Gamblets for opening the complexity-bottleneck of implicit schemes for hyperbolic and parabolic ODEs/PDEs with rough coefficients

TL;DR

This paper introduces a generalized gamblet method that significantly reduces the computational complexity of solving implicit schemes for hyperbolic and parabolic PDEs with rough coefficients, providing rigorous error bounds.

Contribution

It extends gamblets to handle complex PDEs with rough coefficients, enabling near-linear complexity solutions for implicit schemes.

Findings

Achieves near-linear complexity in solving implicit PDE schemes

Provides rigorous a-priori error bounds for the numerical solutions

Develops a multiresolution decomposition adapted to PDE and time-stepping

Abstract

Implicit schemes are popular methods for the integration of time dependent PDEs such as hyperbolic and parabolic PDEs. However the necessity to solve corresponding linear systems at each time step constitutes a complexity bottleneck in their application to PDEs with rough coefficients. We present a generalization of gamblets introduced in \cite{OwhadiMultigrid:2015} enabling the resolution of these implicit systems in near-linear complexity and provide rigorous a-priori error bounds on the resulting numerical approximations of hyperbolic and parabolic PDEs. These generalized gamblets induce a multiresolution decomposition of the solution space that is adapted to both the underlying (hyperbolic and parabolic) PDE (and the system of ODEs resulting from space discretization) and to the time-steps of the numerical scheme.