Adaptive step-size selection for state-space probabilistic differential equation solvers

Abstract

When models are defined implicitly by systems of differential equations with no closed-form solution, small local errors in finite-dimensional solution approximations can propagate into deviations from the true underlying model trajectory. Some recent perspectives in quantifying this uncertainty are based on Bayesian probability modeling: a prior is defined over the unknown solution and updated by conditioning on interrogations of the forward model. Improvement in accuracy via grid refinement must be considered in order for such Bayesian numerical methods to compete with state-of-the-art numerical techniques. We review the principles of Bayesian statistical design and apply these to develop an adaptive probabilistic method to sequentially select time-steps for state-space probabilistic ODE solvers. We investigate the behavior of local error under the adaptive scheme which underlies numerical variable step-size methods. Numerical experiments are used to illustrate the performance of such an adaptive scheme, showing improved accuracy over uniform designs in terms of local error.