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The direct problem of water-wave equations is the problem of determining the surface and its velocity potential, in time T > 0, for a given initial profile and velocity potential, where the profile of the bottom, the bathymetry, is known. In this paper, we study the inverse problem of recovering the shape of the solid bottom boundary of an inviscid, irrotational, incompressible fluid from measurements of a portion of the free surface. In particular, given the water-wave height and its velocity potential on an open set, together with the first time derivative of the free surface, on a single time, we address the identifiability problem. Moreover we compute the derivatives with respect to the shape of the bottom, which allows us to obtain the optimality conditions for this inverse problem.

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In this article, we study the inverse problem of recovering a space-dependent coefficient of the Moore–Gibson–Thompson (MGT) equation from knowledge of the trace of the solution on some open subset of the boundary. We obtain the Lipschitz stability for this inverse problem, and we design a convergent algorithm for the reconstruction of the unknown coefficient. The techniques used are based on Carleman inequalities for wave equations and properties of the MGT equation.

Abstract. We analyze a model optimal control problem for a 2D scalar conservation law: The so-called inverse design problem, the goal being to identify the initial datum leading to a given final time configuration. The presence of shocks is an impediment for classical methods, based on linearization, to be directly applied. We develop an alternating descent method that exploits the generalized linearization that takes into account both the sensitivity of the shock location and of the smooth components of solutions. A numerical implementation is proposed using splitting and finite differences. The descent method we propose is of alternating nature and combines variations taking account of the shock location and those that take care of the smooth components of the solution. The efficiency of the method is illustrated by numerical experiments.