arXiv subjects

Pinki Khatun

Publications and source records attributed to Pinki Khatun.

11 recordsLinked to original sources

arXiv2026

On a Class of Block-Regularized Preconditioned Iterative Methods for Indefinite Least-Squares Problems

This paper proposes a novel block-regularized splitting (BRS) framework for the efficient solution of indefinite least-squares (ILS) problems. Based on a block-wise regularization strategy incorporated into a matrix splitting scheme, we develop a BRS iterative method together with an effective BRS preconditioner. The convergence of the proposed iterative method is rigorously analyzed, and spectral bounds for the BRS-preconditioned matrix are established. To further enhance computational performance, we introduce a relaxed BRS (RBRS) preconditioner, which provides improved spectral properties and significantly accelerates the convergence of Krylov subspace methods. Extensive numerical experiments on both dense and sparse test problems demonstrate that the proposed BRS and RBRS preconditioners consistently outperform existing approaches in terms of iteration count, computational time, and overall efficiency. These results highlight the effectiveness and robustness of the proposed block-regularized splitting framework for solving large-scale ILS problems.

math.NA↗

arXiv2026

Physics-Informed Broad Learning System: An Efficient Backpropagation-Free Framework for Solving Partial Differential Equations

Physics-informed neural networks (PINNs) have emerged as a powerful paradigm for solving partial differential equations (PDEs) by embedding governing physical laws into deep neural networks. However, their reliance on computationally expensive gradient-based optimization and deep architectures often results in slow training, high computational cost, and limited scalability. In this work, we propose a novel physics-informed broad learning system (PI-BLS), the first physics-informed learning framework based on broad RdNNs. The proposed formulation embeds the governing differential operator and the associated initial and boundary constraints directly into a linear output-layer optimization problem, thereby replacing nonlinear gradient-based training with a deterministic least-squares solution obtained via the pseudoinverse. Consequently, the entire learning process is reduced to a single linear optimization stage while preserving the underlying physical constraints. As a result, PI-BLS offers an efficient learning paradigm for a physics-informed learning framework for solving PDEs that eliminates iterative backpropagation while preserving the underlying physical constraints. Experimental results on representative forward PDE benchmarks demonstrate that PI-BLS achieves competitive and often superior performance with reduced training time and model parameters compared with conventional PINNs.

cs.LG↗

arXiv2026

Diophantine Equation $x_1^{3}-x_2^{2}x_1+1=0$ over number fields

Let $\mathbb{K}=\mathbb{Q}(\sqrt{-d})\text{ or }\mathbb{Q}(\sqrt{d})$, where $d$ is a positive square-free integer, and denote by $\mathcal{O}_\mathbb{K}$ the ring of integers of $\mathbb{K}$. I investigate the solution of the equation $x_1^{3}-x_2^{2}x_1+1=0$ where $x_1\in \mathbb{K}$ and $x_2\in\mathcal{O}_\mathbb{K}$. The case $\mathbb{K}=\mathbb{Q}(\sqrt{d})$ for $d\equiv 2,3\pmod{4}$ faces infinite units that require a separate treatment. Using the arithmetic of the quadratic integer rings $\mathbb{Z}[\sqrt{d}]]$, together with norm arguments, divisibility properties, and the explicit structure of its unit group, I prove that the equation has exactly two solutions, namely $(x_1,x_2)=(-1,0)~\text{ and }~(1,\sqrt{2})$ for $\mathbb{K}=\mathbb{Q}(\sqrt{2})$ and one solution $(-1,0)$ for $\mathbb{K}=\mathbb{Q}(\sqrt{d})$ As an application, I consider the family of elliptic curves $C_m:Y^{2}=X^{3}-m^{2}X+1,~ m\in\mathcal{O}_\mathbb{K},$ and deduce that, for every $m\neq0,\sqrt{2}$ the Mordell--Weil group $C_m(\mathbb{K})$ contains no rational point of order two.

math.GM↗

arXiv2025

Structured Backward Errors of Sparse Generalized Saddle Point Problems with Hermitian Block Matrices

In this paper, we derive the structured backward error (BE) for a class of generalized saddle point problems (GSPP) by preserving the sparsity pattern and Hermitian structures of the block matrices. Additionally, we construct the optimal backward perturbation matrices for which the structured BE is achieved. Our analysis also examines the structured BE in cases where the sparsity pattern is not maintained. Through numerical experiments, we demonstrate the reliability of the derived structured BEs and the corresponding optimal backward perturbations. Additionally, the derived structured BEs are used to assess the strong backward stability of numerical methods for solving the GSPP.

math.NA↗

arXiv2025

Partial Condition Numbers for Double Saddle Point Problems

This paper presents a unified framework for investigating the partial condition number (CN) of the solution of double saddle point problems (DSPPs) and provides closed-form expressions for it. This unified framework encompasses the well-known partial normwise CN (NCN), partial mixed CN (MCN) and partial componentwise CN (CCN) as special cases. Furthermore, we derive sharp upper bounds for the partial NCN, MCN and CCN, which are computationally efficient and free of expensive Kronecker products. By applying perturbations that preserve the structure of the block matrices of the DSPPs, we analyze the structured partial NCN, MCN and CCN when the block matrices exhibit linear structures. By leveraging the relationship between DSPP and equality constrained indefinite least squares (EILS) problems, we recover the partial CNs for the EILS problem. Numerical results confirm the sharpness of the derived upper bounds and demonstrate their effectiveness in estimating the partial CNs.

math.NA↗

arXiv2024

Sparsity-Preserving Structured Backward Error Analysis for Double Saddle Point Problems

Backward error (BE) analysis emerges as a powerful tool for assessing the backward stability and strong backward stability of numerical algorithms. In this paper, we explore structured BEs for a class of double saddle point problems (DSPPs) and aim to assess the strong backward stability of numerical algorithms designed to find their solutions. Our investigations preserve the inherent matrix structure and sparsity pattern in the corresponding perturbation matrices and derive explicit formulae for the structured BEs. Moreover, we provide formulae for the structure-preserving minimal perturbation matrices for which the structured BE is attained. Utilizing the relationship between the DSPP and the least squares problem with equality constraints (LSE), we derive the sparsity-preserving BE formula for LSE within our framework. Numerical experiments are performed to test the strong backward stability of various numerical algorithms.

math.NA↗

arXiv2024

Structured Backward Errors for Special Classes of Saddle Point Problems with Applications

In the realm of numerical analysis, the study of structured backward errors (BEs) in saddle point problems (SPPs) has shown promising potential for development. However, these investigations overlook the inherent sparsity pattern of the coefficient matrix of the SPP. Moreover, the existing techniques are not applicable when the block matrices have circulant, Toeplitz, or symmetric-Toeplitz structures and do not even provide structure preserving minimal perturbation matrices for which the BE is attained. To overcome these limitations, we investigate the structured BEs of SPPs when the perturbation matrices exploit the sparsity pattern as well as circulant, Toeplitz, and symmetric-Toeplitz structures. Furthermore, we construct minimal perturbation matrices that preserve the sparsity pattern and the aforementioned structures. Applications of the developed frameworks are utilized to compute BEs for the weighted regularized least squares problem. Finally, numerical experiments are performed to validate our findings, showcasing the utility of the obtained structured BEs in assessing the strong backward stability of numerical algorithms.

math.NA↗

arXiv2024

A Class of Generalized Shift-Splitting Preconditioners for Double Saddle Point Problems

In this paper, we propose a generalized shift-splitting (GSS) preconditioner, along with its two relaxed variants to solve the double saddle point problem (DSPP). The convergence of the associated GSS iterative method is analyzed, and sufficient conditions for its convergence are established. Spectral analyses are performed to derive sharp bounds for the eigenvalues of the preconditioned matrices. Numerical experiments based on examples arising from the PDE-constrained optimization problem and the leaky lid-driven cavity problem demonstrate the effectiveness and robustness of the proposed preconditioners compared with existing state-of-the-art preconditioners.

math.NA↗

arXiv2024

A robust parameterized enhanced shift-splitting preconditioner for three-by-three block saddle point problems

This paper proposes a new parameterized enhanced shift-splitting (PESS) preconditioner to solve the three-by-three block saddle point problem (SPP). Additionally, we introduce a local PESS (LPESS) preconditioner by relaxing the PESS preconditioner. Necessary and sufficient criteria are established for the convergence of the proposed PESS iterative process for any initial guess. Furthermore, we meticulously investigate the spectral bounds of the PESS and LPESS preconditioned matrices. Moreover, empirical investigations have been performed for the sensitivity analysis of the proposed PESS preconditioner, which unveils its robustness. Numerical experiments are carried out to demonstrate the enhanced efficiency and robustness of the proposed PESS and LPESS preconditioners compared to the existing state-of-the-art preconditioners.

math.NA↗

arXiv2023

Condition numbers for the Moore-Penrose inverse and the least squares problem involving rank-structured matrices

Perturbation theory plays a crucial role in sensitivity analysis, which is extensively used to assess the robustness of numerical techniques. To quantify the relative sensitivity of any problem, it becomes essential to investigate structured condition numbers (CNs) via componentwise perturbation theory. This paper addresses and analyzes structured mixed condition number (MCN) and componentwise condition number (CCN) for the Moore-Penrose (M-P) inverse and the minimum norm least squares (MNLS) solution involving rank-structured matrices, which include the Cauchy-Vandermonde (CV) matrices and {1, 1}-quasiseparable (QS) matrices. A general framework has been developed to compute the upper bounds for MCN and CCN of rank deficient parameterized matrices. This framework leads to faster computation of upper bounds of structured CNs for CV and {1, 1}-QS matrices. Furthermore, comparisons of obtained upper bounds are investigated theoretically and experimentally. In addition, the structured effective CNs for the M-P inverse and the MNLS solution of {1, 1}-QS matrices are presented. Numerical tests reveal the reliability of the proposed upper bounds as well as demonstrate that the structured effective CNs are computationally less expensive and can be substantially smaller compared to the unstructured CNs.

math.NA↗

arXiv2023

Structured condition numbers for a linear function of the solution of the generalized saddle point problem

This paper addresses structured normwise, mixed, and componentwise condition numbers (CNs) for a linear function of the solution to the generalized saddle point problem (GSPP). We present a general framework that enables us to measure the structured CNs of the individual components of the solution. Then, we derive their explicit formulae when the input matrices have symmetric, Toeplitz, or some general linear structures. In addition, compact formulae for the unstructured CNs are obtained, which recover previous results on CNs for GSPPs for specific choices of the linear function. Furthermore, applications of the derived structured CNs are provided to determine the structured CNs for the weighted Toeplitz regularized least-squares problems and Tikhonov regularization problems, which retrieves some previous studies in the literature.

math.NA↗