An Explicit Update Scheme for Inverse Parameter and Interface Estimation of Piecewise Constant Coefficients in Linear Elliptic PDEs

We introduce a general and efficient method to recover piecewise constant coefficients occurring in elliptic partial differential equations as well as the interface where these coefficients have jump discontinuities. For this purpose, we use an output least squares approach with level set and augmented Lagrangian methods. Our formulation incorporates the inherent nature of the piecewise constant coefficients, which eliminates the need for a complicated nonlinear solve at every iteration. Instead, we obtain an explicit update formula and therefore vastly speed up computation. We employ our approach to the example problems of Poisson’s equation and linear elasticity and provide the combination of simultaneously recovering coefficients and interface.