On the asymptotical regularization with convex constraints for nonlinear ill-posed problems

Zhong, Min; Wang, Wei; Zhu, Kai

doi:10.1016/j.aml.2022.108247

Cited by 7 publications

(6 citation statements)

References 16 publications

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“…We therefore believe that SAR will be a useful tool for studying biosensor interactions and other real-world inverse problems with deterministic ill-posed forward models. Finally we remark that recently, SAR has been extended for nonlinear operator equations, where another important feature had been explored: it can escape local minimums for nonlinear problems [18].…”

Section: Discussionmentioning

confidence: 99%

“…Proof. The appropriateness of assumption (18) follows from (a) the source conditions of x † can be either range-type as shown in (31) or variational inequalities as discussed in remark 4; (b) φ can be chosen as in example 1 later; (c) the terminating time t * can be selected according to either an a priori rule t * = t * (δ, φ) (see theorem 2 or [27, theorem 1]) or an a posteriori rule (e.g. the discrepancy principle) t * = t * (δ, y δ ) 8 (see theorem 3 or [27, theorem 2]).…”

Section: Uncertainty Quantification Of Sarmentioning

confidence: 99%

“…occurs. Hence, combining ( 16) and (18), we obtain (19) by noting that δ ⃗ ri H −r (R d ) < ∞ for all fixed ⃗ r i .…”

Section: Uncertainty Quantification Of Sarmentioning

confidence: 99%

“…), and x † ∈ R((A * A) p ) for all p < 5/8. The constant C as in(18) can be estimated by C as ≈ (p/e) p ρ + ϑ = 1.0372 (p = 5/8, ρ = 1 and ϑ = 0.6382).…”

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confidence: 99%

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Stochastic asymptotical regularization for linear inverse problems

Zhang¹,

Chen²

2022

Inverse Problems

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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 demonstrate that SAR can quantify the uncertainty in error estimates for inverse problems. We prove the regularizing properties of SAR with regard to mean-square convergence. We also show that SAR is an order-optimal 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 \emph{a priori} and \emph{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 real-world problems, usually obscured by the incomplete mathematical modeling and the ascendence of complex-structured noise.

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Section: Discussionmentioning

confidence: 99%

Section: Uncertainty Quantification Of Sarmentioning

confidence: 99%

“…occurs. Hence, combining ( 16) and (18), we obtain (19) by noting that δ ⃗ ri H −r (R d ) < ∞ for all fixed ⃗ r i .…”

Section: Uncertainty Quantification Of Sarmentioning

confidence: 99%

“…), and x † ∈ R((A * A) p ) for all p < 5/8. The constant C as in(18) can be estimated by C as ≈ (p/e) p ρ + ϑ = 1.0372 (p = 5/8, ρ = 1 and ϑ = 0.6382).…”

mentioning

confidence: 99%

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Stochastic asymptotical regularization for linear inverse problems

Zhang¹,

Chen²

2022

Inverse Problems

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“…where Θ * : X → (−∞,∞] is the Legendre-Fenchel conjugate of Θ. In the recent paper [40], the asymptotical regularization with uniform convex penalty functionals is studied, in which a regularized approximation pair (ξ δ (T),x δ (T)) is obtained by solving the initial value problem…”

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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