No abstract

Abstract. Singular spectral analysis has recently become an attractive tool in a broad range of applications. Its main mechanism of alternating between rank reduction and Hankel projection, however, deserves further mathematical justification. Exploiting a natural parametrization of a general Hankel matrix via its Vandermonde factorization, this paper recasts the numerical scheme for the singular spectral analysis as a semi-linear least squares problem over a compact feasible set. Such a formulation makes it possible to employ global optimization techniques to find the absolute best approximation which then is used to assess the quality of the approximation produced from the singular spectral analysis. What is the main conclusion at the end of this investigation?

In this paper we want to propose practical numerical methods to solve a class of initial-boundary problem of space-time fractional advection-diffusion equations. To start with, an implicit method based on two-sided Grünwald formulae is proposed with a discussion of the stability and consistency. Then, the preconditioned generalized minimal residual (preconditioned GMRES) method and the preconditioned conjugate gradient normal residual (preconditioned CGNR) method, with an easily constructed preconditioner, are developed. Importantly, because the resulting systems are Topelitz-like, the fast Fourier transform can be applied to significantly reduce the computational cost. Numerical experiments are implemented to show the efficiency of our preconditioner, even with cases of variable coefficients.

In the past decade or so, semi-definite programming (SDP) has emerged as a powerful tool capable of handling a remarkably wide range of problems. This article describes an innovative application of SDP techniques to quadratic inverse eigenvalue problems (QIEPs). The notion of QIEPs is of fundamental importance because its ultimate goal of constructing or updating a vibration system from some observed or desirable dynamical behaviors while respecting some inherent feasibility constraints well suits many engineering applications. Thus far, however, QIEPs have remained challenging both theoretically and computationally due to the great variations of structural constraints that must be addressed. Of notable interest and significance are the uniformity and the simplicity in the SDP formulation that solves effectively many otherwise very difficult QIEPs.

In this study, nonnegative matrix factorization is recast as the problem of approximating a polytope on the probability simplex by another polytope with fewer facets. Working on the probability simplex has the advantage that data are limited to a compact set with a known boundary, making it easier to trace the approximation procedure. In particular, the supporting hyperplane that separates a point from a disjoint polytope, a fact asserted by the Hahn-Banach theorem, can be calculated in finitely many steps. This approach leads to a convenient way of computing the proximity map which, in contrast to most existing algorithms where only an approximate map is used, finds the unique and global minimum per iteration. This paper sets up a theoretical framework, outlines a numerical algorithm, and suggests an effective implementation. Testing results strongly evidence that this approach obtains a better low rank nonnegative matrix approximation in fewer steps than conventional methods. Introduction. The problem of nonnegative matrix factorization (NMF) arisesin a large variety of disciplines in sciences and engineering. Its wide range of important applications such as text mining, cheminformatics, factor retrieval, image articulation, dimension reduction, and so on has attracted considerable research efforts. Many different kinds of NMF techniques have been proposed in the literature, notably the popular Lee and Seung iterative update algorithm [17,18]. Successful applications to various models with nonnegative data values are abounding. We can hardly be exhaustive by suggesting [6,11,12,13,16,20,19,21,22,24], and the references contained therein as a partial list of interesting work. The review article [27] contains many other more recent applications and references.The basic idea behind NMF is the typical linear model

Algebraic Riccati equations are encountered in many applications of control and engineering problems, e.g., LQG problems and H ∞ control theory. In this work, we study the properties of one type of discrete-time algebraic Riccati equations. Our contribution is twofold. First, we present sufficient conditions for the existence of a unique positive definite solution. Second, we propose an accelerated algorithm to obtain the positive definite solution with the rate of convergence of any desired order. Numerical experiments strongly support that our approach performs extremely well even in the almost critical case. As a byproduct, we provide show that this method is capable of computing the unique negative definite solution, once it exists.

Nonlinear matrix equations play a crucial role in science and engineering problems. However, solutions of nonlinear matrix equations cannot, in general, be given analytically. One standard way of solving nonlinear matrix equations is to apply the fixed-point iteration with usually only the linear convergence rate. To advance the existing methods, we exploit in this work one type of semigroup property and use this property to propose a technique for solving the equations with the speed of convergence of any desired order. We realize our way by starting with examples of solving the scalar equations and, also, connect this method with some well-known equations including, but not limited to, the Stein matrix equation, the generalized eigenvalue problem, the generalized nonlinear matrix equation, the discrete-time algebraic Riccati equations to express the capacity of this method.

The Euclidean distance matrix for n distinct points in ℝr is generically of rank r + 2. It is shown in this paper via a geometric argument that its nonnegative rank for the case r = 1 is generically n.