Linear-Time Probabilistic Solutions of Boundary Value Problems
This work addresses boundary value problems for researchers and practitioners in computational mathematics and statistics, offering a probabilistic solver with linear-time efficiency, though it appears incremental as it builds on existing methods with tailored priors.
The authors tackled the problem of solving boundary value problems (BVPs) by proposing a fast probabilistic algorithm that computes a posterior distribution over the solution in linear time, achieving quality and cost comparable to non-probabilistic methods while providing uncertainty quantification and other practical benefits.
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.