Related papers: A splitting-based KPIK method for eddy current opt…
The extended Fisher--Kolmogorov (EFK) equation has been used to describe some phenomena in physical, material and biology systems. In this paper, we propose a full-rank splitting scheme and a rank-adaptive splitting approach for this…
In this paper, we propose a low rank approximation method for efficiently solving stochastic partial differential equations. Specifically, our method utilizes a novel low rank approximation of the stiffness matrices, which can significantly…
PDE-constrained optimization problems arise in a broad number of applications such as hyperthermia cancer treatment or blood flow simulation. Discretization of the optimization problem and using a Lagrangian approach result in a large-scale…
In this paper, an efficient parallel splitting method is proposed for the optimal control problem with parabolic equation constraints. The linear finite element is used to approximate the state variable and the control variable in spatial…
In the fields of control theory and machine learning, the dynamic low-rank approximation for large-scale matrices has received substantial attention. Considering large-scale semilinear stiff matrix differential equations, we propose…
We propose a high order adaptive-rank implicit integrators for stiff time-dependent PDEs, leveraging extended Krylov subspaces to efficiently and adaptively populate low-rank solution bases. This allows for the accurate representation of…
In this paper, we propose a novel reduced-rank adaptive filtering algorithm by blending the idea of the Krylov subspace methods with the set-theoretic adaptive filtering framework. Unlike the existing Krylov-subspace-based reduced-rank…
In the present paper, we propose Krylov-based methods for solving large-scale differential Sylvester matrix equations having a low rank constant term. We present two new approaches for solving such differential matrix equations. The first…
Elliptic partial differential equations (PDEs) arise in many areas of computational sciences such as computational fluid dynamics, biophysics, engineering, geophysics and more. They are difficult to solve due to their global nature and…
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…
Conductivity reconstruction in an inverse eddy current problem is considered in the present paper. With the electric field measurement on part of domain boundary, we formulate the reconstruction problem to a constrained optimization problem…
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…
Many scientific and engineering challenges can be formulated as optimization problems which are constrained by partial differential equations (PDEs). These include inverse problems, control problems, and design problems. As a major…
In this paper we will consider distributed Linear-Quadratic Optimal Control Problems dealing with Advection-Diffusion PDEs for high values of the P\'eclet number. In this situation, computational instabilities occur, both for steady and…
Linear matrix equations, such as the Sylvester and Lyapunov equations, play an important role in various applications, including the stability analysis and dimensionality reduction of linear dynamical control systems and the solution of…
This paper develops a new class of Rosenbrock-type integrators based on a Krylov space solution of the linear systems. The new family, called Rosenbrock-Krylov (Rosenbrock-K), is well suited for solving large scale systems of ODEs or…
We consider an effective new method for solving trust-region and norm-regularization problems that arise as subproblems in many optimization applications. We show that the solutions to such subproblems effectively lie in a…
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…
We study a multigrid method for solving large linear systems of equations with tensor product structure. Such systems are obtained from stochastic finite element discretization of stochastic partial differential equations such as the…
In this study, we consider the numerical solution of large systems of linear equations obtained from the stochastic Galerkin formulation of stochastic partial differential equations. We propose an iterative algorithm that exploits the…