Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fast component-by-component construction of lattice algorithms for
  multivariate approximation with POD and SPOD weights   In a recent paper by the same authors, we provided a theoretical foundation
for the component-by-component (CBC) construction of lattice algorithms for
multivariate $L_2$ approximation in the worst case setting, for functions in a
periodic space with general weight parameters. The construction led to an error
bound that achieves the best possible rate of convergence for lattice
algorithms. Previously available literature covered only weights of a simple
form commonly known as product weights. In this paper we address the
computational aspect of the construction. We develop fast CBC construction of
lattice algorithms for special forms of weight parameters, including the
so-called POD weights and SPOD weights which arise from PDE applications,
making the lattice algorithms truly applicable in practice. With $d$ denoting
the dimension and $n$ the number of lattice points, we show that the
construction cost is $\mathcal{O}(d\,n\log(n) + d^2\log(d)\,n)$ for POD
weights, and $\mathcal{O}(d\,n\log(n) + d^3\sigma^2\,n)$ for SPOD weights of
degree $\sigma\ge 2$. The resulting lattice generating vectors can be used in
other lattice-based approximation algorithms, including kernel methods or
splines.

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The categories of this paper are: physics
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The categories of this paper are: solv-int
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The categories of this paper are: dg-ga
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The categories of this paper are: gr-qc
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The categories of this paper are: econ
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: eess
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The categories of this paper are: patt-sol
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The categories of this paper are: q-alg
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The categories of this paper are: cmp-lg
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The categories of this paper are: supr-con
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The categories of this paper are: bayes-an
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The categories of this paper are: funct-an
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The categories of this paper are: mtrl-th