def testOrthogonalization(self):
myRandom = Random(self.mpi_rank, self.mpi_size)
myRandom.normal(1.,self.Q)
_ = self.Q.orthogonalize()
QtQ = self.Q.dot_mv(self.Q)
if self.mpi_rank == 0:
assert np.linalg.norm(QtQ - np.eye(QtQ.shape[0])) < 1e-8
def setUp(self):
mesh = dl.UnitSquareMesh(10, 10)
self.mpi_rank = dl.MPI.rank(mesh.mpi_comm())
self.mpi_size = dl.MPI.size(mesh.mpi_comm())
Vh1 = dl.FunctionSpace(mesh, 'Lagrange', 1)
uh, vh = dl.TrialFunction(Vh1), dl.TestFunction(Vh1)
mh, test_mh = dl.TrialFunction(Vh1), dl.TestFunction(Vh1)
## Set up B
ndim = 2
ntargets = 200
np.random.seed(seed=1)
targets = np.random.uniform(0.1, 0.9, [ntargets, ndim])
B = assemblePointwiseObservation(Vh1, targets)
## Set up Asolver
alpha = dl.Constant(1.0)
varfA = ufl.inner(ufl.grad(uh), ufl.grad(vh))*ufl.dx +\
alpha*ufl.inner(uh,vh)*ufl.dx
A = dl.assemble(varfA)
Asolver = PETScKrylovSolver(A.mpi_comm(), "cg", amg_method())
Asolver.set_operator(A)
Asolver.parameters["maximum_iterations"] = 100
Asolver.parameters["relative_tolerance"] = 1e-12
## Set up C
varfC = ufl.inner(mh, vh) * ufl.dx
C = dl.assemble(varfC)
self.Hop = Hop(B, Asolver, C)
## Set up RHS Matrix M.
varfM = ufl.inner(mh, test_mh) * ufl.dx
self.M = dl.assemble(varfM)
self.Minv = PETScKrylovSolver(self.M.mpi_comm(), "cg", amg_method())
self.Minv.set_operator(self.M)
self.Minv.parameters["maximum_iterations"] = 100
self.Minv.parameters["relative_tolerance"] = 1e-12
myRandom = Random(self.mpi_rank, self.mpi_size)
x_vec = dl.Vector(mesh.mpi_comm())
self.Hop.init_vector(x_vec, 1)
k_evec = 10
p_evec = 50
self.Omega = MultiVector(x_vec, k_evec + p_evec)
self.k_evec = k_evec
myRandom.normal(1., self.Omega)
def testBOrthogonalization(self):
myRandom = Random(self.mpi_rank, self.mpi_size)
myRandom.normal(1.,self.Q)
self.Q.Borthogonalize(self.M)
MQ = MultiVector(self.Q)
MQ.zero()
MatMvMult(self.M, self.Q, MQ)
QtMQ = self.Q.dot_mv(MQ)
if self.mpi_rank == 0:
assert np.linalg.norm(QtMQ - np.eye(QtMQ.shape[0])) < 1e-8
def varianceReductionMC(prior, rqoi, taylor_qoi, nsamples, filename="realizations.txt"):
"""
This function computes Monte Carlo Estimates for forward propagation of uncertainty.
The uncertain parameter satisfies a Gaussian distribution with known mean and covariance
(describes as the inverse of a differential operator).
Convergence of the Monte Carlo estimates is accelerated using a variance reduction
techinque based on a Taylor approximation of the parameter-to-qoi map.
INPUTS:
- prior: an object of type hIPPYlib._Prior that allows to generate samples from the prior distribution
- rqoi: an object of type ReducedQOI that describes the parameter-to-qoi map
- taylor_qoi: an object of type TaylorApproximationQOI that computes the first and second order Taylor
approximation of the qoi
- nsamples: an integer representing the number of samples for the MC estimates
- filename: a string containing the name of the file where the computed qoi
and its Taylor approximations (for each realization of the parameter) are saved
OUTPUTS:
- Sample mean of the quantity of interest q, its Taylor approx q1 and q2, and corrections y1=q-q1 and y2=q-q2.
- MSE (Mean square error) of the standard MC, and the variance reduced MC using q1 and q2.
Note: The variate control MC estimator can be computed off-line by postprocessing the file containing the
values of the computed qoi and Taylor approximations.
"""
noise = dl.Vector()
sample = dl.Vector()
prior.init_vector(noise, "noise")
prior.init_vector(sample, 1)
rank = dl.MPI.rank(noise.mpi_comm())
q_i = np.zeros(nsamples)
q1_i = np.zeros(nsamples)
q2_i = np.zeros(nsamples)
y1_i = np.zeros(nsamples)
y2_i = np.zeros(nsamples)
Eq1_exact = taylor_qoi.expectedValue(order=1)
Eq2_exact = taylor_qoi.expectedValue(order=2)
if rank == 0:
print "nsamples | E[q], E[y1] + E[q1], E[y2] + E[q2]| Var[q] Var[y1] Var[y2]"
fid = open(filename,"w")
for i in range(nsamples):
Random.normal(noise, 1, True)
prior.sample(noise, sample)
q_i[i] = rqoi.reduced_eval(sample)
q1_i[i] = taylor_qoi.eval(sample, order=1)
q2_i[i] = taylor_qoi.eval(sample, order=2)
y1_i[i] = q_i[i] - q1_i[i]
y2_i[i] = q_i[i] - q2_i[i]
if rank == 0:
fid.write("{0:15e} {1:15e} {2:15e}\n".format(q_i[i], q1_i[i], q2_i[i]))
fid.flush()
if ( (i+1) % 10 == 0) or (i+1 == nsamples):
Eq = np.sum(q_i)/float(i+1)
Eq1 = np.sum(q1_i)/float(i+1)
Eq2 = np.sum(q2_i)/float(i+1)
Ey1 = np.sum(y1_i)/float(i+1)
Ey2 = np.sum(y2_i)/float(i+1)
Varq = np.sum(np.power(q_i,2))/float(i) - (float(i+1)/float(i)*Eq*Eq)
Varq1 = np.sum(np.power(q1_i,2))/float(i) - (float(i+1)/float(i)*Eq1*Eq1)
Varq2 = np.sum(np.power(q2_i,2))/float(i) - (float(i+1)/float(i)*Eq2*Eq2)
Vary1 = np.sum(np.power(y1_i,2))/float(i) - (float(i+1)/float(i)*Ey1*Ey1)
Vary2 = np.sum(np.power(y2_i,2))/float(i) - (float(i+1)/float(i)*Ey2*Ey2)
if rank == 0:
print "{0:3} | {1:7e} {2:7e} {3:7e} | {4:7e} {5:7e} {6:7e}".format(
i+1, Eq, Ey1+Eq1_exact, Ey2+Eq2_exact,
Varq, Vary1, Vary2)
Vq1_exact = taylor_qoi.variance(order=1)
Vq2_exact = taylor_qoi.variance(order=1)
if rank == 0:
fid.close()
print "Expected value q1: analytical: ", Eq1_exact, "estimated: ", Eq1
print "Expected value q2: analytical: ", Eq2_exact, "estimated: ", Eq2
print "Variance q1: analytical", Vq1_exact, "estimated: ", Varq1
print "Variance q2: analytical", Vq2_exact, "estimated: ", Varq2
return Eq, Ey1, Ey2, Eq1_exact, Eq2_exact, Varq/nsamples, Vary1/nsamples, Vary2/nsamples