Related papers: Numerical solutions to large-scale differential Ly…
We develop computational methods for approximating the solution of a linear multi-term matrix equation in low rank. We follow an alternating minimization framework, where the solution is represented as a product of two matrices, and…
In the numerical treatment of large-scale Sylvester and Lyapunov equations, projection methods require solving a reduced problem to check convergence. As the approximation space expands, this solution takes an increasing portion of the…
We present numerical solutions for differential equations by expanding the unknown function in terms of Chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema (or zeros) of the…
In this paper, we present numerical procedures to compute solutions of partial differential equations posed on fractals. In particular, we consider the strong form of the equation using standard graph Laplacian matrices and also weak forms…
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…
The paper presents a numerical technique for computing directly the Takens-Bogdanov points in the nonlinear system of differential equations with one constant delay and two parameters. By representing the delay differential equations as…
This paper focuses on the numerical solution of initial value problems for fractional differential equations of linear type. The approach we propose grounds on expressing the solution in terms of some integral weighted by a generalized…
This work considers large-scale Lyapunov matrix equations of the form $AX + XA = \boldsymbol{c}\boldsymbol{c}^T$, where $A$ is a symmetric positive definite matrix and $\boldsymbol{c}$ is a vector. Motivated by the need to solve such…
This paper deals with a special type of Lyapunov functions, namely the solution of Zubov's equation. Such a function can be used to characterize the domain of attraction for systems of ordinary differential equations. We derive and prove an…
We study Bayesian methods for large-scale linear inverse problems, focusing on the challenging task of hyperparameter estimation. Typical hierarchical Bayesian formulations that follow a Markov Chain Monte Carlo approach are possible for…
In this study, a recursive solution technique in conjunction with generalized integrating factors is presented and applied to address first and second order linear differential equations. This approach demonstrates practical utility in…
In this work we study the problem of step size selection for numerical schemes, which guarantees that the numerical solution presents the same qualitative behavior as the original system of ordinary differential equations, by means of tools…
For a system of partial differential equations that has an extended Kovalevskaya form, a reduction procedure is presented that allows one to use a local (point, contact, or higher) symmetry of a system and a symmetry-invariant conservation…
A splitting scheme for backward doubly stochastic differential equations is proposed. The main idea is to decompose a backward doubly stochastic differential equation into a backward stochastic differential equation and a stochastic…
In this paper, our goal is solving backward doubly stochastic differential equation (BDSDE for short) under weak assumptions on the data. The first part of the paper is devoted to the development of some new technical aspects of stochastic…
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…
In this work, we propose a novel backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations (BSDEs), where the deep neural network (DNN) models are trained not only…
We propose a numerical method for computing the Lyapunov exponents of renewal equations (delay equations of Volterra type), consisting first in applying a discrete QR technique to the associated evolution family suitably posed on a Hilbert…
Techniques are developed for decoupling dissipative differential equations. The approach considered is based upon obtaining a sufficient gap in the time dependent linear portion of the equation that corresponds to the linear variational…
The low-rank matrix completion problem can be solved by Riemannian optimization on a fixed-rank manifold. However, a drawback of the known approaches is that the rank parameter has to be fixed a priori. In this paper, we consider the…