Linear-Time Probabilistic Solutions of Boundary Value Problems

• URL: http://arxiv.org/abs/2106.07761v1
• Date: Mon, 14 Jun 2021 21:19:17 GMT
• Title: Linear-Time Probabilistic Solutions of Boundary Value Problems
• Authors: Nicholas Kr\"amer and Philipp Hennig
• Abstract summary: We introduce a Gauss--Markov prior and tailor it specifically to BVPs. This 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.
• Score: 27.70274403550477
• License: http://creativecommons.org/licenses/by/4.0/
• 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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