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This survey explores modern approaches for computing low-rank approximations of high-dimensional matrices by means of the randomized SVD, randomized subspace iteration, and randomized block Krylov iteration. The paper compares the…
Damped wave equations have been used in many real-world fields. In this paper, we study a low-rank solution of the strongly damped wave equation with the damping term, visco-elastic damping term and mass term. Firstly, a second-order finite…
This study focuses on using direct methods (first-discretize-then-optimize) to solve optimal control problems for a class of nonsmooth dynamical systems governed by differential variational inequalities (DVI), called optimal control…
For several classes of mathematical models that yield linear systems, the splitting of the matrix into its Hermitian and skew Hermitian parts is naturally related to properties of the underlying model. This is particularly so for…
This paper develops a new class of exponential-type integrators where all the matrix exponentiations are performed in a single Krylov space of low dimension. The new family, called Lightly Implicit Krylov-Exponential (LIKE), is well suited…
Direct factorization methods for the solution of large, sparse linear systems that arise from PDE discretizations are robust, but typically show poor time and memory scalability for large systems. In this paper, we describe an efficient…
This study explores distributed optimization problems with clique-wise coupling via operator splitting and how we can utilize this framework for performance analysis and enhancement. This framework extends beyond conventional pairwise…
We introduce tensor numerical techniques for solving optimal control problems constrained by elliptic operators in $\mathbb{R}^d$, $d=2,3$, with variable coefficients, which can be represented in a low rank separable form. We construct a…
Developing efficient solvers for large-scale multi-term linear matrix equations remains a central challenge in numerical linear algebra and is still largely unresolved. This paper introduces a methodology leveraging CUR decomposition for…
We consider discontinuous Galerkin methods for an elliptic distributed optimal control problem and we propose multigrid methods to solve the discretized system. We prove that the $W$-cycle algorithm is uniformly convergent in the energy…
We consider a pointwise tracking optimal control problem for a semilinear elliptic partial differential equation. We derive the existence of optimal solutions and analyze first and, necessary and sufficient, second order optimality…
We propose a numerical integrator for determining low-rank approximations to solutions of large-scale matrix differential equations. The considered differential equations are semilinear and stiff. Our method consists of first splitting the…
This paper is dedicated to the investigation of a new numerical method to approximate the optimal stopping problem for a discrete-time continuous state space Markov chain under partial observations. It is based on a two-step discretization…
This manuscript presents an adaptive high order discretization technique for elliptic boundary value problems. The technique is applied to an updated version of the Hierarchical Poincar\'e-Steklov (HPS) method. Roughly speaking, the HPS…
In this paper, we propose an original approach to stochastic control problems. We consider a weak formulation that is written as an optimization (minimization) problem on the space of probability measures. We then introduce a penalized…
There are numerous contexts where one wishes to describe the state of a randomly evolving system. Effective solutions combine models that quantify the underlying uncertainty with available observational data to form relatively optimal…
Optimal control for switch-based dynamical systems is a challenging problem in the process control literature. In this study, we model these systems as hybrid dynamical systems with finite number of unknown switching points and reformulate…
We study the effect of the streamline upwind/Petrov Galerkin (SUPG) stabilized finite element method on the discretization of optimal control problems governed by linear advection-diffusion equations. We compare two approaches for the…
We propose a novel Galerkin discretization scheme for stochastic optimal control problems on an indefinite time horizon. The control problems are linear-quadratic in the controls, but possibly nonlinear in the state variables, and the…
Within the realm of early fault-tolerant quantum computing (EFTQC), quantum Krylov subspace diagonalization (QKSD) has emerged as a promising quantum algorithm for the approximate Hamiltonian diagonalization via projection onto the quantum…