About a deficit in low-order convergence rates on the example of autoconvolution

Abstract

Abstract. We revisit in L 2 -spaces the autoconvolution equation x * x = y with solutions which are real-valued or complex-valued functions x(t) defined on a finite real interval, say t ∈ [0, 1]. Such operator equations of quadratic type occur in physics of spectra, in optics and in stochastics, often as part of a more complex task. Because of their weak nonlinearity deautoconvolution problems are not seen as difficult and hence little attention is paid to them wrongly. In this paper, we will indicate on the example of autoconvolution a deficit in low order convergence rates for regularized solutions of nonlinear ill-posed operator equations F (x) = y with solutions x † in a Hilbert space setting. So for the real-valued version of the deautoconvolution problem, which is locally ill-posed everywhere, the classical convergence rate theory developed for the Tikhonov regularization of nonlinear ill-posed problems reaches its limits if standard source conditions using the range of F (x † ) * fail. On the other hand, convergence rate results based on Hölder source conditions with small Hölder exponent and logarithmic source conditions or on the method of approximate source conditions are not applicable since qualified nonlinearity conditions are required which cannot be shown for the autoconvolution case according to current knowledge. We also discuss the complex-valued version of autoconvolution with full data on [0, 2] and see that ill-posedness must be expected if unbounded amplitude functions are admissible. As a new detail, we present situations of local well-posedness if the domain of the autoconvolution operator is restricted to complex L 2 -functions with a fixed and uniformly bounded modulus function.

Convergence results and low-order rates for nonlinear Tikhonov regularization with oversmoothing penalty term

“…We note that the minimizer of the Tikhonov functional may be nonunique, because T δ α can be, for nonlinear forward operators F , a non-convex functional as a consequence of a non-convex misfit term F u − f δ 2 . If, for example, F u := u ⋆ u represents the autoconvolution operator in X = L 2 (0, 1) (cf., e.g., [5] and references therein) and u = 0, then we have T δ α (u) = T δ α (−u), which illustrates the non-uniqueness phenomenon. On the other hand, it should be mentioned that the properties of Tikhonov regularization in Hilbert spaces are well investigated when the penalty functional in the Tikhonov functional is replaced by u → u − u 2 , cf., e.g., [7,Chapter 10] or [23, Section 3.1] and the references therein, respectively.…”

Case Studies and a Pitfall for Nonlinear Variational Regularization Under Conditional Stability

“…A way to ensure the property that the approximate solutions belong to X p ∩ Q ∩ D(F ) is to use regularized solutions which minimize the Tikhonov functional F (x) − y δ 2 Y + α B s x 2 X , subject to x ∈ Q ∩ D(F ), where s ≥ p is required. Hence, Tikhonov-type regularization is here an auxiliary tool which complements the conditional stability estimate (5) in order to obtain stable approximate solutions measured in the norm of X.…”

Regularization Methods for Ill-Posed Problems

“…One consequence of this specific interplay between F and its derivative for the deautoconvolution problem from (58), which is locally ill-posed everywhere, is the fact that the classical convergence rate theory developed for the Tikhonov regularization of nonlinear ill-posed problems reaches its limits if standard source condition fails. Please refer to [24] for details. On the other hand, convergence rate results based on Hölder source conditions with small Hölder exponent and logarithmic source conditions or on the method of approximate source conditions (cf.…”

“…Take into account L For the SPIDER technology, in particular, the phase function is to be determined from noisy observations of the complex function y, whereas information about the amplitude function can be verified by alternative measurements. In [47] some mathematical studies and a regularization approach for this specific problem have been presented, and further analytic investigations for the specific case of a constant kernel k can be found in [24].…”