Computational Methods for Applied InverseProblems covers many directions in the moderntheory 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, Nystr#m method, numerical differentiation, analytic continuation, perturbationregularization, filtering, optimization and sparse solving methods are fully addressed. This issue attempts to bridge the gap between theoretical studies of ill-posed inverse problemsand practical applications.
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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