On the Choice of the Tikhonov Regularization Parameter and the Discretization Level: A Discrepancy-Based Strategy

Albani, Vinicius; Cezaro, Adriano De; Zubelli, Jorge P.

doi:10.2139/ssrn.2512441

Cited by 3 publications

(2 citation statements)

References 20 publications

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“…Thus, the ensemble size determines a discretization level of the finite dimensional space approximating the truth. Therefore, the aforementioned relation between ensemble size and the optimal choice of regularization parameter may be potentially established by using, for example, ideas from [3] where the relation between discretization levels and the selection of the regularization parameter in Tikhonov regularization has been recently investigated.…”

Section: Discussionmentioning

confidence: 99%

A regularizing iterative ensemble Kalman method for PDE-constrained inverse problems

Iglesias¹

2016

Inverse Problems

111

126

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Abstract. We introduce a derivative-free computational framework for approximating solutions to nonlinear PDE-constrained inverse problems. The general aim is to merge ideas from iterative regularization with ensemble Kalman methods from Bayesian inference to develop a derivative-free stable method easy to implement in applications where the PDE (forward) model is only accessible as a black box (e.g. with commercial software). The proposed regularizing ensemble Kalman method can be derived as an approximation of the regularizing Levenberg-Marquardt (LM) scheme [14] in which the derivative of the forward operator and its adjoint are replaced with empirical covariances from an ensemble of elements from the admissible space of solutions. The resulting ensemble method consists of an update formula that is applied to each ensemble member and that has a regularization parameter selected in a similar fashion to the one in the LM scheme. Moreover, an early termination of the scheme is proposed according to a discrepancy principle-type of criterion. The proposed method can be also viewed as a regularizing version of standard Kalman approaches which are often unstable unless ad-hoc fixes, such as covariance localization, are implemented.The aim of this paper is to provide a detailed numerical investigation of the regularizing and convergence properties of the proposed regularizing ensemble Kalman scheme; the proof of these properties is an open problem. By means of numerical experiments, we investigate the conditions under which the proposed method inherits the regularizing properties of the LM scheme of [14] and is thus stable and suitable for its application in problems where the computation of the Fréchet derivative is not computationally feasible. More concretely, we study the effect of ensemble size, number of measurements, selection of initial ensemble and tunable parameters on the performance of the method. The numerical investigation is carried out with synthetic experiments on two model inverse problems: (i) identification of conductivity on a Darcy flow model and (ii) electrical impedance tomography with the complete electrode model. We further demonstrate the potential application of the method in solving shape identification problems that arises from the aforementioned forward models by means of a level-set approach for the parameterization of unknown geometries.

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Section: Discussionmentioning

confidence: 99%

A regularizing iterative ensemble Kalman method for PDE-constrained inverse problems

Iglesias¹

2016

Inverse Problems

111

126

View full text Add to dashboard Cite

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“…Stability Analysis. In the paper, we may determine a suitable regular parameter > 0 by the discrepancy principle [44,45]; we choose > 0 so that…”

Section: A Novel Fractional Tikhonov Regularization (Nftr)mentioning

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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Finding Near-Optimal Regularization Parameter for Indoor Device-free Localization

Chomu

Atanasovski

Gavrilovska

2016

Wireless Pers Commun

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On the Choice of the Tikhonov Regularization Parameter and the Discretization Level: A Discrepancy-Based Strategy

Cited by 3 publications

References 20 publications

A regularizing iterative ensemble Kalman method for PDE-constrained inverse problems

A regularizing iterative ensemble Kalman method for PDE-constrained inverse problems

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

Finding Near-Optimal Regularization Parameter for Indoor Device-free Localization

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