The book covers many directions in the modern theory of inverse and illposed problems: mathematical physics, optimal inverse design, inverse scattering, inverse vibration, biomedical imaging, oceanography, seismic imaging and remote sensing; methods including standard regularization,parallel computing for multidimensional problems, Nystr6m method,numerical differentiation, analytic continuation, perturbation regularization,filtering, optimization and sparse solving methods are fully addressed.

Preface

Editor's Preface

 Introduction

 S. I. Kabanikhin

 Inverse Problems of Mathematical Physics

 1.1 Introduction

 1.2 Examples of Inverse and Ill-posed Problems

 1.3 Well-posed and Ill-posed Problems

 1.4 The Tikhonov Theorem

 1.5 The Ivanov Theorem: Quasi-solution

 1.6 The Lavrentiev's Method

 1.7 The Tikhonov Regularization Method

 References

II Recent Advances in Regularization Theory and Methods

2 D. V. Lukyanenko and A. G. Yagola

 Using Parallel Computing for Solving Multidimensional

 Ill-posed Problems

 2.1 Introduction

 2.2 Using Parallel Computing

2.2.1 Main idea of parallel computing

2.2.2 Parallel computing limitations

 2.3 Parallelization of Multidimensional Ill-posed Problem

2.3.1 Formulation of the problem and method of solution

2.3.2 Finite-difference approximation of the functional and its gradient

2.3.3 Parallelization of the minimization problem

 2.4 Some Examples of Calculations

 2.5 Conclusions

 References

3 M. T. Nair

 Regularization of Fredholm Integral Equations of the First

 Kind using Nystrom Approximation

 3.1 Introduction

 3.2 NystrSm Method for Regularized Equations

3.2.1 NystrSm approximation of integral operators

3.2.2 Approximation of regularized equation

3.2.3 Solvability of approximate regularized equation

3.2.4 Method of numerical solution

 3.3 Error Estimates

3.3.1 Some preparatory results

3.3.2 Error estimate with respect to

3.3.3 Error estimate with respect to

3.3.4 A modified method

 3.4 Conclusion

 References

4 T. Y. Xiao, H. Zhang and L. L. Hao

 Regularization of Numerical Differentiation: Methods and

 Applications

 4.1 Introduction

 4.2 Regularizing Schemes

4.2.1 Basic settings

4.2.2 Regularized difference method (RDM)

4.2.3 Smoother-Based regularization (SBR)

4.2.4 Mollifier regularization method (MRM)

4.2.5 Tikhonov's variational regularization (TiVR)

4.2.6 Lavrentiev regularization method (LRM)

4.2.7 Discrete regularization method (DRM)

4.2.8 Semi-Discrete Tikhonov regularization (SDTR)

4.2.9 Total variation regularization (TVR)

 4.3 Numerical Comparisons

 4.4 Applied Examples

4.4.1 Simple applied problems

4.4.2 The inverse heat conduct problems (IHCP)

4.4.3 The parameter estimation in new product diffusion model

4.4.4 Parameter identification of sturm-liouville operator

4.4.5 The numerical inversion of Abel transform

4.4.6 The linear viscoelastic stress analysis

 4.5 Discussion and Conclusion

 References

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