arXiv subjects

Asim Ilyas

Publications and source records attributed to Asim Ilyas.

4 recordsLinked to original sources

arXiv2026

Efficient Krylov solvers for inverse source problem in 2D space-time fractional diffusion equation

In this work, we consider a two-dimensional time-space fractional diffusion equation with a variable coefficient and investigate the inverse source problem of reconstructing the source term f(x,y) , after regularizing the problem using the quasi-boundary value method to mitigate ill-posedness. A finite difference discretization results in a large-scale linear system with a multilevel Toeplitz-like block structure. We perform a spectral analysis of the associated matrix sequences, employing tools from Generalized Locally Toeplitz (GLT) theory, and construct efficient preconditioners based on the GLT analysis. The proposed preconditioners preserve the multilevel structure of the discretization matrices and leads to a general eigenvalue clustering around one for the preconditioned sequence. Numerical experiments validate the theoretical findings and demonstrate that the proposed approach significantly accelerates the convergence of the GMRES method in reconstructing the source term in the two-dimensional space-time fractional diffusion equation.

math.NA↗

arXiv2026

Structure-preserving preconditioning of discrete space-fractional diffusion equations with variable coefficient and θ-Method

This paper studies the spectral properties of large matrices and the preconditioning of linear systems, arising from the finite difference discretization of a time-dependent space-fractional diffusion equation with a variable coefficient $a(x)$ defined on $Ω\subset \mathbb{R}^d$, $d=1,2$. The model involves a one-sided Riemann-Liouville fractional derivative multiplied by the function $a(x)$, discretized by the shifted Gr"unwald formula in space and the $θ$-method in time. The resulting all-at-once linear systems exhibit a $(d+1)$-level Toeplitz-like matrix structure, with $d=1,2$ denoting the space dimension, while the additional level is due to the time variable. A preconditioning strategy is developed based on the structural properties of the discretized operator. Using the generalized locally Toeplitz (GLT) theory, we analyze the spectral distribution of the unpreconditioned and preconditioned matrix sequences. The main novelty is that the analysis fully covers the case where the variable coefficient $a$ is nonconstant. Numerical results are provided to support the GLT based theoretical findings, and some possible extensions are briefly discussed.

math.NA↗

arXiv2025

Determining the space dependent coefficients in space-time fractional diffusion equations via Krylov preconditioning

We consider a time-space fractional diffusion equation with a variable coefficient and investigate the inverse problem of reconstructing the source term, after regularizing the problem with the quasiboundary value method to mitigate the ill-posedness. The equation involves a Caputo fractional derivative in the space variable and a tempered fractional derivative in the time variable, both of order in (0, 1). A finite difference approximation leads to a two-by-two block linear system of large dimensions. We conduct a spectral analysis of the associated matrix sequences, employing tools from Generalized Locally Toeplitz (GLT) theory, and construct the preconditioner guided by the GLT analysis. Numerical experiments are reported and commented, followed by concluding remarks.

math.NA↗

arXiv2025

On nth Level Fractional Derivatives: An Equivalent Representation and Applications to Inverse Problem

This work contributes to the theory of nth level fractional derivative, where $n$ is a positive integer. An equivalent representation of 2nd level fractional derivative in terms of Riemann-Liouville fractional derivative is presented. We generalized our result and provide representation of nth level fractional derivative. As an application, we solve an inverse problem defined for a diffusion equation involving 2nd level fractional derivative.

math.AP↗