On the Asymptotical Regularization for Linear Inverse Problems in Presence of White Noise

Lü, Shuai; Niu, Pingping; Werner, Frank

doi:10.1137/20m1330841

Cited by 13 publications

(19 citation statements)

References 31 publications

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“…Consequently, it is necessary to apply regularization methods [7]. Regularization methods can be seen in three categories: variational regularization methods [27], iterative regularization methods [18] and asymptotical regularization methods [3,20,24,29,36,37], in which asymptotical regularization can be considered as a continuous analogue of iterative methods. For example, recalling the Landweber iteration [12] x…”

Section: Introductionmentioning

confidence: 99%

Iterative Runge-Kutta-Type Methods with Convex Penalty for Inverse Problems in Hilbert Spaces

Tong¹,

Wang²,

Fu³

et al. 2023

CSIAM-AM

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An s-stage Runge-Kutta-type iterative method with the convex penalty for solving nonlinear ill-posed problems is proposed and analyzed in this paper. The approach is developed by using a family of Runge-Kutta-type methods to solve the asymptotical regularization method, which can be seen as an ODE with the initial value. The convergence and regularity of the proposed method are obtained under certain conditions. The reconstruction results of the proposed method for some special cases are studied through numerical experiments on both parameter identification in inverse potential problem and diffuse optical tomography. The numerical results indicate that the developed methods yield stable approximations to true solutions, especially the implicit schemes have obvious advantages on allowing a wider range of step length, reducing the iterative numbers, and saving computation time.

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Section: Introductionmentioning

confidence: 99%

Iterative Runge-Kutta-Type Methods with Convex Penalty for Inverse Problems in Hilbert Spaces

Tong¹,

Wang²,

Fu³

et al. 2023

CSIAM-AM

View full text Add to dashboard Cite

show abstract

“…In reality, observations are usually affected by external disturbances, and the inverse problems with noisy observations have been widely studied including the case with deterministic ( [7][8][9][10]) and Gaussian white noises ( [11][12][13][14][15]) in statistical inverse problems. In recent years, with the development of data mining based on online data streams ( [18,19]), 3DVAR and Kalman filtering methods were applied to solve statistical inverse problems by introducing artificial dynamical systems based on real-time observations ( [20][21][22]).…”

Section: Introductionmentioning

confidence: 99%

“…Up to now, there have been a large number of mature research results on classical inverse problems in the deterministic framework, and the existing work on the inverse problems based on statistical and stochastic frameworks can be basically segmented into three categories: (i) statistical inverse problems based on deterministic time-invariant compact forward operators and real-time observations in a separable Hilbert space ( [20][21][22]); (ii) distributed parameter estimation problems in finite-dimensional spaces ( [23,[29][30][31][32][33][34][35][36]); (iii) decentralized online learning problems based on stationary (e.g., independent and identically distributed) observations in RKHS ( [38][39][40][41][42]). There are some basic problems, for example,…”

Section: Introductionmentioning

confidence: 99%

Random Inverse Problems Over Graphs: Decentralized Online Learning

Li¹,

Zhang²

2023

Preprint

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We establish a framework of random inverse problems with real-time observations over graphs, and present a decentralized online learning algorithm based on online data streams, which unifies the distributed parameter estimation in Hilbert space and the least mean square problem in reproducing kernel Hilbert space (RKHS-LMS). We transform the algorithm convergence into the asymptotic stability of randomly time-varying difference equations in Hilbert space with L 2 -bounded martingale difference terms and develop the L 2 -asymptotic stability theory. It is shown that if the network graph is connected and the sequence of forward operators satisfies the infinite-dimensional spatio-temporal persistence of excitation condition, then the estimates of all nodes are mean square and almost surely strongly consistent. By equivalently transferring the distributed learning problem in RKHS to the random inverse problem over graphs, we propose a decentralized online learning algorithm in RKHS based on non-stationary and non-independent online data streams, and prove that the algorithm is mean square and almost surely strongly consistent if the operators induced by the random input data satisfy the infinite-dimensional spatio-temporal persistence of excitation condition.

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“…This continuous framework has recently received much attention in the field of computational mathematics. For instance, in the field of inverse problems, the authors in [19] studied a modified evolution equation, (4), for linear inverse problems (1) in the presence of white noise. In particular, they connected this formulation, i.e.…”

Section: Introductionmentioning

confidence: 99%

Stochastic asymptotical regularization for linear inverse problems

Zhang¹,

Chen²

2022

Preprint

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We introduce Stochastic Asymptotical Regularization (SAR) methods for the uncertainty quantification of the stable approximate solution of ill-posed linear-operator equations, which are deterministic models for numerous inverse problems in science and engineering. We prove the regularizing properties of SAR with regard to meansquare convergence. We also show that SAR is an optimal-order regularization method for linear ill-posed problems provided that the terminating time of SAR is chosen according to the smoothness of the solution. This result is proven for both a priori and a posteriori stopping rules under general range-type source conditions. Furthermore, some converse results of SAR are verified. Two iterative schemes are developed for the numerical realization of SAR, and the convergence analyses of these two numerical schemes are also provided. A toy example and a real-world problem of biosensor tomography are studied to show the accuracy and the advantages of SAR: compared with the conventional deterministic regularization approaches for deterministic inverse problems, SAR can provide the uncertainty quantification of the quantity of interest, which can in turn be used to reveal and explicate the hidden information about realworld problems, usually obscured by the incomplete mathematical modeling and the ascendence of complex-structured noise.

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On the Asymptotical Regularization for Linear Inverse Problems in Presence of White Noise

Cited by 13 publications

References 31 publications

Iterative Runge-Kutta-Type Methods with Convex Penalty for Inverse Problems in Hilbert Spaces

Iterative Runge-Kutta-Type Methods with Convex Penalty for Inverse Problems in Hilbert Spaces

Random Inverse Problems Over Graphs: Decentralized Online Learning

Stochastic asymptotical regularization for linear inverse problems

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