arXiv subjects

Daniel B. Szyld

Publications and source records attributed to Daniel B. Szyld.

14 recordsLinked to original sources

arXiv2026

A block Recycled GMRES method with investigations into aspects of solver performance

We propose a block Krylov subspace version of the GCRO-DR method proposed in [Parks et al.; SISC 2005], which is an iterative method allowing for the efficient minimization of the the residual over an augmented Krylov subspace. We offer a clean derivation of our proposed method and discuss methods of selecting recycling subspaces at restart as well as implementation decisions in the context of high-performance computing. Numerical experiments are split into those demonstrating convergence properties and those demonstrating the data movement and cache efficiencies of the dominant operations of the method, measured using processor monitoring code from Intel.

math.NA↗

arXiv2026

Multiprecision computations with Schwarz methods

Schwarz methods and preconditioners, where the approximate solution of the local problems is performed at a lower precision, i.e., with fewer digits of accuracy than in the underlying (double precision) computation. Conditions for the appropriate round-off criteria for the lower precision are presented. It is found experimentally that for the model problems about 5 digits of accuracy are sufficient to achieve the theoretical restrictions, and thus, single precision suffices for the local solves. Several numerical experiments illustrate the obtained results.

math.NA↗

arXiv2025

Random-sketching Techniques to Enhance the Numerical Stability of Block Orthogonalization Algorithms for s-step GMRES

We integrate random sketching techniques into block orthogonalization schemes needed for s-step GMRES. The resulting block orthogonalization schemes generate the basis vectors whose overall orthogonality error is bounded by machine precision as long as each of the corresponding block vectors are numerically full rank. We implement these randomized block orthogonalization schemes using standard distributed-memory linear algebra kernels for s-step GMRES available in the Trilinos software packages. Our performance results on the Perlmutter supercomputer (with four NVIDIA A100 GPUs per node) demonstrate that these randomized techniques can enhance the numerical stability of the orthogonalization and overall solver, without a significant increase in the execution time.

math.NA↗

arXiv2025

Analysis of Randomized Householder-Cholesky QR Factorization with Multisketching

CholeskyQR2 and shifted CholeskyQR3 are two state-of-the-art algorithms for computing tall-and-skinny QR factorizations since they attain high performance on current computer architectures. However, to guarantee stability, for some applications, CholeskyQR2 faces a prohibitive restriction on the condition number of the underlying matrix to factorize. Shifted CholeskyQR3 is stable but has $50\%$ more computational and communication costs than CholeskyQR2. In this paper, a randomized QR algorithm called Randomized Householder-Cholesky (\texttt{rand\_cholQR}) is proposed and analyzed. Using one or two random sketch matrices, it is proved that with high probability, its orthogonality error is bounded by a constant of the order of unit roundoff for any numerically full-rank matrix, and hence it is as stable as shifted CholeskyQR3. An evaluation of the performance of \texttt{rand\_cholQR} on a NVIDIA A100 GPU demonstrates that for tall-and-skinny matrices, \texttt{rand\_cholQR} with multiple sketch matrices is nearly as fast as, or in some cases faster than, CholeskyQR2. Hence, compared to CholeskyQR2, \texttt{rand\_cholQR} is more stable with almost no extra computational or memory cost, and therefore a superior algorithm both in theory and practice.

math.NA↗

arXiv2024

Two-Stage Block Orthogonalization to Improve Performance of $s$-step GMRES

On current computer architectures, GMRES' performance can be limited by its communication cost to generate orthonormal basis vectors of the Krylov subspace. To address this performance bottleneck, its $s$-step variant orthogonalizes a block of $s$ basis vectors at a time, potentially reducing the communication cost by a factor of $s$. Unfortunately, for a large step size $s$, the solver can generate extremely ill-conditioned basis vectors, and to maintain stability in practice, a conservatively small step size is used, which limits the performance of the $s$-step solver. To enhance the performance using a small step size, in this paper, we introduce a two-stage block orthogonalization scheme. Similar to the original scheme, the first stage of the proposed method operates on a block of $s$ basis vectors at a time, but its objective is to maintain the well-conditioning of the generated basis vectors with a lower cost. The orthogonalization of the basis vectors is delayed until the second stage when enough basis vectors are generated to obtain higher performance. Our analysis shows the stability of the proposed two-stage scheme. The performance is improved because while the same amount of computation as the original scheme is required, most of the communication is done at the second stage of the proposed scheme, reducing the overall communication requirements. Our performance results with up to 192 NVIDIA V100 GPUs on the Summit supercomputer demonstrate that when solving a 2D Laplace problem, the two-stage approach can reduce the orthogonalization time and the total time-to-solution by the respective factors of up to $2.6\times$ and $1.6\times$ over the original $s$-step GMRES, which had already obtained the respective speedups of $2.1\times$ and $1.8\times$ over the standard GMRES. Similar speedups were obtained for 3D problems and for matrices from the SuiteSparse Matrix Collection.

math.NA↗

arXiv2023

Preconditioned Chebyshev BiCG for parameterized linear systems

We consider the problem of approximating the solution to $A(μ) x(μ) = b$ for many different values of the parameter $μ$. Here we assume $A(μ)$ is large, sparse, and nonsingular with a nonlinear dependence on $μ$. Our method is based on a companion linearization derived from an accurate Chebyshev interpolation of $A(μ)$ on the interval $[-a,a]$, $a \in \mathbb{R}$. The solution to the linearization is approximated in a preconditioned BiCG setting for shifted systems, where the Krylov basis matrix is formed once. This process leads to a short-term recurrence method, where one execution of the algorithm produces the approximation to $x(μ)$ for many different values of the parameter $μ\in [-a,a]$ simultaneously. In particular, this work proposes one algorithm which applies a shift-and-invert preconditioner exactly as well as an algorithm which applies the preconditioner inexactly. The competitiveness of the algorithms are illustrated with large-scale problems arising from a finite element discretization of a Helmholtz equation with parameterized material coefficient. The software used in the simulations is publicly available online, and thus all our experiments are reproducible.

math.NA↗

arXiv2022

A posteriori superlinear convergence bounds for block conjugate gradient

In this paper, we extend to the block case, the a posteriori bound showing superlinear convergence of Conjugate Gradients developed in [J. Comput. Applied Math., 48 (1993), pp. 327-341]; that is, we obtain similar bounds, but now for block Conjugate Gradients. We also present a series of computational experiments illustrating the validity of the bound developed here, as well as the bound from [SIAM Review, 47 (2005), pp. 247-272] using angles between subspaces. Using these bounds, we make some observations on the onset of superlinearity, and how this onset depends on the eigenvalue distribution and the block size.

math.NA↗

arXiv2022

On the Convergence of Randomized and Greedy Relaxation Schemes for Solving Nonsingular Linear Systems of Equations

We extend results known for the randomized Gauss-Seidel and the Gauss-Southwell methods for the case of a Hermitian and positive definite matrix to certain classes of non-Hermitian matrices. We obtain convergence results for a whole range of parameters describing the probabilities in the randomized method or the greedy choice strategy in the Gauss-Southwell-type methods. We identify those choices which make our convergence bounds best possible. Our main tool is to use weighted l1-norms to measure the residuals. A major result is that the best convergence bounds that we obtain for the expected values in the randomized algorithm are as good as the best for the deterministic, but more costly algorithms of Gauss-Southwell type. Numerical experiments illustrate the convergence of the method and the bounds obtained. Comparisons with the randomized Kaczmarz method are also presented.

math.NA↗

arXiv2022

On non-Hermitian positive (semi)definite linear algebraic systems arising from dissipative Hamiltonian DAEs

We discuss different cases of dissipative Hamiltonian differential-algebraic equations and the linear algebraic systems that arise in their linearization or discretization. For each case we give examples from practical applications. An important feature of the linear algebraic systems is that the (non-Hermitian) system matrix has a positive definite or semidefinite Hermitian part. In the positive definite case we can solve the linear algebraic systems iteratively by Krylov subspace methods based on efficient three-term recurrences. We illustrate the performance of these iterative methods on several examples. The semidefinite case can be challenging and requires additional techniques to deal with "singular part", while the "positive definite part" can still be treated with the three-term recurrence methods.

math.NA↗

arXiv2020

Asynchronous Richardson iterations

We consider asynchronous versions of the first and second order Richardson methods for solving linear systems of equations. These methods depend on parameters whose values are chosen a priori. We explore the parameter values that can be proven to give convergence of the asynchronous methods. This is the first such analysis for asynchronous second order methods. We find that for the first order method, the optimal parameter value for the synchronous case also gives an asynchronously convergent method. For the second order method, the parameter ranges for which we can prove asynchronous convergence do not contain the optimal parameter values for the synchronous iteration. In practice, however, the asynchronous second order iterations may still converge using the optimal parameter values, or parameter values close to the optimal ones, despite this result. We explore this behavior with a multithreaded parallel implementation of the asynchronous methods.

math.NA↗

arXiv2019

Dynamic jamming of dense suspensions under tilted impact

Dense particulate suspensions can not only increase their viscosity and shear thicken under external forcing, but also jam into a solid-like state that is fully reversible when the force is removed. An impact on the surface of a dense suspension can trigger this jamming process by generating a shear front that propagates into the bulk of the system. Tracking and visualizing such a front is difficult because suspensions are optically opaque and the front can propagate as fast as several meters per second. Recently, high-speed ultrasound imaging has been used to overcome this problem and extract two-dimensional sections of the flow field associated with jamming front propagation. Here we extend this method to reconstruct the three-dimensional flow field. This enables us to investigate the evolution of jamming fronts for which axisymmetry cannot be assumed, such as impact at angles tilted away from the normal to the free surface of the suspension. We find that sufficiently far from solid boundaries the resulting flow field is approximately identical to that generated by normal impact, but rotated and aligned with the angle of impact. However, once the front approaches the solid boundary at the bottom of the container, it generates a squeeze flow that deforms the front profile and causes jamming to proceed in a non-axisymmetric manner.

cond-mat.soft↗

arXiv2015

Preconditioned eigensolvers for large-scale nonlinear Hermitian eigenproblems with variational characterizations. II. Interior eigenvalues

We consider the solution of large-scale nonlinear algebraic Hermitian eigenproblems of the form $T(λ)v=0$ that admit a variational characterization of eigenvalues. These problems arise in a variety of applications and are generalizations of linear Hermitian eigenproblems $Av\!=\!λBv$. In this paper, we propose a Preconditioned Locally Minimal Residual (PLMR) method for efficiently computing interior eigenvalues of problems of this type. We discuss the development of search subspaces, preconditioning, and eigenpair extraction procedure based on the refined Rayleigh-Ritz projection. Extension to the block methods is presented, and a moving-window style soft deflation is described. Numerical experiments demonstrate that PLMR methods provide a rapid and robust convergence towards interior eigenvalues. The approach is also shown to be efficient and reliable for computing a large number of extreme eigenvalues, dramatically outperforming standard preconditioned conjugate gradient methods.

math.NA↗

arXiv2013

Krylov Subspace Recycling for Sequences of Shifted Linear Systems

We study the use of Krylov subspace recycling for the solution of a sequence of slowly-changing families of linear systems, where each family consists of shifted linear systems that differ in the coefficient matrix only by multiples of the identity. Our aim is to explore the simultaneous solution of each family of shifted systems within the framework of subspace recycling, using one augmented subspace to extract candidate solutions for all the shifted systems. The ideal method would use the same augmented subspace for all systems and have fixed storage requirements, independent of the number of shifted systems per family. We show that a method satisfying both requirements cannot exist in this framework. As an alternative, we introduce two schemes. One constructs a separate deflation space for each shifted system but solves each family of shifted systems simultaneously. The other builds only one recycled subspace and constructs approximate corrections to the solutions of the shifted systems at each cycle of the iterative linear solver while only minimizing the base system residual. At convergence of the base system solution, we apply the method recursively to the remaining unconverged systems. We present numerical examples involving systems arising in lattice quantum chromodynamics.

math.NA↗

arXiv2006

Asynchronous iterative computations with Web information retrieval structures: The PageRank case

There are several ideas being used today for Web information retrieval, and specifically in Web search engines. The PageRank algorithm is one of those that introduce a content-neutral ranking function over Web pages. This ranking is applied to the set of pages returned by the Google search engine in response to posting a search query. PageRank is based in part on two simple common sense concepts: (i)A page is important if many important pages include links to it. (ii)A page containing many links has reduced impact on the importance of the pages it links to. In this paper we focus on asynchronous iterative schemes to compute PageRank over large sets of Web pages. The elimination of the synchronizing phases is expected to be advantageous on heterogeneous platforms. The motivation for a possible move to such large scale distributed platforms lies in the size of matrices representing Web structure. In orders of magnitude: $10^{10}$ pages with $10^{11}$ nonzero elements and $10^{12}$ bytes just to store a small percentage of the Web (the already crawled); distributed memory machines are necessary for such computations. The present research is part of our general objective, to explore the potential of asynchronous computational models as an underlying framework for very large scale computations over the Grid. The area of ``internet algorithmics'' appears to offer many occasions for computations of unprecedent dimensionality that would be good candidates for this framework.

cs.DC↗