2003, Vol. 43, No. 1, pp. 001–018 c ○ Kluwer Academic Publishers SOME OBSERVATIONS ON LOCAL LEAST SQUARES Abstract.
BIT Numerical Mathematics(2008)
摘要
In a previous work (2) we introduced a construction to produce biorthogonal multiresolutions from given subdivisions. The approach we employed could be interpreted as 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. (3), on the method of averages to explore how and to what extent this local least squares estimation approaches the full least squares approximation. Two example problem domains are used: 1. Data reduction processes - characterized by the removal of knots from splines, the first stage in the construction of a biorthogonal multiresolution, and the simplification of geometric objects according to subdivision schemes; 2. Approximation processes - characterized by data sampling and discrete least squares. We observe in these examples that 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 independent of the size of the full problem in uniform situations.
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关键词
subdivision,refinement,interpolation,approximation,least squares,projection,subspace angles,left inverse,B-splines
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