def solve_alpha_M_beta_F(self, alpha, beta, b, t):
"""Solve :code:`alpha * M * u + beta * F(u, t) = b` with Dirichlet
conditions.
"""
matrix = alpha * self.M + beta * self.A
# See above for float conversion
right_hand_side = -float(beta) * self.b.copy()
if b:
right_hand_side += b
for bc in self.dirichlet_bcs:
bc.apply(matrix, right_hand_side)
# TODO proper preconditioner for convection
if self.convection:
# Use HYPRE-Euclid instead of ILU for parallel computation.
# However, this PC sometimes fails.
# solver = KrylovSolver('gmres', 'hypre_euclid')
# Fallback:
solver = LUSolver()
else:
solver = KrylovSolver("gmres", "hypre_amg")
solver.parameters["relative_tolerance"] = 1.0e-13
solver.parameters["absolute_tolerance"] = 0.0
solver.parameters["maximum_iterations"] = 100
solver.parameters["monitor_convergence"] = True
solver.set_operator(matrix)
u = Function(self.Q)
solver.solve(u.vector(), right_hand_side)
return u
def assert_solves(
mesh: Mesh,
diffusion: Coefficient,
convection: Optional[Coefficient],
reaction: Optional[Coefficient],
source: Coefficient,
exact: Coefficient,
l2_tol: Optional[float] = 1.0e-8,
h1_tol: Optional[float] = 1.0e-6,
):
eafe_matrix = eafe_assemble(mesh, diffusion, convection, reaction)
pw_linears = FunctionSpace(mesh, "Lagrange", 1)
test_function = TestFunction(pw_linears)
rhs_vector = assemble(source * test_function * dx)
bc = DirichletBC(pw_linears, exact, lambda _, on_bndry: on_bndry)
bc.apply(eafe_matrix, rhs_vector)
solution = Function(pw_linears)
solver = LUSolver(eafe_matrix, "default")
solver.parameters["symmetric"] = False
solver.solve(solution.vector(), rhs_vector)
l2_err: float = errornorm(exact, solution, "l2", 3)
assert l2_err <= l2_tol, f"L2 error too large: {l2_err} > {l2_tol}"
h1_err: float = errornorm(exact, solution, "H1", 3)
assert h1_err <= h1_tol, f"H1 error too large: {h1_err} > {h1_tol}"
def direct_solve(A, b, W, which='umfpack'):
'''inv(A)*b'''
print 'Solving system of size %d' % A.size(0)
# NOTE: umfpack sometimes blows up, MUMPS produces crap more often than not
if isinstance(W, list):
wh = ii_Function(W)
LUSolver(which).solve(A, wh.vector(), b)
print('|b-Ax| from direct solver', (A * wh.vector() - b).norm('linf'))
return wh
wh = Function(W)
LUSolver(which).solve(A, wh.vector(), b)
print('|b-Ax| from direct solver', (A * wh.vector() - b).norm('linf'))
if isinstance(
W.ufl_element(),
(ufl.VectorElement, ufl.TensorElement)) or W.num_sub_spaces() == 1:
return ii_Function([W], [wh])
# Now get components
Wblock = serialize_mixed_space(W)
wh = wh.split(deepcopy=True)
return ii_Function(Wblock, wh)
def _get_constant_pressure(self, W):
w = TrialFunction(W)
w_ = TestFunction(W)
p_ = self.test_functions()["p"]
A, b = assemble_system(inner(w, w_) * dx, p_ * dx)
null_fcn = Function(W)
solver = LUSolver("mumps")
solver.solve(A, null_fcn.vector(), b)
return null_fcn
def __init__(self, *args, **kwargs):
"""
Create solver for
:py:class:`SemiDecoupled <muflon.functions.discretization.SemiDecoupled>`
discretization scheme.
See :py:class:`Solver <muflon.solving.solvers.Solver>` for the list of
valid initialization arguments.
"""
super(SemiDecoupled, self).__init__(*args, **kwargs)
# Extract solution functions
DS = self.data["model"].discretization_scheme()
w_ch, w_ns = DS.solution_ctl()
# Adjust bcs
bcs_ch = []
bcs_ns = []
_bcs = self.data["model"].bcs()
for bc_v in _bcs.get("v", []):
assert isinstance(bc_v, tuple)
assert len(bc_v) == len(w_ns.sub(0))
bcs_ns += [bc for bc in bc_v if bc is not None]
bcs_ns += [bc for bc in _bcs.get("p", [])]
# FIXME: Deal with possible bcs for ``phi`` and ``th``
# Prepare solver for CH part
F = self.data["forms"]["nln"]
J = derivative(F, w_ch)
# Store solvers and collect other data
self.data["problem_ch"] = self.CHProblem(F, bcs_ch, J)
self.data["solver"] = OrderedDict()
self.data["solver"]["CH"] = OrderedDict()
self.data["solver"]["CH"]["lin"] = LUSolver("mumps")
# NOTE: Initialization of nonlinear solver is postponed until setup
self.data["solver"]["NS"] = LUSolver("mumps")
self.data["sol_ch"] = w_ch
self.data["sol_ns"] = w_ns
self.data["bcs_ns"] = bcs_ns
# Preparation for tackling singular systems
if self._flags["fix_p"]:
null_space, null_fcn = DS.build_pressure_null_space()
self.data["null_space"] = null_space
self.data["null_fcn"] = null_fcn
# Store number of iterations
self.iters = OrderedDict()
self.iters["CH"] = [0, 0] # (total, last solve)
self.iters["NS"] = [0, 0] # (total, last solve)
def setUp(self):
mesh = UnitSquareMesh(5, 5, 'crossed')
self.V = FunctionSpace(mesh, 'Lagrange', 5)
self.u = Function(self.V)
self.uM = Function(self.V)
self.uMdiag = Function(self.V)
test = TestFunction(self.V)
trial = TrialFunction(self.V)
m = test * trial * dx
self.M = assemble(m)
self.solver = LUSolver()
self.solver.parameters['reuse_factorization'] = True
self.solver.parameters['symmetric'] = True
self.solver.set_operator(self.M)
self.ones = np.ones(self.V.dim())
def __init__(self, *args, **kwargs):
"""
Create linear solver for :py:class:`FullyDecoupled \
<muflon.functions.discretization.FullyDecoupled>`
discretization scheme.
See :py:class:`Solver <muflon.solving.solvers.Solver>` for the list of
valid initialization arguments.
"""
super(FullyDecoupled, self).__init__(*args, **kwargs)
# Create solvers
solver = OrderedDict()
solver["phi"] = LUSolver("mumps")
solver["chi"] = LUSolver("mumps")
solver["v"] = LUSolver("mumps")
solver["p"] = LUSolver("mumps")
self.data["solver"] = solver
# Initialize flags
self._flags["setup"] = False
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']
if self.Parameters.has_key('m0'):
self.m0 = self.Parameters['m0'].copy(deepcopy=True)
isFunction(self.m0)
else:
self.m0 = Function(self.Vm)
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 = PETScMatrix()
assemble(inner(self.mtrial, self.mtest) * dx, tensor=self.M)
# preconditioner is Gamma^{-1}:
if self.beta > 1e-16:
self.precond = self.gamma * self.R + self.beta * self.M
else:
self.precond = self.gamma * self.R + (1e-14) * self.M
# Discrete operator K:
self.K = self.gamma * self.R + self.beta * self.M
# Get eigenvalues for M:
self.eigsolM = SLEPcEigenSolver(self.M)
self.eigsolM.solve()
# Solver for M^{-1}:
self.solverM = LUSolver()
self.solverM.parameters['reuse_factorization'] = True
self.solverM.parameters['symmetric'] = True
self.solverM.set_operator(self.M)
# Solver for K^{-1}:
self.solverK = LUSolver()
self.solverK.parameters['reuse_factorization'] = True
self.solverK.parameters['symmetric'] = True
self.solverK.set_operator(self.K)
def _set_fluid_default_parameters(self):
fluid = df.Parameters(Physics.FLUID.value)
fluid.add("dt", 1e-3)
fluid.add("dummy_parameter", False)
# Add default parameters from both LU and Krylov solvers
fluid.add(LUSolver.default_parameters())
fluid.add(PETScKrylovSolver.default_parameters())
# Add solver parameters
fluid.add(df.Parameters("Solver"))
fluid["Solver"].add("dummy_parameter", False)
self.add(fluid)
def _set_porous_default_parameters(self):
porous = df.Parameters(Physics.POROUS.value)
porous.add("dt", 1e-3)
porous.add("dummy_parameter", False)
# Add default parameters from both LU and Krylov solvers
porous.add(LUSolver.default_parameters())
porous.add(PETScKrylovSolver.default_parameters())
# Add solver parameters
porous.add(df.Parameters("Solver"))
porous["Solver"].add("dummy_parameter", False)
self.add(porous)
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']
if self.Parameters.has_key('m0'):
self.m0 = self.Parameters['m0'].copy(deepcopy=True)
isFunction(self.m0)
else: self.m0 = Function(self.Vm)
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 = PETScMatrix()
assemble(inner(self.mtrial, self.mtest)*dx, tensor=self.M)
# preconditioner is Gamma^{-1}:
if self.beta > 1e-16: self.precond = self.gamma*self.R + self.beta*self.M
else: self.precond = self.gamma*self.R + (1e-14)*self.M
# Discrete operator K:
self.K = self.gamma*self.R + self.beta*self.M
# Get eigenvalues for M:
self.eigsolM = SLEPcEigenSolver(self.M)
self.eigsolM.solve()
# Solver for M^{-1}:
self.solverM = LUSolver()
self.solverM.parameters['reuse_factorization'] = True
self.solverM.parameters['symmetric'] = True
self.solverM.set_operator(self.M)
# Solver for K^{-1}:
self.solverK = LUSolver()
self.solverK.parameters['reuse_factorization'] = True
self.solverK.parameters['symmetric'] = True
self.solverK.set_operator(self.K)
def setUp(self):
mesh = UnitSquareMesh(5, 5, 'crossed')
self.V = FunctionSpace(mesh, 'Lagrange', 5)
self.u = Function(self.V)
self.uM = Function(self.V)
self.uMdiag = Function(self.V)
test = TestFunction(self.V)
trial = TrialFunction(self.V)
m = test*trial*dx
self.M = assemble(m)
self.solver = LUSolver()
self.solver.parameters['reuse_factorization'] = True
self.solver.parameters['symmetric'] = True
self.solver.set_operator(self.M)
self.ones = np.ones(self.V.dim())
def _set_solid_default_parameters(self):
solid = df.Parameters(Physics.SOLID.value)
solid.add("dt", 1e-3)
solid.add("dummy_parameter", False)
# Add boundary condtion parameters
solid.add(df.Parameters("BoundaryConditions"))
solid["BoundaryConditions"].add("base_bc", "fixed")
solid["BoundaryConditions"].add("lv_pressure", 10.0)
solid["BoundaryConditions"].add("rv_pressure", 0.0)
solid["BoundaryConditions"].add("pericardium_spring", 0.0)
solid["BoundaryConditions"].add("base_spring", 0.0)
# Add default parameters from both LU and Krylov solvers
solid.add(NonlinearVariationalSolver.default_parameters())
solid.add(LUSolver.default_parameters())
solid.add(PETScKrylovSolver.default_parameters())
# Add solver parameters
solid.add(df.Parameters("Solver"))
solid["Solver"].add("dummy_parameter", False)
self.add(solid)
def solve_alpha_M_beta_F(self, alpha, beta, b, t):
'''Solve alpha * M * u + beta * F(u, t) = b for u.
'''
A = alpha * self.M + beta * self.A
# See above for float conversion
right_hand_side = - float(beta) * self.b.copy()
if b:
right_hand_side += b
for bc in self.bcs:
bc.apply(A, b)
# The Krylov solver doesn't converge
solver = LUSolver()
solver.set_operator(A)
u = Function(self.V)
solver.solve(u.vector(), b)
return u
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.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)
# Define weak forms to assemble A, C and E
self._wkforma()
self._wkformc()
self._wkforme()
# 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
# 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
# Counters, tolerances and others
self.nbPDEsolves = 0 # Updated when solve_A called
self.nbfwdsolves = 0 # Counter for plots
self.nbadjsolves = 0 # Counter for plots
self._set_plots(plot)
# MPI:
self.mycomm = mycomm
try:
self.myrank = MPI.rank(self.mycomm)
except:
self.myrank = 0
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):
# Solve for uhat
C.transpmult(mhat, self.rhs.vector())
self.bcadj.apply(self.rhs.vector())
self.solve_A(self.u.vector(), -self.rhs.vector())
# Solve for phat
E.transpmult(mhat, self.rhs.vector())
Etmhat = self.rhs.vector().array()
self.rhs.vector().axpy(1.0, self.ObsOp.incradj(self.u))
self.bcadj.apply(self.rhs.vector())
self.solve_A(self.p.vector(), -self.rhs.vector())
# Compute Hessian*x:
y.axpy(1.0/N, C * self.p.vector())
y.axpy(self.GN/N, E * self.u.vector())
y.axpy(1.0, 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.MG.vector()
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):
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 self.ObsOp.noise: self.noise = 0.0
if self.plot:
self.plotu = PlotFenics(self.plotoutdir)
self.plotu.set_varname('u{0}'.format(self.nbfwdsolves))
if cost: self.misfit = 0.0
for ii, rhs in enumerate(self.RHS):
self.solve_A(self.u.vector(), rhs)
if self.plot: self.plotu.plot_vtk(self.u, ii)
u_obs, noiselevel = self.ObsOp.obs(self.u)
self.U.append(u_obs)
if self.ObsOp.noise: self.noise += noiselevel
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.regul
if self.ObsOp.noise and self.myrank == 0:
print 'Total noise in data misfit={:.5e}\n'.\
format(self.noise*.5/len(self.U))
self.ObsOp.noise = False # Safety
if self.plot: self.plotu.gather_vtkplots()
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
if self.plot:
self.plotp = PlotFenics(self.plotoutdir)
self.plotp.set_varname('p{0}'.format(self.nbadjsolves))
self.Nbsrc = len(self.UD)
if grad: self.MG.vector()[:] = np.zeros(self.lenm)
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())
if self.plot: self.plotp.plot_vtk(self.p, ii)
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(1.0, 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()))
if self.plot: self.plotp.gather_vtkplots()
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)
self.bc.apply(self.A)
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)
self.bc.apply(b)
self.RHS.append(b)
else: raise WrongInstanceError("rhs should be Expression")
def _assemble_solverM(self, Vm):
self.MM = assemble(inner(self.mtrial, self.mtest)*dx)
self.solverM = LUSolver()
self.solverM.parameters['reuse_factorization'] = True
self.solverM.parameters['symmetric'] = True
self.solverM.set_operator(self.MM)
def _set_plots(self, plot):
self.plot = plot
if self.plot:
filename, ext = splitext(sys.argv[0])
self.plotoutdir = filename + '/Plots/'
self.plotvarm = PlotFenics(self.plotoutdir)
self.plotvarm.set_varname('m')
def plotm(self, index):
if self.plot: self.plotvarm.plot_vtk(self.m, index)
def gatherm(self):
if self.plot: self.plotvarm.gather_vtkplots()
# 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.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 = []
class BilaplacianPrior(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']
if self.Parameters.has_key('m0'):
self.m0 = self.Parameters['m0'].copy(deepcopy=True)
isFunction(self.m0)
else:
self.m0 = Function(self.Vm)
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 = PETScMatrix()
assemble(inner(self.mtrial, self.mtest) * dx, tensor=self.M)
# preconditioner is Gamma^{-1}:
if self.beta > 1e-16:
self.precond = self.gamma * self.R + self.beta * self.M
else:
self.precond = self.gamma * self.R + (1e-14) * self.M
# Discrete operator K:
self.K = self.gamma * self.R + self.beta * self.M
# Get eigenvalues for M:
self.eigsolM = SLEPcEigenSolver(self.M)
self.eigsolM.solve()
# Solver for M^{-1}:
self.solverM = LUSolver()
self.solverM.parameters['reuse_factorization'] = True
self.solverM.parameters['symmetric'] = True
self.solverM.set_operator(self.M)
# Solver for K^{-1}:
self.solverK = LUSolver()
self.solverK.parameters['reuse_factorization'] = True
self.solverK.parameters['symmetric'] = True
self.solverK.set_operator(self.K)
def Minvpriordot(self, vect):
"""Here M.Gamma^{-1} = K M^{-1} K"""
mhat = Function(self.Vm)
self.solverM.solve(mhat.vector(), self.K * vect)
return self.K * mhat.vector()
def apply_sqrtM(self, vect):
"""Compute M^{1/2}.vect from Vector() vect"""
sqrtMv = Function(self.Vm)
for ii in range(self.Vm.dim()):
r, c, rx, cx = self.eigsolM.get_eigenpair(ii)
RX = Vector(rx)
sqrtMv.vector().axpy(
np.sqrt(r) * np.dot(rx.array(), vect.array()), RX)
return sqrtMv.vector()
def Ldot(self, vect):
"""Here L = K^{-1} M^{1/2}"""
Lb = Function(self.Vm)
self.solverK.solve(Lb.vector(), self.apply_sqrtM(vect))
return Lb.vector()
class ObjectiveAcoustic(LinearOperator):
"""
Computes data misfit, gradient and Hessian evaluation for the seismic
inverse problem using acoustic wave data
"""
#TODO: add support for multiple sources
# CONSTRUCTORS:
def __init__(self, acousticwavePDE, regularization=None):
"""
Input:
acousticwavePDE should be an instantiation from class AcousticWave
"""
self.PDE = acousticwavePDE
self.PDE.exact = None
self.fwdsource = self.PDE.ftime
self.MG = Function(self.PDE.Vl)
self.MGv = self.MG.vector()
self.Grad = Function(self.PDE.Vl)
self.Gradv = self.Grad.vector()
self.srchdir = Function(self.PDE.Vl)
self.delta_m = Function(self.PDE.Vl)
LinearOperator.__init__(self, self.MG.vector(), self.MG.vector())
self.obsop = None # Observation operator
self.dd = None # observations
if regularization == None: self.regularization = ZeroRegularization()
else: self.regularization = regularization
self.alpha_reg = 1.0
# gradient
self.lamtest, self.lamtrial = TestFunction(self.PDE.Vl), TrialFunction(self.PDE.Vl)
self.p, self.v = Function(self.PDE.V), Function(self.PDE.V)
self.wkformgrad = inner(self.lamtest*nabla_grad(self.p), nabla_grad(self.v))*dx
# incremental rhs
self.lamhat = Function(self.PDE.Vl)
self.ptrial, self.ptest = TrialFunction(self.PDE.V), TestFunction(self.PDE.V)
self.wkformrhsincr = inner(self.lamhat*nabla_grad(self.ptrial), nabla_grad(self.ptest))*dx
# Hessian
self.phat, self.vhat = Function(self.PDE.V), Function(self.PDE.V)
self.wkformhess = inner(self.lamtest*nabla_grad(self.phat), nabla_grad(self.v))*dx \
+ inner(self.lamtest*nabla_grad(self.p), nabla_grad(self.vhat))*dx
# Mass matrix:
weak_m = inner(self.lamtrial,self.lamtest)*dx
Mass = assemble(weak_m)
self.solverM = LUSolver()
self.solverM.parameters['reuse_factorization'] = True
self.solverM.parameters['symmetric'] = True
self.solverM.set_operator(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.abc:
#TODO: should probably be tested in other situations
if self.PDE.lumpD:
print '*** Warning: Damping matrix D is lumped. ',\
'Make sure gradient is consistent.'
self.vD, self.pD, self.p1D, self.p2D = Function(self.PDE.V), \
Function(self.PDE.V), Function(self.PDE.V), Function(self.PDE.V)
self.wkformgradD = inner(0.5*sqrt(self.PDE.rho/self.PDE.lam)\
*self.pD, self.vD*self.lamtest)*self.PDE.ds(1)
self.wkformDprime = inner(0.5*sqrt(self.PDE.rho/self.PDE.lam)\
*self.lamhat*self.ptrial, self.ptest)*self.PDE.ds(1)
self.dp, self.dph, self.vhatD = Function(self.PDE.V), \
Function(self.PDE.V), Function(self.PDE.V)
self.p1hatD, self.p2hatD = Function(self.PDE.V), Function(self.PDE.V)
self.wkformhessD = inner(-0.25*sqrt(self.PDE.rho)/(self.PDE.lam*sqrt(self.PDE.lam))\
*self.lamhat*self.dp, self.vD*self.lamtest)*self.PDE.ds(1) \
+ inner(0.5*sqrt(self.PDE.rho/self.PDE.lam)\
*self.dph, self.vD*self.lamtest)*self.PDE.ds(1)\
+ inner(0.5*sqrt(self.PDE.rho/self.PDE.lam)\
*self.dp, self.vhatD*self.lamtest)*self.PDE.ds(1)
def copy(self):
"""(hard) copy constructor"""
newobj = self.__class__(self.PDE.copy())
setfct(newobj.MG, self.MG)
setfct(newobj.srchdir, self.srchdir)
newobj.obsop = self.obsop
return newobj
# FORWARD PROBLEM + COST:
def solvefwd(self, cost=False):
self.PDE.set_fwd()
self.PDE.ftime = self.fwdsource
self.solfwd,_ = self.PDE.solve()
# observations:
self.Bp = np.zeros((len(self.obsop.PtwiseObs.Points),len(self.solfwd)))
for index, sol in enumerate(self.solfwd):
setfct(self.p, sol[0])
self.Bp[:,index] = self.obsop.obs(self.p)
if cost:
assert not self.dd == None, "Provide observations"
self.misfit = self.obsop.costfct(self.Bp, self.dd, self.PDE.times)
self.cost_reg = self.regularization.cost(self.PDE.lam)
self.cost = self.misfit + self.alpha_reg*self.cost_reg
def solvefwd_cost(self): self.solvefwd(True)
# ADJOINT PROBLEM + GRAD:
#@profile
def solveadj(self, grad=False):
self.PDE.set_adj()
self.obsop.assemble_rhsadj(self.Bp, self.dd, self.PDE.times, self.PDE.bc)
self.PDE.ftime = self.obsop.ftimeadj
self.soladj,_ = self.PDE.solve()
if grad:
self.MGv.zero()
if self.PDE.abc:
self.vD.vector().zero(); self.pD.vector().zero();
self.p1D.vector().zero(); self.p2D.vector().zero();
index = 0
for fwd, adj, fact in \
zip(self.solfwd, reversed(self.soladj), self.factors):
ttf, tta = fwd[1], adj[1]
assert isequal(ttf, tta, 1e-16), \
'tfwd={}, tadj={}, reldiff={}'.format(ttf, tta, abs(ttf-tta)/ttf)
setfct(self.p, fwd[0])
setfct(self.v, adj[0])
self.MGv.axpy(fact, assemble(self.wkformgrad))
# self.MGv.axpy(fact, assemble(self.wkformgrad, \
# form_compiler_parameters={'optimize':True,\
# 'representation':'quadrature'}))
if self.PDE.abc:
if index%2 == 0:
self.p2D.vector().axpy(1.0, self.p.vector())
setfct(self.pD, self.p2D)
#self.MGv.axpy(1.0*0.5*self.invDt, assemble(self.wkformgradD))
self.MGv.axpy(fact*0.5*self.invDt, assemble(self.wkformgradD))
setfct(self.p2D, -1.0*self.p.vector())
setfct(self.vD, self.v)
else:
self.p1D.vector().axpy(1.0, self.p.vector())
setfct(self.pD, self.p1D)
#self.MGv.axpy(1.0*0.5*self.invDt, assemble(self.wkformgradD))
self.MGv.axpy(fact*0.5*self.invDt, assemble(self.wkformgradD))
setfct(self.p1D, -1.0*self.p.vector())
setfct(self.vD, self.v)
index += 1
self.MGv.axpy(self.alpha_reg, self.regularization.grad(self.PDE.lam))
self.solverM.solve(self.Gradv, self.MGv)
def solveadj_constructgrad(self): self.solveadj(True)
# HESSIAN:
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)
# lamhat * grad(p).grad(vtilde)
assert isequal(tt, self.solfwd[index][1], 1e-16)
setfct(self.p, self.solfwd[index][0])
setfct(self.v, self.C*self.p.vector())
# D'.dot(p)
if self.PDE.abc and index > 0:
setfct(self.p, \
self.solfwd[index+1][0] - self.solfwd[index-1][0])
self.v.vector().axpy(.5*self.invDt, self.Dp*self.p.vector())
return -1.0*self.v.vector().array()
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)
# lamhat * grad(ptilde).grad(v)
assert isequal(tt, self.soladj[indexa][1], 1e-16)
setfct(self.v, self.soladj[indexa][0])
setfct(self.vhat, self.C*self.v.vector())
# B* B phat
assert isequal(tt, self.solincrfwd[indexf][1], 1e-16)
setfct(self.phat, self.solincrfwd[indexf][0])
self.vhat.vector().axpy(1.0, self.obsop.incradj(self.phat, tt))
# D'.dot(v)
if self.PDE.abc and indexa > 0:
setfct(self.v, \
self.soladj[indexa-1][0] - self.soladj[indexa+1][0])
self.vhat.vector().axpy(-.5*self.invDt, self.Dp*self.v.vector())
return -1.0*self.vhat.vector().array()
def mult(self, lamhat, y):
"""
mult(self, lamhat, y): return y = Hessian * lamhat
inputs:
y, lamhat = Function(V).vector()
"""
self.regularization.assemble_hessian(lamhat)
setfct(self.lamhat, lamhat)
self.C = assemble(self.wkformrhsincr)
if self.PDE.abc: self.Dp = assemble(self.wkformDprime)
# solve for phat
self.PDE.set_fwd()
self.PDE.ftime = self.ftimeincrfwd
self.solincrfwd,_ = self.PDE.solve()
# solve for vhat
self.PDE.set_adj()
self.PDE.ftime = self.ftimeincradj
self.solincradj,_ = self.PDE.solve()
# Compute Hessian*lamhat
y.zero()
index = 0
if self.PDE.abc:
self.vD.vector().zero(); self.vhatD.vector().zero();
self.p1D.vector().zero(); self.p2D.vector().zero();
self.p1hatD.vector().zero(); self.p2hatD.vector().zero();
for fwd, adj, incrfwd, incradj, fact in \
zip(self.solfwd, reversed(self.soladj), \
self.solincrfwd, reversed(self.solincradj), self.factors):
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.p, fwd[0])
setfct(self.v, adj[0])
setfct(self.phat, incrfwd[0])
setfct(self.vhat, incradj[0])
y.axpy(fact, assemble(self.wkformhess))
if self.PDE.abc:
if index%2 == 0:
self.p2D.vector().axpy(1.0, self.p.vector())
self.p2hatD.vector().axpy(1.0, self.phat.vector())
setfct(self.dp, self.p2D)
setfct(self.dph, self.p2hatD)
y.axpy(1.0*0.5*self.invDt, assemble(self.wkformhessD))
setfct(self.p2D, -1.0*self.p.vector())
setfct(self.p2hatD, -1.0*self.phat.vector())
else:
self.p1D.vector().axpy(1.0, self.p.vector())
self.p1hatD.vector().axpy(1.0, self.phat.vector())
setfct(self.dp, self.p1D)
setfct(self.dph, self.p1hatD)
y.axpy(1.0*0.5*self.invDt, assemble(self.wkformhessD))
setfct(self.p1D, -1.0*self.p.vector())
setfct(self.p1hatD, -1.0*self.phat.vector())
setfct(self.vD, self.v)
setfct(self.vhatD, self.vhat)
index += 1
# add regularization term
y.axpy(self.alpha_reg, self.regularization.hessian(lamhat))
# SETTERS + UPDATE:
def update_PDE(self, parameters): self.PDE.update(parameters)
def update_m(self, lam): self.update_PDE({'lambda':lam})
def set_abc(self, mesh, class_bc_abc, lumpD):
self.PDE.set_abc(mesh, class_bc_abc, lumpD)
def backup_m(self): self.lam_bkup = self.getmarray()
def restore_m(self): self.update_m(self.lam_bkup)
def setsrcterm(self, ftime): self.PDE.ftime = ftime
# GETTERS:
def getmcopyarray(self): return self.lam_bkup
def getmarray(self): return self.PDE.lam.vector().array()
def getMGarray(self): return self.MGv.array()
def __init__(self, acousticwavePDE, regularization=None):
"""
Input:
acousticwavePDE should be an instantiation from class AcousticWave
"""
self.PDE = acousticwavePDE
self.PDE.exact = None
self.fwdsource = self.PDE.ftime
self.MG = Function(self.PDE.Vl)
self.MGv = self.MG.vector()
self.Grad = Function(self.PDE.Vl)
self.Gradv = self.Grad.vector()
self.srchdir = Function(self.PDE.Vl)
self.delta_m = Function(self.PDE.Vl)
LinearOperator.__init__(self, self.MG.vector(), self.MG.vector())
self.obsop = None # Observation operator
self.dd = None # observations
if regularization == None: self.regularization = ZeroRegularization()
else: self.regularization = regularization
self.alpha_reg = 1.0
# gradient
self.lamtest, self.lamtrial = TestFunction(self.PDE.Vl), TrialFunction(self.PDE.Vl)
self.p, self.v = Function(self.PDE.V), Function(self.PDE.V)
self.wkformgrad = inner(self.lamtest*nabla_grad(self.p), nabla_grad(self.v))*dx
# incremental rhs
self.lamhat = Function(self.PDE.Vl)
self.ptrial, self.ptest = TrialFunction(self.PDE.V), TestFunction(self.PDE.V)
self.wkformrhsincr = inner(self.lamhat*nabla_grad(self.ptrial), nabla_grad(self.ptest))*dx
# Hessian
self.phat, self.vhat = Function(self.PDE.V), Function(self.PDE.V)
self.wkformhess = inner(self.lamtest*nabla_grad(self.phat), nabla_grad(self.v))*dx \
+ inner(self.lamtest*nabla_grad(self.p), nabla_grad(self.vhat))*dx
# Mass matrix:
weak_m = inner(self.lamtrial,self.lamtest)*dx
Mass = assemble(weak_m)
self.solverM = LUSolver()
self.solverM.parameters['reuse_factorization'] = True
self.solverM.parameters['symmetric'] = True
self.solverM.set_operator(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.abc:
#TODO: should probably be tested in other situations
if self.PDE.lumpD:
print '*** Warning: Damping matrix D is lumped. ',\
'Make sure gradient is consistent.'
self.vD, self.pD, self.p1D, self.p2D = Function(self.PDE.V), \
Function(self.PDE.V), Function(self.PDE.V), Function(self.PDE.V)
self.wkformgradD = inner(0.5*sqrt(self.PDE.rho/self.PDE.lam)\
*self.pD, self.vD*self.lamtest)*self.PDE.ds(1)
self.wkformDprime = inner(0.5*sqrt(self.PDE.rho/self.PDE.lam)\
*self.lamhat*self.ptrial, self.ptest)*self.PDE.ds(1)
self.dp, self.dph, self.vhatD = Function(self.PDE.V), \
Function(self.PDE.V), Function(self.PDE.V)
self.p1hatD, self.p2hatD = Function(self.PDE.V), Function(self.PDE.V)
self.wkformhessD = inner(-0.25*sqrt(self.PDE.rho)/(self.PDE.lam*sqrt(self.PDE.lam))\
*self.lamhat*self.dp, self.vD*self.lamtest)*self.PDE.ds(1) \
+ inner(0.5*sqrt(self.PDE.rho/self.PDE.lam)\
*self.dph, self.vD*self.lamtest)*self.PDE.ds(1)\
+ inner(0.5*sqrt(self.PDE.rho/self.PDE.lam)\
*self.dp, self.vhatD*self.lamtest)*self.PDE.ds(1)
def _assemble_solverM(self, Vm):
self.MM = assemble(inner(self.mtrial, self.mtest)*dx)
self.solverM = LUSolver()
self.solverM.parameters['reuse_factorization'] = True
self.solverM.parameters['symmetric'] = True
self.solverM.set_operator(self.MM)
def set_solver(self):
"""Reset solver for fwd operator"""
self.solver = LUSolver()
self.solver.parameters['reuse_factorization'] = True
self.solver.set_operator(self.A)
class BilaplacianPrior(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']
if self.Parameters.has_key('m0'):
self.m0 = self.Parameters['m0'].copy(deepcopy=True)
isFunction(self.m0)
else: self.m0 = Function(self.Vm)
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 = PETScMatrix()
assemble(inner(self.mtrial, self.mtest)*dx, tensor=self.M)
# preconditioner is Gamma^{-1}:
if self.beta > 1e-16: self.precond = self.gamma*self.R + self.beta*self.M
else: self.precond = self.gamma*self.R + (1e-14)*self.M
# Discrete operator K:
self.K = self.gamma*self.R + self.beta*self.M
# Get eigenvalues for M:
self.eigsolM = SLEPcEigenSolver(self.M)
self.eigsolM.solve()
# Solver for M^{-1}:
self.solverM = LUSolver()
self.solverM.parameters['reuse_factorization'] = True
self.solverM.parameters['symmetric'] = True
self.solverM.set_operator(self.M)
# Solver for K^{-1}:
self.solverK = LUSolver()
self.solverK.parameters['reuse_factorization'] = True
self.solverK.parameters['symmetric'] = True
self.solverK.set_operator(self.K)
def Minvpriordot(self, vect):
"""Here M.Gamma^{-1} = K M^{-1} K"""
mhat = Function(self.Vm)
self.solverM.solve(mhat.vector(), self.K*vect)
return self.K * mhat.vector()
def apply_sqrtM(self, vect):
"""Compute M^{1/2}.vect from Vector() vect"""
sqrtMv = Function(self.Vm)
for ii in range(self.Vm.dim()):
r, c, rx, cx = self.eigsolM.get_eigenpair(ii)
RX = Vector(rx)
sqrtMv.vector().axpy(np.sqrt(r)*np.dot(rx.array(), vect.array()), RX)
return sqrtMv.vector()
def Ldot(self, vect):
"""Here L = K^{-1} M^{1/2}"""
Lb = Function(self.Vm)
self.solverK.solve(Lb.vector(), self.apply_sqrtM(vect))
return Lb.vector()
def main(module_name, ncases, params, petsc_params):
'''
Run the test case in module with ncases. Optionally store results
in savedir. For some modules there are multiple (which) choices of
preconditioners.
'''
# Unpack
for k, v in params.items():
exec(k + '=v', locals())
RED = '\033[1;37;31m%s\033[0m'
print RED % ('\tRunning %s' % module_name)
module = __import__(module_name) # no importlib in python2.7
# Setup the MMS case
u_true, rhs_data = module.setup_mms(eps)
# Setup the convergence monitor
if log:
params = [('solver', solver), ('precond', str(precond)),
('eps', str(eps))]
path = '_'.join([module_name] + ['%s=%s' % pv for pv in params])
path = os.path.join(save_dir if save_dir else '.', path)
path = '.'.join([path, 'txt'])
else:
path = ''
memory, residuals = [], []
monitor = module.setup_error_monitor(u_true, memory, path=path)
# Sometimes it is usedful to transform the solution before computing
# the error. e.g. consider subdomains
if hasattr(module, 'setup_transform'):
# NOTE: transform take two args for case and the current computed
# solution
transform = module.setup_transform
else:
transform = lambda i, x: x
print '=' * 79
print '\t\t\tProblem eps = %g' % eps
print '=' * 79
for i in ncases:
a, L, W = module.setup_problem(i, rhs_data, eps=eps)
# Assemble blocks
t = Timer('assembly')
t.start()
AA, bb = map(ii_assemble, (a, L))
print '\tAssembled blocks in %g s' % t.stop()
wh = ii_Function(W)
if solver == 'direct':
# Turn into a (monolithic) PETScMatrix/Vector
t = Timer('conversion')
t.start()
AAm, bbm = map(ii_convert, (AA, bb))
print '\tConversion to PETScMatrix/Vector took %g s' % t.stop()
t = Timer('solve')
t.start()
LUSolver('umfpack').solve(AAm, wh.vector(), bbm)
print '\tSolver took %g s' % t.stop()
niters = 1
else:
# Here we define a Krylov solver using PETSc
BB = module.setup_preconditioner(W, precond, eps=eps)
## AA and BB as block_mat
ksp = PETSc.KSP().create()
# Default is minres
if '-ksp_type' not in petsc_params:
petsc_params['-ksp_type'] = 'minres'
opts = PETSc.Options()
for key, value in petsc_params.iteritems():
opts.setValue(key, None if value == 'none' else value)
ksp.setOperators(ii_PETScOperator(AA))
ksp.setNormType(PETSc.KSP.NormType.NORM_PRECONDITIONED)
# ksp.setTolerances(rtol=1E-6, atol=None, divtol=None, max_it=300)
ksp.setConvergenceHistory()
# We attach the wrapped preconditioner defined by the module
ksp.setPC(ii_PETScPreconditioner(BB, ksp))
ksp.setFromOptions()
print ksp.getTolerances()
# Want the iterations to start from random
wh.block_vec().randomize()
# Solve, note the past object must be PETSc.Vec
t = Timer('solve')
t.start()
ksp.solve(as_petsc_nest(bb), wh.petsc_vec())
print '\tSolver took %g s' % t.stop()
niters = ksp.getIterationNumber()
residuals.append(ksp.getConvergenceHistory())
# Let's check the final size of the residual
r_norm = (bb - AA * wh.block_vec()).norm()
# Convergence?
monitor.send((transform(i, wh), niters, r_norm))
# Only send the final
if save_dir:
path = os.path.join(save_dir, module_name)
for i, wh_i in enumerate(wh):
# Renaming to make it easier to save state in Visit/Pareview
wh_i.rename('u', str(i))
File('%s_%d.pvd' % (path, i)) << wh_i
# Plot relative residual norm
if plot:
plt.figure()
[
plt.semilogy(res / res[0], label=str(i))
for i, res in enumerate(residuals, 1)
]
plt.legend(loc='best')
plt.show()
class TestLumpedMass(unittest.TestCase):
def setUp(self):
mesh = UnitSquareMesh(5, 5, 'crossed')
self.V = FunctionSpace(mesh, 'Lagrange', 5)
self.u = Function(self.V)
self.uM = Function(self.V)
self.uMdiag = Function(self.V)
test = TestFunction(self.V)
trial = TrialFunction(self.V)
m = test*trial*dx
self.M = assemble(m)
self.solver = LUSolver()
self.solver.parameters['reuse_factorization'] = True
self.solver.parameters['symmetric'] = True
self.solver.set_operator(self.M)
self.ones = np.ones(self.V.dim())
def test00(self):
""" Create a lumped solver """
myobj = LumpedMatrixSolver(self.V)
def test01_set(self):
""" Set operator """
myobj = LumpedMatrixSolver(self.V)
myobj.set_operator(self.M)
def test01_entries(self):
""" Lump matrix """
myobj = LumpedMatrixSolver(self.V)
myobj.set_operator(self.M)
err = 0.0
for index, ii in enumerate(self.M.array()):
err += abs(ii.sum() - myobj.Mdiag[index])
self.assertTrue(err < index*1e-16)
def test02(self):
""" Invert lumped matrix """
myobj = LumpedMatrixSolver(self.V)
myobj.set_operator(self.M)
err = 0.0
for ii in range(len(myobj.Mdiag.array())):
err += abs(1./myobj.Mdiag[ii] - myobj.invMdiag[ii])
self.assertTrue(err < ii*1e-16)
def test03_basic(self):
""" solve """
myobj = LumpedMatrixSolver(self.V)
myobj.set_operator(self.M)
myobj.solve(self.uMdiag.vector(), myobj.Mdiag)
diff = (myobj.one - self.uMdiag.vector()).array()
self.assertTrue(np.linalg.norm(diff)/np.linalg.norm(myobj.one.array()) < 1e-14)
def test04_mult(self):
""" overloaded * operator """
myobj = LumpedMatrixSolver(self.V)
myobj.set_operator(self.M)
self.uMdiag.vector().axpy(1.0, myobj*myobj.one)
diff = (myobj.Mdiag - self.uMdiag.vector()).array()
self.assertTrue(np.linalg.norm(diff)/np.linalg.norm(myobj.Mdiag.array()) < 1e-14)
def test10(self):
""" Create a lumped solver """
myobj = LumpedMatrixSolverS(self.V)
def test11_set(self):
""" Set operator """
myobj = LumpedMatrixSolverS(self.V)
myobj.set_operator(self.M)
def test11_entries(self):
""" Lump matrix """
myobj = LumpedMatrixSolverS(self.V)
myobj.set_operator(self.M)
Msum = np.dot(self.ones, self.M.array().dot(self.ones))
err = abs(myobj.Mdiag.array().dot(self.ones) - \
Msum) / Msum
self.assertTrue(err < 1e-14, err)
def test12(self):
""" Invert lumped matrix """
myobj = LumpedMatrixSolverS(self.V)
myobj.set_operator(self.M)
err = 0.0
for ii in range(len(myobj.Mdiag.array())):
err += abs(1./myobj.Mdiag[ii] - myobj.invMdiag[ii])
self.assertTrue(err < ii*1e-16)
def test13_basic(self):
""" solve """
myobj = LumpedMatrixSolverS(self.V)
myobj.set_operator(self.M)
myobj.solve(self.uMdiag.vector(), myobj.Mdiag)
diff = myobj.one.array() - self.uMdiag.vector().array()
self.assertTrue(np.linalg.norm(diff)/np.linalg.norm(myobj.one.array()) < 1e-14)
def test14_mult(self):
""" overloaded * operator """
myobj = LumpedMatrixSolverS(self.V)
myobj.set_operator(self.M)
self.uMdiag.vector().axpy(1.0, myobj*myobj.one)
diff = (myobj.Mdiag - self.uMdiag.vector()).array()
self.assertTrue(np.linalg.norm(diff)/np.linalg.norm(myobj.Mdiag.array()) < 1e-14)
class TestLumpedMass(unittest.TestCase):
def setUp(self):
mesh = UnitSquareMesh(5, 5, 'crossed')
self.V = FunctionSpace(mesh, 'Lagrange', 5)
self.u = Function(self.V)
self.uM = Function(self.V)
self.uMdiag = Function(self.V)
test = TestFunction(self.V)
trial = TrialFunction(self.V)
m = test * trial * dx
self.M = assemble(m)
self.solver = LUSolver()
self.solver.parameters['reuse_factorization'] = True
self.solver.parameters['symmetric'] = True
self.solver.set_operator(self.M)
self.ones = np.ones(self.V.dim())
def test00(self):
""" Create a lumped solver """
myobj = LumpedMatrixSolver(self.V)
def test01_set(self):
""" Set operator """
myobj = LumpedMatrixSolver(self.V)
myobj.set_operator(self.M)
def test01_entries(self):
""" Lump matrix """
myobj = LumpedMatrixSolver(self.V)
myobj.set_operator(self.M)
err = 0.0
for index, ii in enumerate(self.M.array()):
err += abs(ii.sum() - myobj.Mdiag[index])
self.assertTrue(err < index * 1e-16)
def test02(self):
""" Invert lumped matrix """
myobj = LumpedMatrixSolver(self.V)
myobj.set_operator(self.M)
err = 0.0
for ii in range(len(myobj.Mdiag.array())):
err += abs(1. / myobj.Mdiag[ii] - myobj.invMdiag[ii])
self.assertTrue(err < ii * 1e-16)
def test03_basic(self):
""" solve """
myobj = LumpedMatrixSolver(self.V)
myobj.set_operator(self.M)
myobj.solve(self.uMdiag.vector(), myobj.Mdiag)
diff = (myobj.one - self.uMdiag.vector()).array()
self.assertTrue(
np.linalg.norm(diff) / np.linalg.norm(myobj.one.array()) < 1e-14)
def test04_mult(self):
""" overloaded * operator """
myobj = LumpedMatrixSolver(self.V)
myobj.set_operator(self.M)
self.uMdiag.vector().axpy(1.0, myobj * myobj.one)
diff = (myobj.Mdiag - self.uMdiag.vector()).array()
self.assertTrue(
np.linalg.norm(diff) / np.linalg.norm(myobj.Mdiag.array()) < 1e-14)
def test10(self):
""" Create a lumped solver """
myobj = LumpedMatrixSolverS(self.V)
def test11_set(self):
""" Set operator """
myobj = LumpedMatrixSolverS(self.V)
myobj.set_operator(self.M)
def test11_entries(self):
""" Lump matrix """
myobj = LumpedMatrixSolverS(self.V)
myobj.set_operator(self.M)
Msum = np.dot(self.ones, self.M.array().dot(self.ones))
err = abs(myobj.Mdiag.array().dot(self.ones) - \
Msum) / Msum
self.assertTrue(err < 1e-14, err)
def test12(self):
""" Invert lumped matrix """
myobj = LumpedMatrixSolverS(self.V)
myobj.set_operator(self.M)
err = 0.0
for ii in range(len(myobj.Mdiag.array())):
err += abs(1. / myobj.Mdiag[ii] - myobj.invMdiag[ii])
self.assertTrue(err < ii * 1e-16)
def test13_basic(self):
""" solve """
myobj = LumpedMatrixSolverS(self.V)
myobj.set_operator(self.M)
myobj.solve(self.uMdiag.vector(), myobj.Mdiag)
diff = myobj.one.array() - self.uMdiag.vector().array()
self.assertTrue(
np.linalg.norm(diff) / np.linalg.norm(myobj.one.array()) < 1e-14)
def test14_mult(self):
""" overloaded * operator """
myobj = LumpedMatrixSolverS(self.V)
myobj.set_operator(self.M)
self.uMdiag.vector().axpy(1.0, myobj * myobj.one)
diff = (myobj.Mdiag - self.uMdiag.vector()).array()
self.assertTrue(
np.linalg.norm(diff) / np.linalg.norm(myobj.Mdiag.array()) < 1e-14)
def solve_alpha_M_beta_F(self, alpha, beta, b, t):
"""Solve :code:`alpha * M * u + beta * F(u, t) = b` with Dirichlet
conditions.
"""
matrix = alpha * self.M + beta * self.A
# See above for float conversion
right_hand_side = -float(beta) * self.b.copy()
if b:
right_hand_side += b
for bc in self.dirichlet_bcs:
bc.apply(matrix, right_hand_side)
# TODO proper preconditioner for convection
if self.convection:
# Use HYPRE-Euclid instead of ILU for parallel computation.
# However, this PC sometimes fails.
# solver = KrylovSolver('gmres', 'hypre_euclid')
# Fallback:
solver = LUSolver()
else:
solver = KrylovSolver("gmres", "hypre_amg")
solver.parameters["relative_tolerance"] = 1.0e-13
solver.parameters["absolute_tolerance"] = 0.0
solver.parameters["maximum_iterations"] = 100
solver.parameters["monitor_convergence"] = True
solver.set_operator(matrix)
u = Function(self.Q)
solver.solve(u.vector(), right_hand_side)
return u
