Related papers: Escaping the Krylov space during finite precision …
The quadratic numerical range $W^2(A)$ is a subset of the standard numerical range of a linear operator which still contains its spectrum. It arises naturally in operators which have a $2 \times 2$ block structure, and it consists of at…
In a quantum many-body system, autocorrelation functions can determine linear responses nearby equilibrium and quantum dynamics far from equilibrium. In this letter, we bring out the connection between the operator complexity and the…
We investigate the regularizing behavior of an iterative Krylov subspace method for the solution of linear inverse problems in precisions lower than double. Recent works have considered the projection of iterated Tikhonov methods using…
In this paper we propose and analyze an algorithm for identifying spectral gaps of a real symmetric matrix $A$ by simultaneously approximating the traces of spectral projectors associated with multiple different spectral slices. Our method…
This paper introduces new solvers for the computation of low-rank approximate solutions to large-scale linear problems, with a particular focus on the regularization of linear inverse problems. Although Krylov methods incorporating explicit…
Krylov subspace methods are a powerful family of iterative solvers for linear systems of equations, which are commonly used for inverse problems due to their intrinsic regularization properties. Moreover, these methods are naturally suited…
Quantum computers provide new avenues to access ground and excited state properties of systems otherwise difficult to simulate on classical hardware. New approaches using subspaces generated by real-time evolution have shown efficiency in…
This paper presents a study of the inherent structural properties of Krylov subspaces, in particular for the self-adjoint class of operators, and how they relate with the important phenomenon of `Krylov solvability' of linear inverse…
Many scientific applications require the evaluation of the action of the matrix function over a vector and the most common methods for this task are those based on the Krylov subspace. Since the orthogonalization cost and memory requirement…
We introduce a framework for repurposing error estimators for source problems to compute an estimator for the gap between eigenspaces and their discretizations. Of interest are eigenspaces of finite clusters of eigenvalues of unbounded…
Randomized algorithms for low-rank approximation of quaternion matrices have gained increasing attention in recent years. However, existing methods overlook pass efficiency, the ability to limit the number of passes over the input…
Preconditioned iterative methods for numerical solution of large matrix eigenvalue problems are increasingly gaining importance in various application areas, ranging from material sciences to data mining. Some of them, e.g., those using…
Variance reduction is a crucial idea for Monte Carlo simulation and the stochastic Lanczos quadrature method is a dedicated method to approximate the trace of a matrix function. Inspired by their advantages, we combine these two techniques…
We consider filtered subspace iteration for approximating a cluster of eigenvalues (and its associated eigenspace) of a (possibly unbounded) selfadjoint operator in a Hilbert space. The algorithm is motivated by a quadrature approximation…
A common challenge faced in quantum physics is finding the extremal eigenvalues and eigenvectors of a Hamiltonian matrix in a vector space so large that linear algebra operations on general vectors are not possible. There are numerous…
An efficient Krylov subspace algorithm for computing actions of the $\varphi$ matrix function for large matrices is proposed. This matrix function is widely used in exponential time integration, Markov chains and network analysis and many…
The computation of resistance distance is pivotal in a wide range of graph analysis applications, including graph clustering, link prediction, and graph neural networks. Despite its foundational importance, efficient algorithms for…
Low-rank approximations of original samples are playing more and more an important role in many recently proposed mathematical models from data science. A natural and initial requirement is that these representations inherit original…
We analyze randomized matrix-free quadrature algorithms for spectrum and spectral sum approximation. The algorithms studied include the kernel polynomial method and stochastic Lanczos quadrature, two widely used methods for these tasks. Our…
Gradient methods are frequently used in large scale image deblurring problems since they avoid the onerous computation of the Hessian matrix of the objective function. Second order information is typically sought by a clever choice of the…