Boundary value problems involving elliptic PDEs such as the Laplace and the Helmholtz equations are ubiquitous in mathematical physics and engineering. Many such problems can be alternatively formulated as integral equations that are mathematically more tractable. However, an integral-equation formulation poses a significant computational challenge: solving large dense linear systems that arise upon discretization. In cases where iterative methods converge rapidly, existing methods that draw on fast summation schemes such as the Fast Multipole Method are highly efficient and well-established. More recently, linear complexity direct solvers that sidestep convergence issues by directly computing an invertible factorization have been developed. However, storage and computation costs are high, which limits their ability to solve large-scale problems in practice. In this work, we introduce a distributed-memory parallel algorithm based on an existing direct solver named "strong recursive skeletonization factorization [1]." Specifically, we apply low-rank compression to certain off-diagonal matrix blocks in a way that minimizes computation and data movement. Compared to iterative algorithms, our method is particularly suitable for problems involving ill-conditioned matrices or multiple righthand sides. Large-scale numerical experiments are presented to show the performance of our Julia implementation.

Summary This paper describes efficient algorithms for computing rank‐revealing factorizations of matrices that are too large to fit in main memory (RAM), and must instead be stored on slow external memory devices such as disks (out‐of‐core or out‐of‐memory). Traditional algorithms for computing rank‐revealing factorizations (such as the column pivoted QR factorization and the singular value decomposition) are very communication intensive as they require many vector‐vector and matrix‐vector operations, which become prohibitively expensive when data is not in RAM. Randomization allows to reformulate new methods so that large contiguous blocks of the matrix are processed in bulk. The paper describes two distinct methods. The first is a blocked version of column pivoted Householder QR, organized as a “left‐looking” method to minimize the number of the expensive write operations. The second method results employs a UTV factorization. It is organized as an algorithm‐by‐blocks to overlap computations and I/O operations. As it incorporates power iterations, it is much better at revealing the numerical rank. Numerical experiments on several computers demonstrate that the new algorithms are almost as fast when processing data stored on slow memory devices as traditional algorithms are for data stored in RAM.

We are interested in solving linear systems arising from three applications: (1) kernel methods in machine learning, (2) discretization of boundary integral equations from mathematical physics, and (3) Schur complements formed in the factorization of many large sparse matrices. The coefficient matrices are often data-sparse in the sense that their off-diagonal blocks have low numerical ranks; specifically, we focus on “hierarchically off-diagonal low-rank (HODLR)” matrices. We introduce algorithms for factorizing HODLR matrices and for applying the factorizations on a GPU. The algorithms leverage the efficiency of batched dense linear algebra, and they scale nearly linearly with the matrix size when the numerical ranks are fixed. The accuracy of the HODLR-matrix approximation is a tunable parameter, so we can construct high-accuracy fast direct solvers or low-accuracy robust preconditioners. Numerical results show that we can solve problems with several millions of unknowns in a couple of seconds on a single GPU.

Randomized singular value decomposition (RSVD) is by now a well-established technique for efficiently computing an approximate singular value decomposition of a matrix. Building on the ideas that underpin RSVD, the recently proposed algorithm “randUTV” computes a full factorization of a given matrix that provides low-rank approximations with near-optimal error. Because the bulk of randUTV is cast in terms of communication-efficient operations such as matrix-matrix multiplication and unpivoted QR factorizations, it is faster than competing rank-revealing factorization methods such as column-pivoted QR in most high-performance computational settings. In this article, optimized randUTV implementations are presented for both shared-memory and distributed-memory computing environments. For shared memory, randUTV is redesigned in terms of an algorithm-by-blocks that, together with a runtime task scheduler, eliminates bottlenecks from data synchronization points to achieve acceleration over the standard blocked algorithm based on a purely fork-join approach. The distributed-memory implementation is based on the ScaLAPACK library. The performance of our new codes compares favorably with competing factorizations available on both shared-memory and distributed-memory architectures.

A randomized algorithm for computing a so-called UTV factorization efficiently is presented. Given a matrix A , the algorithm “randUTV” computes a factorization A = UTV * , where U and V have orthonormal columns, and T is triangular (either upper or lower, whichever is preferred). The algorithm randUTV is developed primarily to be a fast and easily parallelized alternative to algorithms for computing the Singular Value Decomposition (SVD). randUTV provides accuracy very close to that of the SVD for problems such as low-rank approximation, solving ill-conditioned linear systems, and determining bases for various subspaces associated with the matrix. Moreover, randUTV produces highly accurate approximations to the singular values of A . Unlike the SVD, the randomized algorithm proposed builds a UTV factorization in an incremental, single-stage, and noniterative way, making it possible to halt the factorization process once a specified tolerance has been met. Numerical experiments comparing the accuracy and speed of randUTV to the SVD are presented. Other experiments also demonstrate that in comparison to column-pivoted QR, which is another factorization that is often used as a relatively economic alternative to the SVD, randUTV compares favorably in terms of speed while providing far higher accuracy.