Convergence Acceleration for Time Dependent Parametric Multifidelity Models

TL;DR

This paper introduces a novel convergence acceleration method for multifidelity models of parameterized ODEs, leveraging a three-step algorithm with spline interpolation and Richardson extrapolation to improve accuracy efficiently.

Contribution

It provides a new convergence analysis and develops a three-step algorithm that combines multifidelity surrogates, spline interpolation, and sequence transformation for faster convergence.

Findings

Effective acceleration on linear systems

Improved accuracy on nonlinear systems

Convergence estimates validated numerically

Abstract

We present a numerical method for convergence acceleration for multifidelity models of parameterized ordinary differential equations. The hierarchy of models is defined as trajectories computed using different timesteps in a time integration scheme. Our first contribution is in novel analysis of the multifidelity procedure, providing a convergence estimate. Our second contribution is development of a three-step algorithm that uses multifidelity surrogates to accelerate convergence: step one uses a multifidelity procedure at three levels to obtain accurate predictions using inexpensive (large timestep) models. Step two uses high-order splines to construct continuous trajectories over time. Finally, step three combines spline predictions at three levels to infer an order of convergence and compute a sequence transformation prediction (in particular we use Richardson extrapolation) that achieves superior error. We demonstrate our procedure on linear and nonlinear systems of parameterized ordinary differential equations.