Estimates in first order approximations to electromagnetic boundary integral equations on stochastic surfaces

In this paper, we address the problem of computing estimates of the variability of “observables.” Observables are measurable quantities which are defined as the integral of an appropriately chosen electromagnetic field against a (current-) distribution. The latter is obtained by solving a boundary value problem. In the case of an uncertain boundary geometry, the current distribution underlying the observable computation is a stochastic distribution whereas the field evaluated on this distribution to define the observable remains deterministic. The result is a stochastic observable of which the variance provides an interesting measure of the spreading of its values. Here, we develop a technique for explicitly computing the covariance operator of the stochastic distribution corresponding to the boundary value problem with uncertain geometry. The variance of observables can be computed directly from this operator as a bilinear form evaluated on the field defining the observable.