Metric-driven numerical methods

TL;DR

This paper introduces metric-driven numerical methods using Riemannian gradient techniques to efficiently solve multiscale PDE eigenvalue problems, improving convergence and approximation in complex regimes.

Contribution

It presents a novel perspective on metric-driven methods, connecting them to multiscale spaces like LOD, and applies these techniques to quantum condensate simulations.

Findings

Enhanced convergence via Sobolev gradient techniques.

Derivation of multiscale approximation spaces from metric choices.

Successful application to spin-orbit-coupled Bose-Einstein condensates.

Abstract

In this paper, we explore the concept of metric-driven numerical methods as a powerful tool for solving various types of multiscale partial differential equations. Our focus is on computing constrained minimizers of functionals - or, equivalently, by considering the associated Euler-Lagrange equations - the solution of a class of eigenvalue problems that may involve nonlinearities in the eigenfunctions. We introduce metric-driven methods for such problems via Riemannian gradient techniques, leveraging the idea that gradients can be represented in different metrics (so-called Sobolev gradients) to accelerate convergence. We show that the choice of metric not only leads to specific metric-driven iterative schemes, but also induces approximation spaces with enhanced properties, particularly in low-regularity regimes or when the solution exhibits heterogeneous multiscale features. In fact, we recover a well-known class of multiscale spaces based on the Localized Orthogonal Decomposition (LOD), now derived from a new perspective. Alongside a discussion of the metric-driven approach for a model problem, we also demonstrate its application to simulating the ground states of spin-orbit-coupled Bose-Einstein condensates.