Adaptive-Mollifiers–High-Resolution-Recovery-of-Piecewise-Smooth-Data-from-its-Spectral-Information.pdf

c 2001 001 Adaptive Mollifiers – High Resolution Recovery of Piecewise Smooth Data from its Spectral Information Eitan Tadmor Jared Tanner June 27, 2006 To Ron DeVore with Friendship and Appreciation Abstract We discuss the reconstruction of piecewise smooth data from its (pseudo-) spectral informa- tion. Spectral projections enjoy superior resolution provided the data is globally smooth, while the presence of jump discontinuities is responsible for spurious O(1) Gibbs oscillations in the neighborhood of edges and an overall deterioration to the unacceptable first-order convergence rate. The purpose is to regain the superior accuracy in the piecewise smooth case, and this is achieved by mollification. Here we utilize a modified version of the two-parameter family of spectral mollifiers intro- duced by Gottlieb Tadmor [GoTa85]. The ubiquitous one-parameter, finite-order mollifiers are based on dilation. In contrast, our mollifiers achieve their high resolution by an intricate process of high-order cancelation. To this end, we first implement a localization step using edge detection procedure, [GeTa00a, GeTa00b]. The accurate recovery of piecewise smooth data is then carried out in the direction of smoothness away from the edges, and adaptivity is responsible f