Olivier Pironneau

Frederic Alauzet, Olivier Pironneau

Adjoints are used in optimization to speed-up computations, simplify optimality conditions or compute sensitivities. Because time is reversed in adjoint equations with first order time derivatives, boundary conditions and transmission conditions through shocks can be difficult to understand. In this article we analyze the adjoint equations that arise in the context of compressible flows governed by the Euler equations of fluid dynamics. We show that the continuous adjoints and the discrete adjoints computed by automatic differentiation agree numerically; in particular the adjoint is found to be continuous at the shocks and usually discontinuous at contact discontinuities by both.

Hideo Kawarada, Olivier Pironneau

In industrial facilities and household equipments scale formation leads to reduced efficiency and damages. Therefore various devices for anti-scale have been designed for a long time. Recently one of them, an aggregation of ceramic spheres, was proposed to prevent scale formation and its efficiency shown experimentally. The purpose of this paper is to clarify the function of this device by proposing several mathematical models and by pursuing their mathematical and numerical analysis. The first model measures the effect on nucleation of calcite of the electric potential near the surface of a ceramic sphere in natural water. The second model is based on a crystal lattice model and argues that the surface tension energy of the calcite particles is reduced by the polarization energy brought by the ceramic balls. The third model is macroscopic and numerical and studies the effect of two ceramic balls arrangements in a flow of water containing calcite particles.

Mathieu Laurière, Gilles Pagès, Olivier Pironneau

Fishing quotas are unpleasant but efficient to control the productivity of a fishing site. A popular model has a stochastic differential equation for the biomass on which a stochastic dynamic programming or a Hamilton-Jacobi-Bellman algorithm can be used to find the stochastic control -- the fishing quota. We compare the solutions obtained by dynamic programming against those obtained with a neural network which preserves the Markov property of the solution. The method is extended to a similar multi species model to check its robustness in high dimension.

Sébastien Geeraert, Charles-Albert Lehalle, Barak Pearlmutter et al.

Automatic differentiation is involved for long in applied mathematics as an alternative to finite difference to improve the accuracy of numerical computation of derivatives. Each time a numerical minimization is involved, automatic differentiation can be used. In between formal derivation and standard numerical schemes, this approach is based on software solutions applying mechanically the chain rule to obtain an exact value for the desired derivative. It has a cost in memory and cpu consumption. For participants of financial markets (banks, insurances, financial intermediaries, etc), computing derivatives is needed to obtain the sensitivity of its exposure to well-defined potential market moves. It is a way to understand variations of their balance sheets in specific cases. Since the 2008 crisis, regulation demand to compute this kind of exposure to many different case, to be sure market participants are aware and ready to face a wide spectrum of configurations. This paper shows how automatic differentiation provides a partial answer to this recent explosion of computation to perform. One part of the answer is a straightforward application of Adjoint Algorithmic Differentiation (AAD), but it is not enough. Since financial sensitivities involves specific functions and mix differentiation with Monte-Carlo simulations, dedicated tools and associated theoretical results are needed. We give here short introductions to typical cases arising when one use AAD on financial markets.