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Register a new userObservable Error Bounds of the Time-splitting Scheme for Quantum-Classical Molecular Dynamics
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Added by
Albert Tres
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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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.
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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, its error properties remain poorly understood. Here we analyze the error of a dynamical spectral estimation method called the variational approach to conformational dynamics (VAC). We bound the approximation error and estimation error for VAC estimates. Our analysis establishes VACs convergence properties and suggests new strategies for tuning VAC to improve accuracy.
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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 elimination based methods (skeleton approximations) are available. In practice, these bounds are useful for matrices with singular values which decrease quickly. Using the Chebyshev norm, this paper provides improved bounds for the errors of the matrix elements. These bounds are substantially better in the practically relevant cases where the eigenvalues decrease polynomially. Results are proven for general real rectangular matrices. Even stronger bounds are obtained for symmetric positive definite matrices. A simple example is given, comparing these new bounds to earlier ones.
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 cubic nonlinearities. We show that the numerical solution is symplectic and preserves the expected mass for all times. On top of that, for the convergence analysis, some exponential moment bounds for the exact and numerical solutions are proved. This enables us to provide strong orders of convergence as well as orders of convergence in probability and almost surely. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme.
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