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Benchmarking Numerical Algorithms for Harmonic Maps into the Sphere

Sören Bartels, Klaus Böhnlein, Christian Palus, Oliver Sander

TL;DR

Numerically benchmark methods for computing harmonic maps into the unit sphere, with particular focus on harmonic maps with singularities, and observes that while the nonconforming and the conforming discretizations both show similar behavior, the second-order trust-region method needs less iterations than the solver based on gradient flow.

Abstract

We numerically benchmark methods for computing harmonic maps into the unit sphere, with particular focus on harmonic maps with singularities. For the discretization we compare two different approaches, both based on Lagrange finite elements. While the first method enforces the unit-length constraint only at the Lagrange nodes, the other one adds a pointwise projection to fulfill the constraint everywhere. For the solution of the resulting algebraic problems we compare a nonconforming gradient flow with a Riemannian trust-region method. Both are energy-decreasing and can be shown to converge globally to stationary points of the discretized Dirichlet energy. We observe that while the nonconforming and the conforming discretizations both show similar behavior for smooth problems, the nonconforming discretization handles singularities better. On the solver side, the second-order trust-region method converges after few steps, whereas the number of gradient-flow steps increases proportionally to the inverse grid element diameter.

Benchmarking Numerical Algorithms for Harmonic Maps into the Sphere

TL;DR

Numerically benchmark methods for computing harmonic maps into the unit sphere, with particular focus on harmonic maps with singularities, and observes that while the nonconforming and the conforming discretizations both show similar behavior, the second-order trust-region method needs less iterations than the solver based on gradient flow.

Abstract

We numerically benchmark methods for computing harmonic maps into the unit sphere, with particular focus on harmonic maps with singularities. For the discretization we compare two different approaches, both based on Lagrange finite elements. While the first method enforces the unit-length constraint only at the Lagrange nodes, the other one adds a pointwise projection to fulfill the constraint everywhere. For the solution of the resulting algebraic problems we compare a nonconforming gradient flow with a Riemannian trust-region method. Both are energy-decreasing and can be shown to converge globally to stationary points of the discretized Dirichlet energy. We observe that while the nonconforming and the conforming discretizations both show similar behavior for smooth problems, the nonconforming discretization handles singularities better. On the solver side, the second-order trust-region method converges after few steps, whereas the number of gradient-flow steps increases proportionally to the inverse grid element diameter.
Paper Structure (29 sections, 2 theorems, 45 equations, 8 figures, 16 tables, 2 algorithms)

This paper contains 29 sections, 2 theorems, 45 equations, 8 figures, 16 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Harmonic maps into the sphere
  3. Discretizations
  4. Geometrically nonconforming Lagrange finite elements
  5. Geometrically conforming projection-based finite elements
  6. Solvers for the discrete problems
  7. Nonconforming discrete gradient flow
  8. Riemannian trust-region method
  9. Benchmarks problems
  10. Inverse stereographic projection
  11. Radial projection
  12. Multiple singularities
  13. Benchmarking the discretizations
  14. Benchmarking procedure
  15. Inverse stereographic projection
  16. ...and 14 more sections

Key Result

Theorem 4

Let $p=1$ and let $(\mathbf{u}_h)_{h>0} \subset H^1(\Omega; \mathbb{R}^3)$ be a bounded sequence such that each $\mathbf{u}_h \in \mathcal{A}_h^\textup{nc}$ and for all $\boldsymbol{\varphi}_h \in T_h^\textup{nc}(\mathbf{u}_h)$ for functionals $\mathcal{R}_h \in H^1_0(\Omega; \mathbb{R}^3)'$ with $\mathcal{R}_h \to 0$ in $H^1_0(\Omega; \mathbb{R}^3)'$ as $h \to 0$. Then every weak accumulation po

Figures (8)

  • Figure 1: Three first-order projection-based finite element functions on a triangle, with values in $S^1$. The prescribed values at the Lagrange points are enlarged for better visibility.
  • Figure 2: Steps of the discrete gradient flow
  • Figure 3: Discrete solutions $\mathbf{u}_h$ for Problems \ref{['prob:inv_stereo']}, \ref{['prob:singular']}\ref{['subprob:radial33']} and \ref{['prob:singular']}\ref{['subprob:radial22']}
  • Figure 4: Stationary configurations of harmonic maps from $(-\frac{1}{2},\frac{1}{2})^3$ to $S^2$ with multiple singularities, obtained with the nonconforming discretization and Algorithm \ref{['alg:gradflow']} with initial singularity degrees $\kappa = 2,3,4,5$ (left to right). Singularities are located on the horizontal mid-surface, which is colored by the Frobenius norm of the discrete solution gradients.
  • Figure 5: Initial configurations $\mathbf{u}_h^0$ for Problems \ref{['prob:inv_stereo']}, \ref{['prob:singular']}\ref{['subprob:radial33']} and \ref{['prob:singular']}\ref{['subprob:radial22']}. These are perturbations of the problem solutions.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Definition 1: Stationary harmonic map
  • Definition 2: (Locally) minimizing harmonic map
  • Definition 3: Distributional harmonic map
  • Theorem 4: Discrete compactness, Bartels15_book
  • proof : Sketch of the proof
  • Remark 5
  • Lemma 6
  • proof