# Least Squares Data Fitting with Applications

Publication Year: 2012

Published by: Johns Hopkins University Press

#### Preface

This book surveys basic modern techniques for the numerical solution of linear and nonlinear least squares problems and introduces the treatment of large and ill-conditioned problems. The theory is extensively illustrated with examples from engineering, environmental sciences, geophysics and ...

#### 1. The Linear Data Fitting Problem

This chapter gives an introduction to the linear data fitting problem: how it is defined, its mathematical aspects and how it is analyzed. We also give important statistical background that provides insight into the data fitting problem. Anyone with more interest in the subject is encouraged to consult ...

#### 2. The Linear Least Squares Problem

This chapter covers some of the basic mathematical facts of the linear least squares problem (LSQ), as well as some important additional statistical results for data fitting. We introduce the two formulations of the least squares problem: the linear system of normal equations and the optimization ...

#### 3. Analysis of Least Squares Problems

The previous chapter has set the stage for introducing some fundamental tools for analyzing further the LSQ problem, namely, the pseudoinverse and the singular value decomposition. After introducing these concepts, we complete this chapter with a definition of the condition number for ...

#### 4. Direct Methods for Full-Rank Problems

This chapter describes a number of algorithms and techniques for the case where the matrix has full rank, and therefore there is a unique solution to the LSQ problem. We compare the efficiency of the different methods and make suggestions about their use. We start with the least expensive, ...

#### 5. Direct Methods for Rank-Deficient Problems

The methods developed so far break down when the matrix *A* is rank
deficient, i.e., for rank(*A*) = *r* < *n*, in which case there is no unique solution.
Within the linear manifold of solutions, the minimum and basic solutions
are specially useful because: ...

#### 6. Methods for Large-Scale Problems

Large-scale problems are those where the number of variables is such that direct methods, like the ones we have described in earlier chapters, cannot be applied because of storage limitations, accuracy restrictions or computational cost. Usually, large-scale problems have special structure, such as sparseness, which ...

#### 8. Nonlinear Least Squares Problems

So far we have discussed data fitting problems in which all the unknown parameters
appear linearly in the fitting model M(*x*,*t*), leading to linear least
squares problems for which we can, in principle, write down a closed-form
solution. We now turn to nonlinear least squares problems (NLLSQ) for ...

#### 9. Algorithms for Solving Nonlinear LSQ Problems

The classical method of Newton and its variants can be used to solve
the nonlinear least squares problem formulated in the previous chapter.
Newton’s method for optimization is based on a second-order Taylor approximation
of the objective function *f*(*x*) and subsequent minimization of ...

#### 10. Ill-Conditioned Problems

Throughout this book, ill conditioning and rank deficiency have been discussed in several places. For completeness, we will summarize the main ideas now. This chapter therefore surveys some important aspects of LSQ problems with ill-conditioned matrices; for more details and algorithms we ...

#### 12. Nonlinear Least Squares Applications

In this chapter we consider several nonlinear least squares applications in detail, starting with the fast training of neural networks and their use in optimal design to generate surrogates of very expensive functionals. Then we consider several inverse problems related to the design of piezoelectrical ...

E-ISBN-13: 9781421408583

E-ISBN-10: 1421408589

Print-ISBN-13: 9781421407869

Print-ISBN-10: 1421407868

Page Count: 328

Illustrations: 9 halftones, 8 line drawings, 46 graphs

Publication Year: 2012

OCLC Number: 828720556

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