Per Christian Hansen's Rank-Deficient and Discrete Ill-Posed Problems: Numerical PDF

By Per Christian Hansen

Here's an outline of contemporary computational stabilization equipment for linear inversion, with functions to numerous difficulties in audio processing, clinical imaging, seismology, astronomy, and different parts. Rank-deficient difficulties contain matrices which are precisely or approximately rank poor. Such difficulties usually come up in reference to noise suppression and different difficulties the place the objective is to suppress undesirable disturbances of given measurements. Discrete ill-posed difficulties come up in reference to the numerical therapy of inverse difficulties, the place one generally desires to compute information regarding inside homes utilizing external measurements. Examples of inverse difficulties are picture recovery and tomography, the place one must enhance blurred photographs or reconstruct images from uncooked info. This ebook describes new and present numerical equipment for the research and resolution of rank-deficient and discrete ill-posed difficulties. The emphasis is on perception into the stabilizing homes of the algorithms and the potency and reliability of the computations.

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Extra resources for Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion

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This involves O(mn(n — p)) operations. 46). We also need to partition L, T, and x as 42 2. DECOMPOSITIONS AND OTHER TOOLS where Lu € Mpxp, Tu € R(n-^xP, and ari e Rp. Then it is given by which is proved by checking the four conditions in [100]. Hence, the two vectors y — LAx and y = (LA)Tx are given by the relations and which lead to the following algorithms for computing y and y: In the above formulas, W need not have orthonormal columns, although this is the best choice from a numerical point of view.

DECOMPOSITIONS AND OTHER TOOLS where Lu € Mpxp, Tu € R(n-^xP, and ari e Rp. Then it is given by which is proved by checking the four conditions in [100]. Hence, the two vectors y — LAx and y = (LA)Tx are given by the relations and which lead to the following algorithms for computing y and y: In the above formulas, W need not have orthonormal columns, although this is the best choice from a numerical point of view. 49) become particularly simple. , if L represents the Laplacian—then an LU factorization of LU is required.

Then the n singular values <7Zof A are approximations to n singular values of K. Moreover, if we introduce the functions where Uij and Vij are the elements of U and V, then these functions are approximations to n left and right singular functions of K, due to the following results. 1. [178, Theorems 1-5]. , n we have 44 2. DECOMPOSITIONS AND OTHER TOOLS This theorem implies that if 8n —> 0 for n —>• oc, then the approximate singular values crz- converge uniformly in n to the true singular values //i, and the corresponding approximate singular functions Ui and vi converge in the mean to the true singular functions.

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