To Örebro University

The basis of our approach is a simple second-order damped oscillating dynamical system (a damped mass-spring system with some externalforce). By changing the damping and the force the solution will fast approach a stationary solution with minimal energy.

For an optimization problem with constraints we generalize the idea such that the system converges to a stationary solution which is a local minima, most often aiming for the global minima. By using an additional damped system for the constraints, we obtain explicit expressions for the Lagrange parameters. 

The dynamical system is solved by a symplectic algorithm, that is normally especially tailored for conservative systems.

Our method is easy to implement, either with coding or with software for dynamical systems and can be applied to a large number of problems.

Numerical experiments show good convergence properties in general, that can be tuned by parameters for the damping and force of the oscillator. In particular, the dynamical constraints make the iterations less sensitive to initial conditions, which do not have to fulfill the constraints.

Calculations of eigenvalue problems and ground states of non-linear Schrödinger equations (NLSE) will be presented. We have worked on multi-component nonlinear Schrödinger equations in arbitrary dimensions. Following the experimental developments of quantum droplets of ultra-cold atoms, we are now modeling other forms of nonlinearities. This opens for future research in non-convex optimization, and calculations of higher eigenvalues to non-linear equations.