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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.
The spectral deferred correction method is a variant of the deferred correction method for solving ordinary differential equations. A benefit of this method is that is uses low order schemes iteratively to produce a high order approximation. In this
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
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 (repre
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
Classical a posteriori error analysis for differential equations quantifies the error in a Quantity of Interest (QoI) which is represented as a bounded linear functional of the solution. In this work we consider a posteriori error estimates of a quan