The iterated Arnoldi-Tikhonov (iAT) method is a regularization technique particularly suited for solving large-scale ill-posed linear inverse problems. Indeed, it reduces the computational complexity through the projection of the discretized problem into a lower-dimensional Krylov subspace, where the problem is then solved.
This paper studies iAT under an additional hypothesis on the discretized operator. It presents a theoretical analysis of the approximation errors, leading to an a posteriori rule for choosing the regularization parameter. Our proposed rule results in more accurate computed approximate solutions compared to the a posteriori rule recently proposed in arXiv:2311.11823. The numerical results confirm the theoretical analysis, providing accurate computed solutions even when the new assumption is not satisfied.
This paper describes solution methods for linear discrete ill-posed problems defined by third order tensors and the t-product formalism introduced in [M. E. Kilmer and C. D. Martin, Factorization strategies for third order tensors, Linear Algebra Appl., 435 (2011), pp. 641--658]. A t-product Arnoldi (t-Arnoldi) process is defined and applied to reduce a large-scale Tikhonov regularization problem for third order tensors to a problem of small size. The data may be represented by a laterally oriented matrix or a third order tensor, and the regularization operator is a third order tensor. The dis…
We investigate iterated Tikhonov methods coupled with a Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill-posed operator equations. We show that the proposed method is a convergent regularization method. In the case of noisy data we propose a modification, the so called loping iterated Tikhonov-Kaczmarz method, where a sequence of relaxation parameters is introduced and a different stopping rule is used. Convergence analysis for this method is also provided.
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 2D elliptic parameter identification problem (Inverse Potential Problem); II) An i…
We investigate Tikhonov regularization methods for nonlinear ill-posed problems in Banach spaces, where the penalty term is described by Bregman distances. We prove convergence and stability results. Moreover, using appropriate source conditions, we are able to derive rates of convergence in terms of Bregman distances. We also analyze an iterated Tikhonov method for nonlinear problems, where the penalization is given by an appropriate convex functional.
Error Estimates for Arnoldo-Tikhonov Regularization for Ill-Posed Operator Equations
Institute for Industrial Mathematics, Kepler University（工业数学研究所，开普勒大学）；Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（约翰·拉德纳计算与应用数学研究所，奥地利科学院）；Department of Mathematical Sciences, Kent State University（凯特州立大学数学科学系）
The present paper is concerned with developing tensor iterative Krylov subspace methods to solve large multi-linear tensor equations. We use the well-known T-product for two tensors to define tensor global Arnoldi and tensor global Gloub-Kahan bidiagonalization algorithms. Furthermore, we illustrate how tensor-based global approaches can be exploited to solve ill-posed problems arising from recovering blurry multichannel (color) images and videos, using the so-called Tikhonov regularization technique, to provide computable approximate regularized solutions. We also review a generalized cross-v…
The GMRES algorithm of Saad and Schultz (1986) is an iterative method for approximately solving linear systems $A{\bf x}={\bf b}$, with initial guess ${\bf x}_0$ and residual ${\bf r}_0 = {\bf b} - A{\bf x}_0$. The algorithm employs the Arnoldi process to generate the Krylov basis vectors (the columns of $V_k$). It is well known that this process can be viewed as a $QR$ factorization of the matrix $B_k = [\: {\bf r}_0, AV_k\:]$ at each iteration. Despite an ${O}(ε)κ(B_k)$ loss of orthogonality, for unit roundoff $ε$ and condition number $κ$, the modified Gram-Schmidt formulation was shown to b…
