Gabriel Turinici

Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced basis approximations, Galerkin projection onto a space W(sub N) spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation, relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures, methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage, in which, given a new parameter value, we calculate the output of interest and associated error bound, depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.

A Priori Convergence Theory for Reduced-Basis Approximations of Single-Parameter Elliptic Partial Differential Equations

We consider “Lagrangian” reduced-basis methods for single-parameter symmetric coercive elliptic partial differential equations. We show that, for a logarithmic-(quasi-)uniform distribution of sample points, the reduced–basis approximation converges exponentially to the exact solution uniformly in parameter space. Furthermore, the convergence rate depends only weakly on the continuity-coercivity ratio of the operator: thus very low-dimensional approximations yield accurate solutions even for very wide parametric ranges. Numerical tests (reported elsewhere) corroborate the theoretical predictions.

Lyapunov control of bilinear Schrödinger equations

A Lyapunov-based approach for trajectory tracking of the Schrodinger equation is proposed. In the finite dimensional case, convergence is precisely analyzed. Connection between the controllability of the linearized system around the reference trajectory and asymptotic tracking is studied. When the linearized system is controllable, such a feedback ensures almost global asymptotic convergence. When the linearized system is not controllable, the stability of the closed-loop system is not asymptotic. To overcome such lack of convergence, we propose, when the reference trajectory is an eigenstate, a modification based on adiabatic invariance. Simulations illustrate the simplicity and also the interest for trajectory generation.

New formulations of monotonically convergent quantum control algorithms

Most of the numerical simulation in quantum (bilinear) control have used one of the monotonically convergent algorithms of Krotov (introduced by Tannor et al.) or of Zhu and Rabitz. However, until now no explicit relationship has been revealed between the two algorithms in order to understand their common properties. Within this framework, we propose in this paper a unified formulation that comprises both algorithms and that extends to a new class of monotonically convergent algorithms. Numerical results show that the newly derived algorithms behave as well as (and sometimes better than) the well-known algorithms cited above.

Quantum wavefunction controllability

Abstract Theoretical results are presented on the ability to arbitrarily steer about a wavefunction for a quantum system under time-dependent external field control. Criteria on the field free Hamiltonian and the field coupling term in the Hamiltonian are presented that assure full wavefunction controllability. Numerical simulations are given to illustrate the criteria. A discussion on the theoretical and practical relationship between dynamical conservation laws and controllability is also included.

COMPUTATION OF MEAN FIELD EQUILIBRIA IN ECONOMICS

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.

A parareal in time procedure for the control of partial differential equations

Abstract We have proposed in a previous note a time discretization for partial differential evolution equation that allows for parallel implementations. This scheme is here reinterpreted as a preconditioning procedure on an algebraic setting of the time discretization. This allows for extending the parallel methodology to the problem of optimal control for partial differential equations. We report a first numerical implementation that reveals a large interest. To cite this article: Y. Maday, G. Turinici, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 387–392.

On the controllability of bilinear quantum systems

We present in this paper controllability results for quantum systems interacting with lasers. A negative result for infinite dimensional spaces serves as a starting point for a finite dimensional analysis. We show that under physically reasonable hypothesis in such systems we can control the population of the eigenstates. Applications are given for a five-level system.

The Parareal in Time Iterative Solver: a Further Direction to Parallel Implementation

This paper is the basic one of the series resulting from the minisymposium entitled “Recent Advances for the Parareal in Time Algorithm” that was held at DD15. The parareal in time algorithm is presented in its current version (predictor-corrector) and the combination of this new algorithm with other more classical iterative solvers for parallelization which makes it possible to really consider the time direction as fertile ground to reduce the time integration costs.

Monotonic Parareal Control for Quantum Systems

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.