Faustino Maestre

Optimal design under the one-dimensional wave equation

An optimal design problem governed by the wave equation is examined in detail. Specifically, we seek the time-dependent optimal layout of two isotropic materials on a 1-d domain by minimizing a functional depending quadratically on the gradient of the state with coefficients that may depend on space, time and design. As it is typical in this kind of problems, they are ill-posed in the sense that there is not an optimal design. We therefore examine relaxation by using the representation of two-dimensional ((x, t) ∈ IR) divergence free vector fields as rotated gradients. By means of gradient Young measures, we transform the original optimal design problem into a non-convex vector variational problem, for which we can compute an explicit form of the “constrained quasiconvexification ” of the cost density. Moreover, this quasiconvexification is recovered by first or second-order laminates which give us the optimal microstructure at every point. Finally, we analyze the relaxed problem and some numerical experiments are performed. The perspective is similar to the one developed in previous papers for linear elliptic state equations. The novelty here lies in the state equation (the wave equation), and our contribution consists in understanding the differences with respect to elliptic cases.

Homogenization and correctors for the wave equation with periodic coefficients

Using the two-scale convergence method, we study the asymptotic behavior of a wave problem in ℝN with periodic coefficients in the space variable and almost-periodic coefficients in the time one. We obtain a nonlocal corrector and show how this implies that the limit problem is nonlocal in general.

On some inverse problems arising in elastography

This paper deals with some inverse problems for linear N-dimensional wave equations with origin in elastography where we try to identify a coefficient from some extra information on (a part of) the boundary. First, we assume that the total variation of the coefficient is a priori bounded. We reformulate the problem as the minimization of an appropriate function in an appropriate constraint set. We prove that this extremal problem possesses at least one solution, first in the one-dimensional case and then, with the help of some regularity results, in the general case, when N ? 2. In the final section, we consider a related (but different) one-dimensional problem, for which the total variation of the coefficient is not bounded a priori. Using some ideas from Pedregal (2005 ESAIM-COCV 15 357?81) and Maestre et al (2008 Interfaces Free Boundaries 10 87?117), we introduce an equivalent variational formulation. Then, we identify a relaxed problem whose solutions can be viewed as Young measures associated with minimizing sequences.

Optimal Potentials for Problems with Changing Sign Data

In this paper, we consider variational optimal control problems. The state equation is an elliptic partial differential equation of a Schrödinger type, governed by the Laplace operator with a potential, with a right-hand side that may change sign. The control variable is the potential itself that may vary in a suitable admissible class of nonnegative potentials. The cost is an integral functional, linear (but non-monotone) with respect to the state function. We prove the existence of optimal potentials, and we provide some necessary conditions for optimality. Several numerical simulations are shown.

An inverse problem in elastography involving Lamé systems

Abstract This paper deals with some inverse problems for the linear elasticity system with origin in elastography: we try to identify the material coefficients from some extra information on (a part of) the boundary. In our main result, we assume that the total variation of the coefficient matrix is a priori bounded. We reformulate the problem as the minimization of a function in an appropriate constraint set. We prove that this extremal problem possesses at least one solution with the help of some regularity results. Two crucial ingredients are a Meyers-like theorem that holds in the context of linear elasticity and a nonlinear interpolation result by Luc Tartar. We also perform some numerical experiments that provide satisfactory results. To this end, we apply the Augmented Lagrangian algorithm, completed with a limited-memory BFGS subalgorithm. Finally, on the basis of these experiments, we illustrate the influence of the starting guess and the errors in the data on the behavior of the iterates.

A Corrector Result for the Wave Equation with High Oscillating Periodic Coefficients

In the homogenization of a wave problem with oscillating coefficients in the diffusion term it is well known that the corresponding limit equation has the same structure with a diffusion term which agrees with the elliptic homogenized limit. Thus one can think that the oscillations of the solution of the wave equation are similar to the ones of the corresponding elliptic problem and then that the corrector for the elliptic problem is still a corrector for the wave problem. However in a paper by Brahim-Otsmane, Francfort and Murat, 1992, it was proved that this only holds if the initial data are “well posed”. In general, it is necessary to add to the elliptic corrector another term depending on the initial data. In this paper we obtain this term in the case of a wave problem posed in \(\mathbb{R}^{N}\) with periodic coefficients. This term is obtained using the two-scale convergence theory. It oscillates periodically in the space variable but almost periodically in the time one.