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When solving partial differential equations (PDEs) using finite difference or finite element methods, efficient solvers are required for handling large sparse linear systems. In this paper, a recursive sparse LU decomposition for matrices…
In this paper, we develop a novel procedure for low-rank tensor regression, namely \emph{\underline{I}mportance \underline{S}ketching \underline{L}ow-rank \underline{E}stimation for \underline{T}ensors} (ISLET). The central idea behind…
For an overdetermined system $\mathsf{A}\mathsf{x} \approx \mathsf{b}$ with $\mathsf{A}$ and $\mathsf{b}$ given, the least-square (LS) formulation $\min_x \, \|\mathsf{A}\mathsf{x}-\mathsf{b}\|_2$ is often used to find an acceptable…
Matrices arising in scientific applications frequently admit linear low-rank approximations due to smoothness in the physical and/or temporal domain of the problem. In large-scale problems, computing an optimal low-rank approximation can be…
Quantum linear system algorithms (QLSAs) for gate-based quantum computing can provide exponential speedups for solving linear systems but face challenges when applied to finite element problems due to the growth of the condition number with…
This paper studies the low-rank property of the inverse of a class of large-scale structured matrices in the tensor-train (TT) format, which is typically discretized from differential operators. An interesting question that we are concerned…
In this article we consider the iterative schemes to compute the canonical (CP) approximation of quantized data generated by a function discretized on a large uniform grid in an interval on the real line. This paper continues the research…
In the iterative solution of dense linear systems from boundary integral equations or systems involving kernel matrices, the main challenges are the expensive matrix-vector multiplication and the storage cost which are usually tackled by…
In this paper we present a non-recursive direct solver, based on the Bartels-Stewart algorithm, for $N$-dimensional Sylvester tensor equations. The method relies only on Schur decompositions of the coefficient matrices and reduces the…
This paper proposes a theoretical framework to address the reduced biquaternion equality-constrained total least squares (RBTLSE) problem. The objective is to find an approximate solution to the system $AX \approx B$, subject to linear…
Low-rank modeling has many important applications in computer vision and machine learning. While the matrix rank is often approximated by the convex nuclear norm, the use of nonconvex low-rank regularizers has demonstrated better empirical…
In this paper, we propose a low rank approximation method for efficiently solving stochastic partial differential equations. Specifically, our method utilizes a novel low rank approximation of the stiffness matrices, which can significantly…
Quantics Tensor Train (QTT) operations such as matrix product operator contractions are prohibitively expensive for large bond dimensions. We propose an adaptive patching scheme that exploits block-sparse QTT structures to reduce costs…
The linearly constrained matrix rank minimization problem is widely applicable in many fields such as control, signal processing and system identification. The tightest convex relaxation of this problem is the linearly constrained nuclear…
This paper lies in the intersection of several fields: number theory, lattice theory, multilinear algebra, and scientific computing. We adapt existing solution algorithms for tensor eigenvalue problems to the tensor-train framework. As an…
This study aims to solve the over-reliance on the rank estimation strategy in the standard tensor factorization-based tensor recovery and the problem of a large computational cost in the standard t-SVD-based tensor recovery. To this end, we…
We present a tensor-decomposition method to solve the Boltzmann transport equation (BTE) in the Bhatnagar-Gross-Krook approximation. The method represents the six-dimensional BTE as a set of six one-dimensional problems, which are solved…
This paper introduces a novel algorithm to approximate the matrix with minimum nuclear norm among all matrices obeying a set of convex constraints. This problem may be understood as the convex relaxation of a rank minimization problem, and…
We present a fast algorithm for linear least squares problems governed by hierarchically block separable (HBS) matrices. Such matrices are generally dense but data-sparse and can describe many important operators including those derived…
The variational quantum eigensolver (VQE) is one of the most promising algorithms to find eigenvalues and eigenvectors of a given Hamiltonian on noisy intermediate-scale quantum (NISQ) devices. A particular application is to obtain ground…