Not used anymore (wkformhess selects GN Hessian if needed)
Left here for back-compatibility """
setfct(self.m, m_in)
self.H = assemble(self.wkformGNhess)
self.precond = self.H + self.sMass
def hessian(self, mhat):
""" returns the Hessian applied along a direction mhat """
isVector(mhat)
return self.H * mhat
def getprecond(self):
""" Precondition by inverting the TV Hessian """
solver = PETScKrylovSolver('cg', self.amgprecond)
solver.parameters["maximum_iterations"] = 2000
solver.parameters["relative_tolerance"] = 1e-24
solver.parameters["absolute_tolerance"] = 1e-24
solver.parameters["error_on_nonconvergence"] = True
solver.parameters["nonzero_initial_guess"] = False
# used to compare iterative application of preconditioner
# with exact application of preconditioner:
#solver = PETScLUSolver("petsc")
#solver.parameters['symmetric'] = True
#solver.parameters['reuse_factorization'] = True
solver.set_operator(self.precond)
return solver
def init_vector(self, u, dim):
self.sMass.init_vector(u, dim)
def solve(self, F, u, grad = None, H = None):
if grad is None:
print "Using Automatic Differentiation to compute the gradient"
grad = derivative(F,u)
if H is None:
print "Using Automatic Differentiation to compute the Hessian"
H = derivative(grad, u)
rtol = self.parameters["rel_tolerance"]
atol = self.parameters["abs_tolerance"]
gdu_tol = self.parameters["gdu_tolerance"]
max_iter = self.parameters["max_iter"]
c_armijo = self.parameters["c_armijo"]
max_backtrack = self.parameters["max_backtracking_iter"]
prt_level = self.parameters["print_level"]
cg_coarsest_tol = self.parameters["cg_coarse_tolerance"]
Fn = assemble(F)
gn = assemble(grad)
g0_norm = gn.norm("l2")
gn_norm = g0_norm
tol = max(g0_norm*rtol, atol)
du = Vector()
self.converged = False
self.reason = 0
if prt_level > 0:
print "{0:3} {1:15} {2:15} {3:15} {4:15} {5:15} {6:5}".format(
"It", "Energy", "||g||", "(g,du)", "alpha", "tol_cg", "cg_it")
for self.it in range(max_iter):
Hn = assemble(H)
Hn.init_vector(du,1)
solver = PETScKrylovSolver("cg", "petsc_amg")
solver.set_operator(Hn)
solver.parameters["nonzero_initial_guess"] = False
cg_tol = min(cg_coarsest_tol, math.sqrt( gn_norm/g0_norm) )
solver.parameters["relative_tolerance"] = cg_tol
lin_it = solver.solve(du,gn)
self.total_cg_iter += lin_it
du_gn = -du.inner(gn)
if(-du_gn < gdu_tol):
self.converged=True
self.reason = 3
break
u_backtrack = u.copy(deepcopy=True)
alpha = 1.
bk_converged = False
#Backtrack
for j in range(max_backtrack):
u.assign(u_backtrack)
u.vector().axpy(-alpha, du)
Fnext = assemble(F)
if Fnext < Fn + alpha*c_armijo*du_gn:
Fn = Fnext
bk_converged = True
break
alpha = alpha/2.
if not bk_converged:
self.reason = 2
break
gn = assemble(grad)
gn_norm = gn.norm("l2")
if prt_level > 0:
print "{0:3d} {1:15f} {2:15f} {3:15f} {4:15f} {5:15f} {6:5d}".format(
self.it, Fn, gn_norm, du_gn, alpha, cg_tol, lin_it)
if gn_norm < tol:
self.converged = True
self.reason = 1
break
self.final_grad_norm = gn_norm
if prt_level > -1:
print self.termination_reasons[self.reason]
if self.converged:
print "Inexact Newton CG converged in ", self.it, \
"nonlinear iterations and ", self.total_cg_iter, "linear iterations."
else:
print "Inexact Newton CG did NOT converge after ", self.it, \
"nonlinear iterations and ", self.total_cg_iter, "linear iterations."
print "Final norm of the gradient", self.final_grad_norm
print "Value of the cost functional", Fn
mtrue_exp = Expression('1 + 7*(pow(pow(x[0] - 0.5,2) +' + \
' pow(x[1] - 0.5,2),0.5) > 0.2)')
#mtrue = interpolate(mtrue_exp, Vme)
mtrue = interpolate(mtrue_exp, Vm)
f = Expression("1.0")
# Assemble weak form
trial = TrialFunction(V)
test = TestFunction(V)
a_true = inner(mtrue*nabla_grad(trial), nabla_grad(test))*dx
A_true = assemble(a_true)
bc.apply(A_true)
solver = PETScKrylovSolver('cg') # doesn't work with ilu preconditioner
#solver = LUSolver() # doesn't work in parallel !?
#solver.parameters['reuse_factorization'] = True
solver.set_operator(A_true)
# Assemble rhs
L = f*test*dx
b = assemble(L)
bc.apply(b)
# Solve:
u_true = Function(V)
"""
solver.solve(u_true.vector(), b)
if myrank == 0: print 'By hand:\n'
print 'P{0}: max(u)={1}\n'.format(myrank, max(u_true.vector().array()))
MPI.barrier(mycomm)
# Same with object
class ObjectiveAcoustic(LinearOperator):
"""
Computes data misfit, gradient and Hessian evaluation for the seismic
inverse problem using acoustic wave data
"""
# CONSTRUCTORS:
def __init__(self, mpicomm_global, acousticwavePDE, sources, \
sourcesindex, timestepsindex, \
invparam='ab', regularization=None):
"""
Input:
acousticwavePDE should be an instantiation from class AcousticWave
"""
self.mpicomm_global = mpicomm_global
self.PDE = acousticwavePDE
self.PDE.exact = None
self.obsop = None # Observation operator
self.dd = None # observations
self.fwdsource = sources
self.srcindex = sourcesindex
self.tsteps = timestepsindex
self.PDEcount = 0
self.inverta = False
self.invertb = False
if 'a' in invparam:
self.inverta = True
if 'b' in invparam:
self.invertb = True
assert self.inverta + self.invertb > 0
Vm = self.PDE.Vm
V = self.PDE.V
VmVm = createMixedFS(Vm, Vm)
self.ab = Function(VmVm) # used for conversion (Vm,Vm)->VmVm
self.invparam = invparam
self.MG = Function(VmVm)
self.MGv = self.MG.vector()
self.Grad = Function(VmVm)
self.srchdir = Function(VmVm)
self.delta_m = Function(VmVm)
self.m_bkup = Function(VmVm)
LinearOperator.__init__(self, self.MGv, self.MGv)
self.GN = False
if regularization == None:
print '[ObjectiveAcoustic] *** Warning: Using zero regularization'
self.regularization = ZeroRegularization(Vm)
else:
self.regularization = regularization
self.PD = self.regularization.isPD()
self.alpha_reg = 1.0
self.p, self.q = Function(V), Function(V)
self.phat, self.qhat = Function(V), Function(V)
self.ahat, self.bhat = Function(Vm), Function(Vm)
self.ptrial, self.ptest = TrialFunction(V), TestFunction(V)
self.mtest, self.mtrial = TestFunction(Vm), TrialFunction(Vm)
if self.PDE.parameters['lumpM']:
self.Mprime = LumpedMassMatrixPrime(Vm, V, self.PDE.M.ratio)
self.get_gradienta = self.get_gradienta_lumped
self.get_hessiana = self.get_hessiana_lumped
self.get_incra = self.get_incra_lumped
else:
self.wkformgrada = inner(self.mtest*self.p, self.q)*dx
self.get_gradienta = self.get_gradienta_full
self.wkformhessa = inner(self.phat*self.mtest, self.q)*dx \
+ inner(self.p*self.mtest, self.qhat)*dx
self.wkformhessaGN = inner(self.p*self.mtest, self.qhat)*dx
self.get_hessiana = self.get_hessiana_full
self.wkformrhsincra = inner(self.ahat*self.ptrial, self.ptest)*dx
self.get_incra = self.get_incra_full
self.wkformgradb = inner(self.mtest*nabla_grad(self.p), nabla_grad(self.q))*dx
self.wkformgradbout = assemble(self.wkformgradb)
self.wkformrhsincrb = inner(self.bhat*nabla_grad(self.ptrial), nabla_grad(self.ptest))*dx
self.wkformhessb = inner(nabla_grad(self.phat)*self.mtest, nabla_grad(self.q))*dx \
+ inner(nabla_grad(self.p)*self.mtest, nabla_grad(self.qhat))*dx
self.wkformhessbGN = inner(nabla_grad(self.p)*self.mtest, nabla_grad(self.qhat))*dx
# Mass matrix:
self.mmtest, self.mmtrial = TestFunction(VmVm), TrialFunction(VmVm)
weak_m = inner(self.mmtrial, self.mmtest)*dx
self.Mass = assemble(weak_m)
self.solverM = PETScKrylovSolver("cg", "jacobi")
self.solverM.parameters["maximum_iterations"] = 2000
self.solverM.parameters["absolute_tolerance"] = 1e-24
self.solverM.parameters["relative_tolerance"] = 1e-24
self.solverM.parameters["report"] = False
self.solverM.parameters["error_on_nonconvergence"] = True
self.solverM.parameters["nonzero_initial_guess"] = False # True?
self.solverM.set_operator(self.Mass)
# Time-integration factors
self.factors = np.ones(self.PDE.times.size)
self.factors[0], self.factors[-1] = 0.5, 0.5
self.factors *= self.PDE.Dt
self.invDt = 1./self.PDE.Dt
# Absorbing BCs
if self.PDE.parameters['abc']:
assert not self.PDE.parameters['lumpD']
self.wkformgradaABC = inner(
self.mtest*sqrt(self.PDE.b/self.PDE.a)*self.p,
self.q)*self.PDE.ds(1)
self.wkformgradbABC = inner(
self.mtest*sqrt(self.PDE.a/self.PDE.b)*self.p,
self.q)*self.PDE.ds(1)
self.wkformgradaABCout = assemble(self.wkformgradaABC)
self.wkformgradbABCout = assemble(self.wkformgradbABC)
self.wkformincrrhsABC = inner(
(self.ahat*sqrt(self.PDE.b/self.PDE.a)
+ self.bhat*sqrt(self.PDE.a/self.PDE.b))*self.ptrial,
self.ptest)*self.PDE.ds(1)
self.wkformhessaABC = inner(
(self.bhat/sqrt(self.PDE.a*self.PDE.b) -
self.ahat*sqrt(self.PDE.b/(self.PDE.a*self.PDE.a*self.PDE.a)))
*self.p*self.mtest, self.q)*self.PDE.ds(1)
self.wkformhessbABC = inner(
(self.ahat/sqrt(self.PDE.a*self.PDE.b) -
self.bhat*sqrt(self.PDE.a/(self.PDE.b*self.PDE.b*self.PDE.b)))
*self.p*self.mtest, self.q)*self.PDE.ds(1)
def copy(self):
"""(hard) copy constructor"""
newobj = self.__class__(self.PDE.copy())
setfct(newobj.MG, self.MG)
setfct(newobj.Grad, self.Grad)
setfct(newobj.srchdir, self.srchdir)
newobj.obsop = self.obsop
newobj.dd = self.dd
newobj.fwdsource = self.fwdsource
newobj.srcindex = self.srcindex
newobj.tsteps = self.tsteps
return newobj
# FORWARD PROBLEM + COST:
#@profile
def solvefwd(self, cost=False):
self.PDE.set_fwd()
self.solfwd, self.solpfwd, self.solppfwd = [], [], []
self.Bp = []
#TODO: make fwdsource iterable to return source term
Ricker = self.fwdsource[0]
srcv = self.fwdsource[2]
for sii in self.srcindex:
ptsrc = self.fwdsource[1][sii]
def srcterm(tt):
srcv.zero()
srcv.axpy(Ricker(tt), ptsrc)
return srcv
self.PDE.ftime = srcterm
solfwd, solpfwd, solppfwd,_ = self.PDE.solve()
self.solfwd.append(solfwd)
self.solpfwd.append(solpfwd)
self.solppfwd.append(solppfwd)
self.PDEcount += 1
#TODO: come back and parallellize this too (over time steps)
Bp = np.zeros((len(self.obsop.PtwiseObs.Points),len(solfwd)))
for index, sol in enumerate(solfwd):
setfct(self.p, sol[0])
Bp[:,index] = self.obsop.obs(self.p)
self.Bp.append(Bp)
if cost:
assert not self.dd == None, "Provide data observations to compute cost"
self.cost_misfit_local = 0.0
for Bp, dd in izip(self.Bp, self.dd):
self.cost_misfit_local += self.obsop.costfct(\
Bp[:,self.tsteps], dd[:,self.tsteps],\
self.PDE.times[self.tsteps], self.factors[self.tsteps])
self.cost_misfit = MPI.sum(self.mpicomm_global, self.cost_misfit_local)
self.cost_misfit /= len(self.fwdsource[1])
self.cost_reg = self.regularization.costab(self.PDE.a, self.PDE.b)
self.cost = self.cost_misfit + self.alpha_reg*self.cost_reg
if DEBUG:
print 'cost_misfit={}, cost_reg={}'.format(\
self.cost_misfit, self.cost_reg)
def solvefwd_cost(self): self.solvefwd(True)
# ADJOINT PROBLEM + GRADIENT:
#@profile
def solveadj(self, grad=False):
self.PDE.set_adj()
self.soladj, self.solpadj, self.solppadj = [], [], []
for Bp, dd in zip(self.Bp, self.dd):
self.obsop.assemble_rhsadj(Bp, dd, self.PDE.times, self.PDE.bc)
self.PDE.ftime = self.obsop.ftimeadj
soladj,solpadj,solppadj,_ = self.PDE.solve()
self.soladj.append(soladj)
self.solpadj.append(solpadj)
self.solppadj.append(solppadj)
self.PDEcount += 1
if grad:
self.MG.vector().zero()
MGa_local, MGb_local = self.MG.split(deepcopy=True)
MGav_local, MGbv_local = MGa_local.vector(), MGb_local.vector()
t0, t1 = self.tsteps[0], self.tsteps[-1]+1
for solfwd, solpfwd, solppfwd, soladj in \
izip(self.solfwd, self.solpfwd, self.solppfwd, self.soladj):
for fwd, fwdp, fwdpp, adj, fact in \
izip(solfwd[t0:t1], solpfwd[t0:t1], solppfwd[t0:t1],\
soladj[::-1][t0:t1], self.factors[t0:t1]):
setfct(self.q, adj[0])
if self.inverta:
# gradient a
setfct(self.p, fwdpp[0])
MGav_local.axpy(fact, self.get_gradienta())
if self.invertb:
# gradient b
setfct(self.p, fwd[0])
assemble(form=self.wkformgradb, tensor=self.wkformgradbout)
MGbv_local.axpy(fact, self.wkformgradbout)
if self.PDE.parameters['abc']:
setfct(self.p, fwdp[0])
if self.inverta:
assemble(form=self.wkformgradaABC, tensor=self.wkformgradaABCout)
MGav_local.axpy(0.5*fact, self.wkformgradaABCout)
if self.invertb:
assemble(form=self.wkformgradbABC, tensor=self.wkformgradbABCout)
MGbv_local.axpy(0.5*fact, self.wkformgradbABCout)
MGa, MGb = self.MG.split(deepcopy=True)
MPIAllReduceVector(MGav_local, MGa.vector(), self.mpicomm_global)
MPIAllReduceVector(MGbv_local, MGb.vector(), self.mpicomm_global)
setfct(MGa, MGa.vector()/len(self.fwdsource[1]))
setfct(MGb, MGb.vector()/len(self.fwdsource[1]))
self.MG.vector().zero()
if self.inverta:
assign(self.MG.sub(0), MGa)
if self.invertb:
assign(self.MG.sub(1), MGb)
if DEBUG:
print 'grad_misfit={}, grad_reg={}'.format(\
self.MG.vector().norm('l2'),\
self.regularization.gradab(self.PDE.a, self.PDE.b).norm('l2'))
self.MG.vector().axpy(self.alpha_reg, \
self.regularization.gradab(self.PDE.a, self.PDE.b))
try:
self.solverM.solve(self.Grad.vector(), self.MG.vector())
except:
# if |G|<<1, first residuals may diverge
# caveat: Hope that ALL processes throw an exception
pseudoGradnorm = np.sqrt(self.MGv.inner(self.MGv))
if pseudoGradnorm < 1e-8:
print '*** Warning: Increasing divergence_limit for Mass matrix solver'
self.solverM.parameters["divergence_limit"] = 1e6
self.solverM.solve(self.Grad.vector(), self.MG.vector())
else:
print '*** Error: Problem with Mass matrix solver'
sys.exit(1)
def solveadj_constructgrad(self): self.solveadj(True)
def get_gradienta_lumped(self):
return self.Mprime.get_gradient(self.p.vector(), self.q.vector())
def get_gradienta_full(self):
return assemble(self.wkformgrada)
# HESSIAN:
#@profile
def ftimeincrfwd(self, tt):
""" Compute rhs for incremental forward at time tt """
try:
index = int(np.where(isequal(self.PDE.times, tt, 1e-14))[0])
except:
print 'Error in ftimeincrfwd at time {}'.format(tt)
print np.min(np.abs(self.PDE.times-tt))
sys.exit(0)
# bhat: bhat*grad(p).grad(qtilde)
# assert isequal(tt, self.solfwdi[index][1], 1e-16)
setfct(self.p, self.solfwdi[index][0])
self.q.vector().zero()
self.q.vector().axpy(1.0, self.C*self.p.vector())
# ahat: ahat*p''*qtilde:
setfct(self.p, self.solppfwdi[index][0])
self.q.vector().axpy(1.0, self.get_incra(self.p.vector()))
# ABC:
if self.PDE.parameters['abc']:
setfct(self.phat, self.solpfwdi[index][0])
self.q.vector().axpy(0.5, self.Dp*self.phat.vector())
return -1.0*self.q.vector()
#@profile
def ftimeincradj(self, tt):
""" Compute rhs for incremental adjoint at time tt """
try:
indexf = int(np.where(isequal(self.PDE.times, tt, 1e-14))[0])
indexa = int(np.where(isequal(self.PDE.times[::-1], tt, 1e-14))[0])
except:
print 'Error in ftimeincradj at time {}'.format(tt)
print np.min(np.abs(self.PDE.times-tt))
sys.exit(0)
# B* B phat
# assert isequal(tt, self.solincrfwd[indexf][1], 1e-16)
setfct(self.phat, self.solincrfwd[indexf][0])
self.qhat.vector().zero()
self.qhat.vector().axpy(1.0, self.obsop.incradj(self.phat, tt))
if not self.GN:
# bhat: bhat*grad(ptilde).grad(v)
# assert isequal(tt, self.soladji[indexa][1], 1e-16)
setfct(self.q, self.soladji[indexa][0])
self.qhat.vector().axpy(1.0, self.C*self.q.vector())
# ahat: ahat*ptilde*q'':
setfct(self.q, self.solppadji[indexa][0])
self.qhat.vector().axpy(1.0, self.get_incra(self.q.vector()))
# ABC:
if self.PDE.parameters['abc']:
setfct(self.phat, self.solpadji[indexa][0])
self.qhat.vector().axpy(-0.5, self.Dp*self.phat.vector())
return -1.0*self.qhat.vector()
def get_incra_full(self, pvector):
return self.E*pvector
def get_incra_lumped(self, pvector):
return self.Mprime.get_incremental(self.ahat.vector(), pvector)
#@profile
def mult(self, abhat, y):
"""
mult(self, abhat, y): return y = Hessian * abhat
inputs:
y, abhat = Function(V).vector()
"""
setfct(self.ab, abhat)
ahat, bhat = self.ab.split(deepcopy=True)
setfct(self.ahat, ahat)
setfct(self.bhat, bhat)
if not self.inverta:
self.ahat.vector().zero()
if not self.invertb:
self.bhat.vector().zero()
self.C = assemble(self.wkformrhsincrb)
if not self.PDE.parameters['lumpM']: self.E = assemble(self.wkformrhsincra)
if self.PDE.parameters['abc']: self.Dp = assemble(self.wkformincrrhsABC)
t0, t1 = self.tsteps[0], self.tsteps[-1]+1
# Compute Hessian*abhat
self.ab.vector().zero()
yaF_local, ybF_local = self.ab.split(deepcopy=True)
ya_local, yb_local = yaF_local.vector(), ybF_local.vector()
# iterate over sources:
for self.solfwdi, self.solpfwdi, self.solppfwdi, \
self.soladji, self.solpadji, self.solppadji \
in izip(self.solfwd, self.solpfwd, self.solppfwd, \
self.soladj, self.solpadj, self.solppadj):
# incr. fwd
self.PDE.set_fwd()
self.PDE.ftime = self.ftimeincrfwd
self.solincrfwd,solpincrfwd,self.solppincrfwd,_ = self.PDE.solve()
self.PDEcount += 1
# incr. adj
self.PDE.set_adj()
self.PDE.ftime = self.ftimeincradj
solincradj,_,_,_ = self.PDE.solve()
self.PDEcount += 1
# assemble Hessian-vect product:
for fwd, adj, fwdp, incrfwdp, \
fwdpp, incrfwdpp, incrfwd, incradj, fact \
in izip(self.solfwdi[t0:t1], self.soladji[::-1][t0:t1],\
self.solpfwdi[t0:t1], solpincrfwd[t0:t1], \
self.solppfwdi[t0:t1], self.solppincrfwd[t0:t1],\
self.solincrfwd[t0:t1], solincradj[::-1][t0:t1], self.factors[t0:t1]):
# ttf, tta, ttf2 = incrfwd[1], incradj[1], fwd[1]
# assert isequal(ttf, tta, 1e-16), 'tfwd={}, tadj={}, reldiff={}'.\
# format(ttf, tta, abs(ttf-tta)/ttf)
# assert isequal(ttf, ttf2, 1e-16), 'tfwd={}, tadj={}, reldiff={}'.\
# format(ttf, ttf2, abs(ttf-ttf2)/ttf)
setfct(self.q, adj[0])
setfct(self.qhat, incradj[0])
if self.invertb:
# Hessian b
setfct(self.p, fwd[0])
setfct(self.phat, incrfwd[0])
if self.GN:
yb_local.axpy(fact, assemble(self.wkformhessbGN))
else:
yb_local.axpy(fact, assemble(self.wkformhessb))
if self.inverta:
# Hessian a
setfct(self.p, fwdpp[0])
setfct(self.phat, incrfwdpp[0])
ya_local.axpy(fact, self.get_hessiana())
if self.PDE.parameters['abc']:
if not self.GN:
setfct(self.p, incrfwdp[0])
if self.inverta:
ya_local.axpy(0.5*fact, assemble(self.wkformgradaABC))
if self.invertb:
yb_local.axpy(0.5*fact, assemble(self.wkformgradbABC))
setfct(self.p, fwdp[0])
setfct(self.q, incradj[0])
if self.inverta:
ya_local.axpy(0.5*fact, assemble(self.wkformgradaABC))
if self.invertb:
yb_local.axpy(0.5*fact, assemble(self.wkformgradbABC))
if not self.GN:
setfct(self.q, adj[0])
if self.inverta:
ya_local.axpy(0.25*fact, assemble(self.wkformhessaABC))
if self.invertb:
yb_local.axpy(0.25*fact, assemble(self.wkformhessbABC))
yaF, ybF = self.ab.split(deepcopy=True)
MPIAllReduceVector(ya_local, yaF.vector(), self.mpicomm_global)
MPIAllReduceVector(yb_local, ybF.vector(), self.mpicomm_global)
self.ab.vector().zero()
if self.inverta:
assign(self.ab.sub(0), yaF)
if self.invertb:
assign(self.ab.sub(1), ybF)
y.zero()
y.axpy(1.0/len(self.fwdsource[1]), self.ab.vector())
if DEBUG:
print 'Hess_misfit={}, Hess_reg={}'.format(\
y.norm('l2'),\
self.regularization.hessianab(self.ahat.vector(),\
self.bhat.vector()).norm('l2'))
y.axpy(self.alpha_reg, \
self.regularization.hessianab(self.ahat.vector(), self.bhat.vector()))
def get_hessiana_full(self):
if self.GN:
return assemble(self.wkformhessaGN)
else:
return assemble(self.wkformhessa)
def get_hessiana_lumped(self):
if self.GN:
return self.Mprime.get_gradient(self.p.vector(), self.qhat.vector())
else:
return self.Mprime.get_gradient(self.phat.vector(), self.q.vector()) +\
self.Mprime.get_gradient(self.p.vector(), self.qhat.vector())
def assemble_hessian(self):
self.regularization.assemble_hessianab(self.PDE.a, self.PDE.b)
# SETTERS + UPDATE:
def update_PDE(self, parameters): self.PDE.update(parameters)
def update_m(self, medparam):
""" medparam contains both med parameters """
setfct(self.ab, medparam)
a, b = self.ab.split(deepcopy=True)
self.update_PDE({'a':a, 'b':b})
def backup_m(self):
""" back-up current value of med param a and b """
assign(self.m_bkup.sub(0), self.PDE.a)
assign(self.m_bkup.sub(1), self.PDE.b)
def restore_m(self):
""" restore backed-up values of a and b """
a, b = self.m_bkup.split(deepcopy=True)
self.update_PDE({'a':a, 'b':b})
def mediummisfit(self, target_medium):
"""
Compute medium misfit at current position
"""
assign(self.ab.sub(0), self.PDE.a)
assign(self.ab.sub(1), self.PDE.b)
try:
diff = self.ab.vector() - target_medium.vector()
except:
diff = self.ab.vector() - target_medium
Md = self.Mass*diff
self.ab.vector().zero()
self.ab.vector().axpy(1.0, Md)
Mda, Mdb = self.ab.split(deepcopy=True)
self.ab.vector().zero()
self.ab.vector().axpy(1.0, diff)
da, db = self.ab.split(deepcopy=True)
medmisfita = np.sqrt(da.vector().inner(Mda.vector()))
medmisfitb = np.sqrt(db.vector().inner(Mdb.vector()))
return medmisfita, medmisfitb
def compare_ab_global(self):
"""
Check that med param (a, b) are the same across all proc
"""
assign(self.ab.sub(0), self.PDE.a)
assign(self.ab.sub(1), self.PDE.b)
ab_recv = self.ab.vector().copy()
normabloc = np.linalg.norm(self.ab.vector().array())
MPIAllReduceVector(self.ab.vector(), ab_recv, self.mpicomm_global)
ab_recv /= MPI.size(self.mpicomm_global)
diff = ab_recv - self.ab.vector()
reldiff = np.linalg.norm(diff.array())/normabloc
assert reldiff < 2e-16, 'Diff in (a,b) across proc: {:.2e}'.format(reldiff)
# GETTERS:
def getmbkup(self): return self.m_bkup.vector()
def getMG(self): return self.MGv
def getprecond(self):
if self.PC == 'prior':
return self.regularization.getprecond()
elif self.PC == 'bfgs':
return self.bfgsop
else:
print 'Wrong keyword for choice of preconditioner'
sys.exit(1)
# SOLVE INVERSE PROBLEM
#@profile
def inversion(self, initial_medium, target_medium, parameters_in=[], \
boundsLS=None, myplot=None):
"""
Solve inverse problem with that objective function
parameters:
solverNS = solver for Newton system ('steepest', 'Newton', 'BFGS')
retolgrad = relative tolerance for stopping criterion (grad)
abstolgrad = absolute tolerance for stopping criterion (grad)
tolcost = tolerance for stopping criterion (cost)
maxiterNewt = max nb of Newton iterations
nbGNsteps = nb of Newton steps with GN Hessian
maxtolcg = max value of the tolerance for CG solver
checkab = nb of steps in-between check of param
inexactCG = [bool] inexact CG solver or exact CG
isprint = [bool] print results to screen
avgPC = [bool] average Preconditioned step over all proc in CG
PC = choice of preconditioner ('prior', or 'bfgs')
"""
parameters = {}
parameters['solverNS'] = 'Newton'
parameters['reltolgrad'] = 1e-10
parameters['abstolgrad'] = 1e-14
parameters['tolcost'] = 1e-24
parameters['maxiterNewt'] = 100
parameters['nbGNsteps'] = 10
parameters['maxtolcg'] = 0.5
parameters['checkab'] = 10
parameters['inexactCG'] = True
parameters['isprint'] = False
parameters['avgPC'] = True
parameters['PC'] = 'prior'
parameters['BFGS_damping'] = 0.2
parameters['memory_limit'] = 50
parameters['H0inv'] = 'Rinv'
parameters.update(parameters_in)
solverNS = parameters['solverNS']
isprint = parameters['isprint']
maxiterNewt = parameters['maxiterNewt']
reltolgrad = parameters['reltolgrad']
abstolgrad = parameters['abstolgrad']
tolcost = parameters['tolcost']
nbGNsteps = parameters['nbGNsteps']
checkab = parameters['checkab']
avgPC = parameters['avgPC']
if parameters['inexactCG']:
maxtolcg = parameters['maxtolcg']
else:
maxtolcg = 1e-12
if solverNS == 'BFGS':
maxtolcg = -1.0
self.PC = parameters['PC']
# BFGS (preconditioner or solver):
if self.PC == 'bfgs' or solverNS == 'BFGS':
self.bfgsop = BFGS_operator(parameters)
H0inv = self.bfgsop.parameters['H0inv']
else:
self.bfgsop = []
self.PDEcount = 0 # reset
if isprint:
print '\t{:12s} {:10s} {:12s} {:12s} {:12s} {:16s}\t\t\t {:10s} {:12s} {:10s} {:10s}'.format(\
'iter', 'cost', 'misfit', 'reg', '|G|', 'medmisf', 'a_ls', 'tol_cg', 'n_cg', 'PDEsolves')
a0, b0 = initial_medium.split(deepcopy=True)
self.update_PDE({'a':a0, 'b':b0})
self._plotab(myplot, 'init')
Mab = self.Mass*target_medium.vector()
self.ab.vector().zero()
self.ab.vector().axpy(1.0, Mab)
Ma, Mb = self.ab.split(deepcopy=True)
at, bt = target_medium.split(deepcopy=True)
atnorm = np.sqrt(at.vector().inner(Ma.vector()))
btnorm = np.sqrt(bt.vector().inner(Mb.vector()))
alpha = -1.0 # dummy value for print outputs
self.solvefwd_cost()
for it in xrange(maxiterNewt):
MGv_old = self.MGv.copy()
self.solveadj_constructgrad()
gradnorm = np.sqrt(self.MGv.inner(self.Grad.vector()))
if it == 0: gradnorm0 = gradnorm
medmisfita, medmisfitb = self.mediummisfit(target_medium)
self._plotab(myplot, str(it))
self._plotgrad(myplot, str(it))
# Stopping criterion (gradient)
if gradnorm < gradnorm0*reltolgrad or gradnorm < abstolgrad:
print '{:12d} {:12.4e} {:12.2e} {:12.2e} {:11.4e} {:10.2e} ({:4.1f}%) {:10.2e} ({:4.1f}%)'.\
format(it, self.cost, self.cost_misfit, self.cost_reg, gradnorm,\
medmisfita, 100.0*medmisfita/atnorm, medmisfitb, 100.0*medmisfitb/btnorm),
print '{:11.3f} {:12.2} {:10} {:10d}'.format(\
alpha, "", "", self.PDEcount)
if isprint:
print '\nGradient sufficiently reduced'
print 'Optimization converged'
return
# Assemble Hessian of regularization for nonlinear regularization:
self.assemble_hessian()
# Update BFGS approx (s, y, H0)
if self.PC == 'bfgs' or solverNS == 'BFGS':
if it > 0:
s = self.srchdir.vector() * alpha
y = self.MGv - MGv_old
theta = self.bfgsop.update(s, y)
else:
theta = 1.0
if H0inv == 'Rinv':
self.bfgsop.set_H0inv(self.regularization.getprecond())
elif H0inv == 'Minv':
print 'H0inv = Minv? That is not a good idea'
sys.exit(1)
# Compute search direction and plot
tolcg = min(maxtolcg, np.sqrt(gradnorm/gradnorm0))
self.GN = (it < nbGNsteps) # use GN or full Hessian?
# most time spent here:
if avgPC:
cgiter, cgres, cgid = compute_searchdirection(self,
{'method':solverNS, 'tolcg':tolcg,\
'max_iter':250+1250*(self.GN==False)},\
comm=self.mpicomm_global, BFGSop=self.bfgsop)
else:
cgiter, cgres, cgid = compute_searchdirection(self,
{'method':solverNS, 'tolcg':tolcg,\
'max_iter':250+1250*(self.GN==False)}, BFGSop=self.bfgsop)
# addt'l safety: zero-out entries of 'srchdir' corresponding to
# param that are not inverted for
if not self.inverta*self.invertb:
srcha, srchb = self.srchdir.split(deepcopy=True)
if not self.inverta:
srcha.vector().zero()
assign(self.srchdir.sub(0), srcha)
if not self.invertb:
srchb.vector().zero()
assign(self.srchdir.sub(1), srchb)
self._plotsrchdir(myplot, str(it))
if isprint:
print '{:12d} {:12.4e} {:12.2e} {:12.2e} {:11.4e} {:10.2e} ({:4.1f}%) {:10.2e} ({:4.1f}%)'.\
format(it, self.cost, self.cost_misfit, self.cost_reg, gradnorm,\
medmisfita, 100.0*medmisfita/atnorm, medmisfitb, 100.0*medmisfitb/btnorm),
print '{:11.3f} {:12.2e} {:10d} {:10d}'.format(\
alpha, tolcg, cgiter, self.PDEcount)
# Backtracking line search
cost_old = self.cost
statusLS, LScount, alpha = bcktrcklinesearch(self, parameters, boundsLS)
cost = self.cost
# Perform line search for dual variable (TV-PD):
if self.PD:
self.regularization.update_w(self.srchdir.vector(), alpha)
if it%checkab == 0:
self.compare_ab_global()
# Stopping criterion (LS)
if not statusLS:
if isprint:
print '\nLine search failed'
print 'Optimization aborted'
return
# Stopping criterion (cost)
if np.abs(cost-cost_old)/np.abs(cost_old) < tolcost:
if isprint:
print '\nCost function stagnates'
print 'Optimization aborted'
return
if isprint:
print '\nMaximum number of Newton iterations reached'
print 'Optimization aborted'
# PLOTS:
def _plotab(self, myplot, index):
""" plot media during inversion """
if not myplot == None:
if self.invparam == 'a' or self.invparam == 'ab':
myplot.set_varname('a'+index)
myplot.plot_vtk(self.PDE.a)
if self.invparam == 'b' or self.invparam == 'ab':
myplot.set_varname('b'+index)
myplot.plot_vtk(self.PDE.b)
def _plotgrad(self, myplot, index):
""" plot grad during inversion """
if not myplot == None:
if self.invparam == 'a':
myplot.set_varname('Grad_a'+index)
myplot.plot_vtk(self.Grad)
elif self.invparam == 'b':
myplot.set_varname('Grad_b'+index)
myplot.plot_vtk(self.Grad)
elif self.invparam == 'ab':
Ga, Gb = self.Grad.split(deepcopy=True)
myplot.set_varname('Grad_a'+index)
myplot.plot_vtk(Ga)
myplot.set_varname('Grad_b'+index)
myplot.plot_vtk(Gb)
def _plotsrchdir(self, myplot, index):
""" plot srchdir during inversion """
if not myplot == None:
if self.invparam == 'a':
myplot.set_varname('srchdir_a'+index)
myplot.plot_vtk(self.srchdir)
elif self.invparam == 'b':
myplot.set_varname('srchdir_b'+index)
myplot.plot_vtk(self.srchdir)
elif self.invparam == 'ab':
Ga, Gb = self.srchdir.split(deepcopy=True)
myplot.set_varname('srchdir_a'+index)
myplot.plot_vtk(Ga)
myplot.set_varname('srchdir_b'+index)
myplot.plot_vtk(Gb)
# SHOULD BE REMOVED:
def set_abc(self, mesh, class_bc_abc, lumpD):
self.PDE.set_abc(mesh, class_bc_abc, lumpD)
def init_vector(self, x, dim):
self.Mass.init_vector(x, dim)
def getmcopyarray(self): return self.getmcopy().array()
def getMGarray(self): return self.MGv.array()
def setsrcterm(self, ftime): self.PDE.ftime = ftime
def solve(self, F, u, grad=None, H=None):
if grad is None:
print("Using Symbolic Differentiation to compute the gradient")
grad = derivative(F, u)
if H is None:
print("Using Symbolic Differentiation to compute the Hessian")
H = derivative(grad, u)
rtol = self.parameters["rel_tolerance"]
atol = self.parameters["abs_tolerance"]
gdu_tol = self.parameters["gdu_tolerance"]
max_iter = self.parameters["max_iter"]
c_armijo = self.parameters["c_armijo"]
max_backtrack = self.parameters["max_backtracking_iter"]
prt_level = self.parameters["print_level"]
cg_coarsest_tol = self.parameters["cg_coarse_tolerance"]
Fn = assemble(F)
gn = assemble(grad)
g0_norm = gn.norm("l2")
gn_norm = g0_norm
tol = max(g0_norm * rtol, atol)
du = Vector()
self.converged = False
self.reason = 0
if prt_level > 0:
print(
"{0:>3} {1:>15} {2:>15} {3:>15} {4:>15} {5:>15} {6:>7}".format(
"It", "Energy", "||g||", "(g,du)", "alpha", "tol_cg",
"cg_it"))
for self.it in range(max_iter):
Hn = assemble(H)
Hn.init_vector(du, 1)
solver = PETScKrylovSolver("cg", "petsc_amg")
solver.set_operator(Hn)
solver.parameters["nonzero_initial_guess"] = False
cg_tol = min(cg_coarsest_tol, math.sqrt(gn_norm / g0_norm))
solver.parameters["relative_tolerance"] = cg_tol
lin_it = solver.solve(du, gn)
self.total_cg_iter += lin_it
du_gn = -du.inner(gn)
if (-du_gn < gdu_tol):
self.converged = True
self.reason = 3
break
u_backtrack = u.copy(deepcopy=True)
alpha = 1.
bk_converged = False
#Backtrack
for j in range(max_backtrack):
u.assign(u_backtrack)
u.vector().axpy(-alpha, du)
Fnext = assemble(F)
if Fnext < Fn + alpha * c_armijo * du_gn:
Fn = Fnext
bk_converged = True
break
alpha = alpha / 2.
if not bk_converged:
self.reason = 2
break
gn = assemble(grad)
gn_norm = gn.norm("l2")
if prt_level > 0:
print("{0:3d} {1:15e} {2:15e} {3:15e} {4:15e} {5:15e} {6:7d}".
format(self.it, Fn, gn_norm, du_gn, alpha, cg_tol,
lin_it))
if gn_norm < tol:
self.converged = True
self.reason = 1
break
self.final_grad_norm = gn_norm
if prt_level > -1:
print(self.termination_reasons[self.reason])
if self.converged:
print( "Inexact Newton CG converged in ", self.it, \
"nonlinear iterations and ", self.total_cg_iter, "linear iterations." )
else:
print( "Inexact Newton CG did NOT converge after ", self.it, \
"nonlinear iterations and ", self.total_cg_iter, "linear iterations.")
print("Final norm of the gradient", self.final_grad_norm)
print("Value of the cost functional", Fn)
def _pressure_poisson(self,
p1, p0,
mu, ui,
divu,
p_bcs=None,
p_n=None,
rotational_form=False,
tol=1.0e-10,
verbose=True
):
'''Solve the pressure Poisson equation
- \Delta phi = -div(u),
boundary conditions,
for
\nabla p = u.
'''
P = p1.function_space()
p = TrialFunction(P)
q = TestFunction(P)
a2 = dot(grad(p), grad(q)) * dx
L2 = -divu * q * dx
if p0:
L2 += dot(grad(p0), grad(q)) * dx
if p_n:
n = FacetNormal(P.mesh())
L2 += dot(n, p_n) * q * ds
if rotational_form:
L2 -= mu * dot(grad(div(ui)), grad(q)) * dx
if p_bcs:
solve(a2 == L2, p1,
bcs=p_bcs,
solver_parameters={
'linear_solver': 'iterative',
'symmetric': True,
'preconditioner': 'hypre_amg',
'krylov_solver': {'relative_tolerance': tol,
'absolute_tolerance': 0.0,
'maximum_iterations': 100,
'monitor_convergence': verbose}
})
else:
# If we're dealing with a pure Neumann problem here (which is the
# default case), this doesn't hurt CG if the system is consistent,
# cf.
#
# Iterative Krylov methods for large linear systems,
# Henk A. van der Vorst.
#
# And indeed, it is consistent: Note that
#
# <1, rhs> = \sum_i 1 * \int div(u) v_i
# = 1 * \int div(u) \sum_i v_i
# = \int div(u).
#
# With the divergence theorem, we have
#
# \int div(u) = \int_\Gamma n.u.
#
# The latter term is 0 iff inflow and outflow are exactly the same
# at any given point in time. This corresponds with the
# incompressibility of the liquid.
#
# In turn, this hints towards penetrable boundaries to require
# Dirichlet conditions on the pressure.
#
A = assemble(a2)
b = assemble(L2)
#
# In principle, the ILU preconditioner isn't advised here since it
# might destroy the semidefiniteness needed for CG.
#
# The system is consistent, but the matrix has an eigenvalue 0.
# This does not harm the convergence of CG, but when
# preconditioning one has to take care that the preconditioner
# preserves the kernel. ILU might destroy this (and the
# semidefiniteness). With AMG, the coarse grid solves cannot be LU
# then, so try Jacobi here.
# <http://lists.mcs.anl.gov/pipermail/petsc-users/2012-February/012139.html>
#
prec = PETScPreconditioner('hypre_amg')
PETScOptions.set('pc_hypre_boomeramg_relax_type_coarse', 'jacobi')
solver = PETScKrylovSolver('cg', prec)
solver.parameters['absolute_tolerance'] = 0.0
solver.parameters['relative_tolerance'] = tol
solver.parameters['maximum_iterations'] = 100
solver.parameters['monitor_convergence'] = verbose
# Create solver and solve system
A_petsc = as_backend_type(A)
b_petsc = as_backend_type(b)
p1_petsc = as_backend_type(p1.vector())
solver.set_operator(A_petsc)
try:
solver.solve(p1_petsc, b_petsc)
except RuntimeError as error:
info('')
# Check if the system is indeed consistent.
#
# If the right hand side is flawed (e.g., by round-off errors),
# then it may have a component b1 in the direction of the null
# space, orthogonal the image of the operator:
#
# b = b0 + b1.
#
# When starting with initial guess x0=0, the minimal achievable
# relative tolerance is then
#
# min_rel_tol = ||b1|| / ||b||.
#
# If ||b|| is very small, which is the case when ui is almost
# divergence-free, then min_rel_to may be larger than the
# prescribed relative tolerance tol.
#
# Use this as a consistency check, i.e., bail out if
#
# tol < min_rel_tol = ||b1|| / ||b||.
#
# For computing ||b1||, we use the fact that the null space is
# one-dimensional, i.e., b1 = alpha e, and
#
# e.b = e.(b0 + b1) = e.b1 = alpha ||e||^2,
#
# so alpha = e.b/||e||^2 and
#
# ||b1|| = |alpha| ||e|| = e.b / ||e||
#
e = Function(P)
e.interpolate(Constant(1.0))
evec = e.vector()
evec /= norm(evec)
alpha = b.inner(evec)
normB = norm(b)
info('Linear system convergence failure.')
info(error.message)
message = ('Linear system not consistent! '
'<b,e> = %g, ||b|| = %g, <b,e>/||b|| = %e, tol = %e.') \
% (alpha, normB, alpha/normB, tol)
info(message)
if tol < abs(alpha) / normB:
info('\int div(u) = %e' % assemble(divu * dx))
#n = FacetNormal(Q.mesh())
#info('\int_Gamma n.u = %e' % assemble(dot(n, u)*ds))
#info('\int_Gamma u[0] = %e' % assemble(u[0]*ds))
#info('\int_Gamma u[1] = %e' % assemble(u[1]*ds))
## Now plot the faulty u on a finer mesh (to resolve the
## quadratic trial functions).
#fine_mesh = Q.mesh()
#for k in range(1):
# fine_mesh = refine(fine_mesh)
#V1 = FunctionSpace(fine_mesh, 'CG', 1)
#W1 = V1*V1
#uplot = project(u, W1)
##uplot = Function(W1)
##uplot.interpolate(u)
#plot(uplot, title='u_tentative')
#plot(uplot[0], title='u_tentative[0]')
#plot(uplot[1], title='u_tentative[1]')
plot(divu, title='div(u_tentative)')
interactive()
exit()
raise RuntimeError(message)
else:
exit()
raise RuntimeError('Linear system failed to converge.')
except:
exit()
return
def _pressure_poisson(self, p1, p0,
mu, ui,
u,
p_bcs=None,
rotational_form=False,
tol=1.0e-10,
verbose=True
):
'''Solve the pressure Poisson equation
-1/r \div(r \nabla (p1-p0)) = -1/r div(r*u),
boundary conditions,
for
\nabla p = u.
'''
r = Expression('x[0]', degree=1, domain=self.W.mesh())
Q = p1.function_space()
p = TrialFunction(Q)
q = TestFunction(Q)
a2 = dot(r * grad(p), grad(q)) * 2 * pi * dx
# The boundary conditions
# n.(p1-p0) = 0
# are implicitly included.
#
# L2 = -div(r*u) * q * 2*pi*dx
div_u = 1/r * (r * u[0]).dx(0) + u[1].dx(1)
L2 = -div_u * q * 2*pi*r*dx
if p0:
L2 += r * dot(grad(p0), grad(q)) * 2*pi*dx
# In the Cartesian variant of the rotational form, one makes use of the
# fact that
#
# curl(curl(u)) = grad(div(u)) - div(grad(u)).
#
# The same equation holds true in cylindrical form. Hence, to get the
# rotational form of the splitting scheme, we need to
#
# rotational form
if rotational_form:
# If there is no dependence of the angular coordinate, what is
# div(grad(div(u))) in Cartesian coordinates becomes
#
# 1/r div(r * grad(1/r div(r*u)))
#
# in cylindrical coordinates (div and grad are in cylindrical
# coordinates). Unfortunately, we cannot write it down that
# compactly since u_phi is in the game.
# When using P2 elements, this value will be 0 anyways.
div_ui = 1/r * (r * ui[0]).dx(0) + ui[1].dx(1)
grad_div_ui = as_vector((div_ui.dx(0), div_ui.dx(1)))
L2 -= r * mu * dot(grad_div_ui, grad(q)) * 2*pi*dx
#div_grad_div_ui = 1/r * (r * grad_div_ui[0]).dx(0) \
# + (grad_div_ui[1]).dx(1)
#L2 += mu * div_grad_div_ui * q * 2*pi*r*dx
#n = FacetNormal(Q.mesh())
#L2 -= mu * (n[0] * grad_div_ui[0] + n[1] * grad_div_ui[1]) \
# * q * 2*pi*r*ds
if p_bcs:
solve(
a2 == L2, p1,
bcs=p_bcs,
solver_parameters={
'linear_solver': 'iterative',
'symmetric': True,
'preconditioner': 'amg',
'krylov_solver': {'relative_tolerance': tol,
'absolute_tolerance': 0.0,
'maximum_iterations': 100,
'monitor_convergence': verbose}
}
)
else:
# If we're dealing with a pure Neumann problem here (which is the
# default case), this doesn't hurt CG if the system is consistent,
# cf. :cite:`vdV03`. And indeed it is consistent if and only if
#
# \int_\Gamma r n.u = 0.
#
# This makes clear that for incompressible Navier-Stokes, one
# either needs to make sure that inflow and outflow always add up
# to 0, or one has to specify pressure boundary conditions.
#
# If the right-hand side is very small, round-off errors may impair
# the consistency of the system. Make sure the system we are
# solving remains consistent.
A = assemble(a2)
b = assemble(L2)
# Assert that the system is indeed consistent.
e = Function(Q)
e.interpolate(Constant(1.0))
evec = e.vector()
evec /= norm(evec)
alpha = b.inner(evec)
normB = norm(b)
# Assume that in every component of the vector, a round-off error
# of the magnitude DOLFIN_EPS is present. This leads to the
# criterion
# |<b,e>| / (||b||*||e||) < DOLFIN_EPS
# as a check whether to consider the system consistent up to
# round-off error.
#
# TODO think about condition here
#if abs(alpha) > normB * DOLFIN_EPS:
if abs(alpha) > normB * 1.0e-12:
divu = 1 / r * (r * u[0]).dx(0) + u[1].dx(1)
adivu = assemble(((r * u[0]).dx(0) + u[1].dx(1)) * 2 * pi * dx)
info('\int 1/r * div(r*u) * 2*pi*r = %e' % adivu)
n = FacetNormal(Q.mesh())
boundary_integral = assemble((n[0] * u[0] + n[1] * u[1])
* 2 * pi * r * ds)
info('\int_Gamma n.u * 2*pi*r = %e' % boundary_integral)
message = ('System not consistent! '
'<b,e> = %g, ||b|| = %g, <b,e>/||b|| = %e.') \
% (alpha, normB, alpha / normB)
info(message)
# Plot the stuff, and project it to a finer mesh with linear
# elements for the purpose.
plot(divu, title='div(u_tentative)')
#Vp = FunctionSpace(Q.mesh(), 'CG', 2)
#Wp = MixedFunctionSpace([Vp, Vp])
#up = project(u, Wp)
fine_mesh = Q.mesh()
for k in range(1):
fine_mesh = refine(fine_mesh)
V = FunctionSpace(fine_mesh, 'CG', 1)
W = V * V
#uplot = Function(W)
#uplot.interpolate(u)
uplot = project(u, W)
plot(uplot[0], title='u_tentative[0]')
plot(uplot[1], title='u_tentative[1]')
#plot(u, title='u_tentative')
interactive()
exit()
raise RuntimeError(message)
# Project out the roundoff error.
b -= alpha * evec
#
# In principle, the ILU preconditioner isn't advised here since it
# might destroy the semidefiniteness needed for CG.
#
# The system is consistent, but the matrix has an eigenvalue 0.
# This does not harm the convergence of CG, but when
# preconditioning one has to make sure that the preconditioner
# preserves the kernel. ILU might destroy this (and the
# semidefiniteness). With AMG, the coarse grid solves cannot be LU
# then, so try Jacobi here.
# <http://lists.mcs.anl.gov/pipermail/petsc-users/2012-February/012139.html>
#
prec = PETScPreconditioner('hypre_amg')
from dolfin import PETScOptions
PETScOptions.set('pc_hypre_boomeramg_relax_type_coarse', 'jacobi')
solver = PETScKrylovSolver('cg', prec)
solver.parameters['absolute_tolerance'] = 0.0
solver.parameters['relative_tolerance'] = tol
solver.parameters['maximum_iterations'] = 100
solver.parameters['monitor_convergence'] = verbose
# Create solver and solve system
A_petsc = as_backend_type(A)
b_petsc = as_backend_type(b)
p1_petsc = as_backend_type(p1.vector())
solver.set_operator(A_petsc)
solver.solve(p1_petsc, b_petsc)
# This would be the stump for Epetra:
#solve(A, p.vector(), b, 'cg', 'ml_amg')
return
class TVPD():
""" Total variation using primal-dual Newton """
def __init__(self, parameters):
"""
TV regularization in primal-dual format
Input parameters:
* k = regularization parameter
* eps = regularization constant (see above)
* rescaledradiusdual = radius of dual set
* exact = use full TV (bool)
* PCGN = use GN Hessian to precondition (bool);
not recommended for performance but can help avoid num.instability
* print (bool)
"""
self.parameters = {}
self.parameters['k'] = 1.0
self.parameters['eps'] = 1e-2
self.parameters['rescaledradiusdual'] = 1.0
self.parameters['exact'] = False
self.parameters['PCGN'] = False
self.parameters['print'] = False
self.parameters['correctcost'] = True
self.parameters['amg'] = 'default'
assert parameters.has_key('Vm')
self.parameters.update(parameters)
self.Vm = self.parameters['Vm']
k = self.parameters['k']
eps = self.parameters['eps']
exact = self.parameters['exact']
amg = self.parameters['amg']
self.m = Function(self.Vm)
testm = TestFunction(self.Vm)
trialm = TrialFunction(self.Vm)
# WARNING: should not be changed.
# As it is, code only works with DG0
if self.parameters.has_key('Vw'):
Vw = self.parameters['Vw']
else:
Vw = FunctionSpace(self.Vm.mesh(), 'DG', 0)
self.wx = Function(Vw)
self.wxrs = Function(Vw) # re-scaled dual variable
self.wxhat = Function(Vw)
self.gwx = Function(Vw)
self.wy = Function(Vw)
self.wyrs = Function(Vw) # re-scaled dual variable
self.wyhat = Function(Vw)
self.gwy = Function(Vw)
self.wxsq = Vector()
self.wysq = Vector()
self.normw = Vector()
self.factorw = Vector()
testw = TestFunction(Vw)
trialw = TrialFunction(Vw)
normm = inner(nabla_grad(self.m), nabla_grad(self.m))
TVnormsq = normm + Constant(eps)
TVnorm = sqrt(TVnormsq)
if self.parameters['correctcost']:
meshtmp = UnitSquareMesh(self.Vm.mesh().mpi_comm(), 10, 10)
Vtmp = FunctionSpace(meshtmp, 'CG', 1)
x = SpatialCoordinate(meshtmp)
correctioncost = 1. / assemble(sqrt(4.0 * x[0] * x[0]) * dx)
else:
correctioncost = 1.0
self.wkformcost = Constant(k * correctioncost) * TVnorm * dx
if exact:
sys.exit(1)
# self.w = nabla_grad(self.m)/TVnorm # full Hessian
# self.Htvw = inner(Constant(k) * nabla_grad(testm), self.w) * dx
self.misfitwx = inner(testw, self.wx * TVnorm - self.m.dx(0)) * dx
self.misfitwy = inner(testw, self.wy * TVnorm - self.m.dx(1)) * dx
self.Htvx = assemble(inner(Constant(k) * testm.dx(0), trialw) * dx)
self.Htvy = assemble(inner(Constant(k) * testm.dx(1), trialw) * dx)
self.massw = inner(TVnorm * testw, trialw) * dx
mpicomm = self.Vm.mesh().mpi_comm()
invMwMat, VDM, VDM = setupPETScmatrix(Vw, Vw, 'aij', mpicomm)
for ii in VDM.dofs():
invMwMat[ii, ii] = 1.0
invMwMat.assemblyBegin()
invMwMat.assemblyEnd()
self.invMwMat = PETScMatrix(invMwMat)
self.invMwd = Vector()
self.invMwMat.init_vector(self.invMwd, 0)
self.invMwMat.init_vector(self.wxsq, 0)
self.invMwMat.init_vector(self.wysq, 0)
self.invMwMat.init_vector(self.normw, 0)
self.invMwMat.init_vector(self.factorw, 0)
u = Function(Vw)
uflrank = len(u.ufl_shape)
if uflrank == 0:
ones = ("1.0")
elif uflrank == 1:
ones = (("1.0", "1.0"))
else:
sys.exit(1)
u = interpolate(Constant(ones), Vw)
self.one = u.vector()
self.wkformAx = inner(testw, trialm.dx(0) - \
self.wx * inner(nabla_grad(self.m), nabla_grad(trialm)) / TVnorm) * dx
self.wkformAxrs = inner(testw, trialm.dx(0) - \
self.wxrs * inner(nabla_grad(self.m), nabla_grad(trialm)) / TVnorm) * dx
self.wkformAy = inner(testw, trialm.dx(1) - \
self.wy * inner(nabla_grad(self.m), nabla_grad(trialm)) / TVnorm) * dx
self.wkformAyrs = inner(testw, trialm.dx(1) - \
self.wyrs * inner(nabla_grad(self.m), nabla_grad(trialm)) / TVnorm) * dx
kovsq = Constant(k) / TVnorm
self.wkformGNhess = kovsq * inner(nabla_grad(trialm),
nabla_grad(testm)) * dx
factM = 1e-2 * k
M = assemble(inner(testm, trialm) * dx)
self.sMass = M * factM
self.Msolver = PETScKrylovSolver('cg', 'jacobi')
self.Msolver.parameters["maximum_iterations"] = 2000
self.Msolver.parameters["relative_tolerance"] = 1e-24
self.Msolver.parameters["absolute_tolerance"] = 1e-24
self.Msolver.parameters["error_on_nonconvergence"] = True
self.Msolver.parameters["nonzero_initial_guess"] = False
self.Msolver.set_operator(M)
if amg == 'default': self.amgprecond = amg_solver()
else: self.amgprecond = amg
if self.parameters['print']:
print '[TVPD] TV regularization -- primal-dual method',
if self.parameters['PCGN']:
print ' -- PCGN',
print ' -- k={}, eps={}'.format(k, eps)
print '[TVPD] preconditioner = {}'.format(self.amgprecond)
print '[TVPD] correction cost with factor={}'.format(
correctioncost)
def isTV(self):
return True
def isPD(self):
return True
def cost(self, m):
""" evaluate the cost functional at m """
setfct(self.m, m)
return assemble(self.wkformcost)
def costvect(self, m_in):
return self.cost(m_in)
def _assemble_invMw(self):
""" Assemble inverse of matrix Mw,
weighted mass matrix in dual space """
# WARNING: only works if Mw is diagonal (e.g, DG0)
Mw = assemble(self.massw)
Mwd = get_diagonal(Mw)
as_backend_type(self.invMwd).vec().pointwiseDivide(\
as_backend_type(self.one).vec(),\
as_backend_type(Mwd).vec())
self.invMwMat.set_diagonal(self.invMwd)
def grad(self, m):
""" compute the gradient at m """
setfct(self.m, m)
self._assemble_invMw()
self.gwx.vector().zero()
self.gwx.vector().axpy(1.0, assemble(self.misfitwx))
normgwx = norm(self.gwx.vector())
self.gwy.vector().zero()
self.gwy.vector().axpy(1.0, assemble(self.misfitwy))
normgwy = norm(self.gwy.vector())
if self.parameters['print']:
print '[TVPD] |gw|={}'.format(np.sqrt(normgwx**2 + normgwy**2))
return self.Htvx*(self.wx.vector() - self.invMwd*self.gwx.vector()) \
+ self.Htvy*(self.wy.vector() - self.invMwd*self.gwy.vector())
#return assemble(self.Htvw) - self.Htv*(self.invMwd*self.gw.vector())
def gradvect(self, m_in):
return self.grad(m_in)
def assemble_hessian(self, m):
""" build Hessian matrix at given point m """
setfct(self.m, m)
self._assemble_invMw()
self.Ax = assemble(self.wkformAx)
Hxasym = MatMatMult(self.Htvx, MatMatMult(self.invMwMat, self.Ax))
Hx = (Hxasym + Transpose(Hxasym)) * 0.5
Axrs = assemble(self.wkformAxrs)
Hxrsasym = MatMatMult(self.Htvx, MatMatMult(self.invMwMat, Axrs))
Hxrs = (Hxrsasym + Transpose(Hxrsasym)) * 0.5
self.Ay = assemble(self.wkformAy)
Hyasym = MatMatMult(self.Htvy, MatMatMult(self.invMwMat, self.Ay))
Hy = (Hyasym + Transpose(Hyasym)) * 0.5
Ayrs = assemble(self.wkformAyrs)
Hyrsasym = MatMatMult(self.Htvy, MatMatMult(self.invMwMat, Ayrs))
Hyrs = (Hyrsasym + Transpose(Hyrsasym)) * 0.5
self.H = Hx + Hy
self.Hrs = Hxrs + Hyrs
PCGN = self.parameters['PCGN']
if PCGN:
HGN = assemble(self.wkformGNhess)
self.precond = HGN + self.sMass
else:
self.precond = self.Hrs + self.sMass
def hessian(self, mhat):
return self.Hrs * mhat
def compute_what(self, mhat):
""" Compute update direction for what, given mhat """
self.wxhat.vector().zero()
self.wxhat.vector().axpy(
1.0, self.invMwd * (self.Ax * mhat - self.gwx.vector()))
normwxhat = norm(self.wxhat.vector())
self.wyhat.vector().zero()
self.wyhat.vector().axpy(
1.0, self.invMwd * (self.Ay * mhat - self.gwy.vector()))
normwyhat = norm(self.wyhat.vector())
if self.parameters['print']:
print '[TVPD] |what|={}'.format(
np.sqrt(normwxhat**2 + normwyhat**2))
def update_w(self, mhat, alphaLS, compute_what=True):
""" update dual variable in direction what
and update re-scaled version """
if compute_what: self.compute_what(mhat)
self.wx.vector().axpy(alphaLS, self.wxhat.vector())
self.wy.vector().axpy(alphaLS, self.wyhat.vector())
# rescaledradiusdual=1.0: checked empirically to be max radius acceptable
rescaledradiusdual = self.parameters['rescaledradiusdual']
# wx**2
as_backend_type(self.wxsq).vec().pointwiseMult(\
as_backend_type(self.wx.vector()).vec(),\
as_backend_type(self.wx.vector()).vec())
# wy**2
as_backend_type(self.wysq).vec().pointwiseMult(\
as_backend_type(self.wy.vector()).vec(),\
as_backend_type(self.wy.vector()).vec())
# |w|
self.normw = self.wxsq + self.wysq
as_backend_type(self.normw).vec().sqrtabs()
# |w|/r
as_backend_type(self.normw).vec().pointwiseDivide(\
as_backend_type(self.normw).vec(),\
as_backend_type(self.one*rescaledradiusdual).vec())
# max(1.0, |w|/r)
# as_backend_type(self.factorw).vec().pointwiseMax(\
# as_backend_type(self.one).vec(),\
# as_backend_type(self.normw).vec())
count = pointwiseMaxCount(self.factorw, self.normw, 1.0)
# rescale wx and wy
as_backend_type(self.wxrs.vector()).vec().pointwiseDivide(\
as_backend_type(self.wx.vector()).vec(),\
as_backend_type(self.factorw).vec())
as_backend_type(self.wyrs.vector()).vec().pointwiseDivide(\
as_backend_type(self.wy.vector()).vec(),\
as_backend_type(self.factorw).vec())
minf = self.factorw.min()
maxf = self.factorw.max()
if self.parameters['print']:
# print 'min(factorw)={}, max(factorw)={}'.format(minf, maxf)
print '[TVPD] perc. dual entries rescaled={:.2f} %, min(factorw)={}, max(factorw)={}'.format(\
100.*float(count)/self.factorw.size(), minf, maxf)
def getprecond(self):
""" Precondition by inverting the TV Hessian """
solver = PETScKrylovSolver('cg', self.amgprecond)
solver.parameters["maximum_iterations"] = 2000
solver.parameters["relative_tolerance"] = 1e-24
solver.parameters["absolute_tolerance"] = 1e-24
solver.parameters["error_on_nonconvergence"] = True
solver.parameters["nonzero_initial_guess"] = False
# used to compare iterative application of preconditioner
# with exact application of preconditioner:
#solver = PETScLUSolver("petsc")
#solver.parameters['symmetric'] = True
#solver.parameters['reuse_factorization'] = True
solver.set_operator(self.precond)
return solver
def init_vector(self, u, dim):
self.sMass.init_vector(u, dim)
def solve(W, P,
mu,
u_bcs, p_bcs,
f,
verbose=True,
tol=1.0e-10
):
# Some initial sanity checks.
assert mu > 0.0
WP = MixedFunctionSpace([W, P])
# Translate the boundary conditions into the product space.
# This conditional loop is able to deal with conditions of the kind
#
# DirichletBC(W.sub(1), 0.0, right_boundary)
#
new_bcs = []
for k, bcs in enumerate([u_bcs, p_bcs]):
for bc in bcs:
space = bc.function_space()
C = space.component()
if len(C) == 0:
new_bcs.append(DirichletBC(WP.sub(k),
bc.value(),
bc.domain_args[0]))
elif len(C) == 1:
new_bcs.append(DirichletBC(WP.sub(k).sub(int(C[0])),
bc.value(),
bc.domain_args[0]))
else:
raise RuntimeError('Illegal number of subspace components.')
# Define variational problem
(u, p) = TrialFunctions(WP)
(v, q) = TestFunctions(WP)
# Build system.
# The sign of the div(u)-term is somewhat arbitrary since the right-hand
# side is 0 here. We can either make the system symmetric or positive-
# definite.
# On a second note, we have
#
# \int grad(p).v = - \int p * div(v) + \int_\Gamma p n.v.
#
# Since, we have either p=0 or n.v=0 on the boundary, we could as well
# replace the term dot(grad(p), v) by -p*div(v).
#
a = mu * inner(grad(u), grad(v))*dx \
- p * div(v) * dx \
- q * div(u) * dx
#a = mu * inner(grad(u), grad(v))*dx + dot(grad(p), v) * dx \
# - div(u) * q * dx
L = dot(f, v)*dx
A, b = assemble_system(a, L, new_bcs)
if has_petsc():
# For an assortment of preconditioners, see
#
# Performance and analysis of saddle point preconditioners
# for the discrete steady-state Navier-Stokes equations;
# H.C. Elman, D.J. Silvester, A.J. Wathen;
# Numer. Math. (2002) 90: 665-688;
# <http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.145.3554>.
#
# Set up field split.
W = SubSpace(WP, 0)
P = SubSpace(WP, 1)
u_dofs = W.dofmap().dofs()
p_dofs = P.dofmap().dofs()
prec = PETScPreconditioner()
prec.set_fieldsplit([u_dofs, p_dofs], ['u', 'p'])
PETScOptions.set('pc_type', 'fieldsplit')
PETScOptions.set('pc_fieldsplit_type', 'additive')
PETScOptions.set('fieldsplit_u_pc_type', 'lu')
PETScOptions.set('fieldsplit_p_pc_type', 'jacobi')
## <http://scicomp.stackexchange.com/questions/7288/which-preconditioners-and-solver-in-petsc-for-indefinite-symmetric-systems-sho>
#PETScOptions.set('pc_type', 'fieldsplit')
##PETScOptions.set('pc_fieldsplit_type', 'schur')
##PETScOptions.set('pc_fieldsplit_schur_fact_type', 'upper')
#PETScOptions.set('pc_fieldsplit_detect_saddle_point')
##PETScOptions.set('fieldsplit_u_pc_type', 'lsc')
##PETScOptions.set('fieldsplit_u_ksp_type', 'preonly')
#PETScOptions.set('pc_type', 'fieldsplit')
#PETScOptions.set('fieldsplit_u_pc_type', 'hypre')
#PETScOptions.set('fieldsplit_u_ksp_type', 'preonly')
#PETScOptions.set('fieldsplit_p_pc_type', 'jacobi')
#PETScOptions.set('fieldsplit_p_ksp_type', 'preonly')
## From PETSc/src/ksp/ksp/examples/tutorials/ex42-fsschur.opts:
#PETScOptions.set('pc_type', 'fieldsplit')
#PETScOptions.set('pc_fieldsplit_type', 'SCHUR')
#PETScOptions.set('pc_fieldsplit_schur_fact_type', 'UPPER')
#PETScOptions.set('fieldsplit_p_ksp_type', 'preonly')
#PETScOptions.set('fieldsplit_u_pc_type', 'bjacobi')
## From
##
## Composable Linear Solvers for Multiphysics;
## J. Brown, M. Knepley, D.A. May, L.C. McInnes, B. Smith;
## <http://www.computer.org/csdl/proceedings/ispdc/2012/4805/00/4805a055-abs.html>;
## <http://www.mcs.anl.gov/uploads/cels/papers/P2017-0112.pdf>.
##
#PETScOptions.set('pc_type', 'fieldsplit')
#PETScOptions.set('pc_fieldsplit_type', 'schur')
#PETScOptions.set('pc_fieldsplit_schur_factorization_type', 'upper')
##
#PETScOptions.set('fieldsplit_u_ksp_type', 'cg')
#PETScOptions.set('fieldsplit_u_ksp_rtol', 1.0e-6)
#PETScOptions.set('fieldsplit_u_pc_type', 'bjacobi')
#PETScOptions.set('fieldsplit_u_sub_pc_type', 'cholesky')
##
#PETScOptions.set('fieldsplit_p_ksp_type', 'fgmres')
#PETScOptions.set('fieldsplit_p_ksp_constant_null_space')
#PETScOptions.set('fieldsplit_p_pc_type', 'lsc')
##
#PETScOptions.set('fieldsplit_p_lsc_ksp_type', 'cg')
#PETScOptions.set('fieldsplit_p_lsc_ksp_rtol', 1.0e-2)
#PETScOptions.set('fieldsplit_p_lsc_ksp_constant_null_space')
##PETScOptions.set('fieldsplit_p_lsc_ksp_converged_reason')
#PETScOptions.set('fieldsplit_p_lsc_pc_type', 'bjacobi')
#PETScOptions.set('fieldsplit_p_lsc_sub_pc_type', 'icc')
# Create Krylov solver with custom preconditioner.
solver = PETScKrylovSolver('gmres', prec)
solver.set_operator(A)
else:
# Use the preconditioner as recommended in
# <http://fenicsproject.org/documentation/dolfin/dev/python/demo/pde/stokes-iterative/python/documentation.html>,
#
# prec = inner(grad(u), grad(v))*dx - p*q*dx
#
# although it doesn't seem to be too efficient.
# The sign on the last term doesn't matter.
prec = mu * inner(grad(u), grad(v))*dx \
- p*q*dx
M, _ = assemble_system(prec, L, new_bcs)
#solver = KrylovSolver('tfqmr', 'amg')
solver = KrylovSolver('gmres', 'amg')
solver.set_operators(A, M)
solver.parameters['monitor_convergence'] = verbose
solver.parameters['report'] = verbose
solver.parameters['absolute_tolerance'] = 0.0
solver.parameters['relative_tolerance'] = tol
solver.parameters['maximum_iterations'] = 500
# Solve
up = Function(WP)
solver.solve(up.vector(), b)
# Get sub-functions
u, p = up.split()
return u, p
class ObjectiveFunctional(LinearOperator):
"""
Provides data misfit, gradient and Hessian information for the data misfit
part of a time-independent symmetric inverse problem.
"""
__metaclass__ = abc.ABCMeta
# Instantiation
def __init__(self, V, Vm, bc, bcadj, \
RHSinput=[], ObsOp=[], UD=[], Regul=[], Data=[], plot=False, \
mycomm=None):
# Define test, trial and all other functions
self.trial = TrialFunction(V)
self.test = TestFunction(V)
self.mtrial = TrialFunction(Vm)
self.mtest = TestFunction(Vm)
self.rhs = Function(V)
self.m = Function(Vm)
self.mcopy = Function(Vm)
self.srchdir = Function(Vm)
self.delta_m = Function(Vm)
self.MG = Function(Vm)
self.MGv = self.MG.vector()
self.Grad = Function(Vm)
self.Gradnorm = 0.0
self.lenm = len(self.m.vector().array())
self.u = Function(V)
self.ud = Function(V)
self.diff = Function(V)
self.p = Function(V)
# Store other info:
self.ObsOp = ObsOp
self.UD = UD
self.reset() # Initialize U, C and E to []
self.Data = Data
self.GN = 1.0 # GN = 0.0 => GN Hessian; = 1.0 => full Hessian
# Define weak forms to assemble A, C and E
self._wkforma()
self._wkformc()
self._wkforme()
# Operators and bc
LinearOperator.__init__(self, self.delta_m.vector(), \
self.delta_m.vector())
self.bc = bc
self.bcadj = bcadj
self._assemble_solverM(Vm)
self.assemble_A()
self.assemble_RHS(RHSinput)
self.Regul = Regul
self.regparam = 1.0
if Regul != []:
self.PD = self.Regul.isPD()
# Counters, tolerances and others
self.nbPDEsolves = 0 # Updated when solve_A called
self.nbfwdsolves = 0 # Counter for plots
self.nbadjsolves = 0 # Counter for plots
# MPI:
self.mycomm = mycomm
def copy(self):
"""Define a copy method"""
V = self.trial.function_space()
Vm = self.mtrial.function_space()
newobj = self.__class__(V, Vm, self.bc, self.bcadj, [], self.ObsOp, \
self.UD, self.Regul, self.Data, False)
newobj.RHS = self.RHS
newobj.update_m(self.m)
return newobj
def mult(self, mhat, y):
"""mult(self, mhat, y): do y = Hessian * mhat
member self.GN sets full Hessian (=1.0) or GN Hessian (=0.0)"""
N = self.Nbsrc # Number of sources
y[:] = np.zeros(self.lenm)
for C, E in zip(self.C, self.E):
C.transpmult(mhat, self.rhs.vector())
if self.bcadj is not None:
self.bcadj.apply(self.rhs.vector())
self.solve_A(self.u.vector(), -self.rhs.vector())
E.transpmult(mhat, self.rhs.vector())
Etmhat = self.rhs.vector().array()
self.rhs.vector().axpy(1.0, self.ObsOp.incradj(self.u))
if self.bcadj is not None:
self.bcadj.apply(self.rhs.vector())
self.solve_A(self.p.vector(), -self.rhs.vector())
y.axpy(1.0 / N, C * self.p.vector())
y.axpy(self.GN / N, E * self.u.vector())
y.axpy(self.regparam, self.Regul.hessian(mhat))
# Getters
def getm(self):
return self.m
def getmarray(self):
return self.m.vector().array()
def getmcopyarray(self):
return self.mcopy.vector().array()
def getVm(self):
return self.mtrial.function_space()
def getMGarray(self):
return self.MG.vector().array()
def getMGvec(self):
return self.MGv
def getGradarray(self):
return self.Grad.vector().array()
def getGradnorm(self):
return self.Gradnorm
def getsrchdirarray(self):
return self.srchdir.vector().array()
def getsrchdirvec(self):
return self.srchdir.vector()
def getsrchdirnorm(self):
return np.sqrt(
(self.MM * self.getsrchdirvec()).inner(self.getsrchdirvec()))
def getgradxdir(self):
return self.gradxdir
def getcost(self):
return self.cost, self.misfit, self.regul
def getprecond(self):
return self.Regul.getprecond()
# Prec = PETScKrylovSolver("richardson", "amg")
# Prec.parameters["maximum_iterations"] = 1
# Prec.parameters["error_on_nonconvergence"] = False
# Prec.parameters["nonzero_initial_guess"] = False
# Prec.set_operator(self.Regul.get_precond())
# return Prec
def getMass(self):
return self.MM
# Setters
def setsrchdir(self, arr):
self.srchdir.vector()[:] = arr
def setgradxdir(self, valueloc):
"""Sum all local results for Grad . Srch_dir"""
try:
valueglob = MPI.sum(self.mycomm, valueloc)
except:
valueglob = valueloc
self.gradxdir = valueglob
# Solve
def solvefwd(self, cost=False):
"""Solve fwd operators for given RHS"""
self.nbfwdsolves += 1
if cost: self.misfit = 0.0
self.U = []
self.C = []
for ii, rhs in enumerate(self.RHS):
self.solve_A(self.u.vector(), rhs)
u_obs, noiselevel = self.ObsOp.obs(self.u)
self.U.append(u_obs)
if cost:
self.misfit += self.ObsOp.costfct(u_obs, self.UD[ii])
self.C.append(assemble(self.c))
if cost:
self.misfit /= len(self.U)
self.regul = self.Regul.cost(self.m)
self.cost = self.misfit + self.regparam * self.regul
def solvefwd_cost(self):
"""Solve fwd operators for given RHS and compute cost fct"""
self.solvefwd(True)
def solveadj(self, grad=False):
"""Solve adj operators"""
self.nbadjsolves += 1
self.Nbsrc = len(self.UD)
if grad:
self.MG.vector().zero()
self.E = []
for ii, C in enumerate(self.C):
self.ObsOp.assemble_rhsadj(self.U[ii], self.UD[ii], \
self.rhs, self.bcadj)
self.solve_A(self.p.vector(), self.rhs.vector())
self.E.append(assemble(self.e))
if grad:
self.MG.vector().axpy(1.0 / self.Nbsrc, C * self.p.vector())
if grad:
self.MG.vector().axpy(self.regparam, self.Regul.grad(self.m))
self.solverM.solve(self.Grad.vector(), self.MG.vector())
self.Gradnorm = np.sqrt(self.Grad.vector().inner(self.MG.vector()))
def solveadj_constructgrad(self):
"""Solve adj operators and assemble gradient"""
self.solveadj(True)
# Assembler
def assemble_A(self):
"""Assemble operator A(m)"""
self.A = assemble(self.a)
if self.bc is not None:
self.bc.apply(self.A)
compute_eigfenics(self.A, 'eigA.txt')
self.set_solver()
def solve_A(self, b, f):
"""Solve system of the form A.b = f,
with b and f in form to be used in solver."""
self.solver.solve(b, f)
self.nbPDEsolves += 1
def assemble_RHS(self, RHSin):
"""Assemble RHS for fwd solve"""
if RHSin == []: self.RHS = None
else:
self.RHS = []
for rhs in RHSin:
if isinstance(rhs, Expression):
L = rhs * self.test * dx
b = assemble(L)
if self.bc is not None:
self.bc.apply(b)
self.RHS.append(b)
elif isinstance(rhs, GenericVector):
self.RHS.append(rhs)
else:
raise WrongInstanceError(
"rhs should be an Expression or a GenericVector")
def _assemble_solverM(self, Vm):
self.MM = assemble(inner(self.mtrial, self.mtest) * dx)
self.solverM = PETScKrylovSolver('cg', 'jacobi')
self.solverM.parameters["maximum_iterations"] = 1000
self.solverM.parameters["relative_tolerance"] = 1e-12
self.solverM.parameters["error_on_nonconvergence"] = True
self.solverM.parameters["nonzero_initial_guess"] = False
# self.solverM = LUSolver()
# self.solverM.parameters['reuse_factorization'] = True
# self.solverM.parameters['symmetric'] = True
self.solverM.set_operator(self.MM)
# Update param
def update_Data(self, Data):
"""Update Data member"""
self.Data = Data
self.assemble_A()
self.reset()
def update_m(self, m):
"""Update values of parameter m"""
if isinstance(m, np.ndarray):
self.m.vector()[:] = m
elif isinstance(m, Function):
self.m.assign(m)
elif isinstance(m, float):
self.m.vector()[:] = m
elif isinstance(m, int):
self.m.vector()[:] = float(m)
else:
raise WrongInstanceError('Format for m not accepted')
self.assemble_A()
self.reset()
def backup_m(self):
self.mcopy.assign(self.m)
def restore_m(self):
self.update_m(self.mcopy)
def reset(self):
"""Reset U, C and E"""
self.U = []
self.C = []
self.E = []
def set_solver(self):
"""Reset solver for fwd operator"""
#self.solver = LUSolver()
#self.solver.parameters['reuse_factorization'] = True
self.solver = PETScKrylovSolver("cg", "amg")
self.solver.parameters["maximum_iterations"] = 1000
self.solver.parameters["relative_tolerance"] = 1e-12
self.solver.parameters["error_on_nonconvergence"] = True
self.solver.parameters["nonzero_initial_guess"] = False
self.solver.set_operator(self.A)
def addPDEcount(self, increment=1):
"""Increase 'nbPDEsolves' by 'increment'"""
self.nbPDEsolves += increment
def resetPDEsolves(self):
self.nbPDEsolves = 0
# Additional methods for compatibility with CG solver:
def init_vector(self, x, dim):
"""Initialize vector x to be compatible with parameter
Does not work in dolfin 1.3.0"""
self.MM.init_vector(x, 0)
def init_vector130(self):
"""Initialize vector x to be compatible with parameter"""
return Vector(Function(self.mcopy.function_space()).vector())
# Abstract methods
@abc.abstractmethod
def _wkforma(self):
self.a = []
@abc.abstractmethod
def _wkformc(self):
self.c = []
@abc.abstractmethod
def _wkforme(self):
self.e = []
def inversion(self, initial_medium, target_medium, mpicomm, \
parameters_in=[], myplot=None):
""" solve inverse problem with that objective function """
parameters = {'tolgrad':1e-10, 'tolcost':1e-14, 'maxnbNewtiter':50, \
'maxtolcg':0.5}
parameters.update(parameters_in)
maxnbNewtiter = parameters['maxnbNewtiter']
tolgrad = parameters['tolgrad']
tolcost = parameters['tolcost']
tolcg = parameters['maxtolcg']
mpirank = MPI.rank(mpicomm)
self.update_m(initial_medium)
self._plotm(myplot, 'init')
if mpirank == 0:
print '\t{:12s} {:10s} {:12s} {:12s} {:12s} {:10s} \t{:10s} {:12s} {:12s}'.format(\
'iter', 'cost', 'misfit', 'reg', '|G|', 'medmisf', 'a_ls', 'tol_cg', 'n_cg')
dtruenorm = np.sqrt(target_medium.vector().\
inner(self.MM*target_medium.vector()))
self.solvefwd_cost()
for it in xrange(maxnbNewtiter):
self.solveadj_constructgrad() # compute gradient
if it == 0: gradnorm0 = self.Gradnorm
diff = self.m.vector() - target_medium.vector()
medmisfit = np.sqrt(diff.inner(self.MM * diff))
if mpirank == 0:
print '{:12d} {:12.4e} {:12.2e} {:12.2e} {:11.4e} {:10.2e} ({:4.2f})'.\
format(it, self.cost, self.misfit, self.regul, \
self.Gradnorm, medmisfit, medmisfit/dtruenorm),
self._plotm(myplot, str(it))
self._plotgrad(myplot, str(it))
if self.Gradnorm < gradnorm0 * tolgrad or self.Gradnorm < 1e-12:
if mpirank == 0:
print '\nGradient sufficiently reduced -- optimization stopped'
break
# Compute search direction:
tolcg = min(tolcg, np.sqrt(self.Gradnorm / gradnorm0))
self.assemble_hessian() # for regularization
cgiter, cgres, cgid, tolcg = compute_searchdirection(
self, 'Newt', tolcg)
self._plotsrchdir(myplot, str(it))
# Line search:
cost_old = self.cost
statusLS, LScount, alpha = bcktrcklinesearch(self, 12)
if mpirank == 0:
print '{:11.3f} {:12.2e} {:10d}'.format(alpha, tolcg, cgiter)
if self.PD: self.Regul.update_w(self.srchdir.vector(), alpha)
if np.abs(self.cost - cost_old) / np.abs(cost_old) < tolcost:
if mpirank == 0:
if tolcg < 1e-14:
print 'Cost function stagnates -- optimization aborted'
break
tolcg = 0.001 * tolcg
def assemble_hessian(self):
self.Regul.assemble_hessian(self.m)
def _plotm(self, myplot, index):
""" plot media during inversion """
if not myplot == None:
myplot.set_varname('m' + index)
myplot.plot_vtk(self.m)
def _plotgrad(self, myplot, index):
""" plot grad during inversion """
if not myplot == None:
myplot.set_varname('Grad_m' + index)
myplot.plot_vtk(self.Grad)
def _plotsrchdir(self, myplot, index):
""" plot srchdir during inversion """
if not myplot == None:
myplot.set_varname('srchdir_m' + index)
myplot.plot_vtk(self.srchdir)
def compute_pressure(
P,
p0,
mu,
ui,
u,
my_dx,
p_bcs=None,
rotational_form=False,
tol=1.0e-10,
verbose=True,
):
"""Solve the pressure Poisson equation
.. math::
\\begin{align}
-\\frac{1}{r} \\div(r \\nabla (p_1-p_0)) =
-\\frac{1}{r} \\div(r u),\\\\
\\text{(with boundary conditions)},
\\end{align}
for :math:`\\nabla p = u`.
The pressure correction is based on the update formula
.. math::
\\frac{\\rho}{dt} (u_{n+1}-u^*)
+ \\begin{pmatrix}
\\text{d}\\phi/\\text{d}r\\\\
\\text{d}\\phi/\\text{d}z\\\\
\\frac{1}{r} \\text{d}\\phi/\\text{d}\\theta
\\end{pmatrix}
= 0
with :math:`\\phi = p_{n+1} - p^*` and
.. math::
\\frac{1}{r} \\frac{\\text{d}}{\\text{d}r} (r u_r^{(n+1)})
+ \\frac{\\text{d}}{\\text{d}z} (u_z^{(n+1)})
+ \\frac{1}{r} \\frac{\\text{d}}{\\text{d}\\theta} (u_{\\theta}^{(n+1)})
= 0
With the assumption that u does not change in the direction
:math:`\\theta`, one derives
.. math::
- \\frac{1}{r} \\div(r \\nabla \\phi) =
\\frac{1}{r} \\frac{\\rho}{dt} \\div(r (u_{n+1} - u^*))\\\\
- \\frac{1}{r} \\langle n, r \\nabla \\phi\\rangle =
\\frac{1}{r} \\frac{\\rho}{dt} \\langle n, r (u_{n+1} - u^*)\\rangle
In its weak form, this is
.. math::
\\int r \\langle\\nabla\\phi, \\nabla q\\rangle \\,2 \\pi =
- \\frac{\\rho}{dt} \\int \\div(r u^*) q \\, 2 \\pi
- \\frac{\\rho}{dt} \\int_{\\Gamma}
\\langle n, r (u_{n+1}-u^*)\\rangle q \\, 2\\pi.
(The terms :math:`1/r` cancel with the volume elements :math:`2\\pi r`.)
If the Dirichlet boundary conditions are applied to both :math:`u^*` and
:math:`u_n` (the latter in the velocity correction step), the boundary
integral vanishes.
If no Dirichlet conditions are given (which is the default case), the
system has no unique solution; one eigenvalue is 0. This however, does not
hurt CG convergence if the system is consistent, cf. :cite:`vdV03`. And
indeed it is consistent if and only if
.. math::
\\int_\\Gamma r \\langle n, u\\rangle = 0.
This condition makes clear that for incompressible Navier-Stokes, one
either needs to make sure that inflow and outflow always add up to 0, or
one has to specify pressure boundary conditions.
Note that, when using a multigrid preconditioner as is done here, the
coarse solver must be chosen such that it preserves the nullspace of the
problem.
"""
W = ui.function_space()
r = SpatialCoordinate(W.mesh())[0]
p = TrialFunction(P)
q = TestFunction(P)
a2 = dot(r * grad(p), grad(q)) * 2 * pi * my_dx
# The boundary conditions
# n.(p1-p0) = 0
# are implicitly included.
#
# L2 = -div(r*u) * q * 2*pi*my_dx
div_u = 1 / r * (r * u[0]).dx(0) + u[1].dx(1)
L2 = -div_u * q * 2 * pi * r * my_dx
if p0:
L2 += r * dot(grad(p0), grad(q)) * 2 * pi * my_dx
# In the Cartesian variant of the rotational form, one makes use of the
# fact that
#
# curl(curl(u)) = grad(div(u)) - div(grad(u)).
#
# The same equation holds true in cylindrical form. Hence, to get the
# rotational form of the splitting scheme, we need to
#
# rotational form
if rotational_form:
# If there is no dependence of the angular coordinate, what is
# div(grad(div(u))) in Cartesian coordinates becomes
#
# 1/r div(r * grad(1/r div(r*u)))
#
# in cylindrical coordinates (div and grad are in cylindrical
# coordinates). Unfortunately, we cannot write it down that
# compactly since u_phi is in the game.
# When using P2 elements, this value will be 0 anyways.
div_ui = 1 / r * (r * ui[0]).dx(0) + ui[1].dx(1)
grad_div_ui = as_vector((div_ui.dx(0), div_ui.dx(1)))
L2 -= r * mu * dot(grad_div_ui, grad(q)) * 2 * pi * my_dx
# div_grad_div_ui = 1/r * (r * grad_div_ui[0]).dx(0) \
# + (grad_div_ui[1]).dx(1)
# L2 += mu * div_grad_div_ui * q * 2*pi*r*dx
# n = FacetNormal(Q.mesh())
# L2 -= mu * (n[0] * grad_div_ui[0] + n[1] * grad_div_ui[1]) \
# * q * 2*pi*r*ds
p1 = Function(P)
if p_bcs:
solve(
a2 == L2,
p1,
bcs=p_bcs,
solver_parameters={
"linear_solver": "iterative",
"symmetric": True,
"preconditioner": "hypre_amg",
"krylov_solver": {
"relative_tolerance": tol,
"absolute_tolerance": 0.0,
"maximum_iterations": 100,
"monitor_convergence": verbose,
},
},
)
else:
# If we're dealing with a pure Neumann problem here (which is the
# default case), this doesn't hurt CG if the system is consistent,
# cf. :cite:`vdV03`. And indeed it is consistent if and only if
#
# \int_\Gamma r n.u = 0.
#
# This makes clear that for incompressible Navier-Stokes, one
# either needs to make sure that inflow and outflow always add up
# to 0, or one has to specify pressure boundary conditions.
#
# If the right-hand side is very small, round-off errors may impair
# the consistency of the system. Make sure the system we are
# solving remains consistent.
A = assemble(a2)
b = assemble(L2)
# Assert that the system is indeed consistent.
e = Function(P)
e.interpolate(Constant(1.0))
evec = e.vector()
evec /= norm(evec)
alpha = b.inner(evec)
normB = norm(b)
# Assume that in every component of the vector, a round-off error
# of the magnitude DOLFIN_EPS is present. This leads to the
# criterion
# |<b,e>| / (||b||*||e||) < DOLFIN_EPS
# as a check whether to consider the system consistent up to
# round-off error.
#
# TODO think about condition here
# if abs(alpha) > normB * DOLFIN_EPS:
if abs(alpha) > normB * 1.0e-12:
# divu = 1 / r * (r * u[0]).dx(0) + u[1].dx(1)
adivu = assemble(((r * u[0]).dx(0) + u[1].dx(1)) * 2 * pi * my_dx)
info("\\int 1/r * div(r*u) * 2*pi*r = {:e}".format(adivu))
n = FacetNormal(P.mesh())
boundary_integral = assemble((n[0] * u[0] + n[1] * u[1]) * 2 * pi * r * ds)
info("\\int_Gamma n.u * 2*pi*r = {:e}".format(boundary_integral))
message = (
"System not consistent! "
"<b,e> = {:g}, ||b|| = {:g}, <b,e>/||b|| = {:e}.".format(
alpha, normB, alpha / normB
)
)
info(message)
# # Plot the stuff, and project it to a finer mesh with linear
# # elements for the purpose.
# plot(divu, title='div(u_tentative)')
# # Vp = FunctionSpace(Q.mesh(), 'CG', 2)
# # Wp = MixedFunctionSpace([Vp, Vp])
# # up = project(u, Wp)
# fine_mesh = Q.mesh()
# for k in range(1):
# fine_mesh = refine(fine_mesh)
# V = FunctionSpace(fine_mesh, 'CG', 1)
# W = V * V
# # uplot = Function(W)
# # uplot.interpolate(u)
# uplot = project(u, W)
# plot(uplot[0], title='u_tentative[0]')
# plot(uplot[1], title='u_tentative[1]')
# # plot(u, title='u_tentative')
# interactive()
# exit()
raise RuntimeError(message)
# Project out the roundoff error.
b -= alpha * evec
#
# In principle, the ILU preconditioner isn't advised here since it
# might destroy the semidefiniteness needed for CG.
#
# The system is consistent, but the matrix has an eigenvalue 0.
# This does not harm the convergence of CG, but when
# preconditioning one has to make sure that the preconditioner
# preserves the kernel. ILU might destroy this (and the
# semidefiniteness). With AMG, the coarse grid solves cannot be LU
# then, so try Jacobi here.
# <http://lists.mcs.anl.gov/pipermail/petsc-users/2012-February/012139.html>
#
prec = PETScPreconditioner("hypre_amg")
from dolfin import PETScOptions
PETScOptions.set("pc_hypre_boomeramg_relax_type_coarse", "jacobi")
solver = PETScKrylovSolver("cg", prec)
solver.parameters["absolute_tolerance"] = 0.0
solver.parameters["relative_tolerance"] = tol
solver.parameters["maximum_iterations"] = 100
solver.parameters["monitor_convergence"] = verbose
# Create solver and solve system
A_petsc = as_backend_type(A)
b_petsc = as_backend_type(b)
p1_petsc = as_backend_type(p1.vector())
solver.set_operator(A_petsc)
solver.solve(p1_petsc, b_petsc)
return p1
def test_poisson(k):
# Polynomial order and mesh resolution
nx_list = [4, 8, 16]
# Error list
error_u_l2, error_u_h1 = [], []
for nx in nx_list:
mesh = UnitSquareMesh(nx, nx)
# Define FunctionSpaces and functions
V = FunctionSpace(mesh, "DG", k)
Vbar = FunctionSpace(mesh,
FiniteElement("CG", mesh.ufl_cell(), k)["facet"])
u_soln = Expression("sin(pi*x[0])*sin(pi*x[1])",
degree=k + 1,
domain=mesh)
f = Expression("2*pi*pi*sin(pi*x[0])*sin(pi*x[1])", degree=k + 1)
u, v = Function(V), TestFunction(V)
ubar, vbar = Function(Vbar), TestFunction(Vbar)
n = FacetNormal(mesh)
h = CellDiameter(mesh)
alpha = Constant(6 * k * k)
penalty = alpha / h
def facet_integral(integrand):
return integrand('-') * dS + integrand('+') * dS + integrand * ds
u_flux = ubar
F_v_flux = grad(u) + penalty * outer(u_flux - u, n)
residual_local = inner(grad(u), grad(v)) * dx
residual_local += facet_integral(inner(outer(u_flux - u, n), grad(v)))
residual_local -= facet_integral(inner(F_v_flux, outer(v, n)))
residual_local -= f * v * dx
residual_global = facet_integral(inner(F_v_flux, outer(vbar, n)))
a_ll = derivative(residual_local, u)
a_lg = derivative(residual_local, ubar)
a_gl = derivative(residual_global, u)
a_gg = derivative(residual_global, ubar)
l_l = -residual_local
l_g = -residual_global
bcs = [DirichletBC(Vbar, u_soln, "on_boundary")]
# Initialize static condensation assembler
assembler = AssemblerStaticCondensation(a_ll, a_lg, a_gl, a_gg, l_l,
l_g, bcs)
A_g, b_g = PETScMatrix(), PETScVector()
assembler.assemble_global_lhs(A_g)
assembler.assemble_global_rhs(b_g)
for bc in bcs:
bc.apply(A_g, b_g)
solver = PETScKrylovSolver()
solver.set_operator(A_g)
PETScOptions.set("ksp_type", "preonly")
PETScOptions.set("pc_type", "lu")
PETScOptions.set("pc_factor_mat_solver_type", "mumps")
solver.set_from_options()
solver.solve(ubar.vector(), b_g)
assembler.backsubstitute(ubar._cpp_object, u._cpp_object)
# Compute L2 and H1 norms
e_u_l2 = assemble((u - u_soln)**2 * dx)**0.5
e_u_h1 = assemble(grad(u - u_soln)**2 * dx)**0.5
if mesh.mpi_comm().rank == 0:
error_u_l2.append(e_u_l2)
error_u_h1.append(e_u_h1)
if mesh.mpi_comm().rank == 0:
iterator_list = [1.0 / float(nx) for nx in nx_list]
conv_u_l2 = compute_convergence(iterator_list, error_u_l2)
conv_u_h1 = compute_convergence(iterator_list, error_u_h1)
# Optimal rate of k + 1 - tolerance
assert np.all(conv_u_l2 >= (k + 1.0 - 0.15))
# Optimal rate of k - tolerance
assert np.all(conv_u_h1 >= (k - 0.1))
def _compute_pressure(p0,
alpha,
rho,
dt,
mu,
div_ui,
p_bcs=None,
p_function_space=None,
rotational_form=False,
tol=1.0e-10,
verbose=True):
'''Solve the pressure Poisson equation
- \\Delta phi = -div(u),
boundary conditions,
for p with
\\nabla p = u.
'''
#
# The following is based on the update formula
#
# rho/dt (u_{n+1}-u*) + \nabla phi = 0
#
# with
#
# phi = (p_{n+1} - p*) + chi*mu*div(u*)
#
# and div(u_{n+1})=0. One derives
#
# - \nabla^2 phi = rho/dt div(u_{n+1} - u*),
# - n.\nabla phi = rho/dt n.(u_{n+1} - u*),
#
# In its weak form, this is
#
# \int \grad(phi).\grad(q)
# = - rho/dt \int div(u*) q - rho/dt \int_Gamma n.(u_{n+1}-u*) q.
#
# If Dirichlet boundary conditions are applied to both u* and u_{n+1} (the
# latter in the final step), the boundary integral vanishes.
#
# Assume that on the boundary
# L2 -= inner(n, rho/k (u_bcs - ui)) * q * ds
# is zero. This requires the boundary conditions to be set for ui as well
# as u_final.
# This creates some problems if the boundary conditions are supposed to
# remain 'free' for the velocity, i.e., no Dirichlet conditions in normal
# direction. In that case, one needs to specify Dirichlet pressure
# conditions.
#
if p0:
P = p0.function_space()
else:
P = p_function_space
p1 = Function(P)
p = TrialFunction(P)
q = TestFunction(P)
a2 = dot(grad(p), grad(q)) * dx
L2 = -alpha * rho / dt * div_ui * q * dx
L2 += dot(grad(p0), grad(q)) * dx
if rotational_form:
L2 -= mu * dot(grad(div_ui), grad(q)) * dx
if p_bcs:
solve(a2 == L2,
p1,
bcs=p_bcs,
solver_parameters={
'linear_solver': 'iterative',
'symmetric': True,
'preconditioner': 'hypre_amg',
'krylov_solver': {
'relative_tolerance': tol,
'absolute_tolerance': 0.0,
'maximum_iterations': 100,
'monitor_convergence': verbose,
'error_on_nonconvergence': True
}
})
else:
# If we're dealing with a pure Neumann problem here (which is the
# default case), this doesn't hurt CG if the system is consistent, cf.
#
# Iterative Krylov methods for large linear systems,
# Henk A. van der Vorst.
#
# And indeed, it is consistent: Note that
#
# <1, rhs> = \sum_i 1 * \int div(u) v_i
# = 1 * \int div(u) \sum_i v_i
# = \int div(u).
#
# With the divergence theorem, we have
#
# \int div(u) = \int_\Gamma n.u.
#
# The latter term is 0 if and only if inflow and outflow are exactly
# the same at any given point in time. This corresponds with the
# incompressibility of the liquid.
#
# Another lesson from this:
# If the mesh has penetration boundaries, you either have to specify
# the normal component of the velocity such that \int(n.u) = 0, or
# specify Dirichlet conditions for the pressure somewhere.
#
A = assemble(a2)
b = assemble(L2)
# If the right hand side is flawed (e.g., by round-off errors), then it
# may have a component b1 in the direction of the null space,
# orthogonal to the image of the operator:
#
# b = b0 + b1.
#
# When starting with initial guess x0=0, the minimal achievable
# relative tolerance is then
#
# min_rel_tol = ||b1|| / ||b||.
#
# If ||b|| is very small, which is the case when ui is almost
# divergence-free, then min_rel_to may be larger than the prescribed
# relative tolerance tol. This happens, for example, when the time
# steps is very small.
# Sanitation of right-hand side is easy with
#
# e = Function(P)
# e.interpolate(Constant(1.0))
# evec = e.vector()
# evec /= norm(evec)
# print(b.inner(evec))
# b -= b.inner(evec) * evec
#
# However it's hard to decide when the right-hand side is inconsistent
# because of round-off errors in previous steps, or because the system
# is actually inconsistent (insufficient boundary conditions or
# something like that). Hence, don't do anything and rather try to
# fight the cause for round-off.
# In principle, the ILU preconditioner isn't advised here since it
# might destroy the semidefiniteness needed for CG.
#
# The system is consistent, but the matrix has an eigenvalue 0. This
# does not harm the convergence of CG, but with preconditioning one has
# to make sure that the preconditioner preserves the kernel. ILU might
# destroy this (and the semidefiniteness). With AMG, the coarse grid
# solves cannot be LU then, so try Jacobi here.
# <http://lists.mcs.anl.gov/pipermail/petsc-users/2012-February/012139.html>
#
# TODO clear everything; possible in FEniCS 2017.1
# <https://fenicsproject.org/qa/12916/clear-petscoptions>
# PETScOptions.clear()
prec = PETScPreconditioner('hypre_amg')
PETScOptions.set('pc_hypre_boomeramg_relax_type_coarse', 'jacobi')
solver = PETScKrylovSolver('cg', prec)
solver.parameters['absolute_tolerance'] = 0.0
solver.parameters['relative_tolerance'] = tol
solver.parameters['maximum_iterations'] = 1000
solver.parameters['monitor_convergence'] = verbose
solver.parameters['error_on_nonconvergence'] = True
# Create solver and solve system
A_petsc = as_backend_type(A)
b_petsc = as_backend_type(b)
p1_petsc = as_backend_type(p1.vector())
solver.set_operator(A_petsc)
solver.solve(p1_petsc, b_petsc)
return p1