Least Squares Data Fitting with Applications

Cover

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Title Page, Copyright

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pp. iii-iv

Contents

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pp. v-vii

Preface

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pp. ix-xi

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 ...

Symbols and Acronyms

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pp. xii-xv

1. The Linear Data Fitting Problem

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pp. 1-23

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

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pp. 25-45

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

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pp. 47-63

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

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pp. 65-90

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

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pp. 91-103

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

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pp. 105-119

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 ...

7. Additional Topics in Least Squares

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pp. 121-145

In this chapter we collect some more specialized topics, such as problems with constraints, sensitivity analysis, total least squares and compressed sensing. ...

8. Nonlinear Least Squares Problems

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pp. 147-162

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

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pp. 163-190

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

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pp. 191-202

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 ...

11. Linear Least Squares Applications

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pp. 203-230

We now present in detail two larger applications of linear least squares for fitting of temperature profiles and geological surface modeling. Both of them use splines, so for completeness, we start with a summary of their definitions and basic ...

12. Nonlinear Least Squares Applications

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pp. 231-261

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 ...

Appendix A: Sensitivity Analysis

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pp. 263-265

Appendix B: Linear Algebra Background

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pp. 267-270

Appendix C: Advanced Calculus Background

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pp. 271-274

Appendix D: Statistics

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pp. 275-280

References

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pp. 281-300

Index

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pp. 301-305