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We derive a second-order ordinary differential equation (ODE) which is the limit of Nesterov's accelerated gradient method. This ODE exhibits approximate equivalence to Nesterov's scheme and thus can serve as a tool for analysis. We show…
Improved local numerical solution for the ADER-DG numerical method with a local DG predictor for solving the initial value problem for a first-order ODE system is proposed. The improved local numerical solution demonstrates convergence…
Probabilistic solvers for ordinary differential equations (ODEs) provide efficient quantification of numerical uncertainty associated with simulation of dynamical systems. Their convergence rates have been established by a growing body of…
The numerical solution methods for partial differential equation (PDE) solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods…
Some continuous optimization methods can be connected to ordinary differential equations (ODEs) by taking continuous limits, and their convergence rates can be explained by the ODEs. However, since such ODEs can achieve any convergence rate…
Convergence analysis of accelerated first-order methods for convex optimization problems are presented from the point of view of ordinary differential equation solvers. A new dynamical system, called Nesterov accelerated gradient flow, has…
We propose a high order numerical homogenization method for dissipative ordinary differential equations (ODEs) containing two time scales. Essentially, only first order homogenized model globally in time can be derived. To achieve a high…
This paper presents a sequence of deferred correction (DC) schemes built recursively from the implicit midpoint scheme for the numerical solution of general first order ordinary differential equations (ODEs). It is proven that each scheme…
The objective of this paper is to prove the convergence of a linear implicit multi-step numerical method for ordinary differential equations. The algorithm is obtained via Taylor approximations. The convergence is proved following the…
A key appeal of the recently proposed Neural Ordinary Differential Equation (ODE) framework is that it seems to provide a continuous-time extension of discrete residual neural networks. As we show herein, though, trained Neural ODE models…
A development of an inverse first-order divided difference operator for functions of several variables is presented. Two generalized derivative-free algorithms builded up from Ostrowski's method for solving systems of nonlinear equations…
For the ordinary differential equation (ODE) $\dot{x}(t) = f(t,x)$, $x(0) = x_0$, $t\geq 0$, $x\in R^d$, assume $f$ to be at least continuous in $t$ and locally Lipshitz in $x$, and if necessary, several times continuously differentiable in…
The numerical methods for differential equation solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods have the restricted class of…
We study first-order optimization methods obtained by discretizing ordinary differential equations (ODEs) corresponding to Nesterov's accelerated gradient methods (NAGs) and Polyak's heavy-ball method. We consider three discretization…
The framework of Baikov-Gazizov-Ibragimov approximate symmetries has proven useful for many examples where a small perturbation of an ordinary differential equation (ODE) destroys its local symmetry group. For the perturbed model, some of…
Designing and analyzing optimization methods via continuous-time models expressed as ordinary differential equations (ODEs) is a promising approach for its intuitiveness and simplicity. A key concern, however, is that the convergence rates…
Stochastic differential equations (SDEs) offer powerful and accessible mathematical models for capturing both deterministic and probabilistic aspects of dynamic behavior across a wide range of physical, financial, and social systems.…
Systems of differential-algebraic equations (DAEs) represent a widespread formalism in the modeling of constrained mechanical systems and electrical networks. Due to the automatic, object-oriented generation of the equations of motion and…
The integral-algebraic equation (IAE) is a mixed system of first-kind and second-kind Volterra integral equations (VIEs). This paper mainly focuses on the discontinuous Galerkin (DG) method to solve index-2 IAEs. First, the convergence…
Differential Equations are among the most important Mathematical tools used in creating models in the science, engineering, economics, mathematics, physics, aeronautics, astronomy, dynamics, biology, chemistry, medicine, environmental…