Related papers: Parallel Cholesky Factorization for Banded Matrice…
We introduce a parallel algorithm to construct a preconditioner for solving a large, sparse linear system where the coefficient matrix is a Laplacian matrix (a.k.a., graph Laplacian). Such a linear system arises from applications such as…
The dominant cost in solving least-square problems using Newton's method is often that of factorizing the Hessian matrix over multiple values of the regularization parameter ($\lambda$). We propose an efficient way to interpolate the…
As it has become common to use many computer cores in routine applications, finding good ways to parallelize popular algorithms has become increasingly important. In this paper, we present a parallelization scheme for Markov chain Monte…
The multi-scale factor models are particularly appealing for analyzing matrix- or tensor-valued data, due to their adaptiveness to local geometry and intuitive interpretation. However, the reliance on the binary tree for recursive…
OpenMP parallelization of multiple precision Taylor series method is proposed. A very good parallel performance scalability and parallel efficiency inside one computation node of a CPU-cluster is observed. We explain the details of the…
We propose a solution strategy for linear systems arising in interior method optimization, which is suitable for implementation on hardware accelerators such as graphical processing units (GPUs). The current gold standard for solving these…
Due to the advent of multicore architectures and massive parallelism, the tiled Cholesky factorization algorithm has recently received plenty of attention and is often referenced by practitioners as a case study. It is also implemented in…
We introduce the sparsified Cholesky and sparsified multigrid algorithms for solving systems of linear equations. These algorithms accelerate Gaussian elimination by sparsifying the nonzero matrix entries created by the elimination process.…
The solution of a sparse system of linear equations is ubiquitous in scientific applications. Iterative methods, such as the Preconditioned Conjugate Gradient method (PCG), are normally chosen over direct methods due to memory and…
Many important real-world applications, such as System Identification with Gaussian Processes, involve solving linear systems with symmetric positive-definite matrices. The iterative CG method and direct solvers based on the Cholesky…
The modified Cholesky decomposition is commonly used for precision matrix estimation given a specified order of random variables. However, the order of variables is often not available or cannot be pre-determined. In this work, we propose…
It is well known that matrices with low Hessenberg-structured displacement rank enjoy fast algorithms for certain matrix factorizations. We show how $n\times n$ principal finite sections of the Gram matrix for the orthogonal polynomial…
A fast algorithm for inverse Cholesky factorization is proposed, to compute a triangular square-root of the estimation error covariance matrix for Vertical Bell Laboratories Layered Space-Time architecture (V-BLAST). It is then applied to…
We explore the trade-offs of performing linear algebra using Apache Spark, compared to traditional C and MPI implementations on HPC platforms. Spark is designed for data analytics on cluster computing platforms with access to local disks…
This paper introduces an efficient and generic framework for finite-element simulations under an implicit time integration scheme. Being compatible with generic constitutive models, a fast matrix assembly method exploits the fact that…
Incomplete factorization is a powerful preconditioner for Krylov subspace methods for solving large-scale sparse linear systems. Existing incomplete factorization techniques, including incomplete Cholesky and incomplete LU factorizations,…
As multicore systems continue to gain ground in the High Performance Computing world, linear algebra algorithms have to be reformulated or new algorithms have to be developed in order to take advantage of the architectural features on these…
In recent years, randomized algorithms have established themselves as fundamental tools in computational linear algebra, with applications in scientific computing, machine learning, and quantum information science. Many randomized matrix…
This article proposes and analyzes several variants of the randomized Cholesky QR factorization of a matrix $X$. Instead of computing the R factor from $X^T X$, as is done by standard methods, we obtain it from a small, efficiently…
The increasing number of processing elements and decreas- ing memory to core ratio in modern high-performance platforms makes efficient strong scaling a key requirement for numerical algorithms. In order to achieve efficient scalability on…