Abstract:
Probabilistic numerics has emerged as a promising approach for quantifying and propagating numerical error in computational simulations. Given a numerical problem, probabilistic numerical methods compute not only a point estimate for the solution, but they provide a full posterior distribution over the quantity of interest. This distribution quantifies the numerical approximation error of the method in a structured manner.
For ordinary differential equations (ODEs), probabilistic numerical methods based on Gaussian filtering and smoothing have been introduced as a particularly promising class of methods. These so-called ODE filters scale linearly in the number of time steps, they satisfy well-known stability properties, and they converge to the true solution with polynomial rates, all while also providing a posterior distribution over the ODE solution. But despite these properties, ODE filters have not yet reached the same level of computational efficiency and versatility as their non-probabilistic counterparts.
In this thesis, we address these limitations and establish ODE filtering as a flexible, efficient, and feature-rich framework for probabilistic numerical simulation and inference. To achieve this, we examine the different building blocks of these solvers, including the underlying prior model, information operator, and approximate inference scheme, and we show how they can be adjusted to further improve the performance and utility of these methods.
Our contributions broadly fall into four categories. First, we consider particular classes of differential equations and we leverage their structure to develop specialized solvers with improved stability and efficiency for semi-linear ODEs, higher-order ODEs, Hamiltonian dynamical systems, and differential-algebraic equations. Second, we leverage structure of the underlying state-space model and develop efficient ODE filters for high-dimensional problems. Third, we investigate the underlying inference algorithm and its time discretization, and we present a step-size adaptation scheme as well as a parallel-in-time ODE filter, both of which can provide significant speed-ups. Fourth, we apply ODE filters to parameter inference problems and we develop new numerical-error-aware algorithms for robust parameter inference in ODEs. In addition, to make these methods accessible to a broader audience and to facilitate their application in practice, we develop ProbNumDiffEq.jl, an efficient, accessible, and feature-rich open-source software library for probabilistic numerical ODE solvers.
In summary, this thesis improves the computational efficiency, stability, and versatility of ODE filters and reduces the gap between classic and probabilistic numerical methods. Our contributions thereby establish ODE filtering as a flexible and efficient framework for probabilistic numerical simulation and inference and pave the way for the broader adoption of probabilistic numerics in scientific computing and engineering applications.
