Damped Dynamical Systems for Solving Equations and Optimization Problems

Mårten Gulliksson (), Magnus Ögren, Anna Oleynik and Ye Zhang
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Mårten Gulliksson: School of Engineering and Technology, Mathematics
Magnus Ögren: School of Engineering and Technology, Mathematics
Anna Oleynik: University of Bergen, Department of Mathematics
Ye Zhang: Chemnitz University of Technology, Faculty of Mathematics

Chapter 82 in Handbook of the Mathematics of the Arts and Sciences, 2021, pp 2171-2215 from Springer

Abstract: Abstract We present an approach for solving optimization problems with or without constrains which we call Dynamical Functional Particle Method (DFMP). The method consists of formulating the optimization problem as a second order damped dynamical system and then applying symplectic method to solve it numerically. In the first part of the chapter, we give an overview of the method and provide necessary mathematical background. We show that DFPM is a stable, efficient, and given the optimal choice of parameters, competitive method. Optimal parameters are derived for linear systems of equations, linear least squares, and linear eigenvalue problems. A framework for solving nonlinear problems is developed and numerically tested. In the second part, we adopt the method to several important applications such as image analysis, inverse problems for partial differential equations, and quantum physics. At the end, we present open problems and share some ideas of future work on generalized (nonlinear) eigenvalue problems, handling constraints with reflection, global optimization, and nonlinear ill-posed problems.

Keywords: Optimization; Damped dynamical systems; Convex problems; Eigenvalue problems; Image analysis; Inverse problems; Quantum physics; Schrödinger equation (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/978-3-319-57072-3_32

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