Begell House Inc.
International Journal for Uncertainty Quantification
IJUQ
2152-5080
6
1
2016
GLOBAL SENSITIVITY ANALYSIS: AN EFFICIENT NUMERICAL METHOD FOR APPROXIMATING THE TOTAL SENSITIVITY INDEX
1-17
Matieyendou
Lamboni
University of Guyane, Department DFRST, 2091 route de Baduel, 97346 Cayenne Cedex,
French Guiana (present address); 228-UMR Espace-Dev, 275 route de Montabo, 97323 Cayenne Cedex, French Guiana (present address); EC-Joint Research Centre, Institute for Environment and Sustainability, Via Fermi 2749, 21027 Ispra, Italy
Variance-based sensitivity analysis and multivariate sensitivity analysis aim to apportion the variability of model output(s) into input factors and their interactions. Total sensitivity index (TSI) gives for each input its overall contribution, including the effects of its interactions with all the other inputs, in the variability of the model output(s). We investigate a numerical approximation of TSIs mainly based upon quadrature rules and quasi-Monte Carlo. The estimation of a TSI relies on the estimation of a total effect function (TEF), which allows for computing the TSI values by taking its variance. First, the paper derives the specific formula for the computation of the TEF, including the theoretical properties of the approximation, and second, it gives an overview of its application in many situations. Our approach gives the exact estimation of TSIs for a class of exact quadrature rules (especially for polynomial functions) and an interesting approximation for other functions. Numerical tests show the faster convergence rate of our approach and their usefulness in practice.
AN EFFICIENT MESH-FREE IMPLICIT FILTER FOR NONLINEAR FILTERING PROBLEMS
19-33
Feng
Bao
Department of Computational and Applied Mathematics, Oak Ridge National Laboratory, One Bethel Valley Road, P.O. Box 2008, MS-6164, Oak Ridge, Tennessee 37831-6164, USA
Yanzhao
Cao
Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849; School of Mathematics, Sun Yat Sun University, China
Clayton G.
Webster
Department of Computational and Applied Mathematics, Oak Ridge National Laboratory, One Bethel Valley Road, P.O. Box 2008, MS-6164, Oak Ridge, Tennessee 37831-6164, USA
Guannan
Zhang
Department of Computational and Applied Mathematics, Oak Ridge National Laboratory, One Bethel Valley Road, P.O. Box 2008, MS-6164, Oak Ridge, Tennessee 37831-6164, USA
In this paper, we propose a mesh-free approximation method for the implicit filter developed in Bao et al., Commun. Comput. Phys., 16(2):382-402, 2014, which is a novel numerical algorithm for nonlinear filtering problems. The implicit filter approximates conditional distributions in the optimal filter over a deterministic state space grid and is developed from samples of the current state obtained by solving the state equation implicitly. The purpose of the mesh-free approximation is to improve the efficiency of the implicit filter in moderately high-dimensional problems. The construction of the algorithm includes generation of random state space points and a mesh-free interpolation method. Numerical experiments show the effectiveness and efficiency of our algorithm.
A METROPOLIS-HASTINGS METHOD FOR LINEAR INVERSE PROBLEMS WITH POISSON LIKELIHOOD AND GAUSSIAN PRIOR
35-55
Johnathan M.
Bardsley
Department of Mathematical Sciences, The University of Montana, Missoula, Montana 59812-0864, USA
Aaron
Luttman
Signal Processing and Data Analysis, National Security Technologies, LLC, P.O. Box 98521, M/S NLV078, Las Vegas, Nevada, 89193-8521, USA
Poisson noise models arise in a wide range of linear inverse problems in imaging. In the Bayesian setting, the Poisson likelihood function together with a Gaussian prior yields a posterior density that is not of a well-known form and is thus difficult to sample from, especially for large-scale problems. In this work, we present a method for computing samples from posterior density functions with Poisson likelihood and Gaussian prior, using a Gaussian approximation of the posterior as an independence proposal within a Metropolis−Hastings framework. We consider a class of Gaussian priors, some of which are edge-preserving, and which we motivate using Markov random fields. We present two sampling algorithms: one which samples the unknown image alone, leaving the prior scaling (or regularization) parameter alone, and another which samples both the unknown image and the prior scaling parameter. For this paper, we make the assumption that our unknown image is sufficiently positive that proposed samples are always positive, allowing us to ignore the nonnegativity constraint. Results are demonstrated on synthetic data−including a synthetic X-ray radiograph generated from a radiation transport code−and on real images used to calibrate a pulsed power high-energy X-ray source at the U.S. Department of Energy's Nevada National Security Site.
ROBUST UNCERTAINTY QUANTIFICATION USING PRECONDITIONED LEAST-SQUARES POLYNOMIAL APPROXIMATIONS WITH l1-REGULARIZATION
57-77
Jan
Van Langenhove
Sorbonne Universités, UPMC Univ Paris 06, UMR 7190, Institut Jean le Rond d'Alembert, F-75005, Paris, France;
CNRS, UMR 7190, Institut Jean le Rond d'Alembert, F-75005, Paris, France
D.
Lucor
LIMSI, CNRS, Université Paris-Saclay, Campus Universitaire bat 508, Rue John von Neumann, F-91405 Orsay cedex, France
A.
Belme
Sorbonne Universités, UPMC Univ Paris 06, UMR 7190, Institut Jean le Rond d'Alembert, F-75005, Paris, France;
CNRS, UMR 7190, Institut Jean le Rond d'Alembert, F-75005, Paris, France
We propose a noniterative robust numerical method for the nonintrusive uncertainty quantification of multivariate stochastic problems with reasonably compressible polynomial representations. The approximation is robust to data outliers or noisy evaluations which do not fall under the regularity assumption of a stochastic truncation error but pertains to a more complete error model, capable of handling interpretations of physical/computational model (or measurement) errors. The method relies on the cross-validation of a pseudospectral projection of the response on generalized Polynomial Chaos approximation bases; this allows an initial model selection and assessment yielding a preconditioned response. We then apply a l1-penalized regression to the preconditioned response variable. Nonlinear test cases have shown this approximation to be more effective in reducing the effect of scattered data outliers than standard compressed sensing techniques and of comparable efficiency to iterated robust regression techniques.
REFINED LATINIZED STRATIFIED SAMPLING: A ROBUST SEQUENTIAL SAMPLE SIZE EXTENSION METHODOLOGY FOR HIGH-DIMENSIONAL LATIN HYPERCUBE AND STRATIFIED DESIGNS
79-97
Michael D.
Shields
Department of Civil Engineering, Johns Hopkins University, Baltimore, Maryland
A robust sequential sampling method, refined latinized stratified sampling, for simulation-based uncertainty quantification and reliability analysis is proposed. The method combines the benefits of the two leading approaches, hierarchical Latin hypercube sampling (HLHS) and refined stratified sampling, to produce a method that significantly reduces the variance of statistical estimators for very high-dimensional problems. The method works by hierarchically creating sample designs that are both Latin and stratified. The intermediate sample designs are then produced using the refined stratified sampling method. This causes statistical estimates to converge at rates that are equal to or better than HLHS while affording maximal flexibility in sample size extension (one-at-a-time or n-at-a-time sampling are possible) that does not exist in HLHS—which grows the sample size exponentially. The properties of the method are highlighted for several very high-dimensional problems, demonstrating the method has the distinct benefit of rapid convergence for transformations of all kinds.