A Simple, Bias-free Approximation of Covariance Functions by the Multilevel Monte Carlo Method Having Nearly Optimal Complexity

We develop simple and bias-free Monte Carlo and multilevel Monte Carlo approximations to covariance functions of sufficiently regular random fields in tensor products of Hilbert spaces. We investigate approximating the covariance function by means of full tensor product approximations, and additionally derive sparse tensor product approximation variants to overcome the curse of dimensionality and yield essentially optimal complexity, i.e., up to logarithmic factors. A priori convergence and work estimates for the different variants are given and subsequently compared theoretically and numerically, where experiments for an exemplary elliptic diffusion problem validate the theoretical findings.

Automated summary

In this study, we develop simple and bias-free Monte Carlo and multilevel Monte Carlo approximations for covariance functions of sufficiently regular random fields in tensor products of Hilbert spaces. We investigate the approximation of the covariance function using full tensor product approximations and derive sparse tensor product approximation variants to overcome the curse of dimensionality and achieve essentially optimal complexity. A priori convergence and work estimates for the different variants are provided and compared theoretically and numerically, with experimental results validating the theoretical findings.