Related papers: Backward error analysis for multisymplectic discre…
We have researched the condition for symplectic discretization to preserve local boundedness for the space of 2-dimensional Hamiltonian dynamical systems in this paper.
Symmetry-preserving (mimetic) discretization aims to preserve certain properties of a continuous differential operator in its discrete counterpart. For these discretizations, stability and (discrete) conservation of mass, momentum and…
This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the Multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and…
In this work, we investigate the numerical approximation of the second order non-autonomous semilnear parabolic partial differential equation (PDE) using the finite element method. To the best of our knowledge, only the linear case is…
In [5] the authors suggested a new algorithm for the numerical approximation of a BSDE by merging the cubature method with the first order discretization developed by [3] and [16]. Though the algorithm presented in [5] compared…
Machine learning methods are widely used in the natural sciences to model and predict physical systems from observation data. Yet, they are often used as poorly understood "black boxes," disregarding existing mathematical structure and…
This paper deals with the asymptotic behavior and FEM error analysis of a class of strongly damped wave equations using a semidiscrete finite element method in spatial directions combined with a finite difference scheme in the time…
In this work, we develop a class of high-order multiderivative time integration methods that is able to preserve certain functionals discretely. Important ingredients are the recently developed Hermite-Birkhoff-Predictor-Corrector methods…
We present a new class of exponential integrators for ordinary differential equations. They are locally exact, i.e., they preserve the linearization of the original system at every point. Their construction consists in modifying existing…
We present a mathematical and numerical investigation to the shrinkingdimer saddle dynamics for finding any-index saddle points in the solution landscape. Due to the dimer approximation of Hessian in saddle dynamics, the local Lipschitz…
Multi-symplectic diamond schemes proposed by McLachlan and Wilkins (2015) provide a framework for the numerical integration of Hamiltonian partial differential equations, combining local implicitness with high-order accuracy and discrete…
This work focuses on numerical solutions of optimal control problems. A time discretization error representation is derived for the approximation of the associated value function. It concerns Symplectic Euler solutions of the Hamiltonian…
In this paper, we consider stochastic Runge-Kutta methods for stochastic Hamiltonian partial differential equations and present some sufficient conditions for multisymplecticity of stochastic Runge-Kutta methods of stochastic Hamiltonian…
Existing model reduction techniques for high-dimensional models of conservative partial differential equations (PDEs) encounter computational bottlenecks when dealing with systems featuring non-polynomial nonlinearities. This work presents…
Structure-preserving algorithms for solving conservative PDEs with added linear dissipation are generalized to systems with time-dependent damping/driving terms. This study is motivated by several PDE models of physical phenomena, such as…
This paper is concerned with the adaptive numerical treatment of stochastic partial differential equations. Our method of choice is Rothe's method. We use the implicit Euler scheme for the time discretization. Consequently, in each step, an…
We propose a multi-index algorithm for the Monte Carlo (MC) discretization of a linear, elliptic PDE with affine-parametric input. We prove an error vs. work analysis which allows a multi-level finite-element approximation in the physical…
The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators allows for formal statistical quantification of the error due to discretisation in the numerical context. Competing…
Mathematical descriptions of flow phenomena usually come in the form of partial differential equations. The differential operators used in these equations may have properties such as symmetry, skew-symmetry, positive or negative…
There are many evolution partial differential equations which can be cast into Hamiltonian form. Conservation laws of these equations are related to one-parameter Hamiltonian symmetries admitted by the PDEs. The same result holds for…