Related papers: Numerical Eigenvalue Optimization by Shape-Variati…

We carry on our study of the connection between two shape optimization problems with spectral cost. On the one hand, we consider the optimal design problem for the survival threshold of a population living in a heterogenous habitat…

We investigate shape optimization for the principal eigenvalue of the Pucci extremal operator \[ \left\{ \begin{aligned} -\mathcal{M}^+_{\lambda,\Lambda}(D^{2}u)&=\mu^{+}_{1}(\Omega)u &&\text{in }\Omega,\\ u &=0 &&\text{on }\partial\Omega,…

We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the…

We study three types of fourth-order Steklov eigenvalue problems. For the first two of them, we derive the asymptotic expansion of their spectra on Euclidean annular domains $\mathbb{B}^n_1\setminus \overline{\mathbb{B}^n_\epsilon}$ as…

In this work we study a general shape optimization problem where the state equation is given in terms of a nonlocal operator. Examples of the problems considered are monotone combinations of fractional eigenvalues. Moreover, we also analyze…

We consider a shape optimization problem for the first mixed Steklov-Dirichlet eigenvalues of domains bounded by two balls in two-point homogeneous space. We give a geometric proof which is motivated by Newton's shell theorem

We study the eigenvalue problem for the complex Monge-Amp\`ere operator in bounded hyperconvex domains in $\C^n$, where the right-hand side is a non-pluripolar positive Borel measure. We establish the uniqueness of eigenfunctions in the…

We propose and analyze an efficient spectral-Galerkin approximation for the Maxwell transmission eigenvalue problem in spherical geometry. Using a vector spherical harmonic expansion, we reduce the problem to a sequence of equivalent…

We present a new approach to compute selected eigenvalues and eigenvectors of the two-parameter eigenvalue problem. Our method requires computing generalized eigenvalue problems of the same size as the matrices of the initial two-parameter…

We consider the multiphase shape optimization problem $$\min\Big\{\sum_{i=1}^h\lambda_1(\Omega_i)+\alpha|\Omega_i|:\ \Omega_i\ \hbox{open},\ \Omega_i\subset D,\ \Omega_i\cap\Omega_j=\emptyset\Big\},$$ where $\alpha>0$ is a given constant…

This paper sets up an approach for shape optimization problems constrained by variational inequalities (VI) in an appropriate shape space. In contrast to classical VI, where no explicit dependence on the domain is given, VI constrained…

In this article, we study numerical approximation of eigenvalue problems of the Schr\"{o}dinger operator $\displaystyle -\Delta u + \frac{c^2}{|x|^2}u$. There are three stages in our investigation: We start from a ball of any dimension, in…

A multigrid method is proposed in this paper to solve eigenvalue problems by the finite element method based on the shifted-inverse power iteration technique. With this scheme, solving eigenvalue problem is transformed to a series of…

A method to compute guaranteed lower bounds to the eigenvalues of the Maxwell system in two or three space dimensions is proposed as a generalization of the method of Liu and Oishi [SIAM J. Numer. Anal., 51, 2013] for the Laplace operator.…

Kernel-based methods in Numerical Analysis have the advantage of yielding optimal recovery processes in the "native" Hilbert space $\calh$ in which they are reproducing. Continuous kernels on compact domains have an expansion into…

For the eigenvalue problem of the Steklov differential operator, by following Liu's approach, an algorithm utilizing the conforming finite element method (FEM) is proposed to provide guaranteed lower bounds for the eigenvalues. The proposed…

In this paper two types of multgrid methods, i.e., the Rayleigh quotient iteration and the inverse iteration with fixed shift, are developed for solving the Maxwell eigenvalue problem with discontinuous relative magnetic permeability and…

Shape optimization with respect to eigenvalues of a cavity plays an important role in the design of new resonators or in the optimization of existing ones. In our paper, we propose a gradient-based optimization scheme, which we enhance with…

We formulate an optimization problem for the dependence of the eigenvalues of Maxwell's equations in a cavity upon variation of the electric permittivity and we prove a corresponding Maximum Principle.

The purpose of this study is twofold. First, we revisit a shape optimization reformulation of a prototypical shape inverse problem and briefly propose a simple yet efficient numerical approach for solving the corresponding minimization…