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1998 | Book

Inverse Problems Read chapter Read first chapter

Inverse Problems for Partial Differential Equations

Author: Victor Isakov

Publisher: Springer New York

Book Series : Applied Mathematical Sciences

Included in: Professional Book Archive

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About this book

This book describes the contemporary state of the theory and some numerical aspects of inverse problems in partial differential equations. The topic is of sub­ stantial and growing interest for many scientists and engineers, and accordingly to graduate students in these areas. Mathematically, these problems are relatively new and quite challenging due to the lack of conventional stability and to nonlinearity and nonconvexity. Applications include recovery of inclusions from anomalies of their gravitational fields; reconstruction of the interior of the human body from exterior electrical, ultrasonic, and magnetic measurements, recovery of interior structural parameters of detail of machines and of the underground from similar data (non-destructive evaluation); and locating flying or navigated objects from their acoustic or electromagnetic fields. Currently, there are hundreds of publica­ tions containing new and interesting results. A purpose of the book is to collect and present many of them in a readable and informative form. Rigorous proofs are presented whenever they are relatively short and can be demonstrated by quite general mathematical techniques. Also, we prefer to present results that from our point of view contain fresh and promising ideas. In some cases there is no com­ plete mathematical theory, so we give only available results. We do not assume that a reader possesses an enormous mathematical technique. In fact, a moderate knowledge of partial differential equations, of the Fourier transform, and of basic functional analysis will suffice.

Table of Contents

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
$$ Ax = y $$
(2.0)
linear) continuous operator acting from a subset X of a Banach space into a subset Y of another Banach space, and x ∈ X is to be found given y. We discuss solvability of this equation when A −1 does not exist by outlining basic results of the theory created in the 1960s by Ivanov, John, Lavrent’ev, and Tikhonov. In Section 2.1 we give definitions of well- and ill-posedness, together with important illustrational examples. In Section 2.2 we describe a class of equations (2.0) that can be numerically solved in a stable way. Section 2.3 is to the variational construction of algorithms of solutions by minimizing Tikhonov stabilizing fimctionals. In Section 2.4 we show that stability estimates for equation (2.0) imply convergence rates for numerical algorithms and discuss the relation between convergence of these algorithms and the existence of a solution to (2.0). The final section, Section 2.5, describes some iterative regularization algorithms.
Victor Isakov
3. Uniqueness and Stability in the Cauchy Problem
Abstract
In this chapter we formulate and in many cases prove results on 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 logarithmic convexity. In Section 3.2 we formulate general conditional Carleman estimates, and we apply the results to the general Cauchy problem. We also formulate a global version of Holmgren’s theorem and the recent result of Tataru on nonanalytic equations. In Section 3.3 we consider elliptic and parabolic equations of second order, construct for them pseudoconvex weight functions and obtain complete and general uniqueness and stability results for the Cauchy problem. Section 3.4 is devoted to substantially less understood hyperbolic equations and Schrödinger-type equations. Here, for some particular but interesting domains we also give appropriate weight functions and obtain a quite explicit description of uniqueness domains for lateral Cauchy problems. Additional information to Sections 3.2–3.4 can be found in the book of Zuily [Z].
Victor Isakov
4. Elliptic Equations. Single Boundary Measurements
Abstract
In this chapter we consider the elliptic second-order differential equation
$$ Au = f\;{\text{in}}\;\Omega ,\;f = {f_0} - \sum\limits_{j = 1}^n {{\partial _j}{f_j}} $$
(4.0.1)
with the Dirichlet boundary data
$$ u = g\;{\text{on}}\;\partial \Omega . $$
(4.0.2)
Victor Isakov
5. Elliptic Equations: Many Boundary Measurements
Abstract
We consider the Dirichlet problem (4.0.1), (4.0.2). We assume that for any Dirichlet data g we are given the Neumann data h; in other words, we know the results of all possible boundary measurements, or the so-called Dirichlet-to-Neumann operator Λ: H (1/2)(∂Ω) → H (−1/2)(∂Ω), whichmapsthe Dirichlet data g into the Neumann data h. From Theorem 4.1 the operator Λ is well-defined and continuous, provided that Ω is a bounded domain with Lipschitz ∂Ω. In Sections 5.1, 5.4, 5.7 we consider scalar a, b = 0, c = 0. The study of this problem was initiated by the paper of Calderon [C], who studied the inverse problem linearized around a constant and suggested a fruitful approach, which was extended by Sylvester and Uhlmann in their fundamental paper [SyU2], where the uniqueness problem was completely solved in the three-dimensional case.
Victor Isakov
6. Scattering problems
Abstract
The stationary incoming wave u of frequency k is a solution to the perturbed Helmholtz equation (scattering by medium)
$$ Au - {k^2}u = 0in{\mathbb{R}^3} $$
(6.0.1)
(A is the elliptic operator − div(a ∇) + b · ∇ + c with ℜb, div b = 0, and ℑc ≤ 0, which coincides with the Laplace operator outside a ball B and which possesses the uniqueness of continuation property) or to the Helmholtz equation (scattering by an obstacle)
$$ \Delta u + {k^2}u = 0\;in\;{D_e} = {\mathbb{R}^3}\backslash \bar D $$
(6.0.2)
with the Dirichlet boundary data
$$ u = 0\;{\text{on}}\;\partial D\;({\text{soft}}\;{\text{obstacle}}\;D). $$
(6.0.3d)
o the Neumann boundary data
$$ {\partial _v}u = 0\;{\text{on}}\;\partial D\;({\text{hard}}\;{\text{obstacle}}\;D). $$
(6.0.3n)
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 information about first arrival times of a wave). While there has been significant progress in the classical Radon problem when manifolds are hyperplanes and the weight function is unity, the situation is not quite clear even when the weight function is monotone along, say, straight lines in the plane case (attenuation). We give a brief review of this area, referring for more information to the book of Natterer [Nat].
Victor Isakov
8. Hyperbolic Problems
Abstract
In this chapter we are interested in finding coefficients of the second-order hyperbolic operator
$$ {a_0}\partial _t^2u + Au = f\;in\;Q = \Omega \times (0,T) $$
(8.0.1)
given the initial data
$$ u = {u_0},\;{\partial _t}u = {u_1}\;on\;\Omega \times \left\{ 0 \right\}, $$
(8.0.2)
the Neumann lateral data
$$ av \cdot \nabla u = h\;{\text{on}}\;{\Gamma _1} \times (0,T), $$
(8.0.3)
and the additional lateral data
$$ u = g\;{\text{on}}\;{\Gamma _0} \times (0,T). $$
(8.0.4)
Victor Isakov
9. Inverse parabolic problems
Abstract
In this chapter we consider the second-order parabolic equation
$$ {a_0}{\partial _t}u - div(a\nabla u) + b\nabla u + cu = f{\mkern 1mu} in{\mkern 1mu} Q = \Omega \times (0,{\mkern 1mu} T), $$
(9.0.1)
where Ω is a bounded domain the space ℝ n with the C 2-smooth boundary ∂Ω. In Section 9.5 we study the nonlinear equation
$$ {a_0}(x,u){u_t} - \Delta u + c(x,{\mkern 1mu} t,{\mkern 1mu} u) = 0{\mkern 1mu} in{\mkern 1mu} Q. $$
(9.0.1n)
Victor Isakov
10. Some Numerical Methods
Abstract
In this chapter we will briefly review some popular numerical methods widely used in practice. It is of course 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
Metadata
Title
Inverse Problems for Partial Differential Equations
Author
Victor Isakov
Copyright Year
1998
Publisher
Springer New York
Electronic ISBN
978-1-4899-0030-2
Print ISBN
978-1-4899-0032-6
DOI
https://doi.org/10.1007/978-1-4899-0030-2