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This article is concerned with the numerical solution of multiobjective control problems associated with nonlinear partial differential equations and more precisely the Burgers equation. For this kind of problems, we look for the Nash equilibrium, which is the solution to a noncooperative game. To compute the solution of the problem, we use a combination of finite-difference methods for the time discretization, finite-element methods for the space discretization, and a quasi-Newton BFGS algorithm for the iterative solution of the discrete control problem. Finally, we apply the above methodology to the solution of several tests problems. To be able to compare our results with existing results in the literature, we discuss first a single-objective control problem, already investigated by other authors. Finally, we discuss the multiobjective case.

This article is concerned with the numerical solution of multiobjective control problems associated with linear partial differential equations. More precisely, for such problems, we look for the Nash equilibrium, which is the solution to a noncooperative game. First, we study the continuous case. Then, to compute the solution of the problem, we combine finite-difference methods for the time discretization, finite-element methods for the space discretization, and conjugate-gradient algorithms for the iterative solution of the discrete control problems. Finally, we apply the above methodology to the solution of several tests problems.

In these few pages, we have attempt to describe the main trenbds of our multigrid research program. The adaptative MG method is already an efficient 2-D solver, which in several cases is twice as faster as an implicit time-stepping previously derived [12]. The agglomerating MG method is still in an experimental phase but seems very promising.

The problem of finding an optimal, smooth, convex, blunt body shape having minimum integral convective heat flux for 2D laminar supersonic viscous gas flow for large Reynolds number under condition, that the equilibrium temperature of the body surface does not exceed the preassigned limit value, is considered. This heat-flux optimal-shape-design problem is solved by means of the asynchrone parallel floating-point genetic algorithms via calculation of the heat-flux distribution along the body surface in the framework of 2D boundary layer theory. As a test task, the inverse problem that consists in finding the body shape that realizes target heat-flux distribution for given flow condition was considered. The influence of the number of processors on the algorithm properties was investigated.

The following sections contain the different contributions from the ECARP consortium to the resolution of the workshop test cases defined in section II. Each contribution contains a theoretical description of the optimization technique used by each partner and the results obtained by using this techniques for the resolution of some test cases.

In this paper we address the problem of solving the compressible Navier Stokes equations. The development of space programs in Europe and the U.S. has generated renewed interest in aerodynamic ~ aerothermal design. Significant progress has been made recently in the Finite Element solutions of the Euler equations or their coupling with species convection equations for reactive flows and the difficulties associated mainly accurate and stable shock capturing capabilities are well known. When solving the Navier Stokes equations, additionnal features have to he computed: a precise representation of the houndary layer yielding pressure, skin friction and heat transfer coefficients is required along with recirculation, vortex, wake and shock interaction structures. This is critical for a number of aerospace applications dealing with strongly separated flows, including the simulation of flight at high angle of attack or the evaluation of the efficiency of body flaps. We will outline two Finite Element methodologies coupled with mesh adaptation that are being developed to accurately capture those features. One is based on a Galerkin Least-square formulation which brings good stability properties and makes this approach especially attractive for high Mach number calculations (space applications). The other one relies on accurate centered space discretbation and makes use of compatible finite element approximation and is advocated for aerodynamic shape design at subsonic and transonic regimes(aircraft application). For both methods, the grid quality control is still a major open problem even in the context of triangular Finite Element unstructured meshes; in order to capture the essential features of the flow, the mesh has to take into account not only the geometry hut also he fine enough in those regions where the solution needs accuracy. The mesh adaption used in this paper relies on refinement and node movement procedures provided by physical or geometrical criteria. Numerical flow simulation around reference simplified 2D geometries and 3D industrial computations originating from aircraft aerospace applications are presented for the development and validation of the above methodologies.

In this paper we present a noncooperative game theoretical model for the well known problem of experimental design. A virtual player decides the design variables of an experiment and all the players solve a Nash equilibrium problem by optimizing suitable payoff functions. We consider the case where the design variables are the coordinates of n points in a region of the plane and we look for the optimal configuration of the points under some constraints. Arising from a concrete situation, concerning the ARGO-YBJ experiments, the goal is to find the optimal configuration of the detector, consisting of a single layer of resistive plate counters. Theoretical and computational results are presented for this location problem.

Interface boundary conditions are the key ingredient to design efficient domain decomposition methods. However, convergence cannot be obtained for any method in a number of iterations less than the number of subdomains minus one in the case of a one-way splitting. This optimal convergence can be obtained with generalized Robin type boundary conditions associated with an operator equal to the Schur complement of the outer domain. Since the Schur complement is too expensive to compute exactly, a new approach based on the computation of the exact Schur complement for a small patch around each interface node is presented for the two-Lagrange multiplier FETI method.