Optimal Choice of a Truncation Level for the Truncated SVD Solution of Linear First Kind Integral Equations When Data are Noisy

Abstract

Given error contaminated discrete data $z_i = \int_0^1 {k(s_i ,t)f(t)dt + \varepsilon _i } $, $i = 1, \cdots ,n$, we apply the truncated singular value decomposition to find an approximate solution to the Fredholm first kind integral equation $(Kf)(x) = \int_0^1 {k(s,t)f(t)dt = g(s)} $. We define an optimal truncation level m and apply generalized cross validation to choose an estimate of this optimal truncation level which depends on the data. Convergence rates for $\|f_{m;n} - f\|$ , where $f_{m;n} $ denotes the approximation obtained from the data with this optimal truncation level, are obtained. A numerical example is included.