[Paper Review] A particular class of solutions of a system of eikonal equations

Thierry Champion, Gisella Croce|arXiv (Cornell University)|May 3, 2010

Numerical methods in inverse problems15 references3 citations

TL;DR

This paper introduces a variational selection method for solutions of a system of eikonal equations where the gradient magnitude is almost everywhere 1. By minimizing functionals based on the Hausdorff measure of gradient discontinuity sets—weighted or unweighted—it identifies a distinguished class of solutions, with existence and uniqueness results under specific geometric conditions on the domain, and provides a recursive selection procedure for multiple minimizers.

ABSTRACT

In this article we study a system of eikonal equations. Our aim is to isolate the solutions which minimise the discontinuity set of the gradient.

Motivation & Objective

• To identify a particular class of solutions among the infinitely many solutions of a system of eikonal equations with |∂u/∂xi| = 1 a.e.
• To overcome the lack of convexity in the solution set, which prevents standard minimization of Lp or BV-type functionals.
• To develop a variational criterion based on the discontinuity sets of the partial derivatives ∂u/∂xi to select preferred solutions.
• To establish existence of minimizers for two types of functionals: one based on the (N−1)-dimensional Hausdorff measure of jump sets, and another with a weight function h.
• To relate the selected solutions to viscosity solutions when they exist, particularly in dimensions N=1 and N=2.

Proposed method

• Define the solution set S(Ω) as all u ∈ W^{1,∞}(Ω) ∩ C₀(Ω) satisfying |∂u/∂xi| = 1 a.e. in Ω for i=1,…,N.
• Introduce the functional F(v) = ∑_{i=1}^N ℋ^{N−1}(J_{∂v/∂xi}) to minimize the total jump set measure of the partial derivatives.
• For general domains, use a weighted functional F_h(v) = ∑_{i=1}^N ∫_{J_{∂v/∂xi}} h(x) dℋ^{N−1}(x), with h ∈ C₀(Ω), h ≥ 0.
• Apply direct methods of the calculus of variations using compactness and lower semicontinuity in BV and SBV spaces.
• Use sequential compactness in BV and SBV (via Theorems 6.1–6.3) and lower semicontinuity (Theorem 6.4) to prove existence of minimizers.
• Propose a recursive selection method using L²-orthogonal basis functions to distinguish among multiple minimizers of F_h.

Experimental results

Research questions

• RQ1 Can a unique solution be selected from the infinite family of solutions to the eikonal system |∂u/∂xi| = 1 a.e.?
• RQ2 Does minimizing the total (N−1)-dimensional Hausdorff measure of the jump sets of ∂u/∂xi yield a well-posed variational problem?
• RQ3 How does the minimizer of the weighted functional F_h relate to viscosity solutions when they exist?
• RQ4 Can multiple minimizers of F_h be distinguished using additional variational criteria?
• RQ5 Is there a systematic method to select a single solution from the set of minimizers of F_h?

Key findings

• For domains Ω with boundary composed of finitely many faces whose normals lie in E = {x : |x_i| = 1/√N for all i}, the functional F(v) = ∑_{i=1}^N ℋ^{N−1}(J_{∂v/∂xi}) admits a minimizer in S(Ω).
• For general Lipschitz domains, the unweighted functional F is generally infinite on S(Ω), so a weighted functional F_h with h ∈ C₀(Ω) is introduced and shown to admit a minimizer.
• When N=1 or N=2, the minimizer of F_h coincides with the unique viscosity solution of the eikonal system, if it exists.
• The set of minimizers of F_h may contain more than two functions; a recursive selection method using an L²-orthogonal basis is proposed to isolate a single solution.
• The minimization process relies on compactness in BV and SBV spaces, with weak* convergence of the Lebesgue and jump parts of the derivatives.
• The method ensures that any sequence minimizing F_h has a subsequence converging to a minimizer, via the direct method and semicontinuity of the jump variation.