Lavrent'ev regularization of ill-posed problems containing nonlinear near-to-monotone operators with application to autoconvolution equation

Abstract

In this paper the method of Lavrent'ev for regularization of ill-posed problems with near-to-monotone operators is considered. An a priori rule of choice of the parameter of regularization is presented and the corresponding error estimate is derived. The theory is applied to the autoconvolution equation. Numerical examples are provided.

Local Regularization for the Nonlinear Inverse Autoconvolution Problem

“…We propose several regularization methods and provide a theoretical basis for their convergence; of note is that this class of methods does not require an initial guess of the unknown solution. Our numerical results confirm effectiveness of the methods, with results comparing favorably to numerical examples found in the literature for the autoconvolution problem (e.g., [13] for examples using Tikhonov regularization with total variation constraints, and [16] for examples using the method of Lavrent'ev); this especially seems to be true when it comes to the recovery of sharp features in the unknown solution. We also show the effectiveness of our method in cases not covered by the theory.…”

“…In order for easy comparison with numerical tests in the literature for existing methods (a Tikhonov regularization approach using total variation constraints in [13] and a Lavrent'ev approach in [16]), we demonstrate the recovery of both a continuous x and a discontinuousx using Methods 1 and 2 above. We select our true solution x ahead of time, then generate the data function f by integration, f (t) = t 0x…”

“…Therefore, the kernel a R [σ](·) is nonnegative, nonincreasing and convex. Using Lemma 2 of [16] for σ ≥ σ 0 , we have B σ R (x) accretive on L 2 (0, 1), from which it follows that B R (x) is accretive in L 2 (0, 1) with a weighted e −2σt -norm. A similar statement may be made for B R (x) using a weighted L 2 (0, R) norm, from which it follows that…”

“…In [16], Janno applied the Lavrent'ev regularization method to the autoconvolution equation and provided both theoretical convergence estimates and numerical examples. The advantage of Lavrent'ev method is that it preserves the causal nature of Volterra problems and therefore leads to a fast sequential method.…”

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

“…This operator equation of quadratic type occurs in physics of spectra, in optics and in stochastics, often as part of a more complex task (see, e.g., [2,6,35]). A series of studies on deautoconvolution and regularization have been published for the setting (2.1), see for example [7,24,25,32]. Some first basic mathematical analysis of the autoconvolution equation can already be found in the paper [14].…”