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Hierarchical Reconstruction Method f...

University of Maryland, College Park., Applied Mathematics and Scientific Computation.

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Hierarchical Reconstruction Method for Solving Ill-posed Linear Inverse Problems.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Hierarchical Reconstruction Method for Solving Ill-posed Linear Inverse Problems./
Author:	Zhong, Ming.
Published:	Ann Arbor : ProQuest Dissertations & Theses, : 2016,
Description:	122 p.
Notes:	Source: Dissertation Abstracts International, Volume: 77-10(E), Section: B.
Contained By:	Dissertation Abstracts International77-10B(E).
Subject:	Applied mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=10128674
ISBN:	9781339866758

Hierarchical Reconstruction Method for Solving Ill-posed Linear Inverse Problems.
Zhong, Ming.

Hierarchical Reconstruction Method for Solving Ill-posed Linear Inverse Problems. - Ann Arbor : ProQuest Dissertations & Theses, 2016 - 122 p.

Source: Dissertation Abstracts International, Volume: 77-10(E), Section: B.

Thesis (Ph.D.)--University of Maryland, College Park, 2016.

We present a detailed analysis of the application of a multi-scale Hierarchical Reconstruction method for solving a family of ill-posed linear inverse problems. When the observations on the unknown quantity of interest and the observation operators are known, these inverse problems are concerned with the recovery of the unknown from its observations. Although the observation operators we consider are linear, they are inevitably ill-posed in various ways. We recall in this context the classical Tikhonov regularization method with a stabilizing function which targets the specific ill-posedness from the observation operators and preserves desired features of the unknown. Having studied the mechanism of the Tikhonov regularization, we propose a multi-scale generalization to the Tikhonov regularization method, so-called the Hierarchical Reconstruction (HR) method. First introduction of the HR method can be traced back to the Hierarchical Decomposition method in Image Processing. The HR method successively extracts information from the previous hierarchical residual to the current hierarchical term at a finer hierarchical scale. As the sum of all the hierarchical terms, the hierarchical sum from the HR method provides an reasonable approximate solution to the unknown, when the observation matrix satisfies certain conditions with specific stabilizing functions. When compared to the Tikhonov regularization method on solving the same inverse problems, the HR method is shown to be able to decrease the total number of iterations, reduce the approximation error, and offer self control of the approximation distance between the hierarchical sum and the unknown, thanks to using a ladder of finitely many hierarchical scales. We report numerical experiments supporting our claims on these advantages the HR method has over the Tikhonov regularization method.

ISBN: 9781339866758Subjects--Topical Terms:

2122814
Applied mathematics.

Hierarchical Reconstruction Method for Solving Ill-posed Linear Inverse Problems.
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300 $a 122 p.
500 $a Source: Dissertation Abstracts International, Volume: 77-10(E), Section: B.
500 $a Adviser: Eitan Tadmor.
502 $a Thesis (Ph.D.)--University of Maryland, College Park, 2016.
520 $a We present a detailed analysis of the application of a multi-scale Hierarchical Reconstruction method for solving a family of ill-posed linear inverse problems. When the observations on the unknown quantity of interest and the observation operators are known, these inverse problems are concerned with the recovery of the unknown from its observations. Although the observation operators we consider are linear, they are inevitably ill-posed in various ways. We recall in this context the classical Tikhonov regularization method with a stabilizing function which targets the specific ill-posedness from the observation operators and preserves desired features of the unknown. Having studied the mechanism of the Tikhonov regularization, we propose a multi-scale generalization to the Tikhonov regularization method, so-called the Hierarchical Reconstruction (HR) method. First introduction of the HR method can be traced back to the Hierarchical Decomposition method in Image Processing. The HR method successively extracts information from the previous hierarchical residual to the current hierarchical term at a finer hierarchical scale. As the sum of all the hierarchical terms, the hierarchical sum from the HR method provides an reasonable approximate solution to the unknown, when the observation matrix satisfies certain conditions with specific stabilizing functions. When compared to the Tikhonov regularization method on solving the same inverse problems, the HR method is shown to be able to decrease the total number of iterations, reduce the approximation error, and offer self control of the approximation distance between the hierarchical sum and the unknown, thanks to using a ladder of finitely many hierarchical scales. We report numerical experiments supporting our claims on these advantages the HR method has over the Tikhonov regularization method.
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