Related papers: High Order Three Part Split Symplectic Integration…
We propose a high-order numerical methodology for computing the ground state and time evolution of the two-dimensional Gross-Pitaevskii equation with harmonic trapping potential. The ground state is obtained by combining normalized gradient…
The paper intends to lay out the first steps towards constructing a unified framework to understand the symplectic and spectral theory of finite dimensional integrable Hamiltonian systems. While it is difficult to know what the best…
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…
The superiority of symplectic methods for stochastic Hamiltonian systems has been widely recognized, yet the probabilistic mechanism behind this superiority remains incompletely understood. This paper studies the superiority of symplectic…
The efficient evaluation of high-dimensional integrals is of importance in both theoretical and practical fields of science, such as data science, statistical physics, and machine learning. However, exact computation methods suffer from the…
The explicit split-operator algorithm is often used for solving the linear and nonlinear time-dependent Schr\"{o}dinger equations. However, when applied to certain nonlinear time-dependent Schr\"{o}dinger equations, this algorithm loses…
In this paper we consider splitting methods for nonlinear ordinary differential equations in which one of the (partial) flows that results from the splitting procedure can not be computed exactly. Instead, we insert a well-chosen state…
This study proposes a high-order multi-scale method tailored for time-dependent nonlinear thermo-electro-mechanical coupling problems of composite structures with highly spatial heterogeneity, which incorporate temperature-dependent…
A new coupling argument is introduced to establish Driver's integration by parts formula and shift Harnack inequality. Unlike known coupling methods where two marginal processes with different starting points are constructed to move…
We assess the applicability and efficiency of time-adaptive high-order splitting methods applied for the numerical solution of (systems of) nonlinear parabolic problems under periodic boundary conditions. We discuss in particular several…
Compared to other symplectic integrators (the Wisdom and Holman map and its higher order generalizations) that also take advantage of the hierarchical nature of the motion of the planets around the central star, our methods require solving…
A parallel implementation of coupled spin-lattice dynamics in the LAMMPS molecular dynamics package is presented. The equations of motion for both spin only and coupled spin-lattice dynamics are first reviewed, including a detailed account…
We present the symplectic algorithm in the Lagrangian formalism for the Hamiltonian systems by virtue of the noncommutative differential calculus with respect to the discrete time and the Euler--Lagrange cohomological concepts. We also show…
This letter studies symmetric and symplectic exponential integrators when applied to numerically computing nonlinear Hamiltonian systems. We first establish the symmetry and symplecticity conditions of exponential integrators and then show…
Accelerated gradient methods have had significant impact in machine learning -- in particular the theoretical side of machine learning -- due to their ability to achieve oracle lower bounds. But their heuristic construction has hindered…
Geometric numerical integration has recently been exploited to design symplectic accelerated optimization algorithms by simulating the Lagrangian and Hamiltonian systems from the variational framework introduced in Wibisono et al. In this…
In this paper, a symplectic model reduction technique, proper symplectic decomposition (PSD) with symplectic Galerkin projection, is proposed to save the computational cost for the simplification of large-scale Hamiltonian systems while…
Hamiltonian systems are one of the most important class of dynamical systems with a geometric structure called symplecticity and the numerical algorithms which can preserve such geometric structure are of interest. In this article we study…
Dissipation and irreversibility are central to most physical processes, yet they lead to non-unitary dynamics that are challenging to realise on quantum processors. High-order operator splitting is an attractive approach for simulating…
Multiphysics problems involving two or more coupled physical phenomena are ubiquitous in science and engineering. This work develops a new partitioned exponential approach for the time integration of multiphysics problems. After a possible…