Sparse Probabilistic Richardson Extrapolation

TL;DR

This paper introduces a sparse probabilistic Richardson extrapolation method that reduces simulation costs in multi-parameter numerical methods by exploiting sparsity, with strong theoretical guarantees and empirical validation.

Contribution

It develops a sparsity-exploiting approach to probabilistic Richardson extrapolation, significantly lowering the simulation complexity for multi-tolerance parameter problems.

Findings

Reduces the number of simulations needed in multi-parameter extrapolation.

Provides sharp theoretical guarantees for the proposed method.

Demonstrates empirical effectiveness in numerical experiments.

Abstract

Almost every numerical task can be cast as extrapolation with respect to the fidelity or tolerance parameters of a consistent numerical method. This perspective enables probabilistic uncertainty quantification and optimal experimental design functionality to be deployed, and also unlocks the potential for the convergence of numerical methods to be accelerated. Recent work established Probabilistic Richardson Extrapolation as a proof-of-concept, demonstrating how parallel multi-fidelity simulation can be used to accelerate simulation from a whole-heart model. However, the number of simulations was required to increase super-exponentially in $d$, the number of tolerance parameters appearing in the numerical method. This paper develops a refined notion of 'extrapolation dimension', drastically reducing this simulation requirement when multiple tolerance parameters feature in the numerical method. Sparsity-exploiting methodology is developed that is simultaneously simpler and more powerful compared to earlier work, and this is accompanied by sharp theoretical guarantees and substantial empirical support.