arXiv subjects

Muhammad Faisal Khan

Publications and source records attributed to Muhammad Faisal Khan.

7 recordsLinked to original sources

arXiv2026

Block preconditioning for all-at-once variable-coefficient fractional evolution equations via the GLT analysis

We study a class of nonlocal evolutionary partial differential equations with weakly singular temporal kernel and spatially variable diffusion coefficient. The model is posed on $\Omega \subset \mathbb{R}$, and involves a left-sided Riemann--Liouville fractional derivative in space multiplied by a variable coefficient $a(x)$. The temporal derivative is approximated by an $L1$ type scheme, while the spatial operator is discretized by finite difference techniques, resulting in large scale all at once linear systems with a twolevel Toeplitz like structure. We develop and analyze a block lower triangular strategy that mimics the structure of the coefficient matrix while simplifying its components for computational efficiency. The analysis is carried out at the level of matrix sequences by means of generalized locally Toeplitz (GLT) theory. Within this framework, we characterize the asymptotic spectral distribution of the discretized operators and use the associated GLT symbol to guide the construction of the structured approximation. Numerical experiments using the GMRES solver demonstrate that the proposed preconditioning strategy significantly improves convergence rates, robustness, and scalability for large-scale problems. Open problems and possible extensions are briefly discussed at the end of the present work.

math.NA↗

arXiv2026

Neutral-current neutrino-nucleus scattering off I (127) and Cs (133): Coherent and incoherent contributions with electroweak refinements for odd-A nuclei

Calculations of neutral-current neutrino-nucleus scattering cross sections are important for interpreting low- and intermediate-energy neutrino data, where terrestrial measurements remain limited. The first observation of coherent elastic neutrino-nucleus scattering (CEvNS) in 2017 with a CsI[Na] detector at the Spallation Neutron Source reported results consistent with Standard Model expectations, motivating improved theoretical descriptions beyond the purely coherent regime. In this work, we calculate neutral-current scattering cross sections for 127I and 133Cs within a unified analytical framework that consistently incorporates coherent elastic, incoherent excitation, and spin-dependent axial contributions relevant for odd-A nuclei. A consistent distinction between nuclear and nucleon form factors is maintained throughout the formalism. The incoherent contribution is evaluated using structure-function methods, providing a physically motivated decomposition of the total cross section. Electroweak effects are included through momentum-transfer-dependent sin^2(theta_W) corrections in the MS-bar scheme together with flavor-dependent neutrino charge-radius contributions implemented consistently in the vector couplings. Cross sections are presented as functions of neutrino energy and for decay-at-rest neutrino spectra in the low-to-intermediate energy region where elastic and quasi-elastic processes dominate. Inclusion of incoherent and axial contributions enhances the total cross section near Enu ~ 10 MeV, while the incoherent component becomes dominant around Enu ~ 50 MeV. Expected interaction rates for decay-at-rest neutrinos are of order 0.1 events kg^-1 yr^-1 near a 40 keV recoil threshold. The results provide a systematic assessment of subleading contributions relevant for CsI-based detectors and astrophysical neutrino applications.

hep-ph↗

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

GLT matrix-sequences and few emblematic applications

This thesis advances the spectral theory of structured matrix-sequences within the framework of Generalized Locally Toeplitz (GLT) $*$-algebras, focusing on the geometric mean of Hermitian positive definite (HPD) GLT sequences and its applications in mathematical physics. For two HPD sequences $\{A_n\}_n \sim_{\mathrm{GLT}} κ$ and $\{B_n\}_n \sim_{\mathrm{GLT}} ξ$ in the same $d$-level, $r$-block GLT $*$-algebra, we prove that when $κ$ and $ξ$ commute, the geometric mean sequence $\{G(A_n,B_n)\}_n$ is GLT with symbol $(κξ)^{1/2}$, without requiring invertibility of either symbol, settling \cite[Conjecture 10.1]{garoni2017} for $r=1$, $d\ge1$. In degenerate cases, we identify conditions ensuring $\{G(A_n,B_n)\}_n \sim_{\mathrm{GLT}} G(κ,ξ)$. For $r>1$ and non-commuting symbols, numerical evidence shows the sequence still admits a spectral symbol, indicating maximality of the commuting result. Numerical experiments in scalar and block settings confirm the theory and illustrate spectral behaviour. We also sketch the extension to $k\ge2$ sequences via the Karcher mean, obtaining $\{G(A_n^{(1)},\ldots,A_n^{(k)})\}_n \sim_{\mathrm{GLT}} G(κ_1,\ldots,κ_k)$. Finally, we apply the GLT framework to mean-field quantum spin systems, showing that matrices from the quantum Curie--Weiss model form GLT sequences with explicitly computable spectral distributions.

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

Geometric means of HPD GLT matrix-sequences: a maximal result beyond invertibility assumptions on the GLT symbols

In the current work, we consider the study of the spectral distribution of the geometric mean matrix-sequence of two matrix-sequences $\{G(A_n, B_n)\}_n$ formed by Hermitian Positive Definite (HPD) matrices, assuming that the two input matrix-sequences $\{A_n\}_n, \{B_n\}_n$ belong to the same $d$-level $r$-block Generalized Locally Toeplitz (GLT) $\ast$-algebra with $d,r\ge 1$ and with GLT symbols $κ, ξ$. Building on recent results in the literature, we examine whether the assumption that at least one of the input GLT symbols is invertible almost everywhere (a.e.) is necessary. Since inversion is mainly required due to the non-commutativity of the matrix product, it was conjectured that the hypothesis on the invertibility of the GLT symbols can be removed. In fact, we prove the conjectured statement that is \[ \{G(A_n, B_n)\}_n \sim_{\mathrm{GLT}} (κξ)^{1/2} \] when the symbols $κ, ξ$ commute, which implies the important case where $r=1$ and $d \geq 1 $, while the statement is generally false or even not well posed when the symbols are not invertible a.e. and do not commute. In fact, numerical experiments are conducted in the case where the two symbols do not commute, showing that the main results of the present work are maximal. Further numerical experiments, visualizations, and conclusions end the present contribution.

math.NA↗

arXiv2025

GLT hidden structures in mean-field quantum spin systems

This work explores structured matrix sequences arising in mean-field quantum spin systems. We express these sequences within the framework of generalized locally Toeplitz (GLT) $*$-algebras, leveraging the fact that each GLT matrix sequence has a unique GLT symbol. This symbol characterizes both the asymptotic singular value distribution and, for Hermitian or quasi-Hermitian sequences, the asymptotic spectral distribution. Specifically, we analyze two cases of real symmetric matrix sequences stemming from mean-field quantum spin systems and determine their associated distributions using GLT theory. Our study concludes with visualizations and numerical tests that validate the theoretical findings, followed by a discussion of open problems and future directions.

quant-ph↗