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

Given error contaminated discrete data zᵢ = ∫₀¹ k(sᵢ ,t)f(t)dt + ε ᵢ, i = 1, ⋯ ,n, we apply the truncated singular value decomposition to find an approximate solution to the Fredholm first kind integral equation (Kf)(x) = ∫₀¹ 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 \|fm;n - f\| , where fm;n denotes the approximation obtained from the data with this optimal truncation level, are obtained. A numerical example is included.