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Multilevel interpolation of scattered data using \({\mathcal{H}}\)-matrices

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Abstract

Scattered data interpolation can be used to approximate a multivariate function by a linear combination of positive definite radial basis functions (RBFs). In practice, the approximation error stagnates (due to numerical instability) even if the function is smooth and the number of data centers is increased. A smaller approximation error can be obtained using multilevel interpolation on a sequence of nested subsets of the initial set of centers. For the construction of these nested subsets, we compare two thinning algorithms from the literature, a greedy algorithm based on nearest neighbor computations and a Poisson point process. The main novelty of our approach lies in the use of \({\mathcal{H}}\)-matrices both for the solution of linear systems and for the evaluation of residual errors at each level. For the solution of linear systems, we use GMRes combined with a domain decomposition preconditioner. Using \({\mathcal{H}}\)-matrices allows us to solve larger problems more efficiently compared with multilevel interpolation based on dense matrices. Numerical experiments with up to 50,000 scattered centers in two and three spatial dimensions demonstrate that the computational time required for the construction of the multilevel interpolant using \({\mathcal{H}}\)-matrices is of almost linear complexity with respect to the number of centers.

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References

  1. Barba, L. A., Knepley, M.G., Yokota, R.: PetRBF – a parallel \(\mathcal {O}({N})\) algorithm for radial basis function interpolation with Gaussians. Comput. Methods Appl. Mech. Eng. 199(25), 1793–1804 (2010)

    MathSciNet  MATH  Google Scholar 

  2. Bebendorf, M., Rjasanow, S.: Adaptive low-rank approximation of collocation matrices. Computing 70(1), 1–24 (2003)

    Article  MathSciNet  Google Scholar 

  3. Börm, S.: H2lib software library. www.h2lib.org, 2017. University of Kiel

  4. Le Borne, S., Wende, M.: Domain decomposition methods in scattered data interpolation with conditionally positive definite radial basis functions. Computers & Mathematics with Applications 77(4), 1178–1196 (2019)

    Article  MathSciNet  Google Scholar 

  5. Le Borne, S., Wende, M.: Iterative solution of saddle-point systems from radial basis function (RBF) interpolation. SIAM J. Sci. Comput. 41(3), A1706–A1732 (2019)

    Article  MathSciNet  Google Scholar 

  6. Cook, R.: Stochastic sampling in computer graphics. ACM Trans. Graph. (TOG) 5(1), 51–72 (1986)

    Article  Google Scholar 

  7. Devillers, O., Hornus, S., Jamin, C.: dD triangulations. In: CGAL 4.12 user and reference manual. CGAL Editorial Board (2018)

  8. Dippe, M.A., Wold, E.H.: Antialiasing through stochastic sampling. Comput. Graph. (ACM) 19(3), 69–78 (1985)

    Article  Google Scholar 

  9. Dunbar, D., Humphreys, G.: A spatial data structure for fast Poisson-disk sample generation. ACM Trans. Graph. (TOG) 25(3), 503–508 (2006)

    Article  Google Scholar 

  10. Dyn, N., Floater, M. S., Iske, A.: Adaptive thinning for bivariate scattered data. J. Comput. Appl. Math. 145(2), 505–517 (2002)

    Article  MathSciNet  Google Scholar 

  11. Floater, M.S., Iske, A.: Multistep scattered data interpolation using compactly supported radial basis functions. J. Comput. Appl. Math. 73(1), 65–78 (1996)

    Article  MathSciNet  Google Scholar 

  12. Floater, M.S., Iske, A.: Thinning algorithms for scattered data interpolation. BIT Numerical Math. 12(4), 705–720 (1998)

    Article  MathSciNet  Google Scholar 

  13. Franke, R.: Scattered data interpolation: tests of some methods. Math. Comput. 38(157), 181–200 (1982)

    MathSciNet  MATH  Google Scholar 

  14. Hackbusch, W.: : Hierarchical matrices: algorithms and analysis. Springer Series in Computational Mathematics, 1st edn. Springer, Heidelberg (2015)

    Book  Google Scholar 

  15. Halton, J.: On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer. Math. 2(1), 84–90 (1960)

    Article  MathSciNet  Google Scholar 

  16. Iske, A., Le Borne, S., Wende, M.: Hierarchical matrix approximation for kernel-based scattered data interpolation. SIAM J. Sci. Comput. 39(5), A2287–A2316 (2017)

    Article  MathSciNet  Google Scholar 

  17. Wendland, H.: Multiscale analysis in sobolev spaces on bounded domains. Numer. Math. 116(3), 493–517 (2010)

    Article  MathSciNet  Google Scholar 

  18. Wendland, H.: Scattered data approximation. Cambridge monographs on applied and computational mathematics. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  19. Wendland, H.: Multiscale radial basis functions, pp 265–299. Springer International Publishing, Cham (2017)

    Google Scholar 

Download references

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Le Borne, S., Wende, M. Multilevel interpolation of scattered data using \({\mathcal{H}}\)-matrices. Numer Algor 85, 1175–1193 (2020). https://doi.org/10.1007/s11075-019-00860-1

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