Laura Grigori

James Demmel, Laura Grigori, Mark Hoemmen et al.

We present parallel and sequential dense QR factorization algorithms that are both optimal (up to polylogarithmic factors) in the amount of communication they perform, and just as stable as Householder QR. We prove optimality by extending known lower bounds on communication bandwidth for sequential and parallel matrix multiplication to provide latency lower bounds, and show these bounds apply to the LU and QR decompositions. We not only show that our QR algorithms attain these lower bounds (up to polylogarithmic factors), but that existing LAPACK and ScaLAPACK algorithms perform asymptotically more communication. We also point out recent LU algorithms in the literature that attain at least some of these lower bounds.

Amal Khabou, James W. Demmel, Laura Grigori et al.

We present the LU decomposition with panel rank revealing pivoting (LU_PRRP), an LU factorization algorithm based on strong rank revealing QR panel factorization. LU_PRRP is more stable than Gaussian elimination with partial pivoting (GEPP). Our extensive numerical experiments show that the new factorization scheme is as numerically stable as GEPP in practice, but it is more resistant to pathological cases and easily solves the Wilkinson matrix and the Foster matrix. We also present CALU_PRRP, a communication avoiding version of LU_PRRP that minimizes communication. CALU_PRRP is based on tournament pivoting, with the selection of the pivots at each step of the tournament being performed via strong rank revealing QR factorization. CALU_PRRP is more stable than CALU, the communication avoiding version of GEPP. CALU_PRRP is also more stable in practice and is resistant to pathological cases on which GEPP and CALU fail.

James Demmel, Laura Grigori, Mark Hoemmen et al.

We present parallel and sequential dense QR factorization algorithms for tall and skinny matrices and general rectangular matrices that both minimize communication, and are as stable as Householder QR. The sequential and parallel algorithms for tall and skinny matrices lead to significant speedups in practice over some of the existing algorithms, including LAPACK and ScaLAPACK, for example up to 6.7x over ScaLAPACK. The parallel algorithm for general rectangular matrices is estimated to show significant speedups over ScaLAPACK, up to 22x over ScaLAPACK.

Meisam Sharify, Stéphane Gaubert, Laura Grigori

We show that a solution of the optimal assignment problem can be obtained as the limit of the solution of an entropy maximization problem, as a deformation parameter tends to infinity. This allows us to apply entropy maximization algorithms to the optimal assignment problem. In particular, the Sinkhorn algorithm leads to a parallelizable method, which can be used as a preprocessing to handle large dense optimal assignment problems. This parallel preprocessing allows one to delete entries which do not belong to optimal permutations, leading to a reduced instance which becomes solvable with limited memory requirements.

James Demmel, Laura Grigori, Mark Hoemmen et al.

We present parallel and sequential dense QR factorization algorithms that are both optimal (up to polylogarithmic factors) in the amount of communication they perform, and just as stable as Householder QR. Our first algorithm, Tall Skinny QR (TSQR), factors m-by-n matrices in a one-dimensional (1-D) block cyclic row layout, and is optimized for m >> n. Our second algorithm, CAQR (Communication-Avoiding QR), factors general rectangular matrices distributed in a two-dimensional block cyclic layout. It invokes TSQR for each block column factorization.

Laura Grigori, Frédéric Nataf

This paper introduces a new preconditioning technique that is suitable for matrices arising from the discretization of a system of PDEs on unstructured grids. The preconditioner satisfies a so-called filtering property, which ensures that the input matrix is identical with the preconditioner on a given filtering vector. This vector is chosen to alleviate the effect of low frequency modes on convergence and so decrease or eliminate the plateau which is often observed in the convergence of iterative methods. In particular, the paper presents a general approach that allows to ensure that the filtering condition is satisfied in a matrix decomposition. The input matrix can have an arbitrary sparse structure. Hence, it can be reordered using nested dissection, to allow a parallel computation of the preconditioner and of the iterative process.

Stephen Becker, James Folberth, Laura Grigori

The unpivoted and pivoted Householder QR factorizations are ubiquitous in numerical linear algebra. A difficulty with pivoted Householder QR is the communication bottleneck introduced by pivoting. In this paper we propose using random orthogonal systems to quickly mix together the columns of a matrix before computing an unpivoted QR factorization. This method computes a URV factorization which forgoes expensive pivoted QR steps in exchange for mixing in advance, followed by a cheaper, unpivoted QR factorization. The mixing step typically reduces the variability of the column norms, and in certain experiments, allows us to compute an accurate factorization where a plain, unpivoted QR performs poorly. We experiment with linear least-squares, rank-revealing factorizations, and the QLP approximation, and conclude that our randomized URV factorization behaves comparably to a similar randomized rank-revealing URV factorization, but at a fraction of the computational cost. Our experiments provide evidence that our proposed factorization might be rank-revealing with high probability.

Michel Fabrice Serret, Alice Cortinovis, Yijun Dong et al.

The attention mechanism is the computational core of modern Transformer architectures, but its quadratic complexity in the input sequence length is the bottleneck for large-scale inference. This has motivated a rapidly growing body of work aimed at accelerating attention through approximation and reformulation. In this survey, we revisit attention mechanisms through the lens of numerical analysis, with a particular emphasis on tools and perspectives from numerical linear algebra. Our goal is twofold: first, we aim to systematically review and classify fast approximation methods according to the numerical principles they exploit. These include sparsity and clustering approaches, low-rank and subspace projection techniques, randomized sketching methods, and tensor-based decompositions. We also discuss kernel-inspired reformulations of attention and recent architectural variants, such as Latent Attention, that modify the standard softmax formulation to improve efficiency. Second, by presenting these developments within a unified mathematical framework, we aim to bridge the gap between disciplines and highlight opportunities for further contributions from computational mathematics, particularly numerical linear algebra, to the design of scalable attention mechanisms.

Hussam Al Daas, Grey Ballard, Laura Grigori et al.

Sketching is widely used in randomized linear algebra for low-rank matrix approximation, column subset selection, and many other problems, and it has gained significant traction in machine learning applications. However, sketching large matrices often necessitates distributed memory algorithms, where communication overhead becomes a critical bottleneck on modern supercomputing clusters. Despite its growing relevance, distributed-memory parallel strategies for sketching remain largely unexplored. In this work, we establish communication lower bounds for sketching using dense matrices that determine how much data movement is required to perform it in parallel. One important observation of our lower bounds is that no communication is required for a small number of processors. We show that our lower bounds are tight by presenting communication optimal algorithms. Furthermore, we extend our approach to determine communication lower bounds for computations of Nyström approximation where sketching is applied twice. We also introduce novel parallel algorithms whose communication costs are close to the lower bounds. Finally, we implement our algorithms on modern state-of-the-art supercomputing infrastructures which have both CPU- and GPU-equipped systems and demonstrate their parallel scalability.