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
def linesearch(self):
""" Perform inexact backtracking line search """
regularization = self.parameters["regularization"]
# compute new direction for dual variables
if regularization == "TV" and self.Reg.isPD():
self.Reg.compute_dw(self.dg)
# line search for primal variable
self.alpha = self.parameters["alpha0"]
rho = self.parameters["rho"]
c = self.parameters["c"]
self.computecost()
costref = self.cost
cdJdf = ((self.MG).inner(self.dg.vector())) * c
self.LS = False
for ii in xrange(self.parameters["max_backtrack"]):
setfct(self.gtmp, self.g.vector() + self.dg.vector() * self.alpha)
if self.computecost(self.gtmp) < costref + self.alpha * cdJdf:
self.g.vector().axpy(self.alpha, self.dg.vector())
self.LS = True
break
else:
self.alpha *= rho
# update dual variable
if regularization == "TV" and self.Reg.isPD():
self.Reg.update_w(self.alpha)
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 hessianab(self, m1h, m2h):
""" m1h, m2h = Vector(V) """
setfct(self.m1h, m1h)
setfct(self.m2h, m2h)
assign(self.m12h.sub(0), self.m1h)
assign(self.m12h.sub(1), self.m2h)
return self.H * self.m12h.vector()
def iteration_centered(self, tt):
setfct(self.rhs, (self.Dt**2)*self.ftime(tt))
self.rhs.vector().axpy(-1.0, self.MminD*self.u0.vector())
self.rhs.vector().axpy(2.0, self.M*self.u1.vector())
self.rhs.vector().axpy(-self.Dt**2, self.K*self.u1.vector())
if not self.bc == None: self.bc.apply(self.rhs.vector())
self.solverMplD.solve(self.u2.vector(), self.rhs.vector())
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
def linesearch(self, MG):
""" Perform inexact backtracking line search """
alphaLS = self.parametersLS['alpha0']
rhoLS = self.parametersLS['rho']
cLS = self.parametersLS['c']
cost, misfit, reg = self.costmisfitreg()
costref = cost
setfct(self.ucopy, self.u)
MGdu = MG.inner(self.du.vector()) * cLS
success = False
for ii in xrange(self.parametersLS['max_backtrack']):
setfct(self.u, self.ucopy)
self.u.vector().axpy(alphaLS, self.du.vector())
cost, misfit, reg = self.costmisfitreg()
if cost < costref + alphaLS * MGdu:
success = True
break
else:
alphaLS *= rhoLS
if self.Regul.isPD(): self.Regul.update_w(self.du.vector(), alphaLS)
return success, alphaLS, cost, misfit, reg
def hessianab(self, m1h, m2h):
""" m1h, m2h = Vector(V) """
setfct(self.m1, m1h)
setfct(self.m2, m2h)
assign(self.m.sub(0), self.m1)
assign(self.m.sub(1), self.m2)
return self.regTV.hessian(self.m.vector())
def hessianab(self, ahat, bhat):
""" ahat, bhat = Vector(V) """
setfct(self.ahat, ahat)
setfct(self.bhat, bhat)
assign(self.abhat.sub(0), self.ahat)
assign(self.abhat.sub(1), self.bhat)
return self.H * self.abhat.vector()
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 iteration_backward(self, tt):
setfct(self.rhs, self.Dt*self.ftime(tt))
self.rhs.vector().axpy(-self.Dt, self.K*self.u1.vector())
self.rhs.vector().axpy(-1.0, self.D*(self.u1.vector()-self.u0.vector()))
if not self.bc == None: self.bc.apply(self.rhs.vector())
self.solverM.solve(self.sol.vector(), self.rhs.vector())
setfct(self.u2, 2.0*self.u1.vector())
self.u2.vector().axpy(-1.0, self.u0.vector())
self.u2.vector().axpy(self.Dt, self.sol.vector())
def plot_timeseries(self, timeseries, varname, index, skip, function):
""" Plot every 'skip' entry of entry 'index' of a 'timeseries'.
'function' = dl.Function """
self.set_varname(varname)
for ii, sol in enumerate(timeseries):
if ii % skip == 0:
setfct(function, sol[index])
self.plot_vtk(function, ii)
self.gather_vtkplots()
def plot_timeseries(self, timeseries, varname, index, skip, function):
""" Plot every 'skip' entry of entry 'index' of a 'timeseries'.
'function' = dl.Function """
self.set_varname(varname)
for ii, sol in enumerate(timeseries):
if ii%skip == 0:
setfct(function, sol[index])
self.plot_vtk(function, ii)
self.gather_vtkplots()
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 solveTV(self):
""" Solve image denoising pb """
self.u.vector().zero()
normu_true = self.mediummisfit()
# initial state = noisy image
setfct(self.u, self.u_0)
print 'min(u)={}, max(u)={}'.format(\
np.amin(self.u.vector().array()), np.amax(self.u.vector().array()))
cost, misfit, reg = self.costmisfitreg()
costold = cost
MG, normMG = self.gradient()
normMG0 = normMG
print '\t{:12s} {:12s} {:12s} {:12s} {:12s} {:12s} {:12s}\t{:12s} {:12s}'.format(\
'iter', 'cost', 'misfit', 'reg', '||G||', 'a_LS', 'medmisfit', 'tol_cg', 'n_cg')
print '{:12d} {:12.4e} {:12.4e} {:12.4e} {:12.4e} {:12s} {:12.2e} ({:7.2f} %)'.format(\
0, cost, misfit, reg, normMG, '', self.mediummisfit(), 100.*self.mediummisfit()/normu_true)
cgtol = self.parametersLS['cgtol']
maxiter = self.parametersLS['maxiter']
for ii in xrange(maxiter):
if self.inexact:
cgtol = min(cgtol, np.sqrt(normMG / normMG0))
#cgtol = min(0.5, np.sqrt(normMG/normMG0)))
else:
cgtol = 1e-12
cgiter, MGdu = self.searchdirection(MG, cgtol)
success, alphaLS, cost, misfit, reg = self.linesearch(MG)
MG, normMG = self.gradient()
print '{:12d} {:12.4e} {:12.4e} {:12.4e} {:12.4e} {:12.4e} {:12.2e} ({:7.2f} %) {:12.2e} {:12d}'.format(\
ii+1, cost, misfit, reg, normMG, alphaLS, \
self.mediummisfit(), 100.*self.mediummisfit()/normu_true, cgtol, cgiter)
if normMG < min(1e-12, 1e-10 * normMG0):
print 'gradient sufficiently reduced -- optimization converged'
break
if not success:
print 'Line search failed -- optimization aborted'
break
if np.abs(cost - costold) / costold < 1e-14:
if cgtol < 1e-10:
print 'cost functional stagnates -- optimization aborted'
break
else:
cgtol *= 1e-3
costold = cost
print 'min(u)={}, max(u)={}'.format(\
np.amin(self.u.vector().array()), np.amax(self.u.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)
# 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 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 solve(self):
""" General solver method """
if self.timestepper == 'backward':
def iterate(tt): self.iteration_backward(tt)
elif self.timestepper == 'centered':
def iterate(tt): self.iteration_centered(tt)
else:
print "Time stepper not implemented"
sys.exit(1)
if self.verbose: print 'Compute solution'
solout = [] # Store computed solution
# u0:
tt = self.get_tt(0)
if self.verbose: print 'Compute solution -- time {}'.format(tt)
setfct(self.u0, self.u0init)
solout.append([self.u0.vector().array(), tt])
# Compute u1:
if not self.u1init == None: self.u1 = self.u1init
else:
assert(not self.utinit == None)
setfct(self.rhs, self.ftime(tt))
self.rhs.vector().axpy(-self.fwdadj, self.D*self.utinit.vector())
self.rhs.vector().axpy(-1.0, self.K*self.u0.vector())
if not self.bc == None: self.bc.apply(self.rhs.vector())
self.solverM.solve(self.sol.vector(), self.rhs.vector())
setfct(self.u1, self.u0)
self.u1.vector().axpy(self.fwdadj*self.Dt, self.utinit.vector())
self.u1.vector().axpy(0.5*self.Dt**2, self.sol.vector())
tt = self.get_tt(1)
if self.verbose: print 'Compute solution -- time {}'.format(tt)
solout.append([self.u1.vector().array(), tt])
# Iteration
for nn in xrange(2, self.Nt+1):
iterate(tt)
# Advance to next time step
setfct(self.u0, self.u1)
setfct(self.u1, self.u2)
tt = self.get_tt(nn)
if self.verbose:
print 'Compute solution -- time {}, rhs {}'.\
format(tt, np.max(np.abs(self.ftime(tt))))
solout.append([self.u1.vector().array(),tt])
if self.fwdadj > 0.0:
assert isequal(tt, self.Tf, 1e-16), \
'tt={}, Tf={}, reldiff={}'.format(tt, self.Tf, abs(tt-self.Tf)/self.Tf)
else:
assert isequal(tt, self.t0, 1e-16), \
'tt={}, t0={}, reldiff={}'.format(tt, self.t0, abs(tt-self.t0))
return solout, self.computeerror()
def generatedata(self, noisepercent):
""" compute data and add noisepercent (%) of noise """
sigma = noisepercent * np.linalg.norm(self.f_true.vector().array()) / np.sqrt(self.dimV)
print "sigma_noise = ", sigma
np.random.seed(11) # TODO: tmp
eta = sigma * np.random.randn(self.dimV)
self.dn = dl.Function(self.V)
setfct(self.dn, eta)
self.dn.vector().axpy(1.0, self.f_true.vector())
print "min(true)={}, max(true)={}".format(
np.amin(self.f_true.vector().array()), np.amax(self.f_true.vector().array())
)
print "min(noisy)={}, max(noisy)={}".format(
np.amin(self.dn.vector().array()), np.amax(self.dn.vector().array())
)
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())
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 test_gradient(self, f=None, n=5):
""" test gradient with FD approx around point f """
if f == None:
f = self.f_true
pm = [1.0, -1.0]
eps = 1e-5
self.gradient(f)
for nn in xrange(1, n + 1):
expr = dl.Expression("sin(n*pi*x[0]/200)*sin(n*pi*x[1]/100)", n=nn)
df = dl.interpolate(expr, self.V)
MGdf = self.MG.inner(df.vector())
cost = []
for sign in pm:
setfct(self.g, f)
self.g.vector().axpy(sign * eps, df.vector())
cost.append(self.computecost(self.g))
MGFD = (cost[0] - cost[1]) / (2 * eps)
print "n={}:\tMGFD={:.5e}, MGdf={:.5e}, error={:.2e}".format(
nn, MGFD, MGdf, np.abs(MGdf - MGFD) / np.abs(MGdf)
)
def test_hessian(self, f=None, n=5):
""" test Hessian with FD approx around point f """
if f == None:
f = self.f_true
pm = [1.0, -1.0]
eps = 1e-5
self.Hessian(f)
for nn in xrange(1, n + 1):
expr = dl.Expression("sin(n*pi*x[0]/200)*sin(n*pi*x[1]/100)", n=nn)
df = dl.interpolate(expr, self.V)
Hdf = (self.Hess * df.vector()).array()
MG = []
for sign in pm:
setfct(self.g, f)
self.g.vector().axpy(sign * eps, df.vector())
self.gradient(self.g)
MG.append(self.MG.array())
HFD = (MG[0] - MG[1]) / (2 * eps)
print "n={}:\tHFD={:.5e}, Hdf={:.5e}, error={:.2e}".format(
nn, np.linalg.norm(HFD), np.linalg.norm(Hdf), np.linalg.norm(Hdf - HFD) / np.linalg.norm(Hdf)
)
def update_w(self, alpha):
""" update w and re-scale wH """
self.w.vector().axpy(alpha, self.dw.vector())
# project each w (coord-wise) onto unit sphere to get wH
(wx, wy) = self.w.split(deepcopy=True)
wxa, wya = wx.vector().array(), wy.vector().array()
normw = np.sqrt(wxa**2 + wya**2)
factorw = [max(1.0, ii) for ii in normw]
setfct(wx, wxa/factorw)
setfct(wy, wya/factorw)
assign(self.wH.sub(0), wx)
assign(self.wH.sub(1), wy)
# check
(wx,wy) = self.wH.split(deepcopy=True)
wxa, wya = wx.vector().array(), wy.vector().array()
assert np.amax(np.sqrt(wxa**2 + wya**2)) <= 1.0 + 1e-14
# Print results
dualresnorm = assemble(self.dualresnorm)
normgraddm = assemble(self.normgraddm)
print 'line search dual variable: max(|w|)={}, err(w,df)={}, |grad(dm)|={}'.\
format(np.amax(np.sqrt(normw)), np.sqrt(dualresnorm), np.sqrt(normgraddm))
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()
def checkhessfd_med(ObjFctal, Medium, tolgradchk=1e-6, H = [1e-5, 1e-6, 1e-4], doublesided=True):
"""Finite-difference check for the Hessian of an ObjectiveFunctional
object"""
lenm = len(ObjFctal.getmarray())
ObjFctal.backup_m()
MGref = ObjFctal.getMGarray()
if doublesided: factor = [1.0, -1.0]
else: factor = [1.0]
hessxdir = ObjFctal.srchdir
dirfct = ObjFctal.delta_m
for textnb, dirct in zip(range(lenm), Medium):
# Do computations for analytical Hessian:
setfct(dirfct, dirct)
ObjFctal.mult(dirfct.vector(), hessxdir.vector())
normhess = np.linalg.norm(hessxdir.vector().array())
print 'Hessian check -- direction {0}: ||H.x||={1:.5e}'\
.format(textnb+1, normhess)
# Do computations for FD Hessian:
for hh in H:
MG = []
for fact in factor:
ObjFctal.update_m(ObjFctal.getmcopyarray() + fact*hh*dirct)
ObjFctal.solvefwd_cost()
ObjFctal.solveadj_constructgrad()
MG.append(ObjFctal.getMGarray())
if doublesided: FDHessx = (MG[0] - MG[1])/(2.0*hh)
else: FDHessx = (MG[0] - MGref)/hh
# Compute errors:
err = np.linalg.norm(hessxdir.vector().array()-FDHessx)/normhess
if err < tolgradchk:
print '\th={0:.1e}: ||FDH.x||={1:.5e}, error={2:.2e} -> OK!'\
.format(hh, np.linalg.norm(FDHessx), err)
break
else:
print '\th={0:.1e}: ||FDH.x||={1:.5e}, error={2:.2e}'\
.format(hh, np.linalg.norm(FDHessx), err)
# Restore initial value of m:
ObjFctal.restore_m()
ObjFctal.solvefwd_cost()
ObjFctal.solveadj_constructgrad()
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
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 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 cost(self, m_in):
""" returns the cost functional for self.m=m_in """
setfct(self.m, m_in)
return assemble(self.wkformcost)
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 assemble_hessianab(self, a, b):
setfct(self.a, a)
setfct(self.b, b)
self.H = assemble(self.hessian)
self.Hprecond = assemble(self.precond)
def gradab(self, ma_in, mb_in):
""" ma_in, mb_in = Function(V) """
setfct(self.a, ma_in)
setfct(self.b, mb_in)
return assemble(self.grad)
def assemble_GNhessian(self, m_in):
""" Assemble the Gauss-Newton Hessian at m_in """
setfct(self.m, m_in)
self.H = assemble(self.wkformGNhess)
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 gradabvect(self, m1, m2):
setfct(self.tmpm1, m1)
setfct(self.tmpm2, m2)
return self.gradab(self.tmpm1, self.tmpm2)
def grad(self, m_in):
""" returns the gradient (in vector format) evaluated at self.m = m_in """
setfct(self.m, m_in)
self.H = None
return assemble(self.wkformgrad)
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))
def update(self, parameters_m):
assert not self.timestepper == None, "You need to set a time stepping method"
# Time options:
if parameters_m.has_key('t0'): self.t0 = parameters_m['t0']
if parameters_m.has_key('tf'): self.tf = parameters_m['tf']
if parameters_m.has_key('Dt'): self.Dt = parameters_m['Dt']
if parameters_m.has_key('t0') or parameters_m.has_key('tf') or parameters_m.has_key('Dt'):
self.Nt = int(round((self.tf-self.t0)/self.Dt))
self.Tf = self.t0 + self.Dt*self.Nt
self.times = np.linspace(self.t0, self.Tf, self.Nt+1)
assert isequal(self.times[1]-self.times[0], self.Dt, 1e-16), "Dt modified"
self.Dt = self.times[1] - self.times[0]
assert isequal(self.Tf, self.tf, 1e-2), "Final time differs by more than 1%"
if not isequal(self.Tf, self.tf, 1e-12):
print 'Final time modified from {} to {} ({}%)'.\
format(self.tf, self.Tf, abs(self.Tf-self.tf)/self.tf)
# Initial conditions:
if parameters_m.has_key('u0init'): self.u0init = parameters_m['u0init']
if parameters_m.has_key('utinit'): self.utinit = parameters_m['utinit']
if parameters_m.has_key('u1init'): self.u1init = parameters_m['u1init']
if parameters_m.has_key('um1init'): self.um1init = parameters_m['um1init']
# Medium parameters:
setfct(self.lam, parameters_m['lambda'])
if self.verbose: print 'lambda updated '
if self.elastic == True:
setfct(self.mu, parameters_m['mu'])
if self.verbose: print 'mu updated'
if self.verbose: print 'assemble K',
self.K = assemble(self.weak_k)
if self.verbose: print ' -- K assembled'
if parameters_m.has_key('rho'):
setfct(self.rho, parameters_m['rho'])
# Mass matrix:
if self.verbose: print 'rho updated\nassemble M',
Mfull = assemble(self.weak_m)
if self.lump:
self.solverM = LumpedMatrixSolverS(self.V)
self.solverM.set_operator(Mfull, self.bc)
self.M = self.solverM
else:
if mpisize == 1:
self.solverM = LUSolver()
self.solverM.parameters['reuse_factorization'] = True
self.solverM.parameters['symmetric'] = True
else:
self.solverM = KrylovSolver('cg', 'amg')
self.solverM.parameters['report'] = False
self.M = Mfull
if not self.bc == None: self.bc.apply(Mfull)
self.solverM.set_operator(Mfull)
if self.verbose: print ' -- M assembled'
# Matrix D for abs BC
if self.abc == True:
if self.verbose: print 'assemble D',
Mfull = assemble(self.weak_m)
Dfull = assemble(self.weak_d)
if self.lumpD:
self.D = LumpedMatrixSolverS(self.V)
self.D.set_operator(Dfull, None, False)
if self.lump:
self.solverMplD = LumpedMatrixSolverS(self.V)
self.solverMplD.set_operators(Mfull, Dfull, .5*self.Dt, self.bc)
self.MminD = LumpedMatrixSolverS(self.V)
self.MminD.set_operators(Mfull, Dfull, -.5*self.Dt, self.bc)
else:
self.D = Dfull
if self.verbose: print ' -- D assembled'
else: self.D = 0.0
def assemble_hessianab(self, m1, m2):
""" Assemble Hessian matrix for regularization """
setfct(self.m1, m1)
setfct(self.m2, m2)
self.H = assemble(self.hessian)
def assemble_hessianab(self, m1, m2):
setfct(self.m1, m1)
setfct(self.m2, m2)
assign(self.m.sub(0), self.m1)
assign(self.m.sub(1), self.m2)
self.regTV.assemble_hessian(self.m)
def grad(self, uin, udin):
isFunction(uin)
isFunction(udin)
setfct(self.diff, uin.vector() - udin.vector())
return self.W * self.diffv
def costfct_F(self, uin, udin):
isFunction(uin)
isFunction(udin)
setfct(self.diff, uin.vector()-udin.vector())
return 0.5 * (self.W*self.diffv).inner(self.diffv)
def costabvect(self, m1, m2):
setfct(self.tmpm1, m1)
setfct(self.tmpm2, m2)
return self.costab(self.tmpm1, self.tmpm2)
def costab(self, ma_in, mb_in):
setfct(self.a, ma_in)
setfct(self.b, mb_in)
return assemble(self.cost)
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 compute_dw(self, dm):
""" Compute dw """
setfct(self.dm, dm)
b = assemble(self.rhswwk)
solve(self.Mw, self.dw.vector(), b)
def gradab(self, m1, m2):
""" returns gradient at (m1,m2) as a vector """
setfct(self.m1, m1)
setfct(self.m2, m2)
return assemble(self.gradm)
def cost(self, m_in):
""" returns the cost functional for self.m=m_in """
setfct(self.m, m_in)
self.H = None
return assemble(self.wkformcost)
def costab(self, m1, m2):
""" Compute value of cost function at (m1,m2) """
setfct(self.m1, m1)
setfct(self.m2, m2)
return assemble(self.wkformcost)
def assemble_hessian(self, m_in):
""" Assemble the Hessian of TV at m_in """
setfct(self.m, m_in)
self.H = assemble(self.wkformhess)
def cost(self, m):
""" evaluate the cost functional at m """
setfct(self.m, m)
return assemble(self.wkformcost)
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 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)
