def __init__(self,
num_vars,
rank,
num_qoi,
obs=None,
noise_covariance=None,
Amatrix=None):
self.num_vars = num_vars
self.rank = rank
self.num_qoi = num_qoi
if Amatrix is None:
self.Amatrix = get_low_rank_matrix(num_qoi, num_vars, rank)
else:
self.Amatrix = Amatrix
if obs is None:
self.obs = np.zeros(num_qoi)
else:
self.obs = obs
if noise_covariance is None:
self.noise_covariance = np.eye(num_qoi)
else:
self.noise_covariance = noise_covariance
self.noise_covariance_inv = np.linalg.inv(self.noise_covariance)
def test_prior_conditioned_misfit_covariance_operator(self):
num_dims = 3
rank = 2
num_qoi = 2
# define prior
prior_mean = np.zeros((num_dims), float)
prior_covariance = np.eye(num_dims)
prior_hessian = np.eye(num_dims)
prior_density = NormalDensity(prior_mean, covariance=prior_covariance)
# define observations
noise_sigma2 = 0.5
noise_covariance = np.eye(num_qoi)*noise_sigma2
noise_covariance_inv = np.linalg.inv(noise_covariance)
noise_covariance_chol_factor = np.linalg.cholesky(noise_covariance)
truth_sample = prior_density.generate_samples(1)[:, 0]
Amatrix = get_low_rank_matrix(num_qoi, num_dims, rank)
noise = np.dot(
noise_covariance_chol_factor, np.random.normal(0., 1., num_qoi))
obs = np.dot(Amatrix, truth_sample)+noise
# define mistit model
misfit_model = QuadraticMisfitModel(
num_dims, rank, num_qoi, obs, noise_covariance, Amatrix)
# Get analytical mean and covariance
exact_laplace_mean, exact_laplace_covariance = \
laplace_posterior_approximation_for_linear_models(
misfit_model.Amatrix, prior_mean, prior_hessian,
noise_covariance_inv, obs)
objective = LogUnormalizedPosterior(
misfit_model, misfit_model.gradient_set, prior_density.pdf,
prior_density.log_pdf, prior_density.log_pdf_gradient)
map_point, obj_min = find_map_point(objective, prior_mean)
sample = map_point
assert np.allclose(objective.gradient(sample), np.zeros(num_dims))
prior_covariance_sqrt_operator = CholeskySqrtCovarianceOperator(
prior_density.covariance)
misfit_hessian_operator = MisfitHessianVecOperator(
misfit_model, sample, fd_eps=1e-7)
standard_svd_opts = {
'num_singular_values': rank, 'num_extra_samples': 10}
svd_opts = {'single_pass': False, 'standard_opts': standard_svd_opts}
laplace_covariance_sqrt = get_laplace_covariance_sqrt_operator(
prior_covariance_sqrt_operator, misfit_hessian_operator,
svd_opts)
identity = np.eye(num_dims)
laplace_covariance_chol_factor = laplace_covariance_sqrt.apply(
identity)
laplace_covariance = np.dot(
laplace_covariance_chol_factor, laplace_covariance_chol_factor.T)
# print laplace_covariance
# print exact_laplace_covariance
assert np.allclose(laplace_covariance, exact_laplace_covariance)
def setup_quadratic_misfit_problem(prior, rank, noise_sigma2=1):
# Define observations
num_qoi = 2 * rank
#assert num_qoi>=rank
noise_covariance = np.eye(num_qoi) * noise_sigma2
noise_covariance_inv = np.linalg.inv(noise_covariance)
# In high dimensions computing cholesky factor is too expensive.
# That is why we use PDE based operator
noise_covariance_chol_factor = np.linalg.cholesky(noise_covariance)
truth_sample = prior.generate_samples(1)[:, 0]
num_vars = truth_sample.shape[0]
Amatrix = get_low_rank_matrix(num_qoi, num_vars, rank)
noise = np.dot(noise_covariance_chol_factor,
np.random.normal(0., noise_sigma2, num_qoi))
obs = np.dot(Amatrix, truth_sample) + noise
# Define mistit model
misfit_model = QuadraticMisfitModel(num_vars, rank, num_qoi, Amatrix)
return misfit_model, noise_covariance_inv, obs