Adaptive step-size selection for state-space probabilistic differential equation solvers
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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.
KeywordsUncertainty quantification Differential equations Statistical design Numerical methods Data assimilation
This material was developed, in part, at the Prob Num 2018 workshop hosted by the Lloyd’s Register Foundation programme on Data-Centric Engineering at the Alan Turing Institute, UK, and supported by the National Science Foundation, USA. This material was based upon work partially supported by the National Science Foundation under Grant DMS-1127914 to the Statistical and Applied Mathematical Sciences Institute (SAMSI) and the Isaac Newton Institute for Mathematical Sciences’ program on Uncertainty Quantification for Complex Systems: theory and methodologies. Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the above-named funding bodies and research institutions.
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