Fast algorithms for component-by-component construction of rank- lattice rules in shift-invariant reproducing kernel Hilbert spaces |
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Authors: | Dirk Nuyens Ronald Cools |
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Institution: | Katholieke Universiteit Leuven, Department of Computer Science, Celestijnenlaan 200A, B-3001 Heverlee, Belgium ; Katholieke Universiteit Leuven, Department of Computer Science, Celestijnenlaan 200A, B-3001 Heverlee, Belgium |
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Abstract: | We reformulate the original component-by-component algorithm for rank- lattices in a matrix-vector notation so as to highlight its structural properties. For function spaces similar to a weighted Korobov space, we derive a technique which has construction cost , in contrast with the original algorithm which has construction cost . Herein is the number of dimensions and the number of points (taken prime). In contrast to other approaches to speed up construction, our fast algorithm computes exactly the same quantity as the original algorithm. The presented algorithm can also be used to construct randomly shifted lattice rules in weighted Sobolev spaces. |
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Keywords: | Numerical integration quasi--Monte Carlo rank-$1$ lattice rules component-by-component construction fast algorithms |
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