Domain decomposition methods for pseudo spectral approximations

Summary Multidomain pseudo spectral approximations of second order boundary value problems in one dimension are considered. The equation is collocated at the Chebyshev nodes inside each subinterval. Different patching conditions at the interfaces are analyzed. Results of stability and convergence are given.Research supported in part by AFOSR Grant 85-0303     

Updating the singular value decomposition

Summary LetA be anm×n matrix with known singular value decomposition. The computation of the singular value decomposition of a matrixà is considered, whereà is obtained by appending a row or a column toA whenmn or by deleting a row or a column fromA whenm>n. An algorithm is also presented for solving the updated least squares problemà y–b, obtained from the least squares problemAx–b by appending an equation, deleting an equation, appending an unknown, or deleting an unknown.This research was supported by NSF grants MCS 75-06510 and MCS 76-03139     

Updating the principal angle decomposition

A class of fast Householder-based sequential algorithms for updating the Principal Angle Decomposition is introduced. The updated Principal Angle Decomposition is of key importance in the adaptive implementation of several fundamental operations on correlated processes, such as adaptive Wiener filtering, rank-adaptive system identification, and rank and data compression concepts using canonical coordinates. An instructive example of rank-adaptive system identification is examined experimentally.     

Approximation of kernel matrices by circulant matrices and its application in kernel selection methods

This paper focuses on developing fast numerical algorithms for selection of a kernel optimal for a given training data set. The optimal kernel is obtained by minimizing a cost functional over a prescribed set of kernels. The cost functional is defined in terms of a positive semi-definite matrix determined completely by a given kernel and the given sampled input data. Fast computational algorithms are developed by approximating the positive semi-definite matrix by a related circulant matrix so that the fast Fourier transform can apply to achieve a linear or quasi-linear computational complexity for finding the optimal kernel. We establish convergence of the approximation method. Numerical examples are presented to demonstrate the approximation accuracy and computational efficiency of the proposed methods.     

On spectral variations under relatively bounded perturbations

Linear operators in Banach and Hilbert spaces are considered. Bounds for the spectrum are established under relatively bounded perturbations. An application to nonselfadjoint differential operators is discussed.     

Noncommutative spectral decomposition with quasideterminant

We develop a noncommutative analogue of the spectral decomposition with the quasideterminant defined by I. Gelfand and V. Retakh. In this theory, by introducing a noncommutative Lagrange interpolating polynomial and combining a noncommutative Cayley-Hamilton's theorem and an identity given by a Vandermonde-like quasideterminant, we can systematically calculate a function of a matrix even if it has noncommutative entries. As examples, the noncommutative spectral decomposition and the exponential matrices of a quaternionic matrix and of a matrix with entries being harmonic oscillators are given.     

On the spectral kernel of symmetric extensions

In a Hilbert space H we consider closed and symmetric operators A and à with closed ranges such that AÃ. We prove a necessary and sufficient condition for the existence of a closed and symmetric operator B with ABà the range of which is not closed. We show that this condition can be fulfilled and, by the way, we get a counter example to the assertion that the continuous part of the spectral kernel of a symmetric operator is contained in the corresponding part of a symmetric extension, as is claimed in the books of Achieser-Glasmann [1], Neumark [2] and Smirnow [3].     

On spectral variations under bounded real matrix perturbations

Summary In this paper we investigate the set of eigenvalues of a perturbed matrix {ie509-1} whereA is given and ^{n × n}, ||< is arbitrary. We determine a lower bound for thisspectral value set which is exact for normal matricesA with well separated eigenvalues. We also investigate the behaviour of the spectral value set under similarity transformations. The results are then applied tostability radii which measure the distance of a matrixA from the set of matrices having at least one eigenvalue in a given closed instability domain _b.     

Solvability of partial differential equations by meshless kernel methods

This paper first provides a common framework for partial differential equation problems in both strong and weak form by rewriting them as generalized interpolation problems. Then it is proven that any well-posed linear problem in strong or weak form can be solved by certain meshless kernel methods to any prescribed accuracy. The work described in this paper was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 101205). Robert Schaback’s research in Hong Kong was sponsored by DFG and City University of Hong Kong.     

Stability of singular spectral types under decaying perturbations

We look at invariance of a.e. boundary condition spectral behavior under perturbations, W, of half-line, continuum or discrete Schrödinger operators. We extend the results of del Rio, Simon, Stolz from compactly supported W's to suitable short-range W. We also discuss invariance of the local Hausdorff dimension of spectral measures under such perturbations.     

Strongly nonlinear perturbations of nonnegative boundary value problems with kernel

The anisotropic Kepler problem is a one parameter family of Hamiltonian systems recently introduced by Gutzwiller to approximate certain quantum mechanical systems. When the parameter μ = 1, we have the ordinary Kepler or central force problem. This system is regularizable by any of several well known methods. When μ > 1, the kinetic energy of the system becomes anisotropic. This destroys the integrability of the problem and changes the orbit structure of the system dramatically. In this paper, we show that the anisotropy of the kinetic energy also destroys the regularizability of the system, at least for most μ > 1.     

Some progress in spectral methods

In this paper,we review some results on the spectral methods.We frst consider the Jacobi spectral method and the generalized Jacobi spectral method for various problems,including degenerated and singular diferential equations.Then we present the generalized Jacobi quasi-orthogonal approximation and its applications to the spectral element methods for high order problems with mixed inhomogeneous boundary conditions.We also discuss the related spectral methods for non-rectangular domains and the irrational spectral methods for unbounded domains.Next,we consider the Hermite spectral method and the generalized Hermite spectral method with their applications.Finally,we consider the Laguerre spectral method and the generalized Laguerre spectral method for many problems defned on unbounded domains.We also present the generalized Laguerre quasi-orthogonal approximation and its applications to certain problems of non-standard type and exterior problems.     

Filtering in Legendre spectral methods

We discuss the impact of modal filtering in Legendre spectral methods, both on accuracy and stability. For the former, we derive sufficient conditions on the filter to recover high order accuracy away from points of discontinuity. Computational results confirm that less strict necessary conditions appear to be adequate. We proceed to discuss a instability mechanism in polynomial spectral methods and prove that filtering suffices to ensure stability. The results are illustrated by computational experiments.

     

Spectral approximation for compact integral operators by degenerate kernel methods

We establish a new improved error estimate for the solution of the integral equation eigenvalue problem by degenerate kernel methods. In [6] these estimates were proved under the assumption of normality of the original kernel as well as of the approximating degenerate kernel. Now we consider any compact integral operator and a general Banach space situation, in contrast to the Hilbert space setting in [6], This will be done by combining the techniques in [6] with the suitably transformed estimates of [5]. Our results show that degenerate kernel methods have, besides their overall property of furnishing easy approximations to eigenfunctions, for eigenvalues an order of convergence comparable to quadrature methods.     

Solving deconvolution type problems by wavelet decomposition methods

The paper considers an inverse problem associated with equations of the form Kf = g, where K is a convolution-type operator. The aim is to find a solution f for given function g. We construct approximate solutions by applying a wavelet basis that is well adapted to this problem. For this basis we calculate the elementary solutions that are the approximate preimages of the wavelets. The solution for the inverse problem is then constructed as an appropriate finite linear combination of the elementary solutions. Under certain assumptions we estimate the approximation error and discuss the advantages of the proposed scheme.     

Modeling sequence evolution with kernel methods

We model the evolution of biological and linguistic sequences by comparing their statistical properties. This comparison is performed by means of efficiently computable kernel functions, that take two sequences as an input and return a measure of statistical similarity between them. We show how the use of such kernels allows to reconstruct the phylogenetic trees of primates based on the mitochondrial DNA (mtDNA) of existing animals, and the phylogenetic tree of Indo-European and other languages based on sample documents from existing languages. Kernel methods provide a convenient framework for many pattern analysis tasks, and recent advances have been focused on efficient methods for sequence comparison and analysis. While a large toolbox of algorithms has been developed to analyze data by using kernels, in this paper we demonstrate their use in combination with standard phylogenetic reconstruction algorithms and visualization methods.     

方差分量的广义谱分解估计

对于随机效应部分为一般平衡多向分类的线性混合模型,将王松桂(2002)提出的一种称之为谱分解估计的参数估计新方法推广到随机效应设计阵为任意矩阵的含两个方差分量的线性混合模型,给出了方差分量的广义谱分解估计方法,并证明了所得估计的一些统计性质。另外,还就广义谱分解估计类中某些特殊估计和对应的方差分析估计进行了比较,得到了它们相等的充分必要条件。