We consider the problem of lconstructing a multidimensional vector function fln: "* from a finite set of linear measures. These can be irregularly sampled responses of several linear filters. Traditional approaches reconstruct in an a priori given space, e.g., the space of bandlimited functions. Instead, we have chosen to specify a reconstruction that is optimal in the sense of a quadratic plausibility criterion J. First, we plsent the solution of the generalized interpolation problem. Latel; we also consider the approximation plblem, and we show that both lead to the same class of solutions.

For the approximate representation of large data sets over a parameter domain in R², many geological and other applications require the construction of surfaces which have minimal energy, i.e., minimal curvature. One way to achieve this is by solving a fourth order elliptic partial differential equation. Its discretization by a difference scheme makes it particularly easy to incorporate (appropriate approximations of) known data points. In this paper, we investigate the performance of different solution methods for the resulting symmetric linear system of equations since this is the most CPU-demanding step in the scattered data approximation procedure. Specifically, we test first the performance of a preconditioned conjugate gradient method with an SSOR and an RILU preconditioner. However, since the partial differential operator does not contain mixed derivatives, using an alternating-direction-implicit scheme (ADI method) which has been employed successfully in the past for secon...