Parameter Selection for Constrained Solutions to III-Posed Problems
Bert W. Rust
Many physical measurements are modeled by linear integral equations expressing each measurement as the sum of an instrumental smearing of the desired function and a random measuring error. Discretizing the integrals gives an ill-conditioned linear regression model with a matrix whose columns are discrete response functions of the instrument. Linear least squares solutions give wildly oscillating, physically impossible estimates of the function being measured. Such estimates are often stabilized either by truncating the singular value decomposition of the response matrix or by introducing a regularization constraint on the solution vector, In the former case it is necessary to choose a numerical rank for the matrix, and in the latter case to choose the value of the Lagrange multiplier in the constrained minimization. This paper suggests methods for using the statistical properties of the measuring errors and the residuals in making those choices.
deconvolution, ill-conditioned linear systems, ill-posed problems, unfolding
Parameter Selection for Constrained Solutions to III-Posed Problems, Computing Science and Statistics, [online], https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=150795
(Accessed December 3, 2023)