Abstract: Surrogate modeling based on Gaussian processes (GPs) is becoming increasingly popular in analysis of complex problems in science and engineering. However, despite the many studies on GP modeling, few focus on functional inputs. Motivated by an inverse scattering problem in which functional inputs representing the support and material properties of the scatterer are involved in the partial differential equations, we propose a new class of kernel functions for functional inputs of GPs. Based on the proposed GP models, we derive the asymptotic convergence properties of the resulting mean squared prediction errors, and demonstrate the finite-sample performance using numerical examples. In the application to inverse scattering, we construct a surrogate model with functional inputs, which is crucial to recovering the reflective index of an inhomogeneous isotropic scattering region of interest for a given far-field pattern.

Key words and phrases: Computer experiments, functional data analysis, scalar-on-function regression, surrogate model, uncertainty quantification.