Least squares with a quadratic constraint

Summary We present the theory of the linear least squares problem with a quadratic constraint. New theorems characterizing properties of the solutions are given. A numerical application is discussed.     

A new algorithm based on factorization for heterogeneous domain decomposition

Often computational models are too expensive to be solved in the entire domain of simulation, and a cheaper model would suffice away from the main zone of interest. We present for the concrete example of an evolution problem of advection reaction diffusion type a heterogeneous domain decomposition algorithm which allows us to recover a solution that is very close to the solution of the fully viscous problem, but solves only an inviscid problem in parts of the domain. Our new algorithm is based on the factorization of the underlying differential operator, and we therefore call it factorization algorithm. We give a detailed error analysis in one spatial dimension, and show that we can obtain approximations in the viscous region which are much closer to the viscous solution in the entire domain of simulation than approximations obtained by other heterogeneous domain decomposition algorithms from the literature. We illustrate our results with numerical experiments in one and two spatial dimensions.     

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Some models for evaluating the cavity formation energies in liquids

In the mixing process of solutes into solvents, the cavity formation energy is generally introduced into the energy balance. In this study, models for calculating this type of energy are proposed. These models are based on the Hansen's partial solubility parameters δ_d (MPa^1/2) dispersive, δ_p (MPa^1/2) polar and δ_h (MPa^1/2) hydrogen-bonding, and on the internal pressure P_j (J cm^?3) of the solvents.     

Asymptotic properties of the QR factorization of banded Hessenberg–Toeplitz matrices

We consider Givens QR factorization of banded Hessenberg–Toeplitz matrices of large order and relatively small bandwidth. We investigate the asymptotic behaviour of the R factor and Givens rotation when the order of the matrix goes to infinity, and present some interesting convergence properties. These properties can lead to savings in the computation of the exact QR factorization and give insight into the approximate QR factorizations of interest in preconditioning. The properties also reveal the relation between the limit of the main diagonal elements of R and the largest absolute root of a polynomial. Copyright © 2005 John Wiley & Sons, Ltd.     

Optimization of the Hermitian and Skew-Hermitian Splitting Iteration for Saddle-Point Problems

We study the asymptotic rate of convergence of the alternating Hermitian/skew-Hermitian iteration for solving saddle-point problems arising in the discretization of elliptic partial differential equations. By a careful analysis of the iterative scheme at the continuous level we determine optimal convergence parameters for the model problem of the Poisson equation written in div-grad form. We show that the optimized convergence rate for small mesh parameter h is asymptotically 1–O(h ^1/2). Furthermore we show that when the splitting is used as a preconditioner for a Krylov method, a different optimization leading to two clusters in the spectrum gives an optimal, h-independent, convergence rate. The theoretical analysis is supported by numerical experiments.This revised version was published online in October 2005 with corrections to the Cover Date.     

Gram‐Schmidt orthogonalization: 100 years and more

In 1907, Erhard Schmidt published a paper in which he introduced an orthogonalization algorithm that has since become known as the classical Gram‐Schmidt process. Schmidt claimed that his procedure was essentially the same as an earlier one published by J. P. Gram in 1883. The Schmidt version was the first to become popular and widely used. An algorithm related to a modified version of the process appeared in an 1820 treatise by P. S. Laplace. Although related algorithms have been around for almost 200 years, it is the Schmidt paper that led to the popularization of orthogonalization techniques. The year 2007 marked the 100th anniversary of that paper. In celebration of that anniversary, we present a comprehensive survey of the research on Gram‐Schmidt orthogonalization and its related QR factorization. Its application for solving least squares problems and in Krylov subspace methods are also reviewed. Software and implementation aspects are also discussed. Copyright © 2012 John Wiley & Sons, Ltd.