Processus de diffusion: Outils de modélisation, de prévision et de contrôle

Diffusion processes defined by systems of stochastic differential equations are considered to model, forecast and control different physical phenomena.

We begin by generalizing optimal control problems for wear models of a machine by considering a performance criterion that takes the risk sensitivity of the optimizer into account. The optimal control is obtained for two and three-dimensional models from a mathematical expectation for a related uncontrolled process. Explicit solutions are presented.

Next, in order to forecast drainage basin runoff, mathematical models involving diffusion processes are tested against hydrological data obtained from the hydrographic basin of the Saguenay-Lac-St-Jean, located in northeastern Quebec. An integrated Ornstein-Uhlenbeck process is found to give better results than a deterministic model presently in use for one-day ahead estimates.

Finally, we investigate the validity of a new structure for a single neuron, that will eventually be used in multilayer neural networks to perform nonlinear pattern recognition. This new architecture is inspired by biological assumptions involving diffusion processes. It is clearly established that only six parameters are sufficient to solve the XOR problem.