Skip to main content
SpringerLink
Log in

Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering and Quasi-Interpolation: A Unified View

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Notes

  1. The negative order norm \(\Vert \cdot \Vert _{-\ell , \varOmega }\) is the norm associated with \(H^{-\ell }(\varOmega )\) (i.e., the dual space of the Sobolev space \(H^\ell (\varOmega )\)).

  2. The first-order central B-spline is often denoted as \(b_0(x)\), but herein the authors chose to follow the notation used in the previously published definition of SIAC kernels throughout the article.

References

  1. Cockburn, B., Karniadakis, G.E., Shu, C.W. (eds.): Discontinuous Galerkin Methods. Springer, Berlin (2000)

    MATH  Google Scholar 

  2. Cockburn, B., Luskin, M., Shu, C.W., Süli, E.: Enhanced accuracy by post-processing for finite element methods for hyperbolic equations. Mathematics of Computation 72(242), 577–606 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ji, L., van Slingerland, P., Ryan, J.K., Vuik, K.: Superconvergent error estimates for a position-dependent smoothness-increasing accuracy-conserving filter for DG solutions. Math. Comput. 83, 2239–2262 (2014)

    Article  MATH  Google Scholar 

  4. Ji, L., Xu, Y., Ryan, J.K.: Accuracy enhancement of the linear convection–diffusion equation in multiple dimensions. Math. Comput. 81, 1929–1950 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ryan, J.K., Shu, C.W.: One-sided post-processing technique for the discontinuous Galerkin methods. Methods Appl. Anal. 10(2), 295–308 (2003)

    MathSciNet  MATH  Google Scholar 

  6. van Slingerland, P., Ryan, J.K., Vuik, K.: Position-dependent smoothness-increasing accuracy-conserving (SIAC) filtering for improving discontinuous Galerkin solutions. SIAM J. Sci. Comp. 33(2), 802–825 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Mirzaee, H., Ji, L., Ryan, J.K., Kirby, R.M.: Smoothness-increasing accuracy-conserving (SIAC) postprocessing for discontinuous Galerkin solutions over structured triangular meshes. SIAM J. Numer. Anal. 49(5), 1899–1920 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Mirzaee, H., King, J., Ryan, J.K., Kirby, R.M.: Smoothness-increasing accuracy-conserving filters for discontinuous Galerkin solutions over unstructured triangular meshes. SIAM J. Sci. Comput. 35(1), A212–A230 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Mirzaee, H., Ryan, J.K., Kirby, R.M.: Smoothness-increasing accuracy-conserving (SIAC) filters for discontinuous Galerkin solutions: Application to structured tetrahedral meshes. J. Sci. Comp. 58(3), 690–704 (2014)

  10. Walfisch, D., Ryan, J.K., Kirby, R.M., Haimes, R.: One-sided smoothness-increasing accuracy-conserving filtering for enhanced streamline integration through a discontinuous fields. J. Sci. Comp. 38(2), 164–184 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bramble, J.H., Schatz, A.H.: Higher order local accuracy by averaging in the finite element method. Math. Comput. 31(137), 94–111 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  12. Höllig, K.: Finite element methods with B-splines. Soc. Ind. Appl. Math. (2003)

  13. Mirzaee, H., Ryan, J.K., Kirby, R.M.: Efficient implementation of smoothness-increasing accuracy-conserving (SIAC) filters for discontinuous Galerkin solutions. J. Sci. Comp. 52(1), 85–112 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Thomée, V.: High order local approximations to derivatives in the finite element method. Math. Comput. 31(139), 652–660 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  15. Unser, M.: Splines: a perfect fit for signal and image processing. IEEE Signal Process. Mag. 16(6), 22–38 (1999)

    Article  Google Scholar 

  16. Unser, M.: Sampling: 50 years after Shannon. Proc. IEEE 88(4), 569–587 (2000)

    Article  Google Scholar 

  17. Unser, M., Aldroubi, A., Eden, M.: B-Spline signal processing: part I-theory. IEEE Trans. Signal Process. 41(2), 821–833 (1993)

    Article  MATH  Google Scholar 

  18. Vuçini, E., Möller, T., Gröller, M.E.: On visualization and reconstruction from non-uniform point sets using b-splines. Comput. Graph. Forum 28(3), 1007–1014 (2009)

    Article  Google Scholar 

  19. DeBoor, C.: A Practical Guide to Splines. Springer, Berlin (1978)

    Book  Google Scholar 

  20. de Boor, C.: Approximation order without quasi-interpolants. In: Cheney, E., Chui, C., Schumaker, L. (eds.) Approximation Theory VII, pp. 1–18 (1993)

  21. de Boor, C., Höllig, K., Riemenschneider, S.: Box Splines. Springer New York Inc., New York, NY (1993)

    Book  MATH  Google Scholar 

  22. Cohen, E., Riesenfeld, R.F., Elber, G.: Geometric Modeling with Splines: An Introduction. A K Peters, Natick, MA (2001)

    MATH  Google Scholar 

  23. Strang, G., Fix, G.: A fourier analysis of the finite element variational method. In: Constructive Aspects of Functional Analysis, pp. 796–830 (1971)

  24. De Boor, C., Daniel, J.: Splines with Non-Negative B-Spline Coefficients: CNA. Defense Technical Information Center (1973)

  25. Marsden, M.: An identity for spline functions with applications to variation diminishing spline approximation. MRC technical summary report. University of Wisconsin–Madison (1968)

  26. Mirzargar, M.: A Reconstruction Framework for Common Sampling Lattices. Ph.D. thesis, University of Florida, Gainesville, FL (2012)

  27. Mirzargar, M., Entezari, A.: Quasi interpolation with voronoi splines. IEEE Trans. Vis. Comput. Graph. 17(12), 1832–1841 (2011)

    Article  Google Scholar 

  28. Schoenberg, I.J.: Cardinal Spline Interpolation. Society for Industrial and Applied Mathematics, Philadelphia (1973)

  29. Unser, M., Aldroubi, A., Eden, M.: Fast B-Spline transforms for continuous image representation and interpolation. IEEE Trans. Pattern Anal. Mach. Intell. 13(3), 277–285 (1991)

    Article  Google Scholar 

  30. Steffen, M., Curtis, S., Kirby, R.M., Ryan, J.K.: Investigation of smoothness-increasing accuracy-conserving filters for improving streamline integration through discontinuous fields. IEEE Trans. Vis. Comput. Graph. 14(3), 680–692 (2008)

    Article  Google Scholar 

  31. King, J., Mirzaee, H., Ryan, J.K., Kirby, R.M.: Smoothness-increasing accuracy-conserving (SIAC) filtering for discontinuous Galerkin solutions: Improved errors versus higher-order accuracy. J. Sci. Comp. 53(1), 129–149 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mirzaee, H.: Smoothness-Increasing Accuracy-conserving Filters (SIAC) for Discontinuous Galerkin Solutions. Ph.D. thesis, University of Utah, Salt Lake City, UT (2012)

  33. Mirzaee, H., Ryan, J.K., Kirby, R.M.: Quantification of errors introduced in the numerical approximation and implementation of smoothness-increasing accuracy conserving (SIAC) filtering of discontinuous Galerkin (DG) fields. J. Sci. Comp. 45(1–3), 447–470 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Nektar++ (2014). http://www.nektar.info

Download references

Acknowledgments

The authors wish to thank Dr. Hanieh Mirzaee and Mr. James King for helpful discussion and sharing the symmetric SIAC postprocessing code, Xiaozhou Li for confirming the convergence results and Varun Shankar for useful suggestions. The authors are sponsored in part by the Air Force Office of Scientific Research (AFOSR), Computational Mathematics Program (Program Manager: Dr. Fariba Fahroo), under grant number FA9550-12-1-0428 (first and third author) and FA8655-13-1-3017 (second author).

Author information

Authors and Affiliations

  1. Scientific Computing and Imaging (SCI) Institute, University of Utah, Salt Lake City, UT, 84112, USA

    Mahsa Mirzargar

  2. School of Mathematics, University of East Anglia, Norwich, UK

    Jennifer K. Ryan

  3. School of Computing and the Scientific Computing and Imaging (SCI) Institute, University of Utah, Salt Lake City, UT, 84112, USA

    Robert M. Kirby

Authors
  1. Mahsa Mirzargar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jennifer K. Ryan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Robert M. Kirby
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mahsa Mirzargar.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mirzargar, M., Ryan, J.K. & Kirby, R.M. Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering and Quasi-Interpolation: A Unified View. J Sci Comput 67, 237–261 (2016). https://doi.org/10.1007/s10915-015-0081-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-015-0081-9

Keywords

Search

Navigation