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arXiv · 2610.33763

Inverse problems with fractional operators: an analytical study and its numerical linear algebra treatment via the GLT theory

Asim Ilyas
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Abstract

Fractional derivatives model memory effects and nonlocal interactions beyond classical integer-order models. Fractional differential equations (FDEs) thus describe anomalous diffusion, transport in heterogeneous media, and hereditary dynamics. Their nonlocality, however, makes inverse problems challenging: unknown sources, coefficients, or initial conditions must be recovered from indirect and noisy data. Inverse problems for FDEs are usually ill-posed in the Hadamard sense, so regularization, such as the quasi-boundary value method, is required for stable reconstruction. Discretization produces large linear systems whose matrices inherit the nonlocal structure of fractional operators. Understanding the asymptotic spectral properties of these matrix sequences is essential for efficient solvers and preconditioners. Generalized Locally Toeplitz (GLT) theory provides a unified framework for the spectral distribution of large structured matrices and for preconditioner design. This thesis uses GLT theory to analyze matrix sequences from inverse problems for FDEs, construct structure-preserving preconditioners, and study nonlinear operations in the GLT algebra. In particular, it proves maximal distribution results for geometric means of GLT sequences: when the symbols commute, the geometric mean sequence remains GLT without symbol invertibility. This resolves a long-standing conjecture and shows maximality in the noncommuting degenerate case. Overall, the thesis integrates regularization, spectral analysis, and preconditioning into a coherent computational framework for inverse problems in nonlocal fractional dynamics, while enriching structured matrix analysis.

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Asim Ilyas. 2026-10-24. Inverse problems with fractional operators: an analytical study and its numerical linear algebra treatment via the GLT theory. https://arxiv.org/abs/2610.33763

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