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Metric-driven numerical methods

Patrick Henning, Laura Huynh, Daniel Peterseim

TL;DR

The paper develops metric-driven numerical methods for multiscale PDEs by integrating energy-based metrics with Sobolev (and Riemannian) gradients and by enriching approximation spaces via Localized Orthogonal Decomposition (LOD). It shows that an operator-induced energy metric yields both fast, constraint-preserving gradient dynamics and a corresponding metric-driven space that coincides with LOD, enabling robust multiscale approximations without resolving fine scales. These ideas are extended from linear eigenproblems through nonlinear Gross–Pitaevskii equations to complex two-component spin-orbit coupled Bose–Einstein condensates, with rigorous convergence results and practical numerical experiments. The framework delivers substantial improvements in convergence speed and accuracy, particularly for multiscale and low-regularity problems, and demonstrates applicability to physically rich quantum systems such as SO-coupled BEC ground states and supersolid-like density patterns.

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.

Metric-driven numerical methods

TL;DR

The paper develops metric-driven numerical methods for multiscale PDEs by integrating energy-based metrics with Sobolev (and Riemannian) gradients and by enriching approximation spaces via Localized Orthogonal Decomposition (LOD). It shows that an operator-induced energy metric yields both fast, constraint-preserving gradient dynamics and a corresponding metric-driven space that coincides with LOD, enabling robust multiscale approximations without resolving fine scales. These ideas are extended from linear eigenproblems through nonlinear Gross–Pitaevskii equations to complex two-component spin-orbit coupled Bose–Einstein condensates, with rigorous convergence results and practical numerical experiments. The framework delivers substantial improvements in convergence speed and accuracy, particularly for multiscale and low-regularity problems, and demonstrates applicability to physically rich quantum systems such as SO-coupled BEC ground states and supersolid-like density patterns.

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.

Paper Structure

This paper contains 23 sections, 13 theorems, 132 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The concept of metric-driven discretizations
  3. Metric-driven steepest descents and Riemannian Sobolev gradients
  4. Sobolev gradients
  5. Riemannian Sobolev gradients
  6. Metric-driven gradient flow
  7. Metric-driven Riemannian gradient method
  8. Minimization in metric-driven approximation spaces
  9. Derivation of a metric-driven approximation space
  10. Approximation properties and Localized Orthogonal Decomposition
  11. Metric-driven discretizations for the Gross--Pitaevskii equation
  12. Metric-driven steepest descents for the GPE
  13. Minimization of the GPE functional in metric-driven approximation spaces
  14. Application to ground states of spin-orbit coupled Bose--Einstein condensates
  15. Mathematical setting and problem formulation
  16. ...and 8 more sections

Key Result

Theorem 2.1

In the general setting of this section, let $u^0 \in \mathbb{S}$ denote a nonnegative starting value and consider the iterations given by RSG-linear. Then there exists a step size interval $[\tau_{\hbox{\normalfont\tiny min}}, \tau_{\hbox{\normalfont\tiny max}}] \subset (0,2)$ such that for all $\ta where $u^{\ast} \in \mathbb{S}$ is the unique positive minimizer of problem energy-minimization-pro

Figures (4)

  • Figure 1: Comparison of the energy per iteration for the MDRGM, the inverse iteration and the GFDN.
  • Figure 2: Converged ground state density $|{\bf u}|^2$. The left picture shows the density of the first component and the right picture the density of the second component.
  • Figure 3: Convergence of the metric-driven approximation (LOD, left) compared with standard $\mathbb{P}^1$-FEM (right). Shown are the $H^1$ and energy errors with respect to a sequence of uniformly refined triangular meshes of width $H$.
  • Figure 4: Ground state density $|{\bf u}|^2$ obtained with the LOD method on refinement level 4. Left: first component. Right: second component. The main qualitative features of the disk-like structure are already captured at this resolution.

Theorems & Definitions (24)

  • Theorem 2.1
  • Definition 2.2: Metric-driven discretization
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • Definition 3.1: Metric-driven Riemannian gradient method for the GPE
  • Theorem 3.2: Global convergence
  • Theorem 3.3: Local convergence
  • Definition 3.4: Metric-driven approximation space
  • Theorem 3.5
  • ...and 14 more