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E-raamat: Direct and Inverse Problems in Wave Propagation and Applications [De Gruyter e-raamatud]

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Direct and Inverse Problems in Wave Propagation and ApplicationsSuurem pilt
Teised raamatud teemal:
Wiley OnlineRohkem infot De Gruyter e-raamatute kohta
Raamatu kodulehekülg: http://dx.doi.org/10.1515/9783110282283
This book is the third volume of three volume series recording the "Radon Special Semester 2011 on Multiscale Simulation & Analysis in Energy and the Environment" taking place in Linz, Austria, October 3-7, 2011. This book surveys recent developments in the analysis of wave propagation problems. The topics covered include aspects of the forward problem and problems in inverse problems, as well as applications in the earth sciences.

Wave propagation problems are ubiquitous in environmental applications such as seismic analysis, acoustic and electromagnetic scattering. The design of efficient numerical methods for the forward problem, in which the scattered field is computed from known geometric configurations is very challenging due to the multiscale nature of the problems. Even more challenging are inverse problems where material parameters and configurations have to be determined from measurements in conjunction with the forward problem. This book contains review articles covering several state-of-the-art numerical methods for both forward and inverse problems.

This collection of survey articles focusses on the efficient computation of wave propagation and scattering is a core problem in numerical mathematics, which is currently of great research interest and is central to many applications in energy and the environment. Two generic applications which resonate strongly with the central aims of the Radon Special Semester 2011 are forward wave propagation in heterogeneous media and seismic inversion for subsurface imaging. As an example of the first application, modelling of absorption and scattering of radiation by clouds, aerosol and precipitation is used as a tool for interpretation of (e.g.) solar, infrared and radar measurements, and as a component in larger weather/climate prediction models in numerical weather forecasting. As an example of the second application, inverse problems in wave propagation in heterogeneous media arise in the problem of imaging the subsurface below land or marine deposits.

The book records the achievements of Workshop 3 "Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment". It brings together key numerical mathematicians whose interest is in the analysis and computation of wave propagation and scattering problems, and in inverse problems, together with practitioners from engineering and industry whose interest is in the applications of these core problems.
Preface v
Differential electromagnetic imaging
1(50)
Habib Ammari
1 Introduction
1(2)
2 Basic theory of electromagnetic waves
3(9)
2.1 The Helmholtz equation
3(1)
2.2 The Maxwell equations
3(1)
2.3 Fundamental solutions and radiation conditions
4(1)
2.4 Transmission and boundary conditions
5(1)
2.5 Dirichlet and Neumann functions and the Hodge decomposition
6(1)
2.6 Trace theorems and first Green identity
7(1)
2.7 Lippman---Schwinger representation formulas
8(1)
2.8 The Helmholtz---Kirchhoff theorems
9(1)
2.9 Limiting models
10(1)
2.10 The Maxwell equations with axis invariance
11(1)
2.11 The Maxwell equations versus the Helmholtz equation
12(1)
3 Electric and magnetic polarization tensors
12(1)
4 small-volume expansions
13(6)
4.1 The full Maxwell equations
13(4)
4.2 The eddy currents model
17(1)
4.3 The Helmholtz equation
18(1)
4.4 The conductivity equation
18(1)
4.5 Asymptotic formulas in the time domain
18(1)
5 Imaging in the frequency domain
19(5)
5.1 MUSIC-type imaging at a single frequency
20(2)
5.2 Backpropagation type imaging at a single frequency
22(1)
5.3 Imaging with a broad range of frequencies
23(1)
6 Imaging in the time domain
24(9)
6.1 Time-domain imaging with full measurements
24(1)
6.2 Time-domain imaging in a cavity with limited-view data
25(3)
6.3 Time-domain imaging in dissipative media
28(5)
7 Numerical examples of MUSIC reconstructions for the full Maxwell equations
33(5)
8 Shape representations
38(7)
8.1 High-order polarization tensors
38(3)
8.2 Frequency dependent high-order polarization tensors
41(4)
9 Far-field imaging versus near-field imaging
45(2)
10 Open problems
47(4)
Multitrace boundary integral equations
51(50)
Xavier Claeys
Ralf Hiptmair
Carlos Jerez-Hanckes
1 Introduction
51(6)
1.1 Geometry
54(1)
1.2 Transmission problems
54(3)
2 Boundary integral operators
57(7)
2.1 Trace spaces and operators
58(3)
2.2 Potentials
61(1)
2.3 Calderon projectors
62(2)
3 Classical single-trace integral equations
64(11)
3.1 Skeleton trace spaces
65(4)
3.2 A first-kind boundary integral equation
69(3)
3.3 Boundary element Galerkin discretization
72(3)
4 Preconditioning
75(7)
4.1 Operator products
76(1)
4.2 Calderon identities
77(2)
4.3 Operator preconditioning
79(1)
4.4 Stable duality pairing for boundary elements
80(1)
4.5 The challenge
81(1)
5 Global multitrace formulation
82(6)
5.1 Separated subdomains
82(3)
5.2 The gap idea
85(1)
5.3 Properties of global MTF
86(1)
5.4 Galerkin discretization
87(1)
6 Local multitrace formulation
88(13)
6.1 Partial transmission conditions
89(2)
6.2 Local MTF: variational formulation
91(2)
6.3 Local MTF: Stability
93(2)
6.4 Boundary element Galerkin discretization
95(6)
Direct and Inverse Elastic Scattering Problems for Diffraction Gratings
101(34)
Johannes Elschner
Guanghui Hu
1 Introduction
101(2)
2 Mathematical formulation of direct and inverse scattering problems
103(4)
3 Solvability results for direct scattering problems: variational method
107(8)
3.1 An equivalent variational formulation and its Fredholm property
107(2)
3.2 Uniqueness and existence for direct scattering problems
109(2)
3.3 Uniqueness and existence for transmission gratings
111(4)
4 Uniqueness for inverse scattering problems
115(7)
4.1 Inverse scattering of incident pressure waves
116(5)
4.2 Inverse scattering of incident shear waves
121(1)
5 Numerical solution of direct and inverse scattering problems
122(13)
5.1 A discrete Galerkin method for (DP)
122(3)
5.2 A two-step algorithm for (IP)
125(10)
Multigrid methods for Helmholtz problems: A convergent scheme in 1D using standard components
135(52)
Oliver G. Ernst
Martin J. Gander
1 Introduction
135(4)
2 Smoothing
139(13)
2.1 Smoothing analysis
140(2)
2.2 Jacobi smoothing
142(3)
2.3 Two-step Jacobi smoothing
145(7)
3 Coarse-grid correction
152(13)
3.1 The Laplacian
158(1)
3.2 The Helmholtz operator
159(6)
4 Two-grid iteration
165(10)
4.1 The Laplacian
165(2)
4.2 The Helmholtz operator
167(8)
5 Numerical examples
175(7)
5.1 Two-grid experiments
176(2)
5.2 Multigrid experiments, complexity
178(4)
6 Conclusions
182(5)
Explicit local time-stepping methods for time-dependent wave propagation
187(32)
Marcus J. Grote
Teodora Mitkova
1 Introduction
187(3)
2 Finite element discretizations for the wave equation
190(4)
2.1 Continuous Galerkin formulation
190(1)
2.2 Interior penalty discontinuous Galerkin formulation
191(2)
2.3 Nodal discontinuous Galerkin formulation
193(1)
3 Leap-frog-based LTS methods
194(11)
3.1 Second-order method for undamped waves
195(4)
3.2 Fourth-order method for undamped waves
199(3)
3.3 Second-order leap-frog/Crank-Nicolson-based method for damped waves
202(3)
4 Adams-Bashforth-based LTS methods for damped waves
205(6)
5 Numerical results
211(4)
5.1 Stability
211(1)
5.2 Convergence
212(2)
5.3 Two-dimensional example
214(1)
6 Concluding remarks
215(4)
Absorbing boundary conditions and perfectly matched layers in wave propagation problems
219(14)
Frederic Nataf
1 Introduction
219(1)
2 ABC
220(4)
2.1 Exact ABC
221(1)
2.2 Approximation of the exact ABC
222(2)
3 Plane waves analysis of an ABC
224(1)
4 Perfectly matched layers
225(4)
4.1 Helmholtz equation
226(2)
4.2 The wave equation
228(1)
5 Computation of the reflection coefficient of a PML
229(2)
6 Conclusion
231(2)
Dynamic inverse scattering
233(20)
Roland W. E. Potthast
1 Introduction
233(3)
2 Reconstruction of time-dependent pulses by the point-source method
236(2)
3 Time-domain probe method (TDPM)
238(2)
4 Orthogonality sampling
240(1)
5 Dynamic inversion via data assimilation techniques
240(7)
5.1 Three-dimensional variational data assimilation
242(2)
5.2 Cycled probing and sampling method
244(1)
5.3 Partial reconstruction matching scheme
245(2)
6 Numerical examples
247(6)
Boundary integral equations for Helmholtz boundary value and transmission problems
253(40)
Olaf Steinbach
1 Introduction
253(2)
2 Boundary integral equations
255(11)
2.1 Boundary integral operators
255(3)
2.2 Coercivity of boundary integral operators
258(2)
2.3 Injectivity of boundary integral operators
260(4)
2.4 Interior Robin boundary value problem
264(2)
2.5 Boundary integral equations for exterior boundary value problems
266(1)
3 Exterior Dirichlet boundary value problem
266(14)
3.1 Direct boundary integral equations
267(9)
3.2 Indirect boundary integral equations
276(3)
3.3 Regularised combined boundary integral equations
279(1)
4 Transmission problems
280(10)
4.1 Steklov---Poincare operator equations
281(4)
4.2 Combined boundary integral equations
285(5)
5 Conclusions
290(3)
Color plates 293(16)
Index 309
Ivan Graham, University of Bath, UK; Ulrich Langer, University of Linz, Austria; Jens Markus Melenk, Vienna University of Technology, Austria; Mourad Sini, RICAM, Linz, Austria.
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