4,946 research outputs found
Continuation Newton methods with deflation techniques and quasi-genetic evolution for global optimization problems
The global minimum point of an optimization problem is of interest in
engineering fields and it is difficult to be solved, especially for a nonconvex
large-scale optimization problem. In this article, we consider a new memetic
algorithm for this problem. That is to say, we use the continuation Newton
method with the deflation technique to find multiple stationary points of the
objective function and use those found stationary points as the initial seeds
of the evolutionary algorithm, other than the random initial seeds of the known
evolutionary algorithms. Meanwhile, in order to retain the usability of the
derivative-free method and the fast convergence of the gradient-based method,
we use the automatic differentiation technique to compute the gradient and
replace the Hessian matrix with its finite difference approximation. According
to our numerical experiments, this new algorithm works well for unconstrained
optimization problems and finds their global minima efficiently, in comparison
to the other representative global optimization methods such as the multi-start
methods (the built-in subroutine GlobalSearch.m of MATLAB R2021b, GLODS and
VRBBO), the branch-and-bound method (Couenne, a state-of-the-art open-source
solver for mixed integer nonlinear programming problems), and the
derivative-free algorithms (CMA-ES and MCS).Comment: arXiv admin note: substantial text overlap with arXiv:2103.0582
Relative periodic orbits in transitional pipe flow
A dynamical system description of the transition process in shear flows with
no linear instability starts with a knowledge of exact coherent solutions,
among them travelling waves (TWs) and relative periodic orbits (RPOs). We
describe a numerical method to find such solutions in pipe flow and apply it in
the vicinity of a Hopf bifurcation from a TW which looks to be especially
relevant for transition. The dominant structural feature of the RPO solution is
the presence of weakly modulated streaks. This RPO, like the TW from which it
bifurcates, sits on the laminar-turbulent boundary separating initial
conditions which lead to turbulence from those which immediately relaminarise
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