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arxiv: 2606.07979 · v1 · pith:YCSZYGGBnew · submitted 2026-06-06 · 🧮 math.NA · cs.NA· math.AP· math.OC

A New Level Set Formulation for Improved Dirichlet Eigenvalue Minimizers

Atharv Thakur This is my paper

Pith reviewed 2026-06-27 19:44 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.APmath.OC
keywords level set methodsDirichlet eigenvaluesshape optimizationvolume constraintnumerical optimizationspectral geometryeigenvalue minimization
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The pith

A revised level set formulation computes Dirichlet eigenvalue minimizers that are comparable to or better than the best known.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an improved level set approach for finding shapes that minimize Dirichlet eigenvalues while keeping the volume fixed. It overhauls the level set construction and root-finding steps and introduces a regularized version of the objective function. These changes lead to numerical results that match or surpass previous best minimizers in the literature. A sympathetic reader would care because better computational tools help explore open questions in spectral shape optimization. The work also points to several specific subproblems where such experiments could be useful.

Core claim

By overhauling the classical level set construction and root-finding procedures and using a regularized approximation to the objective function, the new formulation yields computational minimizers for Dirichlet eigenvalues under volume constraint that are either comparable to or improvements on the best known from the literature.

What carries the argument

Overhauled level set construction and root-finding procedures with a regularized objective function approximation

Load-bearing premise

The regularized approximation to the standard objective function preserves the location of the true minimizers without introducing significant bias or shifting the optimum.

What would settle it

Applying the method to a standard test case such as the first Dirichlet eigenvalue and obtaining a minimizer whose eigenvalue value is not at least as low as the current best known result from the literature would falsify the improvement claim.

Figures

Figures reproduced from arXiv: 2606.07979 by Atharv Thakur.

Figure 1
Figure 1. Figure 1: Minimizers 1-25 in Row Major Order [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Minimizers 26-50 in Row Major Order [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Uppermost Eigenfunction Zero Sets for the Minimizer of λ13 attempting to minimize Jp for larger p or directly minimize J did not yield superior results. Thus, we can be fairly certain that our computational results are more or less representative of the true shapes — or at least local minima — and limited primarily by computational precision. As a last remark, we note that our results largely avoid a commo… view at source ↗
Figure 4
Figure 4. Figure 4: Difference Between the Dirichlet Counting Function and Weyl Term In the above, ω(n) denotes the volume of the unit ball in R n. George P´olya later conjectured that this term is, in fact, a uniform bound on bounded open domains [18]. For our regions of interest with Ω ⊂ R 2 , this would mean N Dir Ω (λ) < |Ω| 4π λ. Since the minimizers for the Dirichlet eigenvalues are the natural extreme cases for this co… view at source ↗
Figure 5
Figure 5. Figure 5: Weights vs p for k ∈ [5, 6, 7, 8] [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Weights vs p for k ∈ [9, 10, 15, 17, 22] [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
read the original abstract

This paper makes several improvements to existing level set based approaches to computing shape optimizers for the Dirichlet eigenvalues subject to a volume constraint. The most notable changes in formulation include an overhaul of the classical level set construction and root-finding procedures as well the use of a regularized approximation to the standard objective function. Our resulting computational minimizers are either comparable to or improvements on the best known minimizers from the literature. We conclude with a survey of subproblems within the field that may benefit from numerical experiments; these include the existence of cusps on the boundary, the end-behavior of eigenfunction weights in the p-parameterized problem, and the nature of Weyl asymptotics as they relate to the P\'olya conjecture.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes improvements to level-set methods for computing volume-constrained Dirichlet eigenvalue minimizers. It overhauls the classical level-set construction and root-finding procedures and introduces a regularized approximation to the objective function. The central claim is that the resulting computational minimizers are comparable to or better than the best-known values in the literature; the paper concludes with a survey of open subproblems (cusps, p-parameterized eigenfunction weights, Weyl asymptotics and Pólya’s conjecture) that may benefit from further numerical work.

Significance. If the regularization does not displace the true argmin, the revised formulation could supply more reliable numerical evidence for spectral shape optimization problems. The survey of subproblems is a constructive contribution that may help focus future computational studies in the field.

major comments (1)
  1. [Section describing the regularized objective (near the formulation overhaul)] The regularized approximation to the objective function is introduced without an error analysis, convergence statement, or a-posteriori validation (e.g., comparison of minimizers obtained with the regularized versus unregularized functional for the same mesh and parameter values). Because the central claim rests on the computed shapes being genuine improvements rather than artifacts of the approximation, this omission is load-bearing.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the single major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The regularized approximation to the objective function is introduced without an error analysis, convergence statement, or a-posteriori validation (e.g., comparison of minimizers obtained with the regularized versus unregularized functional for the same mesh and parameter values). Because the central claim rests on the computed shapes being genuine improvements rather than artifacts of the approximation, this omission is load-bearing.

    Authors: We agree that the manuscript would benefit from explicit validation of the regularization step to support the claim that the reported minimizers are genuine improvements. A complete a-priori convergence analysis lies beyond the scope of the present computational paper, but we will add a new subsection containing a-posteriori numerical comparisons. For several representative volume constraints and mesh resolutions we will recompute the optimizers both with and without the regularization term (using the same discretization and parameter settings) and report the resulting eigenvalue values together with the Hausdorff distance between the obtained shapes. These tests will confirm that the regularized and unregularized minimizers coincide within discretization error, thereby showing that the regularization does not displace the argmin. revision: yes

Circularity Check

0 steps flagged

No significant circularity; computational method stands on independent numerical comparisons

full rationale

The paper is a computational methods contribution that overhauls level-set construction, root-finding, and introduces a regularized objective for Dirichlet eigenvalue shape optimization. The central claim—that resulting minimizers are comparable or superior to literature values—is presented as an empirical outcome of the new formulation rather than a definitional or fitted tautology. No equations are shown reducing predictions to inputs by construction, no load-bearing self-citations justify uniqueness, and no ansatz is smuggled via prior work. The derivation chain remains self-contained against external benchmarks, consistent with the reader's assessment of score 2.0.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the regularization parameter is implicitly present but not quantified or justified in the provided text.

pith-pipeline@v0.9.1-grok · 5646 in / 1021 out tokens · 15698 ms · 2026-06-27T19:44:28.170702+00:00 · methodology

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Reference graph

Works this paper leans on

20 extracted references · 4 canonical work pages

  1. [1]

    1923 , publisher =

    Faber, Georg , title =. 1923 , publisher =

    1923

  2. [2]

    , journal =

    Krahn, E. , journal =. Über eine von RayIeigh formulierte Minimaleigenschafft des Kreises , url =

  3. [3]

    , title =

    Krahn, E. , title =. Acta et Commentationes Universitatis Dorpatensis , series =

  4. [4]

    Numerical minimization of eigenmodes of a membrane with respect to the domain , url =

    Oudet, Edouard , journal =. Numerical minimization of eigenmodes of a membrane with respect to the domain , url =

  5. [5]

    Archive for Rational Mechanics and Analysis , year=

    Bucur, Dorin , title=. Archive for Rational Mechanics and Analysis , year=. doi:10.1007/s00205-012-0561-0 , url=

    work page doi:10.1007/s00205-012-0561-0

  6. [6]

    Annals of Global Analysis and Geometry , year=

    Berger, Amandine , title=. Annals of Global Analysis and Geometry , year=. doi:10.1007/s10455-014-9446-9 , url=

    work page doi:10.1007/s10455-014-9446-9

  7. [7]

    1993 , publisher =

    An Introduction to -Convergence , author =. 1993 , publisher =

    1993

  8. [8]

    Minimal Convex Combinations of Sequential Laplace--Dirichlet Eigenvalues , volume =

    Osting, Braxton and Kao, C.-Y , year =. Minimal Convex Combinations of Sequential Laplace--Dirichlet Eigenvalues , volume =. SIAM Journal on Scientific Computing , doi =

  9. [9]

    Numerical Optimization of Low Eigenvalues of the Dirichlet and Neumann Laplacians , volume =

    Antunes, Pedro and Freitas, Pedro , year =. Numerical Optimization of Low Eigenvalues of the Dirichlet and Neumann Laplacians , volume =. Journal of Optimization Theory and Applications , doi =

  10. [10]

    Minimal Convex Combinations of Three Sequential Laplace-Dirichlet Eigenvalues , volume =

    Osting, Braxton and Kao, C.-Y , year =. Minimal Convex Combinations of Three Sequential Laplace-Dirichlet Eigenvalues , volume =. Applied Mathematics and Optimization , doi =

  11. [11]

    1992 , publisher=

    Introduction to Shape Optimization: Shape Sensitivity Analysis , author=. 1992 , publisher=

    1992

  12. [12]

    Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations , journal =

    Stanley Osher and James A Sethian , abstract =. Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations , journal =. 1988 , issn =. doi:https://doi.org/10.1016/0021-9991(88)90002-2 , url =

    work page doi:10.1016/0021-9991(88)90002-2 1988

  13. [13]

    doi:10.1007/b98879 , interhash =

    Osher, Stanley and Fedkiw, Ronald , biburl =. doi:10.1007/b98879 , interhash =

    work page doi:10.1007/b98879

  14. [14]

    2017 , eprint=

    Regularity for Shape Optimizers: The Degenerate Case , author=. 2017 , eprint=

    2017

  15. [15]

    Electronic Transactions on Numerical Analysis , volume=

    An implicitly restarted Lanczos method for large symmetric eigenvalue problems , author=. Electronic Transactions on Numerical Analysis , volume=

  16. [16]

    Tessellations in the sciences: Virtues, techniques and applications of geometric tilings , year=

    Alpha shapes-a survey , author=. Tessellations in the sciences: Virtues, techniques and applications of geometric tilings , year=

  17. [17]

    2006 , publisher=

    Extremum problems for eigenvalues of elliptic operators , author=. 2006 , publisher=

    2006

  18. [18]

    Variational methods in shape optimization problems

    Bucur, Dorin and Buttazzo, Giuseppe. Variational methods in shape optimization problems

  19. [19]

    , journal =

    Weyl, H. , journal =. Ueber die asymptotische Verteilung der Eigenwerte , url =

  20. [20]

    , title =

    Polya, G. , title =. 1954 , pages =

    1954