Parallel-in-Time Probabilistic Numerical ODE Solvers
Nathanael Bosch Adrien Corenflos Fatemeh Yaghoobi Filip Tronarp Philipp Hennig Simo Särkkä
Probabilistic numerical solvers for ordinary differential equations (ODEs)
treat the numerical simulation of dynamical systems as problems of Bayesian
state estimation. Aside from producing posterior distributions over ODE
solutions and thereby quantifying the numerical approximation error of the
method itself, one less-often noted advantage of this formalism is the
algorithmic flexibility gained by formulating numerical simulation in the
framework of Bayesian filtering and smoothing. In this paper, we leverage this
flexibility and build on the time-parallel formulation of iterated extended
Kalman smoothers to formulate a parallel-in-time probabilistic numerical ODE
solver. Instead of simulating the dynamical system sequentially in time, as
done by current probabilistic solvers, the proposed method processes all time
steps in parallel and thereby reduces the span cost from linear to logarithmic
in the number of time steps. We demonstrate the effectiveness of our approach
on a variety of ODEs and compare it to a range of both classic and
probabilistic numerical ODE solvers.