The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
Bayesian parameter estimation via variational methods
Statistics and Computing
Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure
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
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
Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning)
Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning)
Statistics and Computing
Stochastic spectral methods for efficient Bayesian solution of inverse problems
Journal of Computational Physics
Journal of Computational Physics
Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems
Journal of Computational Physics
Monte Carlo Strategies in Scientific Computing
Monte Carlo Strategies in Scientific Computing
Robust Stochastic Approximation Approach to Stochastic Programming
SIAM Journal on Optimization
Numerical Methods for Stochastic Computations: A Spectral Method Approach
Numerical Methods for Stochastic Computations: A Spectral Method Approach
Parameter and State Model Reduction for Large-Scale Statistical Inverse Problems
SIAM Journal on Scientific Computing
IEEE Transactions on Signal Processing
Uncertainty Quantification and Weak Approximation of an Elliptic Inverse Problem
SIAM Journal on Numerical Analysis
Sampling-free linear Bayesian update of polynomial chaos representations
Journal of Computational Physics
Journal of Computational Physics
Hi-index | 31.46 |
We present a new approach to Bayesian inference that entirely avoids Markov chain simulation, by constructing a map that pushes forward the prior measure to the posterior measure. Existence and uniqueness of a suitable measure-preserving map is established by formulating the problem in the context of optimal transport theory. We discuss various means of explicitly parameterizing the map and computing it efficiently through solution of an optimization problem, exploiting gradient information from the forward model when possible. The resulting algorithm overcomes many of the computational bottlenecks associated with Markov chain Monte Carlo. Advantages of a map-based representation of the posterior include analytical expressions for posterior moments and the ability to generate arbitrary numbers of independent posterior samples without additional likelihood evaluations or forward solves. The optimization approach also provides clear convergence criteria for posterior approximation and facilitates model selection through automatic evaluation of the marginal likelihood. We demonstrate the accuracy and efficiency of the approach on nonlinear inverse problems of varying dimension, involving the inference of parameters appearing in ordinary and partial differential equations.