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Quantum-classical molecular dynamics, as a partial classical limit of the full quantum Schrodinger equation, is a widely used framework for quantum molecular dynamics. The underlying equations are nonlinear in nature, containing a quantum part (represents the electrons) and a classical part (stands for the nuclei). An accurate simulation of the wave function typically requires a time step comparable to the rescaled Planck constant $h$, resulting in a formidable cost when $hll 1$. We prove an additive observable error bound of Schwartz observables for the proposed time-splitting schemes based on semiclassical analysis, which decreases as $h$ becomes smaller. Furthermore, we establish a uniform-in-$h$ observable error bound, which allows an $mathcal{O}(1)$ time step to accurately capture the physical observable regardless of the size of $h$. Numerical results verify our estimates.
The extended Lagrangian molecular dynamics (XLMD) method provides a useful framework for reducing the computational cost of a class of molecular dynamics simulations with constrained latent variables. The XLMD method relaxes the constraints by introd
Dynamical spectral estimation is a well-established numerical approach for estimating eigenvalues and eigenfunctions of the Markov transition operator from trajectory data. Although the approach has been widely applied in biomolecular simulations, it
We analyze the Lanczos method for matrix function approximation (Lanczos-FA), an iterative algorithm for computing $f(mathbf{A}) mathbf{b}$ when $mathbf{A}$ is a Hermitian matrix and $mathbf{b}$ is a given mathbftor. Assuming that $f : mathbb{C} righ
Due to their importance in both data analysis and numerical algorithms, low rank approximations have recently been widely studied. They enable the handling of very large matrices. Tight error bounds for the computationally efficient Gaussian eliminat
We analyze the qualitative properties and the order of convergence of a splitting scheme for a class of nonlinear stochastic Schrodinger equations driven by additive It^o noise. The class of nonlinearities of interest includes nonlocal interaction cu