Eric Touboul

Making EGO and CMA-ES Complementary for Global Optimization

The global optimization of expensive-to-calculate continuous functions is of great practical importance in engineering. Among the proposed algorithms for solving such problems, Efficient Global Optimization (EGO) and Covariance Matrix Adaptation Evolution Strategy (CMA-ES) are regarded as two state-of-the-art unconstrained continuous optimization algorithms. Their underlying principles and performances are different, yet complementary: EGO fills the design space in an order controlled by a Gaussian process (GP) conditioned by the objective function while CMA-ES learns and samples multi-normal laws in the space of design variables. This paper proposes a new algorithm, called EGO-CMA, which combines EGO and CMA-ES. In EGO-CMA, the EGO search is interrupted early and followed by a CMA-ES search whose starting point, initial step size and covariance matrix are calculated from the already sampled points and the associated conditional GP. EGO-CMA improves the performance of both EGO and CMA-ES in our 2 to 10 dimensional experiments.

Population balances in case of crossing characteristic curves: Application to T-cells immune response

The progression of a cell population where each individual is characterized by the value of an internal variable varying with time (e.g. size, weight, and protein concentration) is typically modeled by a Population Balance Equation, a first order linear hyperbolic partial differential equation. The characteristics described by internal variables usually vary monotonically with the passage of time. A particular difficulty appears when the characteristic curves exhibit different slopes from each other and therefore cross each other at certain times. In particular such crossing phenomenon occurs during T-cells immune response when the concentrations of protein expressions depend upon each other and also when some global protein (e.g. Interleukin signals) is also involved which is shared by all T-cells. At these crossing points, the linear advection equation is not possible by using the classical way of hyperbolic conservation laws. Therefore, a new Transport Method is introduced in this article which allowed us to find the population density function for such processes. The newly developed Transport method (TM) is shown to work in the case of crossing and to provide a smooth solution at the crossing points in contrast to the classical PDF techniques.

Modelling of strongly coupled particle growth and aggregation

The mathematical modelling of the dynamics of particle suspension is based on the population balance equation (PBE). PBE is an integro-differential equation for the population density that is a function of time t, space coordinates and internal parameters. Usually, the particle is characterized by a unique parameter, e.g. the matter volume v. PBE consists of several terms: for instance, the growth rate and the aggregation rate. So, the growth rate is a function of v and t. In classical modelling, the growth and the aggregation are independently considered, i.e. they are not coupled. However, current applications occur where the growth and the aggregation are coupled, i.e. the change of the particle volume with time is depending on its initial value v0, that in turn is related to an aggregation event. As a consequence, the dynamics of the suspension does not obey the classical Von Smoluchowski equation. This paper revisits this problem by proposing a new modelling by using a bivariate PBE (with two internal variables: v and v0) and by solving the PBE by means of a numerical method and Monte Carlo simulations. This is applied to a physicochemical system with a simple growth law and a constant aggregation kernel.