Tuesday, 11 February 2003

Genetic Algorithms in Geophysical Fluid Dynamics (Formerly Paper number 1.1)

The genetic algorithm (GA) is finding wide acceptance in a variety of disciplines. In this paper, its application in geophysical fluid dynamics will be discussed. In particular, we will demonstrate how the GA can be formulated to solve several types of difficult numerical problems.
First, we will look at solutions to nonlinear boundary value problems in fluid dynamics. The key is to formulate the objective, or cost function that the GA uses to determine the state of the optimization as the solution of the partial differential equation. A variety of ways of doing this will be explored. The procedure consists of first expanding the independent variables in terms of orthogonal basis functions, performing a Galerkin projection, and then requiring that the result be minimized over a given set of points. We show the impact of including more basis functions and minimizing over a larger set of points. We explore the usual types of basis functions (such as Fourier series) as well as using Empirical Orthogonal Functions to do the expansion. We demonstrate that this procedure can be applied to a wide range of problems. At the lowest order, we present results of finding cnoidal waves (solitary waves repeated periodically) of the Fifth Order Korteweg de Vries equation. Not only did we find the single cnoidal wave, but we were also able to find solutions corresponding to double and triple cnoidal waves (two and three waves on each period respectively).
A second use of GAs is to solve for solutions of nonlinear inversion problems. For example, empirical models have gained popularity in recent years as an alternative to the more traditional dynamical models. Linear empirical models are easy to produce from data using standard least squares inversion techniques. However, nonlinear models are more difficult to devise due to the introduction of high order tensors to the problems. GAs can resolve this issue through redefining the problem in terms of optimization and directly searching for the propagator matrix given the data. This technique is demonstrated for matching several low order nonlinear systems, such as a spiral curve and the Lorenz equations. We are able to find valid solutions to a nonlinear model that could not be accomplished with the linearized model.

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