Reproducing Kernel Hilbert Spaces for Parametric Partial Differential Equations

In this paper, we present kernel methods for the approximation of quantities of interest which are derived from solutions of parametric partial differential equations. From a priori information on the parameters in the differential equation, we explicitly construct a reproducing kernel Hilbert space containing the quantity of interest as a function of these parameters. Based on this problem-adapted reproducing kernel, we suggest a regularized reconstruction technique from machine learning in order to approximate the quantity of interest from a finite number of point values. We present a deterministic a priori error analysis for this reconstruction process yielding a subexponential convergence order due to the smoothness of the quantity of interest as a function of the parameters. The error estimates explicitly take into account the error of the numerical evaluation of the quantity of interest for fixed sets of parameters. This leads to a coupling condition between this evaluation error which contains the error of the numerical solution of the associated partial differential equation and the error due to the sampling approximation of the quantity of interest.