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Linear inverse problems. The maximum entropy connection. With CD-ROM. (English) Zbl 1255.65110

Series on Advances in Mathematics for Applied Sciences 83. Hackensack, NJ: World Scientific (ISBN 978-981-4338-77-6/hbk; 978-981-4338-78-3/ebook). xxi, 326 p. (2011).
In this monograph, basic aspects of linear inverse problems are examined. Particular emphasis is put on convexly constraint problems with solutions and data that are values of random variables, respectively. In what follows, we summarize some of the key features of this text. In the first two chapters, an introduction and a collection of examples of linear inverse problems are presented. Topics include underdetermined matrix equations with different types of convex constraints like box constraints or quadratic constraints. Cubic spline interpolation and the inversion of the Laplace transform, the Fourier transform and the \(X\)-ray transform from partial data are also under consideration.
Chapter 3 gives an introduction to the basic theory of ill-posed problems and their regularization. Topics include projection methods, quasi-solutions, variational methods and general regularization schemes. In Chapter 4, deterministic and stochastic approaches for the regularization of linear ill-posed problems in Hilbert spaces are discussed briefly. For the deterministic part this includes Tikhonov regularization, the truncated singular value decomposition, and residual principles as a basic tool for choosing the regularization parameter. The stochastic approach includes Gaussian regularization, Bayesian methods and the maximum likelihood method.
The first subject of Chapter 5 are problems of the following two classes: (a) \(Ax \in B_M(y,T)\), \(x \in C\), and (b) \(Ax = y\), \(x \in C\). Here \(A: V \to W\) is a bounded linear operator between Banach spaces \(V\) and \(W\), and \(y \in W\). In addition, \(C\) is some closed convex set in \(V\), and the notation \(B_M(y,T) = \{ \eta \in W: d_M(y,\eta) < T \}\) is used, where \(d_M\) denotes some metric on \(W\). For solving such problems in the mean, in this chapter the maximum entropy method is considered. Here, a probability distribution \(P\) on \(C\) has to be determined such that \(x = \int_C \xi P(d\xi)\) satisfies either (a) or (b) considered above, and \(P\) in addition is required to be absolutely continuous with respect to a given probability distribution \(Q\) on \(C\). Problems of this form can be solved by maximizing the entropy functional \(S(\rho)\) over an appropriate set of functions \(\rho\), and by minimizing the dual entropy functional \(\Sigma(\lambda)\) for \(\lambda \in W^*\). Here, the entropy functional is of the form \(S(\rho) = - \int_C \rho(\xi) \ln(\rho(\xi)) dQ(\xi)\), where \(\rho(\xi) = \tfrac{dP}{dQ}(\xi)\) is the Radon-Nikodym derivative of \(P\) with respect to \(Q\). In addition, the dual entropy functional is given by \(\Sigma(\lambda) = \ln (Z(\lambda)) + \langle \lambda, y \rangle\), \(\lambda \in W^*\), where \(Z(\lambda) = \int_{C} \exp(- \langle \lambda, A \xi \rangle) \, dQ(\xi)\), and \(W^*\) and \(\langle \cdot, \cdot \rangle\) denote the dual space of \(W\) and the corresponding bilinear form on \(W^* \times W\), respectively. Some properties of both entropy functionals are presented in this chapter, and variants of the maximum entropy method are considered to solve problems, e.g., of the form \(Ax = y + n\), \(x \in C\), with random noise \(n\), or \(Ax = y\), \(x \in C_1\), where \(C_1\) is some translation of \(C\).
Chapter 6 deals with the minimum norm solution of underdetermined finite-dimensional problems \(A x = y\), where \(A\) is an \(m \times n\) matrix, with \(m < n\). Here, the Lagrange multiplier method and the Kaczmarz method are considered. Convex constraints of various types are also incorporated, e.g., equality and inequality constraints, box constraints, Euclidean and non-Euclidean norm constraints. The maximum entropy method is considered here for Bernoulli measures \(Q\) and also for uniform measures \(Q\).
In Chapter 7, some numerical examples and moment problems are considered in a deterministic setting and the maximum entropy setting as well. First, several approaches for the determination of the density of the Earth from its given mass and moment of inertia are presented under the assumption of spherical symmetry. In the next section, a test example for numerical experiments is considered. The chapter concludes with some remarks on Hausdorff, Hamburger and Stieltjes moment problems.
The focus of Chapter 8 is the maximum entropy method for unconstrained linear integral equations of the first kind, \(\int_0^1 A(s,t) x(t) dt = y(s)\), with an unknown function \(x \in V = C[0,1]\). The random function approach is to find a measure \(P\) on \(V\) that is absolutely continuous with respect to a given measure \(Q\) on \(V\) and satisfies \(E_P[ \int_0^1 A(s,t) X(t) dt ] = y(s)\), where the random variable \(X: V \to \mathbb{R}\) is given by \((Xw)(t) = w(t)\), and \(E_P\) means expectation with respect to the measure \(P\). In addition, a random measure approach is considered. The basic theory for both approaches is presented in this chapter, with special applications to different measures like Gaussian, Poisson and exponential distribution.
Chapter 9 is devoted to transportation problems, tomography and the reconstruction from marginals by maximum entropy methods. Chapter 10 then deals with the numerical inversion of the Laplace transform. It starts with some basics about Laplace transforms, followed by inversion methods based on an expansion with respect to sine functions, Legendre polynomials and Laguerre polynomials, respectively. A connection with Pick-Nevanlinna interpolation is also presented. Chapter 11 is devoted to the maxentropic characterization of some probability distributions. The book concludes with appendices containing some background material from functional analysis, stochastics, and entropy functionals. A user guide to Matlab software that is provided on an attached CD-ROM is also presented.
This monograph provides a thorough survey of maximum entropy methods for linear inverse problems. It is well written, but the absence of an index makes reading text difficult sometimes. This text can be recommended to experts as well as graduate students interested in the stochastic connection in inverse problems.

MSC:

65J22 Numerical solution to inverse problems in abstract spaces
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
65F22 Ill-posedness and regularization problems in numerical linear algebra
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
65J10 Numerical solutions to equations with linear operators
44A10 Laplace transform
44A12 Radon transform
65R10 Numerical methods for integral transforms
45A05 Linear integral equations
65R20 Numerical methods for integral equations
44A60 Moment problems
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs

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