Direct and Inverse Problems in Wave Propagation and Applications

✍ Scribed by Ivan G. Graham (editor); Ulrich Langer (editor); Jens M. Melenk (editor); Mourad Sini (editor)

• Publisher: De Gruyter
• Year: 2013
• Tongue: English
• Leaves: 328
• Series: Radon Series on Computational and Applied Mathematics; 14
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Preface
Differential electromagnetic imaging
1 Introduction
2 Basic theory of electromagnetic waves
2.1 The Helmholtz equation
2.2 The Maxwell equations
2.3 Fundamental solutions and radiation conditions
2.4 Transmission and boundary conditions
2.5 Dirichlet and Neumann functions and the Hodge decomposition
2.6 Trace theorems and first Green identity
2.7 Lippman–Schwinger representation formulas
2.8 The Helmholtz–Kirchhoff theorems
2.9 Limiting models
2.10 TheMaxwell equations with axis invariance
2.11 The Maxwell equations versus the Helmholtz equation
3 Electric and magnetic polarization tensors
4 small-volume expansions
4.1 The full Maxwell equations
4.2 The eddy currents model
4.3 The Helmholtz equation
4.4 The conductivity equation
4.5 Asymptotic formulas in the time domain
5 Imaging in the frequency domain
5.1 MUSIC-type imaging at a single frequency
5.2 Backpropagation type imaging at a single frequency
5.3 Imaging with a broad range of frequencies
6 Imaging in the time domain
6.1 Time-domain imaging with full viewmeasurements
6.2 Time-domain imaging in a cavity with limited-view data
6.3 Time-domain imaging in dissipative media
7 Numerical examples of MUSIC reconstructions for the full Maxwell equations
8 Shape representations
8.1 High-order polarization tensors
8.2 Frequency dependent high-order polarization tensors
9 Far-field imaging versus near-field imaging
10 Open problems
Multitrace boundary integral equations
1 Introduction
1.1 Geometry
1.2 Transmission problems
2 Boundary integral operators
2.1 Trace spaces and operators
2.2 Potentials
2.3 Calderón projectors
3 Classical single-trace integral equations
3.1 Skeleton trace spaces
3.2 A first-kind boundary integral equation
3.3 Boundary element Galerkin discretization
4 Preconditioning
4.1 Operator products
4.2 Calderón identities
4.3 Operator preconditioning
4.4 Stable duality pairing for boundary elements
4.5 The challenge
5 Global multitrace formulation
5.1 Separated subdomains
5.2 The gap idea
5.3 Properties of global MTF
5.4 Galerkin discretization
6 Local multitrace formulation
6.1 Partial transmission conditions
6.2 Local MTF: variational formulation
6.3 Local MTF: Stability
6.4 Boundary element Galerkin discretization
Direct and Inverse Elastic Scattering Problems for Diffraction Gratings
1 Introduction
2 Mathematical formulation of direct and inverse scattering problems
3 Solvability results for direct scattering problems: variational method
3.1 An equivalent variational formulation and its Fredholmproperty
3.2 Uniqueness and existence for direct scattering problems
3.3 Uniqueness and existence for transmission gratings
4 Uniqueness for inverse scattering problems
4.1 Inverse scattering of incident pressure waves
4.2 Inverse scattering of incident shear waves
5 Numerical solution of direct and inverse scattering problems
5.1 A discrete Galerkin method for (DP)
5.2 A two-step algorithm for (IP)
Multigrid methods for Helmholtz problems: A convergent scheme in 1D using standard components
1 Introduction
2 Smoothing
2.1 Smoothing analysis
2.2 Jacobi smoothing
2.3 Two-step Jacobi smoothing
3 Coarse-grid correction
3.1 The Laplacian
3.2 The Helmholtz operator
4 Two-grid iteration
4.1 The Laplacian
4.2 The Helmholtz operator
5 Numerical examples
5.1 Two-grid experiments
5.2 Multigrid experiments, complexity
6 Conclusions
Explicit local time-steppingmethods for time-dependent wave propagation
1 Introduction
2 Finite element discretizations for the wave equation
2.1 Continuous Galerkin formulation
2.2 Interior penalty discontinuous Galerkin formulation
2.3 Nodal discontinuous Galerkin formulation
3 Leap-frog-based LTS methods
3.1 Second-order method for undamped waves
3.2 Fourth-order method for undamped waves
3.3 Second-order leap-frog/Crank–Nicolson-basedmethod for damped waves
4 Adams–Bashforth-based LTS methods for damped waves
5 Numerical results
5.1 Stability
5.2 Convergence
5.3 Two-dimensional example
6 Concluding remarks
Absorbing boundary conditions and perfectly matched layers in wave propagation problems
1 Introduction
2 ABC
2.1 Exact ABC
2.2 Approximation of the exact ABC
3 Plane waves analysis of an ABC
4 Perfectly matched layers
4.1 Helmholtz equation
4.2 The wave equation
5 Computation of the reflection coefficient of a PML
6 Conclusion
Dynamic inverse scattering
1 Introduction
2 Reconstruction of time-dependent pulses by the point-source method
3 Time-domain probe method (TDPM)
4 Orthogonality sampling
5 Dynamic inversion via data assimilation techniques
5.1 Three-dimensional variational data assimilation
5.2 Cycled probing and samplingmethod
5.3 Partial reconstruction matching scheme
6 Numerical examples
Boundary integral equations for Helmholtz boundary value and transmission problems
1 Introduction
2 Boundary integral equations
2.1 Boundary integral operators
2.2 Coercivity of boundary integral operators
2.3 Injectivity of boundary integral operators
2.4 Interior Robin boundary value problem
2.5 Boundary integral equations for exterior boundary value problems
3 Exterior Dirichlet boundary value problem
3.1 Direct boundary integral equations
3.2 Indirect boundary integral equations
3.3 Regularised combined boundary integral equations
4 Transmission problems
4.1 Steklov–Poincaré operator equations
4.2 Combined boundary integral equations
5 Conclusions
Color plates
Index