Mathematics for Large Scale Computing

Julio Diaz

• Publisher: CRC Press
• Year: 2020
• Print ISBN: 9780824781224
• eBook ISBN: 9781000673357
• Edition: 1
• Topic: Mathematics
• Subtopic: Computer Science General
• Index: Mathematics

During recent years a great deal of interest has been devoted to large scale computing applications. This has occurred in great part because of the introduction of advanced high performance computer architectures. The book contains survey articles as well as chapters on specific research applications, development and analysis of numerical algorithms, and performance evaluation of algorithms on advanced architectures. The effect of specialized architectural features on the performance of large scale computation is also considered by several authors. Several areas of applications are represented, including the numerical solution of partial differential equations, iterative techniques for large structured problems, the numerical solution of boundary value problems for ordinary differential equations, numerical optimization, and numerical quadrature. Mathematical issues in computer architecture are also presented, including the description of grey codes for generalized hypercubes. The results presented in this volume give, in our opinion, a representative picture of today's state of the art in several aspects of large scale computing.

On the Gauss-Broyden Method for Nonlinear Least Squares

ANDREAS GRIEWANK Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois 60439

LAIHUA SHENG Department of Mathematics, Dedman College, Southern Methodist University, Dallas, Texas 75275

This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contracts W-31-109-Eng-38.

1 GENERAL ASSUMPTIONS AND TERMINOLOGY

Nonlinear least squares problems arise in many applications, in particular data fitting and parameter estimation [3]. Mathematically we have the optimization problem

$\mathit{Min}\gamma \text{(}x\text{)}\equiv \frac{1}{2}\Vert g\text{(}x\text{)}\Vert \mathit{for}g:R^n\to R^m.$

Throughout we will assume that the vector function g has a Lipschitz-continuous Jacobian G(x) ≡ g′(x) ∈ R^m×n with full column rank n ≤ m at all points x in some bounded level set

$ℒ\equiv \text{{}x\in R^n:\gamma \text{(}x\text{)}\le \overline{\gamma }\text{}}$

with a non-empty interior ${ℒ}^0$. Finally we will assume that the restriction of g to $ℒ$ is injective, i.e.

$x,z\in ℒ,g\text{(}x\text{)}=g\text{(}z\text{)}\Rightarrow x=z.$

(1.1)

While these regularity assumptions appear quite strong, they do not exclude the possibility that γ(x) attains several local minima and other stationary points in $ℒ$. Throughout let x_* denote a lo...

Table of contents

• Cover
• Halftitle
• Title Page
• Copyright Page
• Preface
• Table of Contents
• Contributors
• On the Gauss-Broyden Method for Nonlinear Least-Squares
• Parallel Adaptive Algorithms for Multiple Integrals
• A Comparison of Hypercube Implementations of Parallel Shooting
• An Asymptotic Induced Numerical Method for the Convection-Diffusion-Reaction Equation
• The Rate of Convergence of the Modified Method of Characteristics for Linear Advection Equations in One Dimension
• A Time-Discretization Procedure for a Mixed Finite Element Approximation of Contamination by Incompressible Nuclear Waste in Porous Media
• Implementation of Finite Element Alternating-Direction Methods for Vector Computers
• Performance of Advanced Scientific Computers for the Efficient Solution of an Elastic Wave Code for Seismic Modeling
• Generalized Gray Codes and Their Properties
• Nested Block Factorization Preconditioned for Convective-Diffusion Problems in Three Dimensions
• Performance of the Chebyshev Iterative Method, GMRES and ORTHOMIN on a Set of Oil-Reservoir Simulation Problems
• A Survey of Spline Collocation Methods for the Numerical Solution of Differential Equations
• Index