Parameter Selection for Constrained Solutions to III-Posed Problems

Abstract Many physical measurements y(t i ) can be modelled bya system of linear, rst kind integral equationsy(t i ) =Z ˘ up ˘ lo K(t i ;˘)x(˘)d˘ + i ; i = 1;2;:::;m ; (1)where x(˘) is the function being measured, the K(t i ;˘)are known instrument response functions, and the  i are random measuring errors. Discretizing the integralsproduces an ill-conditioned linear regression model, andminimizing the sum of squared residuals forces variancethat should properly be relegated to the residuals intothe least squares estimate which becomes a wildly os-cillating, spurious approximation to x(˘). Adopting theprinciple that acceptable residuals should resemble the i leads to three statistical tests for the suitability of anestimate. Stabilized estimates can be obtained eitherby appending a set of regularization constraints to themodel or by truncating the singular value decompositionofthematrix. Usingatestproblemwithknownsolution,it is shown that conventional methods for choosing theregularization parameter yield unacceptable estimates,but that the three statistical tests can be used to choosean optimal value. It is also shown that truncating thedistribution of singular values does not work as well asdiscarding the components of the rotated data vectorthat are overwhelmed by measurement errors, and thatthe three statistical tests can be used to optimize thechoice of the truncation threshold.