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A stochastic collocation Approach to Bayesian Inference in Inverse Problems

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Communications in Computational Physics 2009年第9期6卷 826-847页

作者： Youssef Marzouk Dongbin Xiu Department of Aeronautics&Astronautics Massachusetts Institute of TechnologyCambridgeMA 02139USA Department of Mathematics Purdue UniversityWest LafayetteIN 47907USA

We present an efficient numerical strategy for the Bayesian solution of inverse problems.stochastic collocation methods,based on generalized polynomial chaos(gPC),are used to construct a polynomial approximation of the forward solution over the support of the prior distribution.This approximation then defines a surrogate posterior probability density that can be evaluated repeatedly at minimal computational cost.The ability to simulate a large number of samples from the posterior distribution results in very accurate estimates of the inverse solution and its associated uncertainty.Combined with high accuracy of the gPC-based forward solver,the new algorithm can provide great efficiency in practical applications.A rigorous error analysis of the algorithm is conducted,where we establish convergence of the approximate posterior to the true posterior and obtain an estimate of the convergence rate.It is proved that fast(exponential)convergence of the gPC forward solution yields similarly fast(exponential)convergence of the posterior.The numerical strategy and the predicted convergence rates are then demonstrated on nonlinear inverse problems of varying smoothness and dimension.

关键词： Inverse problems Bayesian inference stochastic collocation generalized polynomial chaos uncertainty quantification

A stochastic collocation Method for Delay Differential Equations with Random Input

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Advances in Applied Mathematics and Mechanics 2014年第4期6卷 403-418页

作者： Tao Zhou Institute of Computational Mathematics Academy of Mathematics and Systems SciencesThe Chinese Academy of SciencesBeijing 100190China

In this work,we concern with the numerical approach for delay differential equations with random coefficients.We first show that the exact solution of the problem considered admits good regularity in the random space,provided that the given data satisfy some reasonable assumptions.A stochastic collocation method is proposed to approximate the solution in the random space,and we use the Legendre spectral collocation method to solve the resulting deterministic delay differential equations.Convergence property of the proposed method is analyzed.It is shown that the numerical method yields the familiar exponential order of convergence in both the random space and the time space.Numerical examples are given to illustrate the theoretical results.

关键词： Delay differential equations stochastic collocation sparse grid legendre spectral method

Variance-Based Global Sensitivity Analysis via Sparse-Grid Interpolation and Cubature

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Communications in Computational Physics 2011年第3期9卷 542-567页

作者： Gregery T.Buzzard Dongbin Xiu Department of Mathematics Purdue UniversityWest LafayetteIN 47907USA

The stochastic collocation method using sparse grids has become a popular choice for performing stochastic computations in high dimensional(random)parameter space.In addition to providing highly accurate stochastic solutions,the sparse grid collocation results naturally contain sensitivity information with respect to the input random parameters.In this paper,we use the sparse grid interpolation and cubature methods of Smolyak together with combinatorial analysis to give a computationally efficient method for computing the global sensitivity values of Sobol’.This method allows for approximation of all main effect and total effect values from evaluation of f on a single set of sparse grids.We discuss convergence of this method,apply it to several test cases and compare to existing methods.As a result which may be of independent interest,we recover an explicit formula for evaluating a Lagrange basis interpolating polynomial associated with the Chebyshev extrema.This allows one to manipulate the sparse grid collocation results in a highly efficient manner.

关键词： stochastic collocation sparse grids sensitivity analysis Smolyak Sobol’

A Compressed Sensing Approach for Partial Differential Equations with Random Input Data

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Communications in Computational Physics 2012年第9期12卷 919-954页

作者： L.Mathelin K.A.Gallivan LIMSI-CNRS BP 13391403 OrsayFrance Mathematics Dpt. 208 Love Building1017 Academic WayFlorida State UniversityTallahassee FL 32306-4510USA

In this paper,a novel approach for quantifying the parametric uncertainty associated with a stochastic problem output is presented.As with Monte-Carlo and stochastic collocation methods,only point-wise evaluations of the stochastic output response surface are required allowing the use of legacy deterministic codes and precluding the need for any dedicated stochastic code to solve the uncertain problem of interest.The new approach differs from these standard methods in that it is based on ideas directly linked to the recently developed compressed sensing theory.The technique allows the retrieval of the modes that contribute most significantly to the approximation of the solution using a minimal amount of information.The generation of this information,via many solver calls,is almost always the bottle-neck of an uncertainty quantification procedure.If the stochastic model output has a reasonably compressible representation in the retained approximation basis,the proposedmethod makes the best use of the available information and retrieves the dominantmodes.Uncertainty quantification of the solution of both a 2-D and 8-D stochastic Shallow Water problem is used to demonstrate the significant performance improvement of the new method,requiring up to several orders of magnitude fewer solver calls than the usual sparse grid-based Polynomial Chaos(Smolyak scheme)to achieve comparable approximation accuracy.

关键词： Uncertainty quantification compressed sensing collocation technique stochastic spectral decomposition Smolyak sparse approximation stochastic collocation