Inversion of potential field data using the finite element method on parallel computers

• We introduce a method of inversion for gravity and magnetic inversion.
• Approach: ‘first optimize then discretize’.
• Requires solution of forward problem, adjoint state problem and evaluation preconditioner.
• Uses finite element method solvers potentially with different meshes for all three problems.
• Weakly scalable for large number of cores.

In this paper we present a formulation of the joint inversion of potential field anomaly data as an optimization problem with partial differential equation (PDE) constraints. The problem is solved using the iterative Broyden–Fletcher–Goldfarb–Shanno (BFGS) method with the Hessian operator of the regularization and cross-gradient component of the cost function as preconditioner. We will show that each iterative step requires the solution of several PDEs namely for the potential fields, for the adjoint defects and for the application of the preconditioner. In extension to the traditional discrete formulation the BFGS method is applied to continuous descriptions of the unknown physical properties in combination with an appropriate integral form of the dot product. The PDEs can easily be solved using standard conforming finite element methods (FEMs) with potentially different resolutions. For two examples we demonstrate that the number of PDE solutions required to reach a given tolerance in the BFGS iteration is controlled by weighting regularization and cross-gradient but is independent of the resolution of PDE discretization and that as a consequence the method is weakly scalable with the number of cells on parallel computers. We also show a comparison with the UBC–GIF GRAV3D code.