Truncating the Singular Value Decomposition for III-Posed Problems

摘要

Discretizing the first-kind integral equations which model many physicalmeasurement processes yields an ill-conditioned linear regression model b = Ax(sup star) + eta, where x(sup star) is a vector representation of the function being measured, A is an instrument response matrix, b is a vector of measurements, and eta is a vector of unknown, random measuring errors. Least squares estimation usually gives a sum of squared residuals much smaller than the expected value and wildly oscillating, physically implausible estimate x(sup star). These symptoms suggest that the least squares estimate captures part of the variance that properly belongs in the residuals. This paper suggests an alternate strategy which uses the variances of the measuring errors to specify a truncation for the elements of the rotated measurement vector U(sup T)b. The paper also develops some new diagnostics for the residuals which are useful not only for choosing the truncation level for the (U(sup T)b)i, but also for assessing the quality of an estimate obtained by any procedure.