Error estimates for Arnoldi–Tikhonov regularization for ill-posed operator equations

Abstract

Most of the literature on the solution of linear ill-posed operator equations, or their discretization, focuses only on the infinite-dimensional setting or only on the solution of the algebraic linear system of equations obtained by discretization. This paper discusses the influence of the discretization error on the computed solution. We consider the situation when the discretization used yields an algebraic linear system of equations with a large matrix. An approximate solution of this system is computed by first determining a reduced system of fairly small size by carrying out a few steps of the Arnoldi process. Tikhonov regularization is applied to the reduced problem and the regularization parameter is determined by the discrepancy principle. Errors incurred in each step of the solution process are discussed. Computed examples illustrate the error bounds derived.

Error estimates for Golub–Kahan bidiagonalization with Tikhonov regularization for ill–posed operator equations

“…We will discuss the effect of this replacement on the computed solution, as well as the effect of the errors in the operator A h and in the right-hand side function y δ . Another approach for determining a low-rank approximation of the operator A h by applying the Arnoldi process instead of the GKB process has been discussed in [23]. This approach is quite different from the one of the present paper and is based on results by Natterer [20].…”

A new interpretation of (Tikhonov) regularization

“…Let A and x † be the infinite-dimensional operator and solution to [1], and consider their discrete, finite dimensional approximations A mn ∈ R m×n and x † n ∈ R n for the discretization levels m, n ∈ N. A mn is compact and has closed range. One can show that if A is ill-posed, x † n = (A T mn A mn ) μ ξ μ n has a solution for all μ > 0, but with ξ μ n → ∞ as n and/or μ go to infinity [25]. In other words, x † n fulfils a source condition with respect to A mn for all μ 0, but with exploding source element.…”

A new interpretation of (Tikhonov) regularization

“…Let A and x † be the infinite-dimensional operator and solution to (1), and consider their discrete, finite dimensional approximations A mn ∈ R m×n and x † n ∈ R n for the discretization levels m, n ∈ N. A mn is compact and has closed range. One can show that if A is ill-posed, x † n = (A T mn A mn ) µ ξ µ n has a solution for all µ > 0, but with ξ µ n → ∞ as n and/or µ go to infinity [33]. In other words, x † n fulfils a source condition with respect to A mn for all µ ≥ 0, but with exploding source element.…”

“…From (32) we would read a saturation µ < κ + 1 2 for the discrepancy principle, since until then x δ α is smoother than x † . As with classical Tikhonov regularization, we would expect an a-priori choice to saturate at µ = κ + 1 see (33). Let as before x α denote the high-order Tikhonov approximation (31) with noise-free data.…”