- Aeronautics and Astronautics
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Characterizing a Bayesian posterior distribution usually requires many posterior evaluations, which can become computationally intractable when evaluating the posterior involves expensive forward simulations. We address this issue with several innovative ways to use transport maps constructed from samples.
Transport maps are simply deterministic transformations between random variables; however, we have these transformations to be a fundamental tool for Bayesian algorithms. Therefore, we first develop efficient methods for constructing transport maps using only samples of a target distribution. Our approach is heavily based on the solution of a convex optimization problem. Using this optimization approach, we then demonstrate how to construct a composition of simple transformations to characterize complex high dimensional distributions. Using these map-construction techniques, we will then develop three new posterior sampling algorithms.
Our first algorithm exploits multiscale structure and offline map construction to generate approximate posterior samples in large-dimensional problems. We will analyze the accuracy of this approach on a 100 dimensional problem before tackling an inverse problem from porous media that has over 10,000 spatially-distributed parameters. Both problems utilize the multiscale finite element method (MsFEM) as a forward solver.
Our second algorithm is an adaptive MCMC algorithm that uses a transport map to define an efficient proposal mechanism. With several examples, we show that this approach effectively reduces inter-sample correlation and in terms of effective samples size, our algorithm can be over an order of magnitude more efficient than standard MCMC algorithms.
Our last algorithm uses extensive offline computation to construct a specially designed transport map for the joint distribution of the data and parameters. This approach is also similar to approximate Bayesian computation in that we do not require posterior evaluations; only samples of the joint prior are required. With an example, we show that can yield excellent posterior approximations in dramatically less online time time than a standard MCMC sampler.