Abstract:

The recovery of coefficients and/or the right-hand side of partial differential equations arises in the mathematical modeling of applied problems across various areas of physics. In geophysics, for instance, the gravitational potential generated by a given mass can be determined by solving a boundary value problem of the Poisson equation. In this article, we consider an inverse gravimetric problem involving the reconstruction of the domain of a homogeneous object with known density. Additional information about anomalous gravity is taken near the Earth's surface. This problem leads to the recovery of a piecewise-constant right-hand side function for the Poisson equation. A Robin type boundary condition is applied within a truncated computational domain to better approximate the far-distant gravitational field. The numerical algorithm employs an auxiliary smooth function to analytically represent the unknown domain's boundary. The smoothness of this function is ensured by solving another boundary value problem for an elliptic equation. We perform an iterative procedure that minimizes the misfit between modeled and observed data using a gradient-based method. The capabilities of the algorithm are demonstrated with numerical results for two- and three-dimensional test problems.