Linear-Time Probabilistic Solutions of Boundary Value Problems

TL;DR

This paper introduces a linear-time probabilistic algorithm for solving boundary value problems that provides uncertainty quantification and integrates seamlessly with statistical modeling workflows.

Contribution

It presents a novel Gauss--Markov prior tailored for BVPs, enabling efficient probabilistic solutions with uncertainty estimates in linear time.

Findings

Achieves solution in linear time comparable to classical methods

Provides uncertainty quantification and adaptive mesh refinement

Compatible with statistical modeling tool-chains

Abstract

We propose a fast algorithm for the probabilistic solution of boundary value problems (BVPs), which are ordinary differential equations subject to boundary conditions. In contrast to previous work, we introduce a Gauss--Markov prior and tailor it specifically to BVPs, which allows computing a posterior distribution over the solution in linear time, at a quality and cost comparable to that of well-established, non-probabilistic methods. Our model further delivers uncertainty quantification, mesh refinement, and hyperparameter adaptation. We demonstrate how these practical considerations positively impact the efficiency of the scheme. Altogether, this results in a practically usable probabilistic BVP solver that is (in contrast to non-probabilistic algorithms) natively compatible with other parts of the statistical modelling tool-chain.