Stochastic inversion of linear first kind integral equations. I: Continuous theory and the stochastic generalized inverse (Q1081964)

The authors consider the solution of the Fredholm integral equation of the first kind \[ (1)\quad \int^{b}_{a}K(s,t)x(t)dt=y(s),\quad c\leq s\leq d, \] where y(s) is contaminated by noise and where a priori data consisting of ensembles of measurements of x(t) and y(s), also noise contaminated, are available. Associated with (1) is a stochastic integral equation \[ (2)\quad z_{\sigma}(s)=\int^{b}_{a}K(s,t)x(t)dt+\sigma \epsilon (s), \] where x(t) is a stochastic process which represents the data ensemble of solutions to (1). Here \(\epsilon\) (s) represents ''continuous white noise'' which is independent of x(t) and serves as a model for measurement error, and \(\sigma\) represents the noise level. The authors address the mathematical and probabilistic structure of statistical inversion applied to (2). They prove that the best linear estimation in this setting is equivalent to regularization in an appropriate reproducing kernel Hilbert space setting. In particular, the autocorrelation function for the stochastic process x(t) plays the role of a reproducing kernel, and the noise level \(\sigma\) acts as the regularization parameter.