Smoothness-Increasing Accuracy-Conserving (SIAC) filtering and quasi interpolation: A unified view

Mahsa Mirzargar (Lead Author), Jennifer Ryan, Robert Kirby

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)
5 Downloads (Pure)

Abstract

Filtering plays a crucial role in postprocessing and analyzing data in scientific and engineering applications. Various application-specific filtering schemes have been proposed based on particular design criteria. In this paper, we focus on establishing the theoretical connection between quasi-interpolation and a class of kernels (based on B-splines) that are specifically designed for the postprocessing of the discontinuous Galerkin (DG) method called Smoothness-Increasing Accuracy-Conserving (SIAC) filtering. SIAC filtering, as the name suggests, aims to increase the smoothness of the DG approximation while conserving the inherent accuracy of the DG solution (superconvergence). Superconvergence properties of SIAC filtering has been studied in the literature. In this paper, we present the theoretical results that establish the connection between SIAC filtering to long-standing concepts in approximation theory such as quasi-interpolation and polynomial reproduction. This connection bridges the gap between the two related disciplines and provides a decisive advancement in designing new filters and mathematical analysis of their properties. In particular, we derive a closed formulation for convolution of SIAC kernels with polynomials. We also compare and contrast cardinal spline functions as an example of filters designed for image processing applications with SIAC filters of the same order, and study their properties.
Original languageEnglish
Pages (from-to)237-261
Number of pages25
JournalJournal of Scientific Computing
Volume67
Issue number1
Early online date9 Aug 2015
DOIs
Publication statusPublished - Apr 2016

Keywords

  • B-splines
  • SIAC filtering
  • quasi-interpolation
  • approximation theory
  • Fourier Analysis

Cite this