Ill-Posed Problems
As previously mentioned, for problems in mathematical physics Hadamard [118] postulated three requirements: a solution should exist, the solution should be unique, and the solution should depend continuously on the data. The third postulate is motivated b
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Ill-Posed Problems
As previously mentioned, for problems in mathematical physics Hadamard [118] postulated three requirements: a solution should exist, the solution should be unique, and the solution should depend continuously on the data. The third postulate is motivated by the fact that in all applications the data will be measured quantities. Therefore, one wants to make sure that small errors in the data will cause only small errors in the solution. A problem satisfying all three requirements is called wellposed. Otherwise, it is called ill-posed. For a long time, research on ill-posed problems was neglected since they were not considered relevant to the proper treatment of applied problems. However, it eventually became apparent that a growing number of important problems fail to be well-posed, for example Cauchy’s problem for the Laplace equation and the initial boundary value problem for the backward heat equation. In particular, a large number of inverse problems for partial differential equations turn out to be ill-posed. Most classical problems where one assumes the partial differential equation, its domain and its initial and/or boundary data completely prescribed are well-posed in a canonical setting. Usually, such problems are referred to as direct problems. However, if the problem consists in determining part of the differential equation or its domain or its initial and/or boundary data then this inverse problem quite often will be ill-posed in any reasonable setting. In this sense, there is a close linkage and interaction between research on inverse problems and ill-posed problems. This chapter is intended as an introduction into the basic ideas on ill-posed problems and regularization methods for their stable approximate solution. We mainly confine ourselves to linear equations of the first kind with compact operators in Hilbert spaces and base our presentation on the singular value decomposition. From the variety of regularization concepts, we will discuss only the spectral cut-off, Tikhonov regularization, the discrepancy principle and quasi-solutions. At the end of the chapter, we will include some material on nonlinear problems. For a more comprehensive study of ill-posed problems, we refer to Baumeister [18], Engl, Hanke and Neubauer [96], Groetsch [111], Kabanikhin [170], Kaltenbacher, Neubauer and Scherzer [171], Kirsch [182], Kress [205], Louis [232], Morozov [249], Tikhonov and Arsenin [316] and Wang, Yagola and Yang [324]. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences 93, DOI 10.1007/978-1-4614-4942-3 4, © Springer Science+Business Media New York 2013
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4 Ill-Posed Problems
4.1 The Concept of Ill-Posedness We will first make Hadamard’s concept of well-posedness more precise. Definition 4.1 Let A : U ⊂ X → V ⊂ Y be an operator from a subset U of a normed space X into a subset V of a normed space Y. The equation A(ϕ) = f
(4.1)
is called well-posed or properly posed if A : U → V is bijective and the inverse operator A−1 :
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