A Budgeted Multi-Level Monte Carlo Method for Full Field Estimates of Multi-PDE Problems

TL;DR

This paper introduces a novel budgeted multi-level Monte Carlo method that efficiently estimates full field solutions of multi-PDE problems with random data, optimizing memory and CPU usage without prior regularity knowledge.

Contribution

It develops a sparse multi-index update algorithm within a budgeted multi-level Monte Carlo framework for full field estimates, maintaining low memory and CPU costs.

Findings

Full field estimates achieved at the same CPU-time as single quantities

Memory usage comparable to deterministic problem formulations

Validated on diverse multi-PDE problems including elliptic and hyperbolic equations

Abstract

We present a high-performance budgeted multi-level Monte Carlo method for estimates on the entire spatial domain of multi-PDE problems with random input data. The method is designed to operate optimally within memory and CPU-time constraints and eliminates the need for a priori knowledge of the problem's regularity and the algorithm's potential memory demand. To achieve this, we build on the budgeted multi-level Monte Carlo framework and enhance it with a sparse multi-index update algorithm operating on a dynamically assembled parallel data structure to enable estimates of the full field solution. We demonstrate numerically and provide mathematical proof that this update algorithm allows computing the full spatial domain estimates at the same CPU-time cost as a single quantity of interest, and that the maximum memory usage is similar to the memory demands of the deterministic formulation of the problem despite solving the stochastic formulation in parallel. We apply the method to a sequence of interlinked PDE problems, ranging from a stochastic partial differential equation for sampling random fields that serve as the diffusion coefficient in an elliptic subsurface flow problem, to a hyperbolic PDE describing mass transport in the resulting flux field.