Logarithmic stability in determination of a 3D viscoelastic coefficient and a numerical example

Maya de Buhan (1, 2) , Axel Osses (2)

1 LJLL - Laboratoire Jacques-Louis Lions
2 CMM - Center for Mathematical Modeling

We prove a Carleman estimate and a logarithmic stability estimate for an inverse problem in three dimensional viscoelasticity. More precisely, we obtain logarithmic stability for the inverse problem of recovering the spatial part of a viscoelastic coefficient of the form p(x)h(t) from a unique measurement on an arbitrary part of the boundary. The main assumptions are: h (0) = 0, h(0) = 0, p is known in a neighborhood of the boundary and regularity and sensitivity of the reference trajectory. We propose a method to solve the problem numerically and illustrate the theoretical result by a numerical example.

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]

Maya de Buhan, Axel Osses. Logarithmic stability in determination of a 3D viscoelastic coefficient and a numerical example. Inverse Problems, 2010, 26, pp.95006. ⟨10.1088/0266-5611/26/9/095006⟩. ⟨hal-01116979⟩