Notes on numerical fluid mechanics

Semiseparable matrices and many other rank-structured matrices have been widely used in developing new fast matrix algorithms. In this paper, we generalize the hierarchically semiseparable (HSS) matrix representations and propose some fast algorithms for HSS matrices. We represent HSS matrices in terms of general binary HSS trees and use simplified postordering notation for HSS forms. Fast HSS algorithms including new HSS structure generation and HSS form Cholesky factorization are developed. Moreover, we provide a new linear complexity explicit ULV factorization algorithm for symmetric positive definite HSS matrices with a low-rank property. The corresponding factors can be used to solve the HSS systems also in linear complexity. Numerical examples demonstrate the efficiency of the algorithms. All these algorithms have nice data locality. They are useful in developing fast-structured numerical methods for large discretized PDEs (such as elliptic equations), integral equations, eigenvalue problems, etc. Some applications are shown. Copyright q 2009 John Wiley & Sons, Ltd.

https://www.math.purdue.edu/~xiaj/work/fasthss.pdf

Fast algorithms for hierarchically semiseparable matrices

Fiber-optic communication systems are nowadays facing serious challenges due to the fast growing demand on capacity from various new applications and services. It is now well recognized that nonlinear effects limit the spectral efficiency and transmission reach of modern fiber-optic communications. Nonlinearity compensation is therefore widely believed to be of paramount importance for increasing the capacity of future optical networks. Recently, there has been steadily growing interest in the application of a powerful mathematical tool—the nonlinear Fourier transform (NFT)—in the development of fundamentally novel nonlinearity mitigation tools for fiber-optic channels. It has been recognized that, within this paradigm, the nonlinear crosstalk due to the Kerr effect is effectively absent, and fiber nonlinearity due to the Kerr effect can enter as a constructive element rather than a degrading factor. The novelty and the mathematical complexity of the NFT, the versatility of the proposed system designs, and the lack of a unified vision of an optimal NFT-type communication system, however, constitute significant difficulties for communication researchers. In this paper, we therefore survey the existing approaches in a common framework and review the progress in this area with a focus on practical implementation aspects. First, an overview of existing key algorithms for the efficacious computation of the direct and inverse NFT is given, and the issues of accuracy and numerical complexity are elucidated. We then describe different approaches for the utilization of the NFT in practical transmission schemes. After that we discuss the differences, advantages, and challenges of various recently emerged system designs employing the NFT, as well as the spectral efficiency estimates available up-to-date. With many practical implementation aspects still being open, our mini-review is aimed at helping researchers assess the perspectives, understand the bottlenecks, and envision the development paths in the upcoming NFT-based transmission technologies.

Nonlinear Fourier transform for optical data processing and transmission: advances and perspectives

We consider an algebraic representation that is useful for matrices with off-diagonal blocks of low numerical rank. A fast and stable solver for linear systems of equations in which the coefficient matrix has this representation is presented. We also present a fast algorithm to construct the hierarchically semiseparable representation in the general case.

A Fast $ULV$ Decomposition Solver for Hierarchically Semiseparable Representations

Semidefinite programs constitute one of the largest classes of optimization problems that can be solved with reasonable efficiency - both in theory and practice. They play a key role in a variety of research areas, such as combinatorial optimization, approximation algorithms, computational complexity, graph theory, geometry, real algebraic geometry and quantum computing. This book is an introduction to selected aspects of semidefinite programming and its use in approximation algorithms. It covers the basics but also a significant amount of recent and more advanced material. There are many computational problems, such as MAXCUT, for which one cannot reasonably expect to obtain an exact solution efficiently, and in such case, one has to settle for approximate solutions. For MAXCUT and its relatives, exciting recent results suggest that semidefinite programming is probably the ultimate tool. Indeed, assuming the Unique Games Conjecture, a plausible but as yet unproven hypothesis, it was shown that for these problems, known algorithms based on semidefinite programming deliver the best possible approximation ratios among all polynomial-time algorithms. This book follows the semidefinite side of these developments, presenting some of the main ideas behind approximation algorithms based on semidefinite programming. It develops the basic theory of semidefinite programming, presents one of the known efficient algorithms in detail, and describes the principles of some others. It also includes applications, focusing on approximation algorithms.

Approximation Algorithms and Semidefinite Programming

Hilbert spaces and basic operator theory: Linear spaces normed spaces first examples Hilbert spaces The dual space Bounded linear operators Spectrum. Fredholm theory of compact operators Self-adjoint operators Functions of operators spectral decomposition Basics of functional analysis: Spectral theory of unitary operators The fundamental theorems and the basic methods Banach algebras Unbounded self-adjoint and symmetric operators in $H$ Solutions to exercises Bibliography Symbols index Subject index.

Functional analysis : an introduction

A modification of the Dewilde-van der Veen method for inversion of finite structured matrices

Let $\mathcal H_n\subset \mathbb C^{n\times n}$ be the class of $n\times n$ Hessenberg matrices $A$ which are rank-one modifications of a unitary matrix, that is, $A=H +\B u\B w^H$, where $H$ is unitary and $\mathbf{u}, \mathbf{w}\in \mathbb C^n$. The class $\mathcal H_n$ includes three well-known subclasses: unitary Hessenberg matrices, companion (Frobenius) matrices, and fellow matrices. The paper presents some novel fast adaptations of the shifted QR algorithm for computing the eigenvalues of a matrix $A\in \mathcal H_n$ where each step can be performed with $O(n)$ flops and $O(n)$ memory space. Numerical experiments confirm the effectiveness and the robustness of these methods.

Fast QR Eigenvalue Algorithms for Hessenberg Matrices Which Are Rank-One Perturbations of Unitary Matrices

https://www.math.uconn.edu/~olshevsky/papers/TCS.pdf

Computations with quasiseparable polynomials and matrices

Separable Type Representations of Matrices and Fast Algorithms

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