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2017 | Buch

Inverse Problems for Partial Differential Equations

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Über dieses Buch

This third edition expands upon the earlier edition by adding nearly 40 pages of new material reflecting the analytical and numerical progress in inverse problems in last 10 years. As in the second edition, the emphasis is on new ideas and methods rather than technical improvements. These new ideas include use of the stationary phase method in the two-dimensional elliptic problems and of multi frequencies\temporal data to improve stability and numerical resolution. There are also numerous corrections and improvements of the exposition throughout.

This book is intended for mathematicians working with partial differential equations and their applications, physicists, geophysicists, and financial, electrical, and mechanical engineers involved with nondestructive evaluation, seismic exploration, remote sensing, and various kinds of tomography.

Review of the second edition:

"The first edition of this excellent book appeared in 1998 and became a standard reference for everyone interested in analysis and numerics of inverse problems in partial differential equations. … The second edition is considerably expanded and reflects important recent developments in the field … . Some of the research problems from the first edition have been solved … ." (Johannes Elschner, Zentralblatt MATH, Vol. 1092 (18), 2006)

Inhaltsverzeichnis

Frontmatter
1. Inverse Problems
Abstract
In this chapter, we formulate basic inverse problems and indicate their applications. The choice of these problems is not random. We think that it represents their interconnections and some hierarchy.
Victor Isakov
2. Ill-Posed Problems and Regularization
Abstract
In this chapter, we consider the equation
Victor Isakov
3. Uniqueness and Stability in the Cauchy Problem
Abstract
In this chapter we formulate and in many cases prove results on the uniqueness and stability of solutions of the Cauchy problem for general partial differential equations. One of the basic tools is Carleman-type estimates. In Section 3.1 we describe the results for a simplest problem of this kind (the backward parabolic equation), where a choice of the weight function in Carleman estimates is obvious, and the method is equivalent to that of the logarithmic convexity. In Section 3.2 we formulate general conditional Carleman estimates and their simplifications for second-order equations, and we apply the results to the general Cauchy problem and give numerous counterexamples showing that the assumptions of positive results are quite sharp.
Victor Isakov
4. Elliptic Equations: Single Boundary Measurements
Abstract
In this chapter we consider the elliptic second-order differential equation \(Au = f\quad \mathrm{in}\;\varOmega,f = f_{0} -\sum \limits _{j=1}^{n}\partial _{j}f_{j}\) with the Dirichlet boundary data \(u = g_{0}\quad \mathrm{on}\;\partial \varOmega.\) We assume that A = div(−a∇) + b ⋅ ∇ + c with bounded and measurable coefficients a (symmetric real-valued (n × n) matrix) and complex-valued b and c in L (Ω). Another assumption is that A is an elliptic operator; i.e., there is ɛ 0 > 0 such that a(x)ξ ⋅ ξ ≥ ɛ 0 | ξ | 2 for any vector \(\xi \in \mathbb{R}^{n}\) and any x ∈ Ω. Unless specified otherwise, we assume that Ω is a bounded domain in \(\mathbb{R}^{n}\) with the boundary of class C 2. However, most of the results are valid for Lipschitz boundaries.
Victor Isakov
5. Elliptic Equations: Many Boundary Measurements
Abstract
We consider the Dirichlet problem (4.0.1), (4.0.2). At first we assume that for any Dirichlet data g 0 we are given the Neumann data g 1; in other words, we know the results of all possible boundary measurements, or the so-called Dirichlet-to-Neumann operator \(\mathrm{\Lambda }: H^{1/2}(\partial \Omega ) \rightarrow H^{-1/2}(\partial \Omega )\), which maps the Dirichlet data g 0 into the Neumann data g 1.
Victor Isakov
6. Scattering Problems and Stationary Waves
Abstract
The stationary incoming wave u with the wave number k is a solution to the perturbed Helmholtz equation (scattering by medium) \(Au - k^{2}u = 0\,\mathrm{in}\,\mathbb{R}^{3}\)
Victor Isakov
7. Integral Geometry and Tomography
Abstract
The problems of integral geometry are to determine a function given (weighted) integrals of this function over a “rich” family of manifolds. These problems are of importance in medical applications (tomography), and they are quite useful for dealing with inverse problems in hyperbolic differential equations (integrals of unknown coefficients over ellipsoids or lines can be obtained from the first terms of the asymptotic expansion of rapidly oscillating solutions and an information about first-arrival times of a wave). There has been significant progress in the classical Radon problem when manifolds are hyperplanes and the weight function is the unity; there are interesting results in the plane case when a family of curves is regular (resembling locally the family of straight lines) or in case of the family of straight lines with an arbitrary regular attenuation. Still there are many interesting open questions about the problem with local data and simultaneous recovery of density of a source and of attenuation. We give a brief review of this area, referring for more information to the books of Natterer [Nat] and Sharafutdinov [Sh].
Victor Isakov
8. Hyperbolic Problems
Abstract
In this chapter, we are interested in finding coefficients of the second-order hyperbolic operator.
Victor Isakov
9. Inverse Parabolic Problems
Abstract
In this chapter, we consider the second-order parabolic equation
$$\displaystyle{ a_{0}\partial _{t}u -\mathrm{div}(a\nabla u) + b \cdot \nabla u + cu = f\,\mathrm{in}\,Q = \varOmega \times (0,T), }$$
where Ω is a bounded domain the space \(\mathbb{R}^{n}\) with the C 2-smooth boundary ∂ Ω.
Victor Isakov
10. Some Numerical Methods
Abstract
In this chapter, we will briefly review some popular numerical methods widely used in practice. Of course it is not a comprehensive collection. We will demonstrate certain methods that are simple and widely used or, in our opinion, interesting and promising both theoretically and numerically. We observe that most of these methods have not been justified and in some cases even not rigorously tested numerically.
Victor Isakov
Backmatter
Metadaten
Titel
Inverse Problems for Partial Differential Equations
verfasst von
Victor Isakov
Copyright-Jahr
2017
Electronic ISBN
978-3-319-51658-5
Print ISBN
978-3-319-51657-8
DOI
https://doi.org/10.1007/978-3-319-51658-5