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Bert W. Rust and Dianne P. O'Leary, ACMD*

Many measured distributions are modeled by a system of integral equations

where the are known, the are measuring errors, and is an unknown function to be determined. Since these integral equations are ubiquitous in measurement processes, methods for solving them are of great interest to NIST and ACMD.

Replacing by a discrete approximation , gives the linear regression model

where is an matrix and is the covariance matrix for the measurements. The corresponding least squares problem is usually so poorly conditioned that the usual methods give wildly oscillating, unphysical solutions.

This project previously produced three algorithms, CLASIC, FERDIT and BRAKET-LS, which give estimates and confidence intervals for the . CLASIC uses a singular value decomposition (SVD) of to implement iterative refinement on the augmented system

with an option for truncating the SVD to suppress the amplification of measurement noise.

The FERDIT algorithm assumes that the solution is nonnegative and iterates on a regularized regression model

where is the regularization parameter, is a positive vector and is a related nonnegative diagonal matrix designed to constrain to the positive orthant. The iteration is implemented with an SVD of the matrix . The bounds obtained are conservative and suboptimal, but are useful as starting estimates for the BRAKET-LS method.

The BRAKET-LS algorithm also assumes a nonnegative solution and uses a parametric quadratic programming procedure to produce rigorous confidence interval bounds defined by

where is a constant chosen to give the desired confidence level. The method requires initial estimates which do not have to bracket the very closely, but the computational effort is considerably reduced by using the suboptimal bounds obtained from the FERDIT method.

Current efforts have been devoted to simplifying and documenting the Fortran subroutines and writing a paper describing the BRAKET-LS software.