Motivated by a mean field games stylized model for the choice of technologies (with externalities and economy of scale), we consider the associated optimization problem and prove an existence result. To complement the theoretical result, we introduce a monotonic algorithm to find the mean field equilibria. We close with some numerical results, including the multiplicity of equilibria describing the possibility of a technological transition.

International audienceReduced basis methods are an efficient tool for significantly reducing the computational complexity of solving parametrized partial differential equations. Originally introduced for elliptic equations, they have been generalized during the last decade to various types of elliptic, parabolic and hyperbolic systems. In this article, we extend the reduction technique to parametrized variational inequalities. Firstly, we propose a reduced basis variational inequality scheme in a saddle-point form and prove existence and uniqueness of the solution. We state some elementary analytical properties of the scheme such as reproduction of solutions, a-priori stability with respect to the data and Lipschitz- continuity with respect to the parameters. Secondly, we provide rigorous a-posteriori error bounds. An offline/online decomposition guarantees an efficient assembling of the reduced scheme, which can be solved by constrained quadratic programming. The reduction scheme is applied to a one-dimensional obstacle problem with a two-dimensional parameter space. The numerical results confirm the theoret- ical ones and demonstrate the efficiency of the reduction technique

We study the control by electromagnetic fields of molecular alignment and orientation, in a linear, rigid rotor model. With the help of a monotonically convergent algorithm, we find that the optimal field is in the microwave part of the spectrum and acts by resonantly exciting the rotation of the molecule progressively from the ground state, i.e., by rotational ladder climbing. This mechanism is present not only when maximizing orientation or alignment, but also when using prescribed target states that simultaneously optimize the efficiency of orientation/alignment and its duration. The extension of the optimization method to consider a finite rotational temperature is also presented.

Following encouraging experimental results in quantum control, numerical simulations have known significant improvements through the introduction of efficient optimization algorithms. Yet, the computational cost still prevents using these procedures for high-dimensional systems often present in quantum chemistry. Using parareal framework, we present here a time parallelization of these schemes which allows us to reduce significantly their computational cost while still finding convenient controls.