Speaker: Bert W. Rust, NIST

Title: Statistical Stabilization of Ill-Posed Problems

Abstract: A first kind Fredholm integral equation expresses a given univariate function as an integral of the product of a known bivariate kernel and an unknown univariate solution. An ill-posed problem is obtained by discretizing such an integral equation and adding random measuring errors to the discrete values of the given function. This produces a linear regression model whose solution vector is an exquisitely unstable function of the random errors. For more than 40 years, numerical analysts have ignored the statistical properties of those errors in devising methods to stabilize the estimation of the solution. The two most commonly used methods, regularization and truncation of the singular value decomposition, alter the known kernel matrix in order to accommodate the unknown random errors. This talk will briefly review those approaches and then present a new method which leaves the matrix unchanged and instead truncates an orthogonal rotation of the vector of measurements. The statistical properties of the measurement errors are used to guide this truncation and to judge whether or not the resulting solution estimate produces residuals with those same statistical properties.

Place: Science and Technology Building I, Room 242

Refreshments will be served before the talk at 3:00 p.m. in Room 222.

Department of Mathematical Sciences
George Mason University
4400 University Drive, MS 3F2
Fairfax, VA 22030-4444
http://math.gmu.edu/
Tel. 703-993-1460, Fax. 703-993-1491