Condition numbers and perturbation analysis for the Tikhonov regularization of discrete ill-posed problems

Chu, Delin; Lin, Lijing; Tan, Roger C. E.; Wei, Yimin

doi:10.1002/nla.702

Cited by 16 publications

(11 citation statements)

References 34 publications

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“…Table 14 displays the iteration numbersm for solving (A + 4 I)y 4 = b+ ▵b 4 , the corresponding CPU times in seconds, and the relative second norm errors for the solution of the perturbed systems (A + j I)y j = b+ ▵ b j , j = 1, 2, 3, 4 using the methods M i , i = 1, 2, … , 10 with Algorithm 1. Table 15 presents the perturbation analysis results obtained by the standard Tikhonov regularization solution for the values of the regularization parameter found by four parameter-choice methods from table IV in the work of Chu et al 9 when n = 64. Analyzing Tables 13-15, we conclude that the proposed methods M 3 , M 6 , M 8 , M 10 with Algorithm 1 give smoother solutions providing smaller relative second norm errors in solving the perturbed systems (A + j I)y j = b+ ▵b j , j = 2, 3, 4 for Example 4.…”

Section: Examplementioning

confidence: 99%

“…One of the methods is the L-curve method, which is in fact a parameterized curve, which has the regularization parameter in case of Tikhonov regularization. Some other different methods are the discrepancy principle, the quasi-optimality criterion, and generalized cross-validation (see the studies by Groetsch, 6 Hansen, 7,8 Chu et al, 9 and Fermin et al 10 and references therein). These techniques aim to find an optimal value of for the realization of the regularization method.…”

Section: Introductionmentioning

confidence: 99%

“…The application of Algorithm 1 is given by Example 4, which is the image reconstruction problem. 8,9,28 The problem is severely ill posed, and we applied Algorithm 1 with the proposed methods of orders p = 7, 13 from Class 1, with orders p = 9, 19 from Class 2 and the same methods from the literature applied in Examples 1, 2, and 3. The obtained results are also compared with the standard LU method with pivoting, standard Tikhonov regularization solution for the values of the regularization parameter found by four parameter-choice methods from Table IV in the work of Chu et al 9 when n = 64, and these results show that the proposed approaches are successful.…”

Section: Introductionmentioning

confidence: 99%

“…+ 2 = 2((k + 1) + 2), which is true for k + 1. Now, let us consider Class 2 (18) methods by induction when k = 1,q 1 = 9, andΦ9 (R m ) = (I + R m )(I + R 2 m )(I + R 5 m ) + R 4 m , m = 0, 1, … ; thus, iteration V m+1 = V mΦq k+1 (R m )performs seven matrix multiplications per step and = 3(1) + 4 = 7 holds true. Assume that it is true for k that…”

mentioning

confidence: 99%

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On the two classes of high‐order convergent methods of approximate inverse preconditioners for solving linear systems

Buranay

Subasi

Iyikal

2017

Numerical Linear Algebra App

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Two classes of methods for approximate matrix inversion with convergence orders p = 3 * 2 k +1 (Class 1) and p = 5 * 2 k −1 (Class 2), k ≥ 1 an integer, are given based on matrix multiplication and matrix addition. These methods perform less number of matrix multiplications compared to the known hyperpower method or pth-order method for the same orders and can be used to construct approximate inverse preconditioners for solving linear systems. Convergence, error, and stability analyses of the proposed classes of methods are provided. Theoretical results are justified with numerical results obtained by using the proposed methods of orders p = 7, 13 from Class 1 and the methods with orders p = 9, 19 from Class 2 to obtain polynomial preconditioners for preconditioning the biconjugate gradient (BICG) method for solving well-and ill-posed problems. From the literature, methods with orders p = 8, 16 belonging to a family developed by the effective representation of the pth-order method for orders p = 2 k , k is integer k ≥ 1, and other recently given high-order convergent methods of orders p = 6, 7, 8, 12 for approximate matrix inversion are also used to construct polynomial preconditioners for preconditioning the BICG method to solve the considered problems. Numerical comparisons are given to show the applicability, stability, and computational complexity of the proposed methods by paying attention to the asymptotic convergence rates. It is shown that the BICG method converges very quickly when applied to solve the preconditioned system. Therefore, the cost of constructing these preconditioners is amortized if the preconditioner is to be reused over several systems of same coefficient matrix with different right sides. KEYWORDS approximate inverse preconditioners, biconjugate gradient method, error bounds, Fredholm integral equation of the first kind, ill-posed problems, well-posed problems Numer Linear Algebra Appl. 2017;24:e2111.wileyonlinelibrary.com/journal/nla

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Section: Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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On the two classes of high‐order convergent methods of approximate inverse preconditioners for solving linear systems

Buranay

Subasi

Iyikal

2017

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“…Chu et al [12] define the non-structured mixed, componentwise, and normwise condition numbers for the Tikhonov regularization and obtain respectively…”

Section: Structured Condition Numbers For the Tikhonov Regularizationmentioning

confidence: 99%

Structured condition numbers of structured Tikhonov regularization problem and their estimations

Diao

Wei

Qiao

2016

Journal of Computational and Applied Mathematics

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Abstract. Both structured componentwise and structured normwise perturbation analysis of the Tikhonov regularization are presented. The structured matrices under consideration include: Toeplitz, Hankel, Vandermonde, and Cauchy matrices. Structured normwise, mixed and componentwise condition numbers for the Tikhonov regularization are introduced and their explicit expressions are derived. For the general linear structure, we prove the structured condition numbers are smaller than their corresponding unstructured counterparts based on the derived expressions. By means of the power method and small sample condition estimation, the fast condition estimation algorithms are proposed. Our estimation methods can be integrated into Tikhonov regularization algorithms that use the generalized singular value decomposition (GSVD). The structured condition numbers and perturbation bounds are tested on some numerical examples and compared with their unstructured counterparts. Our numerical examples demonstrate that the structured mixed condition numbers give sharper perturbation bounds than existing ones, and the proposed condition estimation algorithms are reliable.

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On condition numbers for least squares with quadric inequality constraint

Diao¹

2017

Computers & Mathematics with Applications

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Condition numbers and perturbation analysis for the Tikhonov regularization of discrete ill-posed problems

Cited by 16 publications

References 34 publications

On the two classes of high‐order convergent methods of approximate inverse preconditioners for solving linear systems

On the two classes of high‐order convergent methods of approximate inverse preconditioners for solving linear systems

Structured condition numbers of structured Tikhonov regularization problem and their estimations

On condition numbers for least squares with quadric inequality constraint

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