A Bulirsch-Stoer algorithm using Gaussian processes
This work addresses uncertainty in numerical integration for scientific computing, representing an incremental advance in probabilistic numerics.
The paper tackles the problem of evaluating asymptotic error in numerical integration schemes by embedding Gaussian process regression into adaptive step-size selection and uncertainty quantification, showing it can match traditional polynomial methods in smooth scenarios and provide reasonable solutions in non-polynomial scenarios like chaotic systems.
In this paper, we treat the problem of evaluating the asymptotic error in a numerical integration scheme as one with inherent uncertainty. Adding to the growing field of probabilistic numerics, we show that Gaussian process regression (GPR) can be embedded into a numerical integration scheme to allow for (i) robust selection of the adaptive step-size parameter and; (ii) uncertainty quantification in predictions of putatively converged numerical solutions. We present two examples of our approach using Richardson's extrapolation technique and the Bulirsch-Stoer algorithm. In scenarios where the error-surface is smooth and bounded, our proposed approach can match the results of the traditional polynomial (parametric) extrapolation methods. In scenarios where the error surface is not well approximated by a finite-order polynomial, e.g. in the vicinity of a pole or in the assessment of a chaotic system, traditional methods can fail, however, the non-parametric GPR approach demonstrates the potential to continue to furnish reasonable solutions in these situations.