Optimal partitions for first eigenvalues of the Laplace operator

Bozorgnia, Farid

doi:10.1002/num.21927

Cited by 8 publications

(7 citation statements)

References 13 publications

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“…Among these we mention [27] where the authors use a rearrangement algorithm to find numerical minimizers for spectral graph partitions, [32] where authors present various results concerning graph and plane partitions. In [13] algorithms for minimizing the sum and the maximum of the eigenvalues are provided, but with few explicit examples. In [18] the authors present a model of chemical reaction which leads to a segregation of phases and is in connection with the minimization of the sum of the eigenvalues.…”

Section: Motivationmentioning

confidence: 99%

Minimal partitions for $p$-norms of eigenvalues

Bonnaillie‐Noël

Bogosel

2018

Interfaces Free Bound.

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In this article we are interested in studying partitions of the square, the disk and the equilateral triangle which minimize a p-norm of eigenvalues of the Dirichlet-Laplace operator. The extremal case of the infinity norm, where we minimize the largest fundamental eigenvalue of each cell, is one of our main interests. We propose three numerical algorithms which approximate the optimal configurations and we obtain tight upper bounds for the energy, which are better than the ones given by theoretical results. A thorough comparison of the results obtained by the three methods is given. We also investigate the behavior of the minimal partitions with respect to p. This allows us to see when partitions minimizing the 1-norm and the infinity-norm are different.With a little abuse of notation, we notice thatThe index ∞ is omitted when there is no confusion. The optimization problem we consider is to determine the infimum of the p-energy (1 ≤ p ≤ ∞) among the partitions of P k (Ω):This optimization problem has been a subject of great interest in the last twenty years. Two cases are especially studied: the sum which corresponds to p = 1 and the max, corresponding to p = ∞. General aspects concerning existence results for optimal partitions problems are presented in [15,14]. Existence and regularity results for optimal partitioning problems regarding non-linear eigenvalue problems, containing as a particular case the Dirichlet eigenvalues, are considered in [17]. In [16] the authors consider the minimization of the partitions minimizing the sum of the Dirichlet-Laplace eigenvalues, stating the spectral honeycomb conjecture and initiating many theoretical and numerical works on the subject. In [24] the authors consider the partitions minimizing the maximum of the fundamental eigenvalues and they provide results concerning connections between such optimal partitions and nodal partitions, for particular values of k. More recently, the link between these two optimization problems is taken into consideration in [23]. In particular, a criterion is established to assert that a ∞-minimal k-partition is not a 1-minimal k-partition. This criterion is given in Proposition 3.8 and applied in Section 4.4.There are few cases for which optimal partitions are known explicitly for the spectral quantities we consider here. This motivates the development of numerical algorithms which can find approximations of optimal partitions and suggest candidates as optimal partitions. The case p = 1, corresponding to the sum of the eigenvalues, was considered in [12], where an algorithm based on a relaxation procedure was presented. The algorithm allowed the study of partitions made of several hundreds of cells and shows that it is likely that partitions made of hexagons are a good candidate to being minimal as k → ∞. The numerical minimization of the largest eigenvalue has been considered in [8,5,9,10]. In [8], we exhibit some candidates for the 3-partition of the square and the disk by using a mixed Dirichlet-Neumann approach that will be used in Sec...

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Section: Motivationmentioning

confidence: 99%

Minimal partitions for $p$-norms of eigenvalues

Bonnaillie‐Noël

Bogosel

2018

Interfaces Free Bound.

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“…In this case the proposed algorithm is lack of deep analysis, especially for the case of three and more competing populations. In the recent work by the current author in collaboration [2] the existence and uniqueness of the scheme, which solves the system (2), have been proven, provided all f i (z, s) are nonnegative and nondecreasing with respect to s. It is noteworthy, that the difference schemes with the same spirit as the system (2), have been successfully applied in quadrature domains theory (see [12]) and in optimal partitions theory (see [10]). This makes us to strongly believe that the ideas behind the difference scheme (2) have great opportunities to be applied in different problems, where the segregated geometry arise.…”

Section: Finite Difference Schemementioning

confidence: 99%

“…Recalling the definition of M h and R h for arbitrary l ∈ 1, m, and z ∈ Ω h we get ûl (z) − ûl h (z) + V h (z) ≤ max Ω h V h (z), ûl h (z) − ûl (z) + V h (z) ≤ max Ω h V h (z). (10)…”

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confidence: 99%

Convergence of the finite difference scheme for a general class of the spatial segregation of reaction–diffusion systems

Arakelyan

2018

Computers & Mathematics with Applications

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In this work we prove convergence of the finite difference scheme for equations of stationary states of a general class of the spatial segregation of reaction-diffusion systems with m ≥ 2 components. More precisely, we show that the numerical solution u l h , given by the difference scheme, converges to the l th component u l , when the mesh size h tends to zero, provided u l ∈ C 2 (Ω), for every l = 1, 2, . . . , m. In particular, our proof provides convergence of a difference scheme for the multi-phase obstacle problem.2010 Mathematics Subject Classification. 35R35, 65N06, 65N22, 92D25.

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“…An adaptation of the algorithm in [BBO10] is presented in [BV16] in the case of the multiphase problem where the objective functional is the sum of the fundamental eigenvalues and an area penalization. In [Boz15] a method for minimizing the sum of the eigenvalues and the maximal eigenvalue is proposed. In [BBN16] the authors study the minimization of the largest eigenvalue by minimizing some p-norms of eigenvalues for large p. They propose a grid restriction procedure in the plane with the purpose of obtaining better precision.…”

Section: Introductionmentioning

confidence: 99%

Efficient algorithm for optimizing spectral partitions

Bogosel¹

2018

Applied Mathematics and Computation

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We present an amelioration of current known algorithms for optimal spectral partitioning problems. The idea is to use the advantage of a representation using density functions while decreasing the computational time. This is done by restricting the computation to neighbourhoods of regions where the associated densities are above a certain threshold. The algorithm extends and improves known methods in the plane and on surfaces in dimension 3. It also makes possible to make some of the first computations of volumic 3D spectral partitions on sufficiently large discretizations.

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Optimal partitions for first eigenvalues of the Laplace operator

Cited by 8 publications

References 13 publications

Minimal partitions for $p$-norms of eigenvalues

Minimal partitions for $p$-norms of eigenvalues

Convergence of the finite difference scheme for a general class of the spatial segregation of reaction–diffusion systems

Efficient algorithm for optimizing spectral partitions

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