Linear-Time Probabilistic Solutions of Boundary Value Problems

Nicholas Krämer; Philipp Hennig

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arXiv · 2106.07761

Linear-Time Probabilistic Solutions of Boundary Value Problems

Nicholas Krämer·Philipp Hennig

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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.

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BibTeX RIS

Nicholas Krämer, Philipp Hennig. 2021-06-14. Linear-Time Probabilistic Solutions of Boundary Value Problems. https://arxiv.org/abs/2106.07761

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