Related papers: Krylov subspaces associated with higher-order line…
In this paper we use some basic facts from the theory of (matrix) Lie groups and algebras to show that many of the classical matrix splittings used to construct stationary iterative methods and preconditioniers for Krylov subspace methods…
In this paper, we introduce a unified framework for nonlinear Krylov subspace methods (nlKrylov) to solve systems of nonlinear equations. Building on classical GCR-like/type linear Krylov solvers such as GMRESR, we generalize these…
In quantum many-body systems, time-evolved states typically remain confined to a smaller region of the Hilbert space known as the $\textit{Krylov subspace}$. The time evolution can be mapped onto a one-dimensional problem of a particle…
This work introduces a method for learning low-dimensional models from data of high-dimensional black-box dynamical systems. The novelty is that the learned models are exactly the reduced models that are traditionally constructed with model…
The objective of this paper is to design novel multi-layer neural network architectures for multiscale simulations of flows taking into account the observed data and physical modeling concepts. Our approaches use deep learning concepts…
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…
This paper introduces new solvers for efficiently computing solutions to large-scale inverse problems with group sparsity regularization, including both non-overlapping and overlapping groups. Group sparsity regularization refers to a type…
Two approaches for approximating the solution of large-scale Lyapunov equations are considered: the alternating direction implicit (ADI) iteration and projective methods by Krylov subspaces. A link between them is presented by showing that…
We consider the solution of large stiff systems of ordinary differential equations with explicit exponential Runge--Kutta integrators. These problems arise from semi-discretized semi-linear parabolic partial differential equations on…
This work is concerned with approximating matrix functions for banded matrices, hierarchically semiseparable matrices, and related structures. We develop a new divide-and-conquer method based on (rational) Krylov subspace methods for…
In this note, we consider preconditioned Krylov subspace methods for discrete fluid-structure interaction problems with a nonlinear hyperelastic material model and covering a large range of flows, e.g, water, blood, and air with highly…
We present a novel Krylov subspace method for approximating $L_f(A, E) \vc{b}$, the matrix-vector product of the Fr\'echet derivative $L_f(A, E)$ of a large-scale matrix function $f(A)$ in direction $E$, a task that arises naturally in the…
Network systems consist of subsystems and their interconnections, and provide a powerful framework for analysis, modeling and control of complex systems. However, subsystems may have high-dimensional dynamics, and the amount and nature of…
This paper introduces the concept of abstracted model reduction: a framework to improve the tractability of structure-preserving methods for the complexity reduction of interconnected system models. To effectively reduce high-order,…
While preconditioning is a long-standing concept to accelerate iterative methods for linear systems, generalizations to matrix functions are still in their infancy. We go a further step in this direction, introducing polynomial…
Ground-state energy and matrix element are reconstructed from correlators in lattice QCD by diagonalizing transfer matrix $\hat{T}$ within the Krylov subspace spanned by $\hat{T}^n|\chi\rangle$, where $|\chi\rangle$ is a state generated by…
In this paper, we investigate the optimal $\mathcal{H}_2$ model reduction problem for single-input single-output (SISO) continuous-time linear time-invariant (LTI) systems. A semi-definite relaxation (SDR) approach is proposed to determine…
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…
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…
In many applications throughout science and engineering, model reduction plays an important role replacing expensive large-scale linear dynamical systems by inexpensive reduced order models that capture key features of the original, full…