Related papers: Backward error analysis for conjugate symplectic m…

A fundamental task in numerical computation is the solution of large linear systems. The conjugate gradient method is an iterative method which offers rapid convergence to the solution, particularly when an effective preconditioner is…

Reduced basis methods are popular for approximately solving large and complex systems of differential equations. However, conventional reduced basis methods do not generally preserve conservation laws and symmetries of the full order model.…

We consider numerical approximations of stochastic differential equations by the Euler method. In the case where the SDE is elliptic or hypoelliptic, we show a weak backward error analysis result in the sense that the generator associated…

We analyze composition methods with complex coefficients exhibiting the so-called ``symmetry-conjugate'' pattern in their distribution. In particular, we study their behavior with respect to preservation of qualitative properties when…

We use backward error analysis for differential equations to obtain modified or distorted equations describing the behaviour of the Newmark scheme applied to the transient structural dynamics equation. Based on the newly derived distorted…

Simulation of many-particle system evolution by molecular dynamics takes to decrease integration step to provide numerical scheme stability on the sufficiently large time interval. It leads to a significant increase of the volume of…

In this paper, we develop an optimization-based framework for solving coupled forward-backward stochastic differential equations. We introduce an integral-form objective function and prove its equivalence to the error between consecutive…

We apply symmetry and invariance methods to analyse systems of difference equations. Non trivial symmetries are derived and their exact solutions obtained.

We consider numerical approximations of stochastic Langevin equations by implicit methods. We show a weak backward error analysis result in the sense that the generator associated with the numerical solution coincides with the solution of a…

Butcher series are combinatorial devices used in the study of numerical methods for differential equations evolving on vector spaces. More precisely, they are formal series developments of differential operators indexed over rooted trees,…

Motivated by understanding the behavior of the Alternating Mirror Descent (AMD) algorithm for bilinear zero-sum games, we study the discretization of continuous-time Hamiltonian flow via the symplectic Euler method. We provide a framework…

We address our attention to the numerical time discretization of stochastic Poisson systems via Poisson integrators. The aim of the investigation regards the backward error analysis of such integrators to reveal their ability of being…

We compare the performances of symplectic and non-symplectic integrators for the computation of normal geodesics and conjugate points in sub-Riemannian geometry at the example of the Martinet case. For this case study we consider first the…

The usual explicit finite-difference method of solving partial differential equations is limited in stability because it approximates the exact amplification factor by power-series. By adapting the same exponential-splitting method of…

Hamilton's equations of motion form a fundamental framework in various branches of physics, including astronomy, quantum mechanics, particle physics, and climate science. Classical numerical solvers are typically employed to compute the…

Inverse problems use physical measurements along with a computational model to estimate the parameters or state of a system of interest. Errors in measurements and uncertainties in the computational model lead to inaccurate estimates. This…

Backward Stochastic Differential Equations (BSDEs) have been widely employed in various areas of social and natural sciences, such as the pricing and hedging of financial derivatives, stochastic optimal control problems, optimal stopping…

The backward-backward algorithm is a tool for finding minima of a regularization of the sum of two convex functions in Hilbert spaces. We generalize this setting to Hadamard spaces and prove the convergence of an error-tolerant version of…

Along with the practical success of the discovery of dynamics using deep learning, the theoretical analysis of this approach has attracted increasing attention. Prior works have established the grid error estimation with auxiliary…

While backward error analysis does not generalise straightforwardly to the strong and weak approximation of stochastic differential equations, it extends for the sampling of ergodic dynamics. The calculation of the modified equation relies…