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BIT Numerical Mathematics

, Volume 46, Issue 3, pp 455–477 | Cite as

Some observations on local least squares

  • R. H. BartelsEmail author
  • G. H. Golub
  • F. F. Samavati
Article

Abstract

In previous work we introduced a construction to produce biorthogonal multiresolutions from given subdivisions. The approach involved estimating the solution to a least squares problem by means of a number of smaller least squares approximations on local portions of the data. In this work we use a result by Dahlquist, et al. on the method of averages to make observational comparisons between this local least squares estimation and full least squares approximation. We have explored examples in two problem domains: data reduction and data approximation. We observe that, particularly for design matrices with a repetitive pattern of column entries, the least squares solution is often well estimated by local least squares, that the estimation rapidly improves with the size of the local least squares problems, and that the quality of the estimate is largely independent of the size of the full problem.

Key words

subdivision refinement interpolation approximation least squares projection subspace angles left inverse B-splines  

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References

  1. 1.
    N. Anderson, A generalization of the method of averages for overdetermined linear systems, Math. Comp., 29 (1975), pp. 607–614.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    R. H. Bartels and F. F. Samavati, Reversing subdivision rules: Local linear conditions and observations on inner products, J. Comput. Appl. Math., 119 (2000), pp. 29–67.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    G. Dahlquist, B. Sjöberg, and P. Svensson, Comparison of the method of averages with the method of least squares, Math. Comp., 22 (1968), pp. 833–845.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    C. de Boor, A Practical Guide to Splines, Applied Mathematical Sciences 27, Springer, 1978.Google Scholar
  5. 5.
    N. Dyn, D. Levin, and J. Gregory, A 4-point interpolatory subdivision scheme for curve design, Comput. Aided Geom. Design, 4 (1988), pp. 257–268.CrossRefMathSciNetGoogle Scholar
  6. 6.
    R. N. Goldman and T. Lyche, Knot Insertion and Deletion Algorithms for B-Spline Curves and Surfaces, SIAM, 1993.Google Scholar
  7. 7.
    G. H. Golub and C. F. V. Loan, Matrix Computations, 2nd edn., The Johns Hopkins University Press, 1989.Google Scholar
  8. 8.
    N. Litke, J. Levin, and P. Schröder, Trimming for subdivision surfaces, Comput. Aided Geom. Design, 18 (2001), pp. 463–481.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    F. F. Samavati and R. H. Bartels, Reversing subdivision using local linear conditions: Generating multiresolutions on regular triangular meshes, Technical report, University of Calgary, Computer Science, TR 2002-711-14, December 2002.Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Stanford UniversityStanfordU.S.A.
  3. 3.University of CalgaryCalgaryCanada

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