Adaptive Mollifiers for High Resolution Recovery of Piecewise Smooth Data from its Spectral Information

Details Adaptive Mollifiers for High Resolution Recovery of Piecewise Smooth Data from its Spectral Information Adaptive Mollifiers for High Resolution Recovery of Piecewise Smooth Data from its Spectral Information

2001-12-03

arxiv.org/abs/math/0112017v1

Foundations of Computational Mathematics, Online publication (2001)

Mathematics

Numerical Analysis

Scientific paper

We discuss the reconstruction of piecewise smooth data from its (pseudo-) spectral information. Spectral projections enjoy superior resolution provided the data is globally smooth, while the presence of jump discontinuities is responsible for spurious ${\cal O}(1)$ 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 introduced 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 for the high resolution. The resulting adaptive mollifier greatly accelerates the convergence rate, recovering piecewise analytic data within exponential accuracy while removing spurious oscillations that remained in [GoTa85]. Thus, these adaptive mollifiers offer a robust, general-purpose ``black box'' procedure for accurate post processing of piecewise smooth data.

Affiliated with

Also associated with

No associations

Rating

Adaptive Mollifiers for High Resolution Recovery of Piecewise Smooth Data from its Spectral Information does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Adaptive Mollifiers for High Resolution Recovery of Piecewise Smooth Data from its Spectral Information, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Adaptive Mollifiers for High Resolution Recovery of Piecewise Smooth Data from its Spectral Information will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-235268