## Tractability of approximation for weighted Korobov spaces on classical and quantum computers

Venue: | Found. of Comput. Math |

Citations: | 11 - 1 self |

### BibTeX

@ARTICLE{Novak_tractabilityof,

author = {Erich Novak and Ian H. Sloan},

title = {Tractability of approximation for weighted Korobov spaces on classical and quantum computers},

journal = {Found. of Comput. Math},

year = {}

}

### OpenURL

### Abstract

We study the approximation problem (or problem of optimal recovery in the L2norm) for weighted Korobov spaces with smoothness parameter α. The weights γj of the Korobov spaces moderate the behavior of periodic functions with respect to successive variables. The non-negative smoothness parameter α measures the decay of Fourier coefficients. For α = 0, the Korobov space is the L2 space, whereas for positive α, the Korobov space is a space of periodic functions with some smoothness and the approximation problem corresponds to a compact operator. The periodic functions are defined on [0,1] d and our main interest is when the dimension d varies and may be large. We consider algorithms using two different classes of information. The first class Λ all consists of arbitrary linear functionals. The second class Λ std consists of only function values and this class is more realistic in practical computations. We want to know when the approximation problem is tractable. Tractability means that there exists an algorithm whose error is at most ε and whose information cost is bounded by a polynomial in the dimension d and in ε −1. Strong tractability means that

### Citations

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Citation Context ... whereas for positive α, the Korobov space is a space of periodic functions with some smoothness and the approximation problem corresponds to a compact operator. The periodic functions are defined on =-=[0,1]-=- d and our main interest is when the dimension d varies and may be large. We consider algorithms using two different classes of information. The first class Λ all consists of arbitrary linear function... |

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Citation Context ...approximation as we shall see in Sections 5.2 and 5.3. Let us summarize the known results about the order of eq n (SN, BN p ) for p = ∞ and p = 2. The case p = ∞ is due to [6], [3] (upper bounds) and =-=[13]-=- (lower bounds). The results in the case p = 2 are due to [8]. Further results for arbitrary 1 ≤ p ≤ ∞ can be also found in [8] and [11]. In what follows, by “log” we mean the logarithm to the base 2.... |

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Citation Context ...e functions ak do not depend on f; they form the fixed output basis of the algorithm. Necessary and sufficient conditions on tractability of approximation in the worst case setting easily follow from =-=[12, 27, 28]-=-. With we have: sγ = inf � s > 0 : ∞� j=1 γ s j � < ∞ , 1. Let α ≥ 0. Strong tractability and tractability of approximation in the class Λ all are equivalent, and this holds iff α > 0 and the sum-expo... |

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Citation Context ... of functional evaluations is closely related to the worst case complexity of the approximation problem, see e.g., [24]. This explains our choice of notation. We are ready to define tractability, see =-=[29]-=-. We say that approximation is tractable in the class Λ iff there exist nonnegative numbers C, p and q such that comp wor (ε, Hd, Λ) ≤ C ε −p d q k=1 ∀ ε ∈ (0, 1), ∀ d ∈ N. (4) The essence of tractabi... |

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20 |
Tractability of multivariate integration for weighted Korobov classes
- Sloan, Wo´zniakowski
(Show Context)
Citation Context ...ns f : [0, 1] d → C that belong to Korobov spaces. These are the most studied spaces of periodic functions. Usually, the unweighted case, in which all variables play the same role, is analyzed. As in =-=[12, 23]-=-, in this paper we analyze a more general case of weighted Korobov spaces, in which the successive variables may have diminishing importance. We consider the unit ball of weighted Korobov spaces Hd. H... |

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Citation Context ...f we only allow the use of random bits (coin tossing as a source of randomness) then again we need function values to be continuous linear functionals, which is guaranteed by the condition α > 1, see =-=[16]-=- for a formal definition of such “restricted” Monte Carlo algorithms. We add that it is easy to obtain random bits from a quantum computer while it is not possible to obtain random numbers from [0, 1]... |

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Citation Context ...e functions ak do not depend on f; they form the fixed output basis of the algorithm. Necessary and sufficient conditions on tractability of approximation in the worst case setting easily follow from =-=[12, 27, 28]-=-. With we have: sγ = inf � s > 0 : ∞� j=1 γ s j � < ∞ , 1. Let α ≥ 0. Strong tractability and tractability of approximation in the class Λ all are equivalent, and this holds iff α > 0 and the sum-expo... |

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Citation Context ...and p = 2. The case p = ∞ is due to [6], [3] (upper bounds) and [13] (lower bounds). The results in the case p = 2 are due to [8]. Further results for arbitrary 1 ≤ p ≤ ∞ can be also found in [8] and =-=[11]-=-. In what follows, by “log” we mean the logarithm to the base 2. Theorem 3 There are constants cj > 0 for j ∈ {1, . . .,9} such that for all n, N ∈ N with 2 < n ≤ c1N we have e q n (SN, B N ∞ ) ≍ n−1 ... |

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Citation Context ... average value of �f − An,d(f, ω)�2 L2([0,1] 2 ) with respect to ω according to a probability measure ̺, and then by taking the worst case with respect to f from the unit ball of Hd. It is known, see =-=[15]-=-, that randomization does not help over the worst case setting for the class Λall . That is why, for the class Λall , tractability and strong tractability in the randomized setting are equivalent to t... |

5 |
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Citation Context ...Quantum Setting Our analysis in this section is based on the framework introduced in [8] of quantum algorithms for the approximate solution of problems of analysis. We refer the reader to the surveys =-=[4]-=-, [21], and to the monographs [7], [14], and [20] for general reading on quantum computation. This approach is an extension of the framework of information-based complexity theory (see [24] and, more ... |

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2000): Complexity of linear problems with a fixed output basis
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Citation Context ... elements of L2([0, 1] d ), and the Lk’s are some continuous linear functionals defined on Hd. Observe that the functions ak do not depend on f, they form the fixed output basis of the algorithm, see =-=[18]-=-. For all the algorithms in this paper we use the optimal basis consisting of the eigenvectors of Wd. We assume that Lk ∈ Λ, and consider two classes of information Λ. The first class is Λ = Λ all = H... |

2 |
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Citation Context ...um Setting Our analysis in this section is based on the framework introduced in [8] of quantum algorithms for the approximate solution of problems of analysis. We refer the reader to the surveys [4], =-=[21]-=-, and to the monographs [7], [14], and [20] for general reading on quantum computation. This approach is an extension of the framework of information-based complexity theory (see [24] and, more formal... |

1 | 2001): Path integration on a quantum computer, submitted for publication. See also http://arXiv.org/abs/quant-ph/0109113 - Traub, Wo´zniakowski |

1 |
Quantum summation with an application to integration. J. Complexity 18. See also http://arXiv.org/abs/quant-ph/0105116. 35 S. Heinrich (2001): Quantum integration in Sobolev classes. Preprint. See also http://arXiv.org/abs/quant-ph/0112153
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(Show Context)
Citation Context ... study the quantum setting. We consider quantum algorithms that run on a (hypothetical) quantum computer. Our analysis in this section is based on the framework 6for quantum algorithms introduced in =-=[8]-=- that is relevant for the approximate solution of problems of analysis. We only consider upper bounds for the class Λstd and weighted Korobov spaces with α > 1 and sγ < ∞. We present a quantum algorit... |