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Damped Dynamical Systems for Solving Equations and Optimization Problems

  • Mårten GullikssonEmail author
  • Magnus Ögren
  • Anna Oleynik
  • Ye Zhang
Living reference work entry

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 

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Mårten Gulliksson
    • 1
    Email author
  • Magnus Ögren
    • 1
  • Anna Oleynik
    • 2
  • Ye Zhang
    • 3
  1. 1.MathematicsSchool of Engineering and TechnologyÖrebroSweden
  2. 2.Department of MathematicsUniversity of BergenBergenNorway
  3. 3.Faculty of MathematicsChemnitz University of TechnologyChemnitzGermany

Section editors and affiliations

  • Torsten Lindström
  • Bharath Sriraman
    • 1
  1. 1.Department of Mathematical SciencesThe University of MontanaMissoulaUSA

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