Review of the Merits of the Stochastic Dynamic Equations and the Monte Carlo Approach in Modeling and Understanding Systems (Invited Presentation)

Review of the Merits of the Stochastic Dynamic Equations and the Monte Carlo Approach in Modeling and Understanding Systems

Rex J. Fleming

Global Aerospace, LLC

The Monte Carlo (MC) approach uses a randomly chosen set of points, usually obtained from a random Gaussian distribution, to evaluate the impact of uncertainty in initial conditions for the time integration of a physical system. The Stochastic Dynamic Equation set (SDE) approach begins with an infinite ensemble of initial states represented simply by the variance of the initial conditions. The MC is thus an approximation of the SDE method. However, the SDE has a closure issue. Time derivatives of second moments in the system of equations also require knowledge of third moments on the right hand side (rhs) of the equations. Time derivatives of third moments involve third and fourth moments on the rhs, and so on.

For large model applications such as weather and climate models, the MC method is the only practical approach. The proper programming of Monte Carlo models can make maximum use of the current multi-processor computers. The SDE method, introduced to the meteorological community by Epstein in 1969, allows a perfect blend of true physics and mathematical statistics. The SDE method can quantify the evolution of uncertainty in a physical system in terms of the energetics of that physical system and the ‘uncertain energy' of error growth in the system. The advantages and merits of both methods are shown in simple but effective examples.

There is an advantage to using both methods to arrive at fundamental conclusions. There are examples for some systems of nonlinear equations where the Monte Carlo method and the SDE approach can be used together to produce a more complete picture of the nonlinearity of the system. This leads to a major advantage for the SDE method that is often overlooked. It will be discussed along with the examples.