A simple regularization method for the ill-posed evolution equation

Tuan, Nguyen Huy; Trong, Dang Duc

doi:10.1007/s10587-011-0019-9

Cited by 7 publications

(4 citation statements)

References 21 publications

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“…However, the regularization method, as a way to deal with the illposed problem, has been used by mathematical workers and engineering technicians [36][37][38] …”

Section: The Regularization Methods For Solvingmentioning

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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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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“…However, the regularization method, as a way to deal with the illposed problem, has been used by mathematical workers and engineering technicians [36][37][38] …”

Section: The Regularization Methods For Solvingmentioning

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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“…For some of works on the continuous spectrum of A, we refer the reader to N.Boussetila and F. Rebbani [3,4], Denche and S. Djezzar [7], N.H. Tuan and D.D. Trong [13,14]. As far as we know, we did not find any results concerned with nonlinear backward Cauchy problems with spectrum continuous operator.…”

Section: Introductionmentioning

confidence: 93%

Regularization for a nonlinear backward parabolic problem with continuous spectrum operator

Tuan¹,

Nhat²,

Trong³

2013

MMN

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We study the backward parabolic problem for a nonlinear parabolic equation of the form u t C Au.t/ D f .t; u.t//; u.T / D ', where A is a positive self-adjoint unbounded operator and f is a Lipschitz function. The problem is ill-posed, in the sense that if the solution does exist, it will not depend continuously on the data. To regularize the problem, we use the quasiboundary method and the quasi-reversibility method to establish a modified problem. We present approximated solutions that depend on a small parameter > 0 and give error estimates for our regularization.

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“…In this case

a = 1

b = 0

, the problem () is the backward parabolic and has been studied a lot in recent decades. Namely, we specify the following references: Tikhonov and Arsenin, 5 Showalter, 6 Dench, 7 Clark and Oppenheimer, 8 Triet et al 9 and Tuan 10,11 (the reader may see the references therein). In our article, we consider the case

F false(x, t false) = φ false(t false) f false(x false)

.…”

Section: Introductionmentioning

confidence: 99%

Recovering the source term for parabolic equation with nonlocal integral condition

Phương

Băleanu

Phong

et al. 2021

Math Methods in App Sciences

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The main purpose of this article is to present a Tikhonov method to construct the source function f(x) of the parabolic diffusion equation. This problem is well known to be severely ill‐posed. Therefore, regularization is required. The error estimates between the sought solution and the regularized solution are obtained under an a priori parameter choice rule and an a posteriori parameter choice rule, respectively. One numerical test illustrates that the proposed method is feasible and effective.

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A simple regularization method for the ill-posed evolution equation

Cited by 7 publications

References 21 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

Regularization for a nonlinear backward parabolic problem with continuous spectrum operator

Recovering the source term for parabolic equation with nonlocal integral condition

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