Confidence Intervals for Discrete Approximations to Ill-Posed Problems

摘要

The authors consider the linear regression model obtained by discretizing asystem of first-kind integral equations with random measurement errors in the right hand side. The errors are assumed to have zero means and known variances. The authors consider the problem of estimating confidence intervals for linear functions of the solution vector. For such problems, the least squares solution is a highly unstable function of the measurements, and the classical confidence intervals are too wide to be useful. The solution can often be stabilized by imposing physically motivated, a priori nonnegativity constraints on the solution. The paper will show how to extend the classical confidence interval estimation procedure to accommodate these nonnegativity constraints in order to obtain improved confidence intervals. The technique defines valid confidence intervals even for problems with fewer measurements than unknowns.