The Dynamical Functional Particle Method

TL;DR

The paper introduces the Dynamical Functional Particle Method (DFPM), an efficient algorithm for solving equations by formulating a damped dynamical system with exponential convergence, applicable to nonlinear problems and eigenvalue problems.

Contribution

It presents the DFPM, a novel dynamical system-based algorithm that leverages symplectic integrators and offers exponential convergence for solving equations, including nonlinear and eigenvalue problems.

Findings

DFPM has exponential convergence rate.

Computational complexity for symmetric eigenvalue problems is $ ext{O}(N^{(d+1)/d})$.

DFPM compares favorably with ARPACK, LAPACK, and other methods.

Abstract

We present a new algorithm which is named the Dynamical Functional Particle Method, DFPM. It is based on the idea of formulating a finite dimensional damped dynamical system whose stationary points are the solution to the original equations. The resulting Hamiltonian dynamical system makes it possible to apply efficient symplectic integrators. Other attractive properties of DFPM are that it has an exponential convergence rate, automatically includes a sparse formulation and in many cases can solve nonlinear problems without any special treatment. We study the convergence and convergence rate of DFPM. It is shown that for the discretized symmetric eigenvalue problems the computational complexity is given by $\mathcal{O}(N^{(d+1)/{d}})$, where \emph{d} is the dimension of the problem and \emph{N} is the vector size. An illustrative example of this is made for the 2-dimensional Schr\"odinger equation. Comparisons are made with the standard numerical libraries ARPACK and LAPACK. The conjugated gradient method and shifted power method are tested as well. It is concluded that DFPM is both versatile and efficient.