Related papers: A general framework for Krylov ODE residuals with …
Augmented Krylov subspace methods aid in accelerating the convergence of a standard Krylov subspace method by including additional vectors in the search space. A residual projection framework based on residual (Petrov-) Galerkin constraints…
We consider distributed optimization methods for problems where forming the Hessian is computationally challenging and communication is a significant bottleneck. We leverage randomized sketches for reducing the problem dimensions as well as…
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…
We propose an adaptive randomized truncation estimator for Krylov subspace methods that optimizes the trade-off between the solution variance and the computational cost, while remaining unbiased. The estimator solves a constrained…
In this work, we collect data from runs of Krylov subspace methods and pipelined Krylov algorithms in an effort to understand and model the impact of machine noise and other sources of variability on performance. We find large variability…
We consider the numerical solution of large-scale symmetric differential matrix Riccati equations. Under certain hypotheses on the data, reduced order methods have recently arisen as a promising class of solution strategies, by forming…
Krylov subspace methods are a ubiquitous tool for computing near-optimal rank $k$ approximations of large matrices. While "large block" Krylov methods with block size at least $k$ give the best known theoretical guarantees, block size one…
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…
This paper investigates explicit expressions for the error associated with the block rational Krylov approximation of matrix functions. Two formulas are proposed, both derived from characterizations of the block FOM residual. The first…
The paper is concerned with methods for computing the best low multilinear rank approximation of large and sparse tensors. Krylov-type methods have been used for this problem; here block versions are introduced. For the computation of…
Krylov subspace methods are widely known as efficient algebraic methods for solving large scale linear systems. However, on massively parallel hardware the performance of these methods is typically limited by communication latency rather…
Probabilistic ideas and tools have recently begun to permeate into several fields where they had traditionally not played a major role, including fields such as numerical linear algebra and optimization. One of the key ways in which these…
In classical frameworks as the Euclidean space, positive definite kernels as well as their analytic properties are explicitly available and can be incorporated directly in kernel-based learning algorithms. This is different if the…
The randomized coordinate descent (RCD) method is a classical algorithm with simple, lightweight iterations that is widely used for various optimization problems, including the solution of positive semidefinite linear systems. As a linear…
One of the most computationally expensive steps of the low-rank ADI method for large-scale Lyapunov equations is the solution of a shifted linear system at each iteration. We propose the use of the extended Krylov subspace method for this…
In this paper, we develop algorithms for computing the recurrence coefficients corresponding to multiple orthogonal polynomials on the step-line. We reformulate the problem as an inverse eigenvalue problem, which can be solved using…
We compare two approaches to compute a portion of the spectrum of dense symmetric definite generalized eigenproblems: one is based on the reduction to tridiagonal form, and the other on the Krylov-subspace iteration. Two large-scale…
A well-known problem in computing some matrix functions iteratively is the lack of a clear, commonly accepted residual notion. An important matrix function for which this is the case is the matrix exponential. Suppose the matrix exponential…
Deflation techniques are typically used to shift isolated clusters of small eigenvalues in order to obtain a tighter distribution and a smaller condition number. Such changes induce a positive effect in the convergence behavior of Krylov…
This paper describes practical randomized algorithms for low-rank matrix approximation that accommodate any budget for the number of views of the matrix. The presented algorithms, which are aimed at being as pass efficient as needed, expand…