Convergence rate analysis of the coefficient identification problems in a Kuramoto–Sivashinsky equation

Abstract

The convergence rate results of the mixed total variation (TV) and Lp regularization to the identification of the unknown anti-diffusion coefficient, and the mixed H2-Lp regularization for the identification of the unknown diffusion coefficient in a Kuramoto–Sivashinsky (KS) equation are investigated with some additional observation. Under suitable source conditions, the convergence rates to regularized solutions for the TV-Lp regularization of the anti-diffusion coefficient are obtained. Analogously, for the identification of diffusion coefficient with H2-Lp regularization, the convergence rates are derived with proper source conditions. In addition, simple and easily interpretable source conditions are also established to deduce the same convergence rates under some assumptions to the sought coefficients. For the source conditions in both two coefficient identification problems, no restrictive smallness condition to the source functions is required.