Bayesian Inversion of PDE-Based Problems Using Integrated Nested Laplace Approximations
Sonia Reilly, New York University
The core computational bottleneck in Bayesian inference is the cost of evaluating high-dimensional integrals. Often these integrals are computed using sampling-based methods such as MCMC, or by approximating the posterior with more tractable distributions through variational inference. A more recent alternative for problems which can be expressed as latent Gaussian models is known as Integrated Nested Laplace Approximations (INLA). This approach combines multiple Laplace approximations, or Gaussian approximations around the mode, with sparse matrix algebra to rapidly and accurately compute integrals over the posterior distribution. INLA has been successfully applied to a variety of spatial statistics problems, but has not yet been applied to inverse problems derived from PDE-governed systems. These pose additional difficulties since the forward operator becomes an expensive, non-local PDE solve and precision matrices cannot be built and stored explicitly. In this poster presentation we will demonstrate a proof of concept application of INLA to one such PDE-based inverse problem.