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Topological derivatives for semilinear elliptic equations

Iguernane M., Nazarov S. A., Roche J. R., Sokolowski J., Szulc K.

International Journal of Applied Mathematics and Computer Science

2009

Vol. 19, no 2

191-205

The form of topological derivatives for an integral shape functional is derived for a class of semilinear elliptic equations. The convergence of finite element approximation for the topological derivatives is shown and the error estimates in the L [...] norm are obtained. The results of numerical experiments which confirm the theoretical convergence rate are presented.

Adaptive Newton-like method for shape optimization

Roche J. R.

Control and Cybernetics

2005

Vol. 34, no 1

363-377

The aim of this work is to introduce an adaptive strategy to monitor the rate of convergence of a Newton-like method in numerical shape optimization. Such superlinear iterative algorithms are often computationally intensive and the rate of convergence depends on how accurate the numerical solution of the state equation is. The model concerns a cost function depending on the unknown domain [Omega], and the solution of an integral equation.

Sojourn time of some reflected Brownian motion in the unit disk

Deaconu M., Gradinaru M., Roche J. R.

Probability and Mathematical Statistics

2000

Vol. 20, Fasc. 1

19--38

We study the heat diffusion in a domain with an obstacle inside. More precisely, we are interested in the quantity of heat in so far as a function of the position of the heat source at time 0. This quantity is also equal to the expectation of the sojourn time of the Brownian motion, reflected on the boundary of a small disk contained in the unit disk, and killed on the unit circle. We give the explicit expression of this expectation. This allows us to make some numerical estimates and thus to illustrate the behaviour of this expectation as a function of starting point of the Brownian motion.