Parallel-in-Time Probabilistic Numerical ODE Solvers

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 computational complexity 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.