Local Regularization of the Autoconvolution Problem
Abstract:
We develop a local regularization theory for the nonlinear autoconvolution
problem. Unlike the classic regularization techniques such as Tikhonov
regularization, this theory provides regularization methods that preserve the
causal nature of the autoconvolution problem, allowing for fast sequential
numerical solution. We prove the convergence of the regularized solutions to
the true solution as the noise level in the data shrinks to zero, with a
certain convergence rate. We propose several regularization methods and
provide
theoretic basis for their convergence. Our numerical results confirm
effectiveness of the methods, suggesting superiority of our methods over the
existing ones, especially in recovering sharp features in the solution.