Computational Methods for Large Distributed Parameter Estimation Problems in 3D

Summary This paper considers problems of distributed parameter estimation from data measurements on solutions of diffusive partial differential equations (PDEs). A nonlinear 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. We develop methods for the resulting constrained optimization problem. The method directly addresses the discretized PDE system which defines a critical point of the Lagrangian. This system is strongly coupled when the regularization parameter is small. We then apply such methods for electromagnetic data inversion in 3D, both in frequency and in time domains. This involves developing appropriate discretizations for the forward problems as well as handling very large systems of algebraic equations. Finally, we explore the use of the so-called Huber's norm for the recovery of piecewise smooth model functions. Since our differential operators are compact, there are many different models that give rise to fields that fit practical data sets well.