  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.
