Inverse modeling : an introduction to the theory and methods of inverse problems and data assimilation /

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Bibliographic Details
Author / Creator:	Nakamura, Gen (Professor of mathematics), author.
Imprint:	Bristol : IOP Publishing, [2015]
Description:	1 volume (various pagings) : illustrations (some color) ; 27 cm
Language:	English
Series:	IOP expanding physics IOP expanding physics.
Subject:	Inverse problems (Differential equations) Stochastic processes. Inverse problems (Differential equations) Stochastic processes.
Format:	Print Book
URL for this record:	http://pi.lib.uchicago.edu/1001/cat/bib/11028162

Hidden Bibliographic Details
Other authors / contributors:	Potthast, Roland, author.
ISBN:	9780750312196 075031219X
Notes:	Includes bibliographical references.

Table of Contents:

Preface
Acknowledgements
Author biographies
1. Introduction
1.1. A tour through theory and applications
1.2. Types of inverse problems
1.2.1. The general inverse problem
1.2.2. Source problems
1.2.3. Scattering from obstacles
1.2.4. Dynamical systems inversion
1.2.5. Spectral inverse problems
Bibliography
2. Functional analytic tools
2.1. Normed spaces, elementary topology and compactness
2.1.1. Norms, convergence and the equivalence of norms
2.1.2. Open and closed sets, Cauchy sequences and completeness
2.1.3. Compact and relatively compact sets
2.2. Hilbert spaces, orthogonal systems and Fourier expansion
2.2.1. Scalar products and orthonormal systems
2.2.2. Best approximations and Fourier expansion
2.3. Bounded operators, Neumann series and compactness
2.3.1. Bounded and linear operators
2.3.2. The solution of equations of the second kind and the Neumann series
2.3.3. Compact operators and integral operators
2.3.4. The solution of equations of the second kind and Riesz theory
2.4. Adjoint operators, eigenvalues and singular values
2.4.1. Riesz representation theorem and adjoint operators
2.4.2. Weak, compactness of Hilbert spaces
2.4.3. Eigenvalues, spectrum and the spectral radius of an operator
2.4.4. Spectral theorem for compact self-adjoint operators
2.4.5. Singular value decomposition
2.5. Lax-Milgram and weak solutions to boundary value problems
2.6. The Fréchet derivative and calculus in normed spaces
Bibliography
3. Approaches to regularization
3.1. Classical regularization methods
3.1.1. Ill-posed problems
3.1.2. Regularization schemes
3.1.3. Spectral damping
3.1.4. Tikhonov regularization and spectral cut-off
3.1.5. The minimum norm solution and its properties
3.1.6. Methods for choosing the regularization parameter
3.2. The Moore-Penrose pseudo-inverse and Tikhonov regularization
3.3. Iterative approaches to inverse problems
3.3.1. Newton and quasi-Newton methods
3.3.2. The gradient or Landweber method
3.3.3. Stopping rules and convergence order
Bibliography
4. A stochastic view of inverse problems
4.1. Stochastic estimators based on ensembles and particles
4.2. Bayesian methods
4.3. Markov chain Monte Carlo methods
4.4. Metropolis-Hastings and Gibbs sampler
4.5. Basic stochastic concepts
Bibliography
5. Dynamical systems inversion and data assimilation
5.1. Set-up for data assimilation
5.2. Three-dimensional variational data assimilation (3D-VAR)
5.3. Four-dimensional variational data assimilation (4D-VAR)
5.3.1. Classical 4D-VAR
5.3.2. Ensemble-Based 4D-VAR
5.4. The Kalman filter and Kalman smoother
5.5. Ensemble Kalman filters (EnKFs)
5.6. Particle filters and nonlinear Bayesian data assimilation
Bibliography
6. Programming of numerical algorithms and useful tools
6.1. MATLAB or OCTAVE programming: the butterfly
6.2. Data assimilation made simple
6.3. Ensemble data assimilation in a nutshell
6.4. An integral equation of the first kind, regularization and atmospheric radiance retrievals
6.5. Integro-differential equations and neural fields
6.6. Image processing operators
Bibliography
7. Neural field inversion and kernel reconstruction
7.1. Simulating neural fields
7.2. Integral kernel reconstruction
7.3. A collocation method for kernel reconstruction
7.4. Traveling neural pulses and homogeneous kernels
7.5. Bi-orthogonal basis functions and integral operator inversion
7.6. Dimensional reduction and localization
Bibliography
8. Simulation of waves and fields
8.1. Potentials and potential operators
8.2. Simulation of wave scattering
8.3. The far field and the far field operator
8.4. Reciprocity relations
8.5. The Lax-Phillips method to calculate scattered waves
Bibliography
9. Nonlinear operators
9.1. Domain derivatives for boundary integral operators
9.2. Domain derivatives for boundary value problems
9.3. Alternative approaches to domain derivatives
9.3.1. The variational approach
9.3.2. Implicit function theorem approach
9.4. Gradient and Newton methods for inverse scattering
9.5. Differentiating dynamical systems: tangent linear models
Bibliography
10. Analysis: uniqueness, stability and convergence questions
10.1. Uniqueness of inverse problems
10.2. Uniqueness and stability for inverse obstacle scattering
10.3. Discrete versus continuous problems
10.4. Relation between inverse scattering and inverse boundary value problems
10.5. Stability of cycled data assimilation
10.6. Review of convergence concepts for inverse problems
10.6.1. Convergence concepts in stochastics and in data assimilation
10.6.2. Convergence concepts for reconstruction methods in inverse scattering
Bibliography
11. Source reconstruction and magnetic tomography
11.1. Current simulation
11.1.1. Currents based on the conductivity problem
11.1.2. Simulation via the finite integration technique
11.2. The Biot-Savart operator and magnetic tomography
11.2.1. Uniqueness and non-uniqueness results
11.2.2. Reducing the ill-posedness of the reconstruction by using appropriate subspaces
11.3. Parameter estimation in dynamic magnetic tomography
11.4. Classification methods for inverse problems
Bibliography
12. Field reconstruction techniques
12.1. Series expansion methods
12.1.1. Fourier-Hankel series for field representation
12.1.2. Field reconstruction via exponential functions with an imaginary argument
12.2. Fourier plane-wave methods
12.3. The potential or Kirsch-Kress method
12.4. The point source method
12.5. Duality and equivalence for the potential method and the point source method
Bibliography
13. Sampling methods
13.1. Orthogonality or direct sampling
13.2. The linear sampling method of Colt on and Kirsch
13.3. Kirsch's factorization method
Bibliography
14. Probe methods
14.1. The SSM
14.1.1. Basic ideas and principles
14.1.2. The needle scheme for probe methods
14.1.3. Domain sampling for probe methods
14.1.4. The contraction scheme for probe methods
14.1.5. Convergence analysis for the SSM
14.2. The probing method for near field data by Ikehata
14.2.1. Basic idea and principles
14.2.2. Convergence and equivalence of the probe and SSM
14.3. The multi-wave no-response and range test of Schulz and Potthast
14.4. Equivalence results
14.4.1. Equivalence of SSM and the no-response test
14.4.2. Equivalence of the no-response test and the range test
14.5. The multi-wave enclosure method of Ikehata
Bibliography
15. Analytic continuation tests
15.1. The range test
15.2. The no-response test of Luke-Potthast
15.3. Duality and equivalence for the range test and no-response test
15.4. Ikehata's enclosure method
15.4.1. Oscillating-decaying solutions
15.4.2. Identification of the singular points
Bibliography
16. Dynamical sampling and probe methods
16.1. Linear sampling method for identifying cavities in a heat conductor
16.1.1. Tools and theoretical foundation
16.1.2. Property of potential
16.1.3. The jump relations of K&ast;
16.2. Nakamura's dynamical probe method
16.2.1. Inverse boundary value problem for heat conductors with inclusions
16.2.2. Tools and theoretical foundation
16.2.3. Proof of theorem 16.2.6
16.2.4. Existence of Runge's approximation functions
16.3. The time-domain probe method
16.4. The BC method of Belishev for the wave equation
Bibliography
17. Targeted observations and meta-inverse problems
17.1. A framework for meta-inverse problems
17.2. Framework adaption or zoom
17.3. Inverse source problems
Bibliography
Appendix A.

Inverse modeling : an introduction to the theory and methods of inverse problems and data assimilation /

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