Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
Numerical recipes in C (2nd ed.): the art of scientific computing
Numerical recipes in C (2nd ed.): the art of scientific computing
RKC: an explicit solver for parabolic PDEs
Journal of Computational and Applied Mathematics
Comparison of approximate methods for handling hyperparameters
Neural Computation
A stochastic projection method for fluid flow. I: basic formulation
Journal of Computational Physics
Computational Methods for Inverse Problems
Computational Methods for Inverse Problems
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
Statistical Regularization of Inverse Problems
SIAM Review
A stochastic projection method for fluid flow II.: random process
Journal of Computational Physics
Classes of kernels for machine learning: a statistics perspective
The Journal of Machine Learning Research
Natural Convection in a Closed Cavity under Stochastic Non-Boussinesq Conditions
SIAM Journal on Scientific Computing
Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes
SIAM Journal on Scientific Computing
Monte Carlo Statistical Methods (Springer Texts in Statistics)
Monte Carlo Statistical Methods (Springer Texts in Statistics)
Inverse Problem Theory and Methods for Model Parameter Estimation
Inverse Problem Theory and Methods for Model Parameter Estimation
High-Order Collocation Methods for Differential Equations with Random Inputs
SIAM Journal on Scientific Computing
An adaptive multi-element generalized polynomial chaos method for stochastic differential equations
Journal of Computational Physics
Preconditioning Markov Chain Monte Carlo Simulations Using Coarse-Scale Models
SIAM Journal on Scientific Computing
Wiener Chaos expansions and numerical solutions of randomly forced equations of fluid mechanics
Journal of Computational Physics
Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning)
Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning)
Statistical inverse problems: discretization, model reduction and inverse crimes
Journal of Computational and Applied Mathematics - Special issue: Applied computational inverse problems
Stochastic spectral methods for efficient Bayesian solution of inverse problems
Journal of Computational Physics
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
SIAM Journal on Numerical Analysis
IEEE Transactions on Signal Processing
Identification of Bayesian posteriors for coefficients of chaos expansions
Journal of Computational Physics
Parameter and State Model Reduction for Large-Scale Statistical Inverse Problems
SIAM Journal on Scientific Computing
Sampling-free linear Bayesian update of polynomial chaos representations
Journal of Computational Physics
A Posteriori Error Analysis of Parameterized Linear Systems Using Spectral Methods
SIAM Journal on Matrix Analysis and Applications
Bayesian inference with optimal maps
Journal of Computational Physics
Simulation-based optimal Bayesian experimental design for nonlinear systems
Journal of Computational Physics
Numerical schemes for dynamically orthogonal equations of stochastic fluid and ocean flows
Journal of Computational Physics
Multiparameter Spectral Representation of Noise-Induced Competence in Bacillus Subtilis
IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB)
Journal of Computational Physics
Telescoping strategies for improved parameter estimation of environmental simulation models
Computers & Geosciences
Ensemble level multiscale finite element and preconditioner for channelized systems and applications
Journal of Computational and Applied Mathematics
Journal of Computational Physics
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We consider a Bayesian approach to nonlinear inverse problems in which the unknown quantity is a spatial or temporal field, endowed with a hierarchical Gaussian process prior. Computational challenges in this construction arise from the need for repeated evaluations of the forward model (e.g., in the context of Markov chain Monte Carlo) and are compounded by high dimensionality of the posterior. We address these challenges by introducing truncated Karhunen-Loeve expansions, based on the prior distribution, to efficiently parameterize the unknown field and to specify a stochastic forward problem whose solution captures that of the deterministic forward model over the support of the prior. We seek a solution of this problem using Galerkin projection on a polynomial chaos basis, and use the solution to construct a reduced-dimensionality surrogate posterior density that is inexpensive to evaluate. We demonstrate the formulation on a transient diffusion equation with prescribed source terms, inferring the spatially-varying diffusivity of the medium from limited and noisy data.