LOCAL REGULARIZATION FOR THE NONLINEAR INVERSEAUTOCONVOLUTION PROBLEM

摘要

We develop a local regularization theory for the nonlinear inverse autoconvolutionproblem. Unlike classical regularization techniques such as Tikhonov regularization, this theoryprovides regularization methods that preserve the causal nature of the autoconvolution problem,allowing for fast sequential numerical solution (0(rN2 - r2N) flops, where r N for the methoddiscussed in this paper as applied to the nonlinear problem; in comparison, the cost for Tikhonovregularization applied to a general linear problem is 0(N3) flops). We prove the convergence of theregularized solutions to the true solution as the noise level in the data shrinks to zero and supplyconvergence rates for the cases of both L2 and continuous data. We propose several regularizationmethods and provide a theoretical basis for their convergence; of note is that this class of methods doesnot require an initial guess of the unknown solution. Our numerical results confirm the effectivenessof the methods, with results comparing favorably to numerical examples found in the literature forthe autoconvolution problem (e.g., [G. Fleischer, R. Gorenflo, and B. Hofmann, ZAMM Z. Angela.Math. Mech., 79 (1999), pp. 149-159] for examples using Tikhonov regularization with total variationconstraints and [J. Jarmo, Inverse Problems, 16 (2000), pp. 333-348] for examples using the methodof Lavrent'ev); this especially seems to be true when it comes to the recovery of sharp features in theunknown solution. We also show the effectiveness of our method in cases not covered by the theory.