A space decomposition scheme for maximum eigenvalue functions and its applications

In this paper, we study nonlinear optimization problems involving eigenvalues of symmetric matrices. One of the difficulties in solving these problems is that the eigenvalue functions are not differentiable when the multiplicity of the function is not one. We apply the \({\mathcal {U}}\)-Lagrangian theory to analyze the largest eigenvalue function of a convex matrix-valued mapping which extends the corresponding results for linear mapping in the literature. We also provides the formula of first-and second-order derivatives of the \({\mathcal {U}}\)-Lagrangian under mild assumptions. These theoretical results provide us new second-order information about the largest eigenvalue function along a suitable smooth manifold, and leads to a new algorithmic framework for analyzing the underlying optimization problem.     

Spectral bundle methods for non-convex maximum eigenvalue functions: second-order methods

We study constrained and unconstrained optimization programs for nonconvex maximum eigenvalue functions. We show how second order techniques may be introduced as soon as it is possible to reliably guess the multiplicity of the maximum eigenvalue at a limit point. We examine in which way standard and projected Newton steps may be combined with a nonsmooth first-order method to obtain a globally convergent algorithm with a fair chance to local superlinear or quadratic convergence. Dedicated to R. T. Rockafellar on the occasion of his 70th anniversary     

Spectral bundle methods for non-convex maximum eigenvalue functions: first-order methods

Many challenging problems in automatic control may be cast as optimization programs subject to matrix inequality constraints. Here we investigate an approach which converts such problems into non-convex eigenvalue optimization programs and makes them amenable to non-smooth analysis techniques like bundle or cutting plane methods. We prove global convergence of a first-order bundle method for programs with non-convex maximum eigenvalue functions. Dedicated to R. T. Rockafellar on the occasion of his 70th anniversary     

The space decomposition theory for a class of eigenvalue optimizations

In this paper we study optimization problems involving eigenvalues of symmetric matrices. One of the difficulties with numerical analysis of such problems is that the eigenvalues, considered as functions of a symmetric matrix, are not differentiable at those points where they coalesce. Here we apply the $\mathcal{U}$ -Lagrangian theory to a class of D.C. functions (the difference of two convex functions): the arbitrary eigenvalue function λ _i , with affine matrix-valued mappings, where λ _i is a D.C. function. We give the first-and second-order derivatives of ${\mathcal{U}}$ -Lagrangian in the space of decision variables R ^m when transversality condition holds. Moreover, an algorithm framework with quadratic convergence is presented. Finally, we present an application: low rank matrix optimization; meanwhile, list its $\mathcal{VU}$ decomposition results.     

A multilevel correction method for Stokes eigenvalue problems and its applications

In this paper, a new multilevel correction scheme is proposed to solve Stokes eigenvalue problems by the finite element method. This new scheme contains a series of correction steps, and the accuracy of eigenpair approximation can be improved after each step. In each correction step, we only need to solve a Stokes problem on the corresponding fine finite element space and a Stokes eigenvalue problem on the coarsest finite element space. This correction scheme can improve the efficiency of solving Stokes eigenvalue problems by the finite element method. As applications of this multilevel correction method, a multigrid method and an adaptive finite element technique are introduced for Stokes eigenvalue problems. Some numerical results are given to validate our schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

A two-grid discretization scheme for eigenvalue problems

A two-grid discretization scheme is proposed for solving eigenvalue problems, including both partial differential equations and integral equations. With this new scheme, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid, and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy.

     

A distribution space for Hilbert transform and its applications

In this paper, a new distribution space is constructed and the definition of the classical Hilbert transform is extended to it. It is shown that is the biggest subspace of on which the extended Hilbert transform is a homeomorphism and both the classical Hilbert transform for L ^p functions and the circular Hilbert transform for periodic functions are special cases of the extension. Some characterizations of the space are given and a class of useful nonlinear phase signals is shown to be in . Finally, the applications of the extended Hilbert transform are discussed. This work was supported by the National Natural Science Foundation of China (Grant Nos. 60475042, 10631080)     

A real algorithm for the Hermitian eigenvalue decomposition

Modifying complex plane rotations, we derive a new Jacobi-type algorithm for the Hermitian eigendecomposition, which uses only real arithmetic. When the fast-scaled rotations are incorporated, the new algorithm brings a substantial reduction in computational costs. The new method has the same convergence properties and parallelism as the symmetric Jacobi algorithm. Computational test results show that it produces accurate eigenvalues and eigenvectors and achieves great reduction in computational time.The work of this author was supported in part by the National Science Foundation grant CCR-8813493 and by the University of Minnesota Army High Performance Computing Research Center contract DAAL 03-89-C-0038.The work of this author was supported in part by the University of Minnesota Army High Performance Computing Research Center contract DAAL 03-89-C-0038.     

A second-order bundle method to minimize the maximum eigenvalue function

In this paper we present a nonsmooth algorithm to minimize the maximum eigenvalue of matrices belonging to an affine subspace of n×n symmetric matrices. We show how a simple bundle method, the approximate eigenvalue method can be used to globalize the second-order method developed by M.L. Overton in the eighties and recently revisited in the framework of the ?-Lagrangian theory. With no additional assumption, the resulting algorithm generates a minimizing sequence. A geometrical and constructive proof is given. To prove that quadratic convergence is achieved asymptotically, some strict complementarity and non-degeneracy assumptions are needed. We also introduce new variants of bundle methods for semidefinite programming. Received: February 9, 1998 / Accepted: May 2, 2000?Published online September 20, 2000     

A generalized eigenvalue problem for quasi-orthogonal rational functions

In general, the zeros of an orthogonal rational function (ORF) on a subset of the real line, with poles among ${\{\alpha_1,\ldots,\alpha_n\}\subset(\mathbb{C}_0\cup\{\infty\})}$ , are not all real (unless ${\alpha_n}$ is real), and hence, they are not suitable to construct a rational Gaussian quadrature rule (RGQ). For this reason, the zeros of a so-called quasi-ORF or a so-called para-ORF are used instead. These zeros depend on one single parameter ${\tau\in(\mathbb{C}\cup\{\infty\})}$ , which can always be chosen in such a way that the zeros are all real and simple. In this paper we provide a generalized eigenvalue problem to compute the zeros of a quasi-ORF and the corresponding weights in the RGQ. First, we study the connection between quasi-ORFs, para-ORFs and ORFs. Next, a condition is given for the parameter ?? so that the zeros are all real and simple. Finally, some illustrative and numerical examples are given.     

A two-grid discretization scheme for the Steklov eigenvalue problem

In the paper, a two-grid discretization scheme is discussed for the Steklov eigenvalue problem. With the scheme, the solution of the Steklov eigenvalue problem on a fine grid is reduced to the solution of the Steklov eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid. Using spectral approximation theory, it is shown theoretically that the two-scale scheme is efficient and the approximate solution obtained by the scheme maintains the asymptotically optimal accuracy. Finally, numerical experiments are carried out to confirm the considered theory.     

A capacitary weak type inequality for Sobolev functions and its applications

In this paper a capacitary weak type inequality for Sobolev functions is established and is applied to reprove some well-known results concerning Lebesgue points, Taylor expansions in the -sense, and the Lusin type approximation of Sobolev functions.

     

Mode decomposition for a synchronous state and its applications

Synchronization of coupled dynamical systems including periodic and chaotic systems is investigated both anlaytically and numerically. A novel method, mode decomposition, of treating the stability of a synchronous state is proposed based on the Floquet theory. A rigorous criterion is then derived, which can be applied to arbitrary coupled systems. Two typical numerical examples: coupled Van der Pol systems (corresponding to the case of coupled periodic oscillators) and coupled Lorenz systems (corresponding to the case of chaotic systems) are used to demonstrate the theoretical analysis.     

A minimax principle for nonlinear eigenvalue problems with applications to nonoverdamped systems

The theory of Rayleigh functionals for non-linear eigenvalue problems T(λ) u = 0 is extended to cases where the functional is defined only on a proper subset. The theory applies to problems which do not satisfy an overdamping condition and yields a minimax characterization of eigenvalues. Applications to damped free vibrations of an elastic body are discussed.     

Symmetric-triangular decomposition and its applications part II: Preconditioners for indefinite systems

As an application of the symmetric-triangular (ST) decomposition given by Golub and Yuan (2001) and Strang (2003), three block ST preconditioners are discussed here for saddle point problems. All three preconditioners transform saddle point problems into a symmetric and positive definite system. The condition number of the three symmetric and positive definite systems are estimated. Therefore, numerical methods for symmetric and positive definite systems can be applied to solve saddle point problems indirectly. A numerical example for the symmetric indefinite system from the finite element approximation to the Stokes equation is given. Finally, some comments are given as well. AMS subject classification (2000) 65F10     

Sandwich theorem for quasiconvex functions and its applications

In convex programming, sandwich theorem is very important because it is equivalent to Fenchel duality theorem. In this paper, we investigate a sandwich theorem for quasiconvex functions. Also, we consider some applications for quasiconvex programming.