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

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