Low-Rank Separated Representation Surrogates of High-Dimensional Stochastic Functions: Application in Bayesian Inference

TL;DR

This paper presents a non-intrusive low-rank separated representation method to efficiently create surrogate models for high-dimensional stochastic functions, significantly reducing computational costs in Bayesian inference tasks.

Contribution

It introduces a regularized least-squares approach with error indicators for constructing high-dimensional surrogates, improving efficiency and scalability over existing scalar methods.

Findings

Linear growth in required samples with dimensionality

Quadratic computational cost in number of inputs

Order-of-magnitude cost reduction for vector-valued models

Abstract

This study introduces a non-intrusive approach in the context of low-rank separated representation to construct a surrogate of high-dimensional stochastic functions, e.g., PDEs/ODEs, in order to decrease the computational cost of Markov Chain Monte Carlo simulations in Bayesian inference. The surrogate model is constructed via a regularized alternative least-square regression with Tikhonov regularization using a roughening matrix computing the gradient of the solution, in conjunction with a perturbation-based error indicator to detect optimal model complexities. The model approximates a vector of a continuous solution at discrete values of a physical variable. The required number of random realizations to achieve a successful approximation linearly depends on the function dimensionality. The computational cost of the model construction is quadratic in the number of random inputs, which potentially tackles the curse of dimensionality in high-dimensional stochastic functions. Furthermore, this vector valued separated representation-based model, in comparison to the available scalar-valued case, leads to a significant reduction in the cost of approximation by an order of magnitude equal to the vector size. The performance of the method is studied through its application to three numerical examples including a 41-dimensional elliptic PDE and a 21-dimensional cavity flow.