A Spectral Projected Gradient Algorithm for an Elliptic Inverse Coefficient Problem with a Robin Boundary Condition

Source abstract

An inverse problem of recovering a lower-order coefficient in a second-order elliptic equation with a Robin boundary condition is considered. Uniqueness of the inverse problem solution is established. For the numerical solution of the inverse problem, a difference scheme is constructed by the integro-interpolation method, and a discrete residual functional for the additional boundary data is introduced. A mesh-independent stability estimate is established for the discrete forward problem. The gradient of the functional is computed using the solution of an adjoint problem. To accelerate the iterative process, a spectral projected gradient method with a Barzilai–Borwein parameter and a nonmonotone line search is employed. Model boundary data are computed on a finer grid than the grid used to solve the inverse problem, and the stopping iteration is determined by the discrepancy principle, taking into account the error in the input data and the difference between the boundary values of the solutions computed on the two grids. The numerical experiments compare the proposed method with the fixed-step gradient method and investigate the effect of input-data errors on the recovery of a smooth coefficient and a coefficient defined by a continuous piecewise-linear function. The results demonstrate a substantial acceleration of the iterative process and the possibility of recovering the considered coefficients from perturbed model data.