Inverse modeling: an introduction to the theory and methods of inverse problems and data assimilation
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Bristol, UK
IOP Publishing
[2015]
|
| Series: | IOP expanding physics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | 1 Band (verschiedene Seitenzählungen) |
| ISBN: | 9780750312196 |
Staff View
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| adam_text | Contents
A
Preface xiii
Acknowledgements xv
Author biographies xvi
1 Introduction 1-1
1.1 A tour through theory and applications 1-2
1.2 Types of inverse problems 1-14
1.2.1 The general inverse problem 1-15
1.2.2 Source problems 1-16
1.2.3 Scattering from obstacles 1-17
1.2.4 Dynamical systems inversion 1-19
1.2.5 Spectral inverse problems 1-21
Bibliography 1-22
2 Functional analytic tools 2-1
2.1 Normed spaces, elementary topology and compactness 2-1
2.1.1 Norms, convergence and the equivalence of norms 2-1
2.1.2 Open and closed sets, Cauchy sequences and completeness 2-5
2.1.3 Compact and relatively compact sets 2-7
2.2 Hilbert spaces, orthogonal systems and Fourier expansion 2-10
2.2.1 Scalar products and orthonormal systems 2-10
2.2.2 Best approximations and Fourier expansion 2-13
2.3 Bounded operators, Neumann series and compactness 2-19
2.3.1 Bounded and linear operators 2-19
2.3.2 The solution of equations of the second kind and the 2-25
Neumann series
2.3.3 Compact operators and integral operators 2-27
2.3.4 The solution of equations of the second kind and Riesz theory 2-32
2.4 Adjoint operators, eigenvalues and singular values 2-33
2.4.1 Riesz representation theorem and adjoint operators 2-33
2.4.2 Weak compactness of Hilbert spaces 2-36
2.4.3 Eigenvalues, spectrum and the spectral radius of an operator 2-38
2.4.4 Spectral theorem for compact self-adjoint operators 2-40
2.4.5 Singular value decomposition 2-46
2.5 Lax-Milgram and weak solutions to boundary value problems 2-48
vii
Inverse Modeling
2.6 The Frechet derivative and calculus in normed spaces 2-50
Bibliography 2-55
3 Approaches to regularization 3-1
3.1 Classical regularization methods 3-1
3.1.1 Ill-posed problems 3-1
3.1.2 Regularization schemes 3-2
3.1.3 Spectral damping 3-4
3.1.4 Tikhonov regularization and spectral cut-off 3-7
3.1.5 The minimum norm solution and its properties 3-10
3.1.6 Methods for choosing the regularization parameter 3-14
3.2 The Moore—Penrose pseudo-inverse and Tikhonov regularization 3-20
3.3 Iterative approaches to inverse problems 3-22
3.3.1 Newton and quasi-Newton methods 3-23
3.3.2 The gradient or Landweber method 3-25
3.3.3 Stopping rules and convergence order 3-31
Bibliography 3-34
4 A stochastic view of inverse problems 4-1
4.1 Stochastic estimators based on ensembles and particles 4-1
4.2 Bayesian methods 4-4
4.3 Markov chain Monte Carlo methods 4-6
4.4 Metropolis-Hastings and Gibbs sampler 4-11
4.5 Basic stochastic concepts 4-14
Bibliography 4-19
5 Dynamical systems inversion and data assimilation 5-1
5.1 Set-up for data assimilation 5-3
5.2 Three-dimensional variational data assimilation (3D-VAR) 5-5
5.3 Four-dimensional variational data assimilation (4D-VAR) 5-8
5.3.1 Classical 4D-VAR 5-8
5.3.2 Ensemble-Based 4D-VAR 5-13
5.4 The Kalman filter and Kalman smoother 5-16
5.5 Ensemble Kalman filters (EnKFs) 5-22
5.6 Particle filters and nonlinear Bayesian data assimilation 5-29
Bibliography 5-33
Vlll
Inverse Modeling
6 Programming of numerical algorithms and useful tools 6-1
6.1 MATLAB or OCTAVE programming: the butterfly 6-1
6.2 Data assimilation made simple 6-4
6.3 Ensemble data assimilation in a nutshell 6-8
6.4 An integral equation of the first kind, regularization and 6-9
atmospheric radiance retrievals
6.5 Integro-differential equations and neural fields 6-12
6.6 Image processing operators 6-15
Bibliography 6-18
7 Neural field inversion and kernel reconstruction 7-1
7.1 Simulating neural fields 7-4
7.2 Integral kernel reconstruction 7-8
7.3 A collocation method for kernel reconstruction 7-17
7.4 Traveling neural pulses and homogeneous kernels 7-20
7.5 Bi-orthogonal basis functions and integral operator inversion 7-23
7.6 Dimensional reduction and localization 7-26
Bibliography 7-30
8 Simulation of waves and fields 8-1
8.1 Potentials and potential operators 8-1
8.2 Simulation of wave scattering 8-11
8.3 The far field and the far field operator 8-15
8.4 Reciprocity relations 8-21
8.5 The Lax-Phillips method to calculate scattered waves 8-23
Bibliography 8-26
9 Nonlinear operators 9-1
9.1 Domain derivatives for boundary integral operators 9-1
9.2 Domain derivatives for boundary value problems 9-8
9.3 Alternative approaches to domain derivatives 9-11
9.3.1 The variational approach 9-11
9.3.2 Implicit function theorem approach 9-18
9.4 Gradient and Newton methods for inverse scattering 9-21
9.5 Differentiating dynamical systems: tangent linear models 9-27
Bibliography 9-30
IX
Inverse Modeling
10 Analysis: uniqueness, stability and convergence questions 10-1
10.1 Uniqueness of inverse problems 10-3
10.2 Uniqueness and stability for inverse obstacle scattering 10-4
10.3 Discrete versus continuous problems 10-7
10.4 Relation between inverse scattering and inverse boundary value 10-9
problems
10.5 Stability of cycled data assimilation 10-14
10.6 Review of convergence concepts for inverse problems 10-18
10.6.1 Convergence concepts in stochastics and in data 10-19
assimilation
10.6.2 Convergence concepts for reconstruction methods in inverse 10-21
scattering
Bibliography 10-24
11 Source reconstruction and magnetic tomography 11-1
11.1 Current simulation 11-2
11.1.1 Currents based on the conductivity problem 11-2
11.1.2 Simulation via the finite integration technique 11-4
11.2 The Biot-Savart operator and magnetic tomography 11-8
11.2.1 Uniqueness and non-uniqueness results 11-12
11.2.2 Reducing the ill-posedness of the reconstruction by 11-16
using appropriate subspaces
11.3 Parameter estimation in dynamic magnetic tomography 11-25
11.4 Classification methods for inverse problems 11-28
Bibliography 11-32
12 Field reconstruction techniques 12-1
12.1 Series expansion methods 12-2
12.1.1 Fourier-Hankel series for field representation 12-2
12.1.2 Field reconstruction via exponential functions with an 12-6
imaginary argument
12.2 Fourier plane-wave methods 12-10
12.3 The potential or Kirsch—Kress method 12-12
12.4 The point source method 12-20
12.5 Duality and equivalence for the potential method and the 12-27
point source method
Bibliography 12-29
x
inverse Modeling
13 Sampling methods 13-1
13.1 Orthogonality or direct sampling 13-2
13.2 The linear sampling method of Colton and Kirsch 13-4
13.3 Kirsch’s factorization method 13-10
Bibliography 13-17
14 Probe methods 14-1
14.1 The SSM 14-2
14.1.1 Basic ideas and principles 14-2
14.1.2 The needle scheme for probe methods 14-7
14.1.3 Domain sampling for probe methods 14-9
14.1.4 The contraction scheme for probe methods 14-10
14.1.5 Convergence analysis for the SSM 14-13
14.2 The probing method for near field data by Ikehata 14-16
14.2.1 Basic idea and principles 14-17
14.2.2 Convergence and equivalence of the probe and SSM 14-20
14.3 The multi-wave no-response and range test of Schulz and Potthast 14-21
14.4 Equivalence results 14-26
14.4.1 Equivalence of SSM and the no-response test 14-27
14.4.2 Equivalence of the no-response test and the range test 14-29
14.5 The multi-wave enclosure method of Ikehata 14-31
Bibliography 14-38
15 Analytic continuation tests 15-1
15.1 The range test 15-1
15.2 The no-response test of Luke-Potthast 15-6
15.3 Duality and equivalence for the range test and no-response test 15-10
15.4 Ikehata’s enclosure method 15-11
15.4.1 Oscillating-decaying solutions 15-13
15.4.2 Identification of the singular points 15-18
Bibliography 15-20
16 Dynamical sampling and probe methods 16-1
16.1 Linear sampling method for identifying cavities in a heat conductor 16-2
16.1.1 Tools and theoretical foundation 16-4
16.1.2 Property of potential 16-14
16.1.3 The jump relations of TC* 16-16
xi
Inverse Modeling
16.2 Nakamura’s dynamical probe method 16-17
16.2.1 Inverse boundary value problem for heat conductors with 16-17
inclusions
16.2.2 Tools and theoretical foundation 16-18
16.2.3 Proof of theorem 16.2.6 16-20
16.2.4 Existence of Runge’s approximation functions 16-24
16.3 The time-domain probe method 16-26
16.4 The BC method of Belishev for the wave equation 16-29
Bibliography 16-35
17 Targeted observations and meta-inverse problems 17-1
17.1 A framework for meta-inverse problems 17-1
17.2 Framework adaption or zoom 17-7
17.3 Inverse source problems 17_8
Bibliography 17-13
Appendix A A-l
XII
lOP Expanding Physics
Inverse Modeling
An introduction to the theory and methods
of inverse problems and data assimilation
Gen Nakamura and Roland Potthast
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V · i À
m
This book provides a comprehensive introduction to the techniques, tools and methods for
inverse problems and data assimilation. Inverse problems are widespread today in science and
technology, and this book is written at the interface between mathematics and applications,
designed for students, researchers and developers in mathematics, physics, engineering,
acoustics, electromagnetics, meteorology, biology, environmental and other applied sciences:
3#
Basic analytic questions and tools are introduced, as well as a wide variety of concepts,
methods and approaches to formulate and solve inverse problems, among others classical
regularization and iterative methods, but also more recent approaches like sampling and
probe methods, field reconstruction techniques in acoustics and electromagnetics and source
reconstruction. In the framework of data assimilation, stochastic and deterministic methods
are becoming more and more intertwined - both variational and ensemble data assimilation
methods are introduced, as well as stochastic and deterministic viewpoints and analysis.
Applications range from neural kernel reconstruction to inverse acoustic and electromagnetic
scattering and thermography, from magnetic tomography forfuel cells or in a medical
environment to weather forecasting and earth science. OCTAVE/MATLAB ՛ codes are included,
which serve as a first step towards simulations and more sophisticated inversion or data
assimilation algorithms.
About the authors
Gen Nakamura is currently Harim Professor at Inha University, S. Korea, and Professor Emeritus
of Hokkaido University, Japan. Roland Potthast currently leads the division for data assimilation
at the German Meteorological Service, Frankfurt/Offenbach, with a part-time professorship in
mathematics at the University of Reading, UK.
About Expanding Physics
Expanding Physics publishes high-quality texts from leading voices across the research
landscape on key a*eas in physics and related subject areas.
9 780750 312196
¡opscience.org/books
O ion ebooks
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| id | DE-604.BV044397643 |
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| indexdate | 2024-07-10T07:51:52Z |
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| language | English |
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| spelling | Nakamura, Gen Verfasser (DE-588)1090788657 aut Inverse modeling an introduction to the theory and methods of inverse problems and data assimilation Gen Nakamura and Roland Potthast, Hokkaido University, Japan, Inha University, Korea, University of Reading, United Kingdom, German Meteorological Service (DWD) Bristol, UK IOP Publishing [2015] 1 Band (verschiedene Seitenzählungen) txt rdacontent n rdamedia nc rdacarrier IOP expanding physics Inverses Problem (DE-588)4125161-1 gnd rswk-swf Datenassimilation (DE-588)4803260-8 gnd rswk-swf Inverses Problem (DE-588)4125161-1 s Datenassimilation (DE-588)4803260-8 s DE-604 Potthast, Roland 1967- Verfasser (DE-588)17287856X aut Erscheint auch als Online-Ausgabe 978-0-7503-1218-9 Erscheint auch als Online-Ausgabe, MOBI 978-0-7503-1220-2 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029799771&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029799771&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Nakamura, Gen Potthast, Roland 1967- Inverse modeling an introduction to the theory and methods of inverse problems and data assimilation Inverses Problem (DE-588)4125161-1 gnd Datenassimilation (DE-588)4803260-8 gnd |
| subject_GND | (DE-588)4125161-1 (DE-588)4803260-8 |
| title | Inverse modeling an introduction to the theory and methods of inverse problems and data assimilation |
| title_auth | Inverse modeling an introduction to the theory and methods of inverse problems and data assimilation |
| title_exact_search | Inverse modeling an introduction to the theory and methods of inverse problems and data assimilation |
| title_full | Inverse modeling an introduction to the theory and methods of inverse problems and data assimilation Gen Nakamura and Roland Potthast, Hokkaido University, Japan, Inha University, Korea, University of Reading, United Kingdom, German Meteorological Service (DWD) |
| title_fullStr | Inverse modeling an introduction to the theory and methods of inverse problems and data assimilation Gen Nakamura and Roland Potthast, Hokkaido University, Japan, Inha University, Korea, University of Reading, United Kingdom, German Meteorological Service (DWD) |
| title_full_unstemmed | Inverse modeling an introduction to the theory and methods of inverse problems and data assimilation Gen Nakamura and Roland Potthast, Hokkaido University, Japan, Inha University, Korea, University of Reading, United Kingdom, German Meteorological Service (DWD) |
| title_short | Inverse modeling |
| title_sort | inverse modeling an introduction to the theory and methods of inverse problems and data assimilation |
| title_sub | an introduction to the theory and methods of inverse problems and data assimilation |
| topic | Inverses Problem (DE-588)4125161-1 gnd Datenassimilation (DE-588)4803260-8 gnd |
| topic_facet | Inverses Problem Datenassimilation |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029799771&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029799771&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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