Range-relaxed criteria for choosing the Lagrange multipliers in nonstationary iterated Tikhonov method

Abstract

In this article we propose a novel nonstationary iterated Tikhonov (NIT)-type method for obtaining stable approximate solutions to ill-posed operator equations modeled by linear operators acting between Hilbert spaces. Geometrical properties of the problem are used to derive a new strategy for choosing the sequence of regularization parameters (Lagrange multipliers) for the NIT iteration. Convergence analysis for this new method is provided. Numerical experiments are presented for two distinct applications: (I) a two-dimensional elliptic parameter identification problem (inverse potential problem); and (II) an image-deblurring problem. The results obtained validate the efficiency of our method compared with standard implementations of the NIT method (where a geometrical choice is typically used for the sequence of Lagrange multipliers).

Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model

“…We can obtain the following theorem according to the first-order second-moment theory of reliability analysis [8][9]. We can prove that this distance is an essential indicator in reliability analysis: reliability indicator  .…”

A range-relaxed criteria for choosing the Lagrange multipliers in the iterated Tikhonov Kaczmarz method for solving systems of linear ill-posed equations

“…In the nonstationary iT methods, each λ k is chosen either a priori (e.g., the geometrical choice λ k = q k , q > 1) or a posteriori [5,11]. In this article we focus on the a posteriori strategy investigated in [5], where the authors propose a choice for the Lagrange multipliers, which requires the residual at the next iterate to assume a prescribed value dependent on the current residual and also on the noise level. More precisely, λ k is chosen so that the next iterate has a prescribed residual satisfying δ Ax δ k+1 − y δ Φ( Ax δ k − y δ , δ), where Φ represents a convex combination of Ax δ k − y δ and δ.…”

“…The method proposed and analyzed in this manuscript for solving the ill-posed problem (1) and ( 2) is a Kaczmarz type method, where each step is defined as in the iT method (4) and the choice of Lagrange multipliers proposed in [5] is adopted. This iterative method is defined by…”

“…In this article we propose a nonstationary iterated Tikhonov Kaczmarz (iTK) type method for obtaining regularized approximations of systems of linear ill-posed operator equations. This is a Kaczmarz type method [25], where each step is defined as in the iterated Tikhonov (iT) method [6, section 1.2] with the Lagrange multiplier being chosen as to guarantee the residual of the next iterate to be in a range [5].…”

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In this article we propose and analyze a nonstationary iterated Tikhonov Kaczmarz (iTK) type method for obtaining stable approximate solutions to systems of ill-posed equations modeled by linear operators acting between Hilbert spaces. We generalize for the iTK iteration the criteria proposed in [5] for the iterated Tikhonov method. The goal is to devise an efficient strategy for choosing the Lagrange multipliers in this method.

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