Fast matrix algebra for Bayesian model calibration

Abstract

In Bayesian model calibration, evaluation of the likelihood function usually involves finding the inverse and determinant of a covariance matrix. When Markov Chain Monte Carlo (MCMC) methods are used to sample from the posterior, hundreds of thousands of likelihood evaluations may be required. In this paper, we demonstrate that the structure of the covariance matrix can be exploited, leading to substantial time savings in practice. We also derive two simple equations for approximating the inverse of the covariance matrix in this setting, which can be computed in near-quadratic time. The practical implications of these strategies are demonstrated using a simple numerical case study and the "quack" R package. For a covariance matrix with 1000 rows, application of these strategies for a million likelihood evaluations leads to a speedup of roughly 4000 compared to the naive implementation

Keywords:

• Bayesian model calibration
• inverse
• determinant
• likelihood
• fast

2010 Mathematics Subject Classifications:

• 41-04
• 62-04
• 62C10
• 65C40

Acknowledgement

This work was supported by a Sandia National Laboratories Laboratory Directed Research and Development (LDRD) grant. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525 and SAND2020-12827 J.

Disclosure statement

This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.