Probabilistic Exponential Integrators
- URL: http://arxiv.org/abs/2305.14978v2
- Date: Tue, 19 Dec 2023 15:21:24 GMT
- Title: Probabilistic Exponential Integrators
- Authors: Nathanael Bosch, Philipp Hennig, Filip Tronarp
- Abstract summary: Like standard solvers, they suffer performance penalties for certain stiff systems.
This paper develops a class of probabilistic exponential solvers with favorable properties.
We evaluate the proposed methods on multiple stiff differential equations.
- Score: 36.98314810594263
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: Probabilistic solvers provide a flexible and efficient framework for
simulation, uncertainty quantification, and inference in dynamical systems.
However, like standard solvers, they suffer performance penalties for certain
stiff systems, where small steps are required not for reasons of numerical
accuracy but for the sake of stability. This issue is greatly alleviated in
semi-linear problems by the probabilistic exponential integrators developed in
this paper. By including the fast, linear dynamics in the prior, we arrive at a
class of probabilistic integrators with favorable properties. Namely, they are
proven to be L-stable, and in a certain case reduce to a classic exponential
integrator -- with the added benefit of providing a probabilistic account of
the numerical error. The method is also generalized to arbitrary non-linear
systems by imposing piece-wise semi-linearity on the prior via Jacobians of the
vector field at the previous estimates, resulting in probabilistic exponential
Rosenbrock methods. We evaluate the proposed methods on multiple stiff
differential equations and demonstrate their improved stability and efficiency
over established probabilistic solvers. The present contribution thus expands
the range of problems that can be effectively tackled within probabilistic
numerics.
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