Stable determination of the nonlinear parameter in the non-diffusive Westervelt equation from the Dirichlet-to-Neumann map

Abstract

The Westervelt equation models the propagation of nonlinear acoustic waves in a regime well-suited for applications such as medical ultrasound imaging. In this work, we prove that the nonlinear parameter, as well as the sound speed, can be stably recovered from the Dirichlet-to-Neumann map associated with the non-diffusive Westervelt equation in (1+3)-dimensions. This result is essential for the feasibility of reconstruction methods. The Dirichlet-to-Neumann map encodes boundary measurements by associating a prescribed pressure profile on the boundary with the resulting pressure fluctuations. We prove stability provided the sound speed is a priori known to be close to a reference sound speed and under certain geometrical conditions. We also verify the result through numerical experiments.