Surrogate modeling and uncertainty quantification for inverse problems and dynamical systems

Inverse scattering aims to infer information about a hidden object by using the received scattered waves and training data collected from forward mathematical models. Recent advances in computing have led to increasing attention towards functional inverse inference, which can reveal more detailed properties of a hidden object. However, rigorous studies on functional inverse, including the reconstruction of the functional input and quantification of uncertainty, remain scarce. Motivated by an inverse scattering problem where the objective is to infer the functional input representing the refractive index of a bounded scatterer, a new Bayesian framework will be discussed in the first part of this talk. It contains a surrogate model that takes into account the functional inputs directly through kernel functions, and a Bayesian procedure that infers functional inputs through the posterior distribution. In the second part of this talk, we will introduce a novel procedure that, given sparse data generated from a stationary deterministic nonlinear dynamical system, can characterize specific local and/or global dynamic behavior with rigorous probability guarantees. More precisely, the sparse data is used to construct a statistical surrogate model based on a Gaussian process (GP). The dynamics of the surrogate model is interrogated using combinatorial methods and characterized using algebraic topological invariants (Conley index). The GP predictive distribution provides a lower bound on the confidence that these topological invariants, and hence the characterized dynamics, apply to the unknown dynamical system.