Since first kind integral equations are ubiquitous in measurement processes, they are of great interest to NIST and ACMD. Discretizations of such equations give
where is a vector of measurements, is the covariance matrix for , is a known instrument response matrix, and is a vector to be determined. This problem is usually poorly conditioned, with a wildly oscillating, unphysical solution. Previous efforts have produced the algorithm BRAKET-LS which stabilizes the solution by requiring nonnegativity and computes confidence interval bounds
where determines the confidence level. The method requires initial estimates which do not have to bracket the very closely, but the computational effort is greatly reduced if they do. More recent efforts have been devoted to generating such estimates and to developing diagnostic tests for the residuals generated by the corresponding solutions. The unfolding codes FERDO and FERD have been adapted to create subroutines which calculate good suboptimal initial intervals. FERDO uses a regularization technique which requires the solution to be in the intersection of the measurement uncertainty ellipsoid with a guaranteed ellipsoid derived from the nonnegativity constraints. The suboptimal intervals are based on a convex combination of the two, and the widths of the intervals depend on the choice of the regularization parameter. FERD is based on the duals of the above problems. In the dual for the primal constraints are replaced by lower bias inequalities and for by upper bias inequalities. Beginning with initial estimates which are neither lower nor upper biased, FERD implements an elimination iteration to find feasible solutions which are used as starting estimates for BRAKET-LS. Another algorithm, called OPTIMO, replaces the measurement constraint ellipsoid with a circumscribing box and uses linear programming to generate suboptimal initial intervals. Preliminary tests have yielded good results for smaller problems but failures because of excessive iterations for problems with more than 60 unknowns. New diagnostic tests for all of these methods were obtained by applying time-series methods to the residuals. Future efforts will be devoted to further developments of the OPTIMO algorithm, fine tuning of the FERDO and FERD algorithms, further development of residual diagnostic tests, and documenting the whole package.