Related papers: Towards a mixed-precision ADI method for Lyapunov …
This research investigates using a mixed-precision iterative refinement method using posit numbers instead of the standard IEEE floating-point format. The method is applied to solve a general linear system represented by the equation $Ax =…
This paper presents an effective low-rank generalized alternating direction implicit iteration (R-GADI) method for solving large-scale sparse and stable Lyapunov matrix equations and continuous-time algebraic Riccati matrix equations. The…
Primal-dual algorithms are frequently used for iteratively solving large-scale convex optimization problems. The analysis of such algorithms is usually done on a case-by-case basis, and the resulting guaranteed rates of convergence can be…
We introduce a new ADI-based low rank solver for $AX-XB=F$, where $F$ has rapidly decaying singular values. Our approach results in both theoretical and practical gains, including (1) the derivation of new bounds on singular values for…
Low precision arithmetic, in particular half precision floating point arithmetic, is now available in commercial hardware. Using lower precision can offer significant savings in computation and communication costs with proportional savings…
In the present paper, we consider large-scale differential Lyapunov matrix equations having a low rank constant term. We present two new approaches for the numerical resolution of such differential matrix equations. The first approach is…
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…
In this work we present a low-rank algorithm for computing low-rank approximations of large-scale Lyapunov operator $\varphi$-functions. These computations play a crucial role in implementing of matrix-valued exponential integrators for…
The low-rank alternating direction implicit (ADI) method is an efficient and effective solver for large-scale standard continuous-time algebraic Riccati equations that admit low-rank solutions. However, the existing low-rank ADI algorithm…
A delay Lyapunov matrix corresponding to an exponentially stable system of linear time-invariant delay differential equations can be characterized as the solution of a boundary value problem involving a matrix valued delay differential…
We formulate two classes of first-order algorithms more general than previously studied for minimizing smooth and strongly convex or, respectively, smooth and convex functions. We establish sufficient conditions, via new discrete Lyapunov…
The present paper continues our investigation of an implementation of a least-squares collocation method for higher-index differential-algebraic equations. In earlier papers, we were able to substantiate the choice of basis functions and…
In this manuscript, we study the properties of a family of second-order differential equations with damping, its discretizations and their connections with accelerated optimization algorithms for $m$-strongly convex and $L$-smooth…
Recently, there has been an increasing interest in using tools from dynamical systems to analyze the behavior of simple optimization algorithms such as gradient descent and accelerated variants. This paper strengthens such connections by…
The convergence of an adaptive mixed finite element method for general second order linear elliptic problems defined on simply connected bounded polygonal domains is analyzed in this paper. The main difficulties in the analysis are posed by…
This paper proposes an effective low-rank alternating direction doubling algorithm (R-ADDA) for computing numerical low-rank solutions to large-scale sparse continuous-time algebraic Riccati matrix equations. The method is based on the…
This paper introduces a second-order differential inclusion for unconstrained convex optimization. In continuous level, solution existence in proper sense is obtained and exponential decay of a novel Lyapunov function along with the…
This paper considers large-scale nonsymmetric continuous-time algebraic Riccati equations (NAREs) that admit low-rank solutions. Low-rank alternating direction implicit (ADI) methods have proven to be an efficient approach for solving…
Sequences of parametrized Lyapunov equations can be encountered in many application settings. Moreover, solutions of such equations are often intermediate steps of an overall procedure whose main goal is the computation of…
Multilevel methods represent a powerful approach in numerical solution of partial differential equations. The multilevel structure can also be used to construct estimates for total and algebraic errors of computed approximations. This paper…