A Dimension-Adaptive Multi-Index Monte Carlo Method Applied to a Model of a Heat Exchanger

TL;DR

This paper introduces an adaptive Multi-Index Monte Carlo method that efficiently simulates PDEs with random coefficients, optimizing the Karhunen-Loève expansion and spatial discretization to reduce computational costs.

Contribution

It develops a novel adaptive algorithm for the Multi-Index Monte Carlo method that automatically determines the necessary expansion terms and discretizations, enhancing efficiency in stochastic PDE simulations.

Findings

Achieved significant computational savings in heat exchanger model simulations.

Successfully applied the adaptive method to a lognormal random field in a PDE context.

Demonstrated the method's effectiveness in a simplified heat exchanger model.

Abstract

We present an adaptive version of the Multi-Index Monte Carlo method, introduced by Haji-Ali, Nobile and Tempone (2016), for simulating PDEs with coefficients that are random fields. A classical technique for sampling from these random fields is the Karhunen-Lo\`eve expansion. Our adaptive algorithm is based on the adaptive algorithm used in sparse grid cubature as introduced by Gerstner and Griebel (2003), and automatically chooses the number of terms needed in this expansion, as well as the required spatial discretizations of the PDE model. We apply the method to a simplified model of a heat exchanger with random insulator material, where the stochastic characteristics are modeled as a lognormal random field, and we show consistent computational savings.