A functional strategy for nonlinear functionals

Abstract

The linear functional strategy introduced by the first a uthor in 1986 provided a s hift in the way inverse problems were solved. It is based on the fact that for applications one is interested in specific properties of the solution of an inverse problem. These properties or quantities of interest are usually obtained by applying a functional to the solution of the inverse problem. The linear functional strategy avoided the need to solve the full inverse problem by solving the adjoint problem for the functional instead. The solution to this adjoint problem is a functional which, when applied to the data returns the quantity of interest. In some cases, the adjoint problem can be solved exactly. In any case, the adjoint problem does not need to deal explicitly with data errors. In this paper we review the original approach. It is noted that any method which is able to produce an approximation to the solution of the adjoint problem which is continuous leads to a linear dependence of the error in the quantity of interest with respect to the data error. Most of the paper considers the application of advances in computational and applied mathematics in the last 30 years to the functional strategy. We define a general (nonlinear) functional strategy and illustrate how this problem is solved. We define a generalised adjoint problem for nonlinear functionals and inverse problems. This adjoint problem is shown to be linear. Furthermore, we observe that nonlinear functionals which are Lipschitz continuous are stable with respect to data errors. The solution of the adjoint problem constrained to Lipschitz continuous functionals leads to Tikhonov regularisation. We indicate how to implement the functional strategy for a simple example and provide links to modern functional analysis.