Optimization of spectral functions of Dirichlet–Laplacian eigenvalues

Osting, Braxton

doi:10.1016/j.jcp.2010.07.040

Cited by 25 publications

(14 citation statements)

References 38 publications

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“…Our BFGS implementation in hanso has been used to solve a variety of practical nonsmooth problems, such as a condition geodesic problem [3] and shape optimization for spectral functions of Dirichlet-Laplacian operators [40].…”

Section: Softwarementioning

confidence: 99%

Nonsmooth optimization via quasi-Newton methods

2012

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We investigate the behavior of quasi-Newton algorithms applied to minimize a nonsmooth function f , not necessarily convex. We introduce an inexact line search that generates a sequence of nested intervals containing a set of points of nonzero measure that satisfy the Armijo and Wolfe conditions if f is absolutely continuous along the line. Furthermore, the line search is guaranteed to terminate if f is semi-algebraic. It seems quite difficult to establish a convergence theorem for quasi-Newton methods applied to such general classes of functions, so we give a careful analysis of a special but illuminating case, the Euclidean norm, in one variable using the inexact line search and in two variables assuming that the line search is exact. In practice, we find that when f is locally Lipschitz and semi-algebraic with bounded sublevel sets, the BFGS (Broyden-Fletcher-Goldfarb-Shanno) method with the inexact line search almost always generates sequences whose cluster points are Clarke stationary and with function values converging R-linearly to a Clarke stationary value. We give references documenting the successful use of BFGS in a variety of nonsmooth applications, particularly the design of low-order controllers for linear dynamical systems. We conclude with a challenging open question.

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Section: Softwarementioning

confidence: 99%

Nonsmooth optimization via quasi-Newton methods

2012

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“…Doing this truncation, we don't perturb the eigenvalues too much. B. Osting gives an estimate of this error in [23]. In this way we can represent a good approximation of the boundary of a star convex shape by a finite number of parameters.…”

Section: Introductionmentioning

confidence: 99%

Qualitative and Numerical Analysis of a Spectral Problem with Perimeter Constraint

Bogosel¹,

Oudet²

2016

SIAM J. Control Optim.

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We consider the problem of optimizing the k th eigenvalue of the Dirichlet Laplace operator under perimeter constraint. We provide a new method based on a Γconvergence result for approximating the corresponding optimal shapes. We also give new optimality conditions in the case of multiple eigenvalues. We deduce from previous conditions the fact that optimal shapes never contain flat parts in their boundaries.

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“…There are several available methods for solving eigenvalues problems on general domains, including the finite difference [33], finite element [29,7], boundary integral [12], method of particular solutions [22,6,13,40], and meshless methods [3]. In this paper we use the finite element method [33,12,29,7].…”

Section: Methodsmentioning

confidence: 99%

“…Results min |Ω|=1 λ k k = 1: Minimizer is a ball (Rayleigh-Faber-Krahn); see [25,4] k = 2: Minimizer is union of 2 balls of equal volume (Krahn-Szegö); see [25,4] k = 3: Minimizer is connected and a ball is a local minimizer [49] k = 4: Candidate minimizer is union of two balls with different volumes; see [25] k ≥ 5: Numerical results: k = 3 : 10 [43] and k = 5 : 15 [3] min |Ω|=1 (1 − γ)λ k + γλ k+1 For k = 1, only γ = 1 is disconnected [49] k = 1 and γ ∈ [0, 1]: numerical results [49] k = 1 : 14 and γ = 1 2 : numerical results [2] k = 1 : 5 and γ ∈ [0, 1], present work max Ω λ k λ 1 k = 2: Maximizer is a ball (Ashbaugh-Benguria); see [5] Numerical results: k = 3, [35]; k = 3 : 13, [40]; k = 3 : 14, [2] where Λ(Ω) = {λ k (Ω)} ∞ k=1 are the Laplace-Dirichlet eigenvalues of Ω. (See section 2 for a mathematical formulation.)…”

Section: Problemmentioning

confidence: 99%

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Minimal Convex Combinations of Sequential Laplace--Dirichlet Eigenvalues

Osting¹,

Kao²

2013

SIAM J. Sci. Comput.

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We propose a computational approach based on the level set method to solve shape optimization problems where the objective function is dependent on the Laplace-Dirichlet eigenvalues of the domain. The approach is applied to the parameterized problem of minimizing the convex combination of sequential Laplace-Dirichlet eigenvalues, λ k and λ k+1 . We show that as a function of the combination parameter, the optimal value is nondecreasing, Lipschitz continuous, and concave and that the minimizing set is upper hemicontinuous. The domains which minimize the first few Laplace-Dirichlet eigenvalues are known analytically or have been studied computationally, and it is known that the optimal solution for some values of k have multiple connected components. Our computations reproduce these previous results for the appropriate parameter values and extend these results, effectively capturing intermediate topology changes. The results are also compared to values obtained analytically for rectangular and elliptical shapes and to values for domains with nearly circular boundaries.

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Optimization of spectral functions of Dirichlet–Laplacian eigenvalues

Cited by 25 publications

References 38 publications

Nonsmooth optimization via quasi-Newton methods

Nonsmooth optimization via quasi-Newton methods

Qualitative and Numerical Analysis of a Spectral Problem with Perimeter Constraint

Minimal Convex Combinations of Sequential Laplace--Dirichlet Eigenvalues

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