Skip to main content

Adaptive Regularization of B-Spline Models for Scientific Data

  • Conference paper
  • First Online:
Computational Science – ICCS 2022 (ICCS 2022)

Abstract

B-spline models are a powerful way to represent scientific data sets with a functional approximation. However, these models can suffer from spurious oscillations when the data to be approximated are not uniformly distributed. Model regularization (i.e., smoothing) has traditionally been used to minimize these oscillations; unfortunately, it is sometimes impossible to sufficiently remove unwanted artifacts without smoothing away key features of the data set. In this article, we present a method of model regularization that preserves significant features of a data set while minimizing artificial oscillations. Our method varies the strength of a smoothing parameter throughout the domain automatically, removing artifacts in poorly-constrained regions while leaving other regions unchanged. The behavior of our method is validated on a collection of two- and three-dimensional data sets produced by scientific simulations.

This work is supported by the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research under Contract DE-AC02-06CH11357, Program Manager Margaret Lentz.

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

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    We consider only “clamped” knot sequences in this paper; thus, the first \(p+1\) knots are always 0 and the last \(p+1\) knots are always 1.

  2. 2.

    Here we assume for simplicity that the degree of the B-spline is the same in each dimension, but the degree can vary in practice if desired.

  3. 3.

    For example, \(\partial ^{(2,0)} f = \partial ^2 f / \partial x_1^2\), and \(\partial ^{(0,2)} f = \partial ^2 f / \partial x_2^2\), while \(\partial ^{(1,1)} f = \partial ^2 f / (\partial x_1 \partial x_2)\).

  4. 4.

    We consider the unnormalized sinc function: \(\mathrm {sinc}(x) = \sin (x)/x\), with \(\mathrm {sinc}(0)=1\).

References

  1. Arigovindan, M., Suhling, M., Jansen, C., Hunziker, P., Unser, M.: Full motion and flow field recovery from echo doppler data. IEEE Trans. Med. Imaging 26(1), 31–45 (2006). https://doi.org/10.1109/TMI.2006.884201

    Article  Google Scholar 

  2. Craven, P., Wahba, G.: Smoothing noisy data with spline functions. Numer. Math. 31(4), 377–403 (1978). https://doi.org/10.1007/BF01404567

    Article  MathSciNet  MATH  Google Scholar 

  3. De Boor, C.: A Practical Guide to Splines. Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York (2001)

    Google Scholar 

  4. Duchon, J.: Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In: Schempp, W., Zeller, K. (eds.) Constructive Theory of Functions of Several Variables. LNM, vol. 571, pp. 85–100. Springer, Heidelberg (1977). https://doi.org/10.1007/BFb0086566

  5. Duijndam, A., Schonewille, M., Hindriks, C.: Reconstruction of band-limited signals, irregularly sampled along one spatial direction. Geophysics 64(2), 524–538 (1999). https://doi.org/10.1190/1.1444559

    Article  Google Scholar 

  6. El-Rushaidat, D., Yeh, R., Tricoche, X.M.: Accurate parallel reconstruction of unstructured datasets on rectilinear grids. J. Visualization 24(4), 787–806 (2021). https://doi.org/10.1007/s12650-020-00740-0

    Article  Google Scholar 

  7. Eyring, V., et al.: Overview of the coupled model intercomparison project phase 6 (CMIP6) experimental design and organization. Geosci. Model Dev. 9(5), 1937–1958 (2016). https://doi.org/10.5194/gmd-9-1937-2016

    Article  Google Scholar 

  8. Francis, B., Viswanath, S., Arigovindan, M.: Scattered data approximation by regular grid weighted smoothing. Sādhanā 43(1), 1–16 (2018). https://doi.org/10.1007/s12046-017-0765-y

    Article  MathSciNet  MATH  Google Scholar 

  9. Gu, C.: Cross-validating non-Gaussian data. J. Comput. Graph. Stat. 1(2), 169–179 (1992). https://doi.org/10.1080/10618600.1992.10477012

    Article  Google Scholar 

  10. Gu, C.: Smoothing Spline ANOVA Models. SSS, vol. 297. Springer, New York (2013). https://doi.org/10.1007/978-1-4614-5369-7

  11. Hughes, T., Cottrell, J., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194(39), 4135–4195 (2005). https://doi.org/10.1016/j.cma.2004.10.008

    Article  MathSciNet  MATH  Google Scholar 

  12. Ku, S., Hager, R., Chang, C.S., Kwon, J., Parker, S.E.: A new hybrid-Lagrangian numerical scheme for gyrokinetic simulation of tokamak edge plasma. J. Comput. Phys. 315, 467–475 (2016). https://doi.org/10.1016/j.jcp.2016.03.062

    Article  MathSciNet  MATH  Google Scholar 

  13. Lee, S., Wolberg, G., Shin, S.: Scattered data interpolation with multilevel b-splines. IEEE Trans. Visual Comput. Graphics 3(3), 228–244 (1997). https://doi.org/10.1109/2945.620490

    Article  Google Scholar 

  14. Lin, H., Maekawa, T., Deng, C.: Survey on geometric iterative methods and their applications. Comput. Aided Des. 95, 40–51 (2018). https://doi.org/10.1016/j.cad.2017.10.002

    Article  Google Scholar 

  15. Nashed, Y.S.G., Peterka, T., Mahadevan, V., Grindeanu, I.: Rational approximation of scientific data. In: Rodrigues, J.M.F., et al. (eds.) ICCS 2019. LNCS, vol. 11536, pp. 18–31. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-22734-0_2

    Chapter  Google Scholar 

  16. Peterka, T., Nashed, Y., Grindeanu, I., Mahadevan, V., Yeh, R., Trixoche, X.: Foundations of multivariate functional approximation for scientific data. In: Proceedings of 2018 IEEE Symposium on Large Data Analysis and Visualization (2018). https://doi.org/10.1109/LDAV.2018.8739195

  17. Piegl, L., Tiller, W.: The NURBS Book, 2 edn. VISUALCOMM. Springer, Heidelberg (1997). https://doi.org/10.1007/978-3-642-59223-2

  18. Vio, R., Strohmer, T., Wamsteker, W.: On the reconstruction of irregularly sampled time series. Publ. Astron. Soc. Pac. 112(767), 74 (2000). https://doi.org/10.1086/316495

    Article  Google Scholar 

  19. Wood, S.N.: Thin plate regression splines. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 65(1), 95–114 (2003). https://doi.org/10.1111/1467-9868.00374

    Article  MathSciNet  MATH  Google Scholar 

  20. Yeh, R.: Efficient knot optimization for accurate B-spline-based data approximation. Ph.D. thesis, Purdue University Graduate School (2020)

    Google Scholar 

  21. Yu, Y., Shemon, E., Mahadevan, V.S., Rahaman, R.O.: Sharp multiphysics tutorials. Technical report ANL/NE-16/1, Argonne National Lab. (ANL), Lemont, IL (United States) (2016). https://doi.org/10.2172/1250465

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to David Lenz .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2022 Raine Yeh and UChicago Argonne, LLC, Operator of Argonne National Laboratory, under exclusive license to Springer Nature Switzerland AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Lenz, D., Yeh, R., Mahadevan, V., Grindeanu, I., Peterka, T. (2022). Adaptive Regularization of B-Spline Models for Scientific Data. In: Groen, D., de Mulatier, C., Paszynski, M., Krzhizhanovskaya, V.V., Dongarra, J.J., Sloot, P.M.A. (eds) Computational Science – ICCS 2022. ICCS 2022. Lecture Notes in Computer Science, vol 13350. Springer, Cham. https://doi.org/10.1007/978-3-031-08751-6_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-08751-6_11

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-08750-9

  • Online ISBN: 978-3-031-08751-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics