Fractional Tikhonov regularization for linear discrete ill-posed problems

Hochstenbach, Michiel E.; Reichel, Lothar

doi:10.1007/s10543-011-0313-9

Cited by 81 publications

(64 citation statements)

References 9 publications

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“…This shortcoming led Klann and Ramlau to introduce the fractional Tikhonov regularization method [17]. Subsequently another approach, also referred to as fractional Tikhonov regularization (FTR), was investigated in [21], the model of which is described as follows:…”

Section: Fractional Tikhonov Regularization (Ftr) Methodmentioning

confidence: 99%

“…Furthermore, the FTR method has been applied to many fields and obtained some remarkable achievements [18][19][20]. M. E. Hochstenbach et al [21] used the FTR method to obtain approximate solution of higher quality which is better than the standard TR method in solving linear system equation with a severe ill-posedness. In [22], we can see that the FTR method is significantly better than the standard TR method, which was proved by several numerical examples.…”

Section: Introductionmentioning

confidence: 99%

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A Novel Fractional Tikhonov Regularization Coupled with an Improved Super-Memory Gradient Method and Application to Dynamic Force Identification Problems

Wang

Ren

Liu

2018

Mathematical Problems in Engineering

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This paper presents a novel inverse technique to provide a stable optimal solution for the ill-posed dynamic force identification problems. Due to ill-posedness of the inverse problems, conventional numerical approach for solutions results in arbitrarily large errors in solution. However, in the field of engineering mathematics, there are famous mathematical algorithms to tackle the illposed problem, which are known as regularization techniques. In the current study, a novel fractional Tikhonov regularization (NFTR) method is proposed to perform an effective inverse identification, then the smoothing functional of the ill-posed problem processed by the proposed method is regarded as an optimization problem, and finally a stable optimal solution is obtained by using an improved super-memory gradient (ISMG) method. The result of the present method is compared with that of the standard TR method and FTR method; new findings can be obtained; that is, the present method can improve accuracy and stability of the inverse identification problem, robustness is stronger, and the rate of convergence is faster. The applicability and efficiency of the present method in this paper are demonstrated through a mathematical example and an engineering example.

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Section: Fractional Tikhonov Regularization (Ftr) Methodmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Novel Fractional Tikhonov Regularization Coupled with an Improved Super-Memory Gradient Method and Application to Dynamic Force Identification Problems

Wang

Ren

Liu

2018

Mathematical Problems in Engineering

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“…The feasibility of this approach for fractional Tikhonov and Lavrentiev methods is illustrated in [7,8] and, therefore, will not be discussed further in this paper.…”

Section: Computed Examplesmentioning

confidence: 99%

“…Numerical results reported in [7] show that the solution x µ,α of (1.5) typically is a more accurate approximation of x for a suitable 0 < α < 1 than when α = 0 or α = 1. The fractional Lavrentiev regularization method described in [7] is related to fractional Tikhonov regularization, which is discussed in [8,10]. This paper proposes to replace the regularization matrix µI in (1.5) by a matrix M µ,α ∈ C n×n , such that M µ,α F < µI F and A 1+α + M µ,α has no eigenvalue smaller than µ.…”

Section: Introduction We Consider the Solution Of Linear Systems Of mentioning

confidence: 99%

Lavrentiev-type regularization methods for Hermitian problems

Noschese

Reichel

2014

Calcolo

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Abstract. Lavrentiev regularization is a popular approach to the solution of linear discrete illposed problems with a Hermitian positive semidefinite matrix. This paper describes Lavrentiev-type regularization methods that can be applied to the solution of linear discrete ill-posed problems with a general Hermitian matrix. Fractional Lavrentiev-type methods as well as modifications suggested by the solution of certain matrix nearness problems are described. Computed examples illustrate the competitiveness of modified fractional Lavrentiev-type methods.

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“…Since K † y δ is not a good approximation of x † , we approximate x † with x δ α := Rαy δ where {Rα} is a family of continuous operators depending on a parameter α. A classical example is the Tikhonov regularization defined by Rα = (K * K + αI) −1 K * , where I denotes the identity and K * the adjoint of K. Recently, new Tikhonov based regularization methods have been proposed in [1], [2] and [3], under the name of fractional Tikhonov, to reduce the oversmoothing property of the Tikhonov regularization in standard form, in order to preserve the details of the approximated solution. Their regularization and convergence properties have been previously investigated showing that they are of optimal order.…”

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confidence: 99%

Iterated fractional Tikhonov regularization

Bianchi

Buccini

Donatelli

et al. 2015

Proc Appl Math and Mech

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We consider linear operator equations of the formwhere K : X → Y is a compact linear operator between Hilbert spaces X and Y. We assume y to be attainable, i.e., that problem (1) has a solution x † = K † y of minimal norm. Here K † denotes the (Moore-Penrose) generalized inverse operator of K, which is unbounded when K is compact, with infinite dimensional range. Hence problem (1) is ill-posed and has to be regularized in order to compute a numerical solution. We want to approximate the solution x † of the equation (1) In this paper, we firstly provide a saturation result similar to the well-known saturation result for Tikhonov regularization [6]: indeed, Tikhonov regularization under suitable a-priori assumption and a-priori choice rule, α = α(δ) ∼ c(δ) 2/3 , is of optimal order and the best possible convergence rate obtainable is). On the other hand, let R(K) be the range of K and let Q be the orthogonal projector onto R(K), ifis not closed, and this shows how Tikhonov regularization for an ill-posed problem with compact operator never yields a convergence rate which is faster than O(δ 2 3 ), since it saturates at this rate. Such results motivated us to introduce the iterated versions of fractional Tikhonov methods in the same spirit of the iterated Tikhonov method. We prove that those iterated methods can overcome the afore-mentioned saturation results.Afterwards, inspired by the works [4,5] we introduce the nonstationary variants of our iterated methods. Differently from the nonstationary iterated Tikhonov, we have two nonstationary sequences of parameters. In the noise free case, we give sufficient conditions on these sequences to guarantee the convergence providing also the corresponding convergence rates. In the noise case, we show the stability of the proposed iterative schemes proving that they are regularization methods. For the brief space we have at disposal, we will only cite some of the main results we obtained for one kind of fractional Tikhonov algorithms, that we call weighted Tikhonov. For the results about the other fractional Tikhonov algorithm analysis, proofs, numerical examples and full details, we invite the interested reader to look at the full paper [7].

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Fractional Tikhonov regularization for linear discrete ill-posed problems

Cited by 81 publications

References 9 publications

A Novel Fractional Tikhonov Regularization Coupled with an Improved Super-Memory Gradient Method and Application to Dynamic Force Identification Problems

A Novel Fractional Tikhonov Regularization Coupled with an Improved Super-Memory Gradient Method and Application to Dynamic Force Identification Problems

Lavrentiev-type regularization methods for Hermitian problems

Iterated fractional Tikhonov regularization

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