Some aspects of the behavior of regularized solutions as the amount of smoothing is varied

AbstractWe consider the operator equation of the first kind, Kf = g, where K is a compact linear operator from a (finite or infinite-dimensional) Hilbert space H1 into the Hilbert space H2, and orthogonally decompose g as g = ĝ + g̃, ĝ ϵ R(K) and g̃ ϵ N(K∗), where K∗ is the adjoint of K. Similarly, for the Twomey-Tichonov regularized solution fγ with smoothing parameter γ > 0 and trial element p ϵ H1, we have p = p̂ + p̃, p̂ ϵ R(K∗) and p̃ ϵ N(K). It is known that, if ĝ ϵ R(K) itself, then limγ→0+ fγ = f̂ + p̃, where f̂ is the least-squares solution of minimum norm to Kf = g. (If K is represented by a matrix A, then f̂ = A+g, where A+ is the Moore-Penrose pseudoinverse.) An analogous limiting property holds for the generalized Landweber iteration as the iteration number k becomes infinite. We discuss the variation of these regularized solutions from f̂ + p̃ to p as γ and k are varied under the assumption that ĝ ϵ R(K), and contrast this behavior with that which occurs when ξ ϵ R(K). In this latter case we show that, for a class of regularizations that includes those described above, the regularized solution is unbounded as the smoothing is decreased. This unboundedness affects the strategy that must be employed in estimating f when g is contaminated with measurement error