Harmonic Analysis of a Source Identification Problem

on a region Da UN with known potential 4> and permeability K. In the physical situation, K is determined in a laboratory test of a core sample and can be quite unrepresentative of the real permeability. The potential is more accurately measured as the height of a water column, but is also subject to some measurement error. Consequently, a direct calculation of / by performing the differentiations on the left of (1) could lead to large errors. The alternative method presented here is different from recent source identification methods used by Cannon and DuChateau [1] for parabolic equations in that hard-to-obtain interior data is used. On the other hand, our method treats more general kinds of source terms and leads to an explicit integral formula for the solution. This formula is used to show that the source / depends continuously on K (or (/>) in the Z? sense. This continuous dependence is based on a Fourier multiplier theorem and IF estimates of spherical maximal functions. It has little in common with the method of searching for a solution to an inverse problem in a compact subspace of naturally occurring functions, and/or using a regularization method (cf. Tikhonov [2]). The integral formula for / can be extended to represent the solution / of a corresponding random source identification problem. In this setup K, , and / are defined on D x Q where Q is a sample space with an associated probability measure. Statistical estimation techniques can be used to interpolate the given data for <£ and K in D and to generate pointwise means and variances. These statistical quantities can then be used with the results of this note to give means and estimates of the variances at each point of D for the function f(x, co). This is a major justification of the method.