By Charles L. Byrne
Iterative Optimization in Inverse Problems brings jointly a few vital iterative algorithms for scientific imaging, optimization, and statistical estimation. It accommodates contemporary paintings that has no longer seemed in different books and attracts at the author’s massive learn within the box, together with his lately built category of SUMMA algorithms. with regards to sequential unconstrained minimization tools, the SUMMA category contains a wide selection of iterative algorithms renowned to researchers in numerous parts, akin to facts and picture processing.
Organizing the themes from basic to extra particular, the ebook first supplies an summary of sequential optimization, the subclasses of auxiliary-function equipment, and the SUMMA algorithms. the following 3 chapters current specific examples in additional aspect, together with barrier- and penalty-function tools, proximal minimization, and forward-backward splitting. the writer additionally specializes in fixed-point algorithms for operators on Euclidean area after which extends the dialogue to incorporate distance measures except the standard Euclidean distance. within the ultimate chapters, particular difficulties illustrate using iterative equipment formerly mentioned. so much chapters include workouts that introduce new principles and make the ebook appropriate for self-study.
Unifying a number of likely disparate algorithms, the e-book indicates how one can derive new homes of algorithms through evaluating recognized houses of alternative algorithms. This unifying strategy additionally is helping researchers—from statisticians engaged on parameter estimation to photo scientists processing scanning info to mathematicians fascinated by theoretical and utilized optimization—discover necessary similar algorithms in components outdoor in their expertise.
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Additional info for Iterative Optimization in Inverse Problems
Example text
6 The Lagrangian in Convex Programming . . . . . . . . 7 Infimal Convolution . . . . . . . . . . . . . . . . . . . 8 Moreau’s Proximity-Function Method . . . . . . . . . Basic Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 35 36 36 36 39 39 39 40 40 40 41 41 42 42 43 Barrier-function and penalty-function methods are the best known examples of sequential optimization. In their usual formulations neither fits into the AF class of algorithms.
4 The Method of Auslander and Teboulle . . . . . . . . 5 The EM Algorithm . . . . . . . . . . . . . . . . . . . The SUMMA Class of AF Methods . . . . . . . . . . . . . . . 1 The SUMMA Property . . . . . . . . . . . . . . . . . 2 Auslander and Teboulle Revisited . . . . . . . . . . . . 3 Proximal Minimization . . . . . . . . . . . . . . . . . 4 The IPA . . . . . . . . . . . . . . . . . . . . .
And i = i(k) = k(mod I) + 1, the entries of xk+1 are k = xkj + α−1 xk+1 j i Aij (bi − (Ax )i ). 26) Because the ART uses only a single equation at each step, it has been called a row-action [80] or sequential method. 1 When Ax = b Has Solutions For the consistent case we have the following result concerning the ART. 1 Let Aˆ x = b and let x0 be arbitrary. Let {xk } be generated by the ART. Then the sequence of Euclidean distances {||ˆ x − xk ||2 } is dek creasing and {x } converges to the solution of Ax = b closest to x0 .