Randomized quasi-optimal local approximation spaces in time

TL;DR

This paper introduces a parallelizable method for constructing local approximation spaces in time for PDEs with heterogeneous coefficients, using randomized simulations and singular value decomposition to achieve quasi-optimal convergence.

Contribution

The paper proposes a novel randomized approach to build local in time approximation spaces for PDEs, enabling parallel computation and provable convergence rates.

Findings

Method outperforms proper orthogonal decomposition in experiments.

Capable of effectively approximating advection-dominated problems.

Constructs local approximation spaces with quasi-optimal convergence.

Abstract

We target time-dependent partial differential equations (PDEs) with heterogeneous coefficients in space and time. To tackle these problems, we construct reduced basis/ multiscale ansatz functions defined in space that can be combined with time stepping schemes within model order reduction or multiscale methods. To that end, we propose to perform several simulations of the PDE for few time steps in parallel starting at different, randomly drawn start points, prescribing random initial conditions; applying a singular value decomposition to a subset of the so obtained snapshots yields the reduced basis/ multiscale ansatz functions. This facilitates constructing the reduced basis/ multiscale ansatz functions in an embarrassingly parallel manner. In detail, we suggest using a data-dependent probability distribution based on the data functions of the PDE to select the start points. Each local in time simulation of the PDE with random initial conditions approximates a local approximation space in one time point that is optimal in the sense of Kolmogorov. The derivation of these optimal local approximation spaces which are spanned by the left singular vectors of a compact transfer operator that maps arbitrary initial conditions to the solution of the PDE in a later point of time, is one other main contribution of this paper. By solving the PDE locally in time with random initial conditions, we construct local ansatz spaces in time that converge provably at a quasi-optimal rate and allow for local error control. Numerical experiments demonstrate that the proposed method can outperform existing methods like the proper orthogonal decomposition even in a sequential setting and is well capable of approximating advection-dominated problems.