Constructing Level Sets Using Smoothed Approximate Bayesian Computation

This paper presents a novel approach to level set estimation for any function/simulation with an arbitrary number of continuous inputs and arbitrary numbers of continuous responses. We present a method that uses existing data from computer model simulations to fit a Gaussian process surrogate and use a newly proposed Markov Chain Monte Carlo technique, which we refer to as Smoothed Approximate Bayesian Computation to sample sets of parameters that yield a desired response, which improves on ``hard-clipped" versions of ABC. We prove that our method converges to the correct distribution (i.e. the posterior distribution of level sets, or probability contours) and give results of our method on known functions and a dam breach simulation where the relationship between input parameters and responses of interest is unknown. Two versions of S-ABC are offered based on: 1) surrogating an accurately known target model and 2) surrogating an approximate model, which leads to uncertainty in estimating the level sets. In addition, we show how our method can be extended to multiple responses with an accompanying example. As demonstrated, S-ABC is able to estimate a level set accurately without the use of a predefined grid or signed distance function.