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 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 _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 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 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()
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()
