Damped Dynamical Systems for Solving Equations and Optimization Problems

• 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.

Ämnesord

NATURVETENSKAP  -- Matematik -- Beräkningsmatematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Computational Mathematics (hsv//eng)

Nyckelord

Optimization

damped dynamical systems

convex problems

eigenvalue problems

image analysis

inverse problems

quantum physics

Schrödinger equation

Mathematics

Matematik

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