Metric-driven numerical methods

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.