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Generally, discretization of partial differential equations (PDEs) creates a sequence of linear systems $A_k x_k = b_k, k = 0, 1, 2, ..., N$ with well-known and structured sparsity patterns. Preconditioners are often necessary to achieve…

Quasi-separable matrices are a class of rank-structured matriceswidely used in numerical linear algebra and of growing interestin computer algebra, with applications in e.g. the linearization ofpolynomial matrices. Various representation…

A general framework for solving image inverse problems is introduced in this paper. The approach is based on Gaussian mixture models, estimated via a computationally efficient MAP-EM algorithm. A dual mathematical interpretation of the…

Computation of the large sparse matrix exponential has been an important topic in many fields, such as network and finite-element analysis. The existing scaling and squaring algorithm (SSA) is not suitable for the computation of the large…

Posit has been a promising alternative to the IEEE-754 floating point format for deep learning applications due to its better trade-off between dynamic range and accuracy. However, hardware implementation of posit arithmetic requires…

The b-posit, or bounded posit, is a variation of the posit format designed for high performance computing (HPC) and AI applications. Unlike traditional floating-point formats (floats), posits use variable-length fields for exponent scaling…

Sketching-based preconditioners have been shown to accelerate the solution of dense least-squares problems with coefficient matrices having substantially more rows than columns. The cost of generating these preconditioners can be reduced by…

Persistent homology is a leading tool in topological data analysis (TDA). Many problems in TDA can be solved via homological -- and indeed, linear -- algebra. However, matrices in this domain are typically large, with rows and columns…

The exponentially growing model size drives the continued success of deep learning, but it brings prohibitive computation and memory cost. From the algorithm perspective, model sparsification and quantization have been studied to alleviate…

Iterative sketching and sketch-and-precondition are randomized algorithms used for solving overdetermined linear least-squares problems. When implemented in exact arithmetic, these algorithms produce high-accuracy solutions to least-squares…

Sparse general matrix-matrix multiplication (spGEMM) is an essential component in many scientific and data analytics applications. However, the sparsity pattern of the input matrices and the interaction of their patterns make spGEMM…

We present a sparse matrix permutation from graph theory that gives stable incomplete Lower-Upper (LU) preconditioners necessary for iterative solutions to the steady state density matrix for quantum optomechanical systems. This reordering…

Mixing precisions for performance has been an ongoing trend as the modern hardware accelerators started including new, and mostly lower-precision, data formats. The advantage of using them is a great potential of performance gain and energy…

With the commercial availability of mixed precision hardware, mixed precision GMRES-based iterative refinement schemes have emerged as popular approaches for solving sparse linear systems. Existing analyses of these approaches, however, are…

The problem of exactly summing n floating-point numbers is a fundamental problem that has many applications in large-scale simulations and computational geometry. Unfortunately, due to the round-off error in standard floating-point…

Solving linear systems is a ubiquitous task in science and engineering. Because directly inverting a large-scale linear system can be computationally expensive, iterative algorithms are often used to numerically find the inverse. To…

Iterative solutions of sparse linear systems and sparse eigenvalue problems have a fundamental role in vital fields of scientific research and engineering. The crucial computing kernel for such iterative solutions is the multiplication of a…

High-performance sparse matrix-matrix (SpMM) multiplication is paramount for science and industry, as the ever-increasing sizes of data prohibit using dense data structures. Yet, existing hardware, such as Tensor Cores (TC), is ill-suited…

We present a comprehensive computational study of a class of linear system solvers, called {\it Triangle Algorithm} (TA) and {\it Centering Triangle Algorithm} (CTA), developed by Kalantari \cite{kalantari23}. The algorithms compute an…

Tensor completion refers to the task of estimating the missing data from an incomplete measurement or observation, which is a core problem frequently arising from the areas of big data analysis, computer vision, and network engineering. Due…