Tensor Conjugate-Gradient methods for tensor linear discrete ill-posed problems

Article Mathematics, Applied

Tensor Arnoldi-Tikhonov and GMRES-Type Methods for Ill-Posed Problems with a t-Product Structure

Lothar Reichel et al.

Summary: This paper presents solution methods for linear discrete ill-posed problems defined by third order tensors and the t-product formalism. It introduces a t-product Arnoldi process for reducing large-scale Tikhonov regularization problems to smaller sizes. Numerical examples compare different solution methods and demonstrate the potential advantages of tensorizing over matricizing linear discrete ill-posed problems for third order tensors.

JOURNAL OF SCIENTIFIC COMPUTING (2022)

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Article Mathematics, Applied

The tensor Golub-Kahan-Tikhonov method applied to the solution of ill-posed problems with a t-product structure

Lothar Reichel et al.

Summary: This paper discusses the application of partial tensor Golub-Kahan bidiagonalization in solving large-scale linear discrete ill-posed problems, based on the t-product formalism for third-order tensors proposed by Kilmer and Martin. The methods presented first reduce the given problem to a smaller size using tensor Golub-Kahan bidiagonalization, and then regularize it with Tikhonov's method. The regularization parameter is determined by the discrepancy principle, resulting in fully automatic solution methods.

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS (2022)

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Article Computer Science, Software Engineering

Randomized Kaczmarz for tensor linear systems

Anna Ma et al.

Summary: Researchers extend the randomized Kaczmarz method to solve multi-linear systems and demonstrate experimentally that the tensor randomized Kaczmarz method converges faster than traditional methods. They also establish connections between the proposed algorithm and a previously known extension for matrix linear systems.

BIT NUMERICAL MATHEMATICS (2022)

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Article Mathematics, Interdisciplinary Applications

A fractional Tikhonov regularization method for an inverse backward and source problems in the time-space fractional diffusion equations

Smina Djennadi et al.

Summary: This research focuses on two types of inverse problems for diffusion equations with Caputo fractional derivatives in time and fractional Sturm-Liouville operator for space. The study proves these inverse problems to be ill-posed in the sense of Hadamard when an additional condition at the final time is given. A new fractional Tikhonov regularization method is used for stable solution reconstruction with error estimates obtained between exact and regularized solutions through a-priori and a-posteriori parameter choice rules, along with numerical examples provided for validation.

CHAOS SOLITONS & FRACTALS (2021)

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Article Mathematics

On tensor GMRES and Golub-Kahan methods via the T-product for color image processing

Mohamed El Guide et al.

Electronic Journal of Linear Algebra (2021)

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Article Physics, Multidisciplinary

The Tikhonov regularization method for the inverse source problem of time fractional heat equation in the view of ABC-fractional technique

Smina Djennadi et al.

Summary: This research examines an inverse source problem for a fractional diffusion equation containing the Atangana-Baleanu-Caputo fractional derivative, obtaining an explicit solution set through expansion method and overdetermination condition. Due to the ill-posed nature of the problem, Tikhonov regularization method is applied to stabilize the solution, with a focus on two parameter choice rules. A simulation example is used to validate the presented theoretical results.

PHYSICA SCRIPTA (2021)

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Article Mathematics, Applied