Linear-Time Probabilistic Solutions of Boundary Value Problems
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.
Cite
@article{arxiv.2106.07761,
title = {Linear-Time Probabilistic Solutions of Boundary Value Problems},
author = {Nicholas Krämer and Philipp Hennig},
journal= {arXiv preprint arXiv:2106.07761},
year = {2021}
}