Probabilistic Richardson extrapolation

For over a century, extrapolation methods have provided a powerful tool to improve the convergence order of a numerical method. However, these tools are not well-suited to modern computer codes, where multiple continua are discretized and convergence orders are not easily analysed. To address this challenge, we present a probabilistic perspective on Richardson extrapolation, a point of view that unifies classical extrapolation methods with modern multi-fidelity modelling, and handles uncertain convergence orders by allowing these to be statistically estimated. The approach is developed using Gaussian processes, leading to Gauss-Richardson Extrapolation. Conditions are established under which extrapolation using the conditional mean achieves a polynomial (or even an exponential) speed-up compared to the original numerical method. Further, the probabilistic formulation unlocks the possibility of experimental design, casting the selection of fidelities as a continuous optimization problem, which can then be (approximately) solved. A case study involving a computational cardiac model demonstrates that practical gains in accuracy can be achieved using the GRE method.