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In this paper, we put forward an efficient spectral-Galerkin approximation in view of dimension reduction scheme for transmission eigenvalue problem in polar geometries. Firstly, we turn the original problem into an equivalent fourth order nonlinear eigenvalue problem. Then the fourth order nonlinear eigenvalue problem is transformed into a coupled fourth order linear eigenvalue system by introducing an auxiliary Poisson equation. Secondly, based on polar coordinate transformation, we further reduce the coupled fourth order linear eigenvalue system to a series of equivalent one-dimensional eigenvalue systems. Thirdly, we derive the essential polar condition and introduce the appropriate weighted Sobolev space according to the polar condition, and establish the weak form and the corresponding discrete form. In addition, by utilizing spectral theory of compact operators, we prove the error estimates of approximation eigenvalues and eigenvectors for each one-dimensional eigenvalue system. Finally, we provide ample numerical experiments, and the numerical results show the effectiveness of the algorithm and the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

To keep structures in the restoration problem is very important via coupling the local information of the image with the proposed model. In this paper we propose a local self-adaptive ℓ p -regularization model for p ∈ (0 , 2) based on the total variation scheme, where the choice of p depends on the local structures described by the eigenvalues of the structure tensor. Since the proposed model as the classic ℓ p problem unifies two classes of optimization problems such as the nonconvex and nonsmooth problem when p ∈ (0 , 1) , and the convex and smooth problem when p ∈ (1 , 2) , it is generally challenging to find a ready algorithmic framework to solve it. Here we propose a new and robust numerical method via coupling with the half-quadratic scheme and the alternating direction method of multipliers (ADMM). The convergence of the proposed algorithm is established and the numerical experiments illustrate the possible advantages of the proposed model and numerical methods over some existing variational-based models and methods. [ABSTRACT FROM AUTHOR]

Abstract: For computing the smallest eigenvalues and their corresponding eigenvectors of symmetric generalized eigenproblems simultaneously, Quillen and Ye have introduced a block inverse-free preconditioned Krylov subspace method (Quillen and Ye, 2010) [14]. To accelerate convergence and compute interior eigenpairs, in this paper we present a refined shifted block inverse-free Krylov subspace algorithm based on the block Arnoldi process that generates a -orthogonality basis of the matrix Krylov subspace. It is proved that this algorithm can guarantee the convergence if the corresponding Ritz values converge. Numerical experiments show that the refined algorithm is more efficient than the original approach. [Copyright &y& Elsevier]

Abstract: Designing an acoustic drum can be categorized into a class of eigenvalue optimization problems in the structural engineering area. In this paper, we propose an algorithm that is based on the Gâteaux derivative of the objective function with respect to the density functions and analyze our algorithm in detail. In the algorithm, we deal with the geometry constraint by exchanging the densities of two domains occupied by two kinds of different materials. Finally we apply this algorithm to some practical examples frequently used by the researchers and present some numerical results to show its feasibility, stability and efficiency. [Copyright &y& Elsevier]

Abstract: The block Arnoldi method is one of the most commonly used techniques for large eigenproblems. In this paper, we exploit certain modified Ritz vectors to take the place of Ritz vectors in the thick-restarted block Arnoldi algorithm, and propose a modified thick-restarted block Arnoldi algorithm for large eigenproblems. We then consider how to periodically combine the refined subspace iterative method with the modified thick-restarting block Arnoldi algorithm for computing a few dominant eigenpairs of a large matrix. The resulting algorithm is called a Subspace-Block Arnoldi algorithm. Numerical experiments show the efficiency of our new algorithms. [Copyright &y& Elsevier]

Abstract: In this paper, an alternatively simpler proof to an eigenvalue theorem of a specially structured rank-r updated complex matrix is presented and also its characteristic polynomial is explicitly determined by Leverrier’s algorithm for system. [Copyright &y& Elsevier]

Suppose the spectrum of a symmetric definite linear pencil is known. This paper addresses the question of what can be said about the spectrum when scalar multiples of a rank-one update are added to each matrix in the pencil.The secular equation for this problem is derived, and from it, a certain separation property is found which gives insight into the connection between the eigenvalues before and after modification.In the context of structural dynamics, the result characterises the behaviour of a finite-dimensional vibrating system undergoing mass and stiffness modifications.The result also leads to applications such as a divide and conquer algorithm for the eigenvalues of the modified system (so-called matrix tearing) and spectral shifting. An illustrative example is also given. [Copyright &y& Elsevier]