Grid refinement and scaling for distributed parameter estimation problems

This paper considers problems of distributed parameter estimation from data measurements on solutions of differential equations. A nonlinear least squares functional is minimized to approximately recover the sought parameter function (i.e. the model). This functional consists of a data fitting term, involving the solution of a finite volume or finite element discretization of the forward differential equation, and a Tikhonov-type regularization term, involving the discretization of a mix of model derivatives. The resulting nonlinear optimization problems can be very large and costly to solve. Thus, we seek ways to solve as much of the problem as possible on coarse grids. We propose to search for the regularization parameter first on a coarse grid. Then, a gradual refinement technique to find both the forward and inverse solutions on finer grids is developed. The grid spacing of the model discretization, as well as the relative weight of the entire regularization term, affect the sort of regularization achieved and the algorithm for gradual grid refinement. We thus investigate a number of questions which arise regarding their relationship, including the correct scaling of the regularization matrix. For nonuniform grids we rigorously associate the practice of using unscaled regularization matrices with approximations of a weighted regularization functional. We also discuss interpolation for grid refinement. Our results are demonstrated numerically using synthetic examples in one and three dimensions.