Related papers: Low-synchronization Arnoldi Methods for the Matrix…
The parallel strong-scaling of Krylov iterative methods is largely determined by the number of global reductions required at each iteration. The GMRES and Krylov-Schur algorithms employ the Arnoldi algorithm for nonsymmetric matrices. The…
Exponential integrators that use Krylov approximations of matrix functions have turned out to be efficient for the time-integration of certain ordinary differential equations (ODEs). This holds in particular for linear homogeneous ODEs,…
With the recent realization of exascale performace by Oak Ridge National Laboratory's Frontier supercomputer, reducing communication in kernels like QR factorization has become even more imperative. Low-synchronization Gram-Schmidt methods,…
Communication-avoiding and pipelined variants of Krylov solvers are critical for the scalability of linear system solvers on future exascale architectures. We present low synchronization variants of iterated classical (CGS) and modified…
We present an overview of randomized orthogonalization techniques that construct a well-conditioned basis whose sketch is orthonormal. Randomized orthogonalization has recently emerged as a powerful paradigm for reducing the computational…
One of the limitations of recycled GCRO methods is the large amount of computation required to orthogonalize the basis vectors of the newly generated Krylov subspace for the approximate solution when combined with those of the recycle…
Exponential integrators are time stepping schemes which exactly solve the linear part of a semilinear ODE system. This class of schemes requires the approxima- tion of a matrix exponential in every step, and one successful modern method is…
We develop an algorithm for computing the solution of a large system of linear ordinary differential equations (ODEs) with polynomial inhomogeneity. This is equivalent to computing the action of a certain matrix function on the vector…
In recent years, a great deal of attention has been focused on numerically solving exponential integrators. The important ingredient to the implementation of exponential integrators is the efficient and accurate evaluation of the so called…
We consider Arnoldi like processes to obtain symplectic subspaces for Hamiltonian systems. Large systems are locally approximated by ones living in low dimensional subspaces; we especially consider Krylov subspaces and some extensions. This…
The orthogonalization process is an essential building block in Krylov space methods, which takes up a large portion of the computational time. Commonly used methods, like the Gram-Schmidt method, consider the projection and normalization…
Krylov subspace methods are an essential building block in numerical simulation software. The efficient utilization of modern hardware is a challenging problem in the development of these methods. In this work, we develop Krylov subspace…
This paper presents a new algorithm KIOPS for computing linear combinations of $\varphi$-functions that appear in exponential integrators. This algorithm is suitable for large-scale problems in computational physics where little or no…
Seismic imaging is a major challenge in geophysics with broad applications. It involves solving wave propagation equations with absorbing boundary conditions (ABC) multiple times. This drives the need for accurate and efficient numerical…
We study a stability preserved Arnoldi algorithm for matrix exponential in the time domain simulation of large-scale power delivery networks (PDN), which are formulated as semi-explicit differential algebraic equations (DAEs). The solution…
The numerical integration of stiff equations is a challenging problem that needs to be approached by specialized numerical methods. Exponential integrators form a popular class of such methods since they are provably robust to stiffness and…
Krylov methods provide a fast and highly parallel numerical tool for the iterative solution of many large-scale sparse linear systems. To a large extent, the performance of practical realizations of these methods is constrained by the…
Randomized orthogonal projection methods (ROPMs) can be used to speed up the computation of Krylov subspace methods in various contexts. Through a theoretical and numerical investigation, we establish that these methods produce…
Error estimates for the numerical solution of the master equation are presented. Estimates are based on adjoint methods. We find that a good estimate can often be computed without spending computational effort on a dual problem. Estimates…
Randomized Krylov subspace methods that employ the sketch-and-solve paradigm to substantially reduce orthogonalization cost have recently shown great promise in speeding up computations for many core linear algebra tasks (e.g., solving…