Optimal integration error on anisotropic classes for restricted Monte Carlo and quantum algorithms

Ye, Peixin; Hu, Xuemei

doi:10.1016/j.jat.2007.04.013

Cited by 10 publications

(10 citation statements)

References 21 publications

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“…Conditioned on G j 1 the random variable G j 2 +n is uniformly distributed on D (q) . Combining this with (22) and (23) shows that conditioned on G j 1 the random variable G j 1 + G j 2 +n + 2 −(q+1) mod 1 is uniformly distributed on D (q) . Hence the random variable G j 1 + G j 2 +n + 2 −(q+1) mod 1 is uniformly distributed on D (q) .…”

Section: Corollary 16mentioning

confidence: 66%

“…In most of the papers on randomized algorithms for continuous problems, uniformly distributed random numbers from [0, 1] are assumed to be available. Random bit Monte Carlo algorithms are studied for the classical, finite-dimensional quadrature problem to approximate [0,1] d f (x) dx in, e.g., Gao et al [8], Heinrich et al [12], Novak [14,15,16], Traub and Woźniakowski [20], Ye and Hu [22]. See Novak and Pfeiffer [17] for a related approach to integral equations.…”

Section: Introductionmentioning

confidence: 99%

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Random bit multilevel algorithms for stochastic differential equations

Giles

Hefter

Mayer

et al. 2019

Journal of Complexity

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We study the approximation of expectations E(f (X)) for solutions X of SDEs and functionals f : C([0, 1], R r ) → R by means of restricted Monte Carlo algorithms that may only use random bits instead of random numbers. We consider the worst case setting for functionals f from the Lipschitz class w.r.t. the supremum norm. We construct a random bit multilevel Euler algorithm and establish upper bounds for its error and cost. Furthermore, we derive matching lower bounds, up to a logarithmic factor, that are valid for all random bit Monte Carlo algorithms, and we show that, for the given quadrature problem, random bit Monte Carlo algorithms are at least almost as powerful as general randomized algorithms.

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Section: Corollary 16mentioning

confidence: 66%

Section: Introductionmentioning

confidence: 99%

Random bit multilevel algorithms for stochastic differential equations

Giles

Hefter

Mayer

et al. 2019

Journal of Complexity

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“…The corresponding restricted randomized algorithms are called bit Monte Carlo algorithms. A non-adaptive version of these was considered in [11,14,3,17]. Most frequently used is the case of uniform distributions on [0, 1].…”

Section: Restricted Randomized Algorithms In a General Settingmentioning

confidence: 99%

“…Restricted Monte Carlo algorithms were considered in [12,13,16,11,14,3,17,4,5,6]. Restriction usually means that the algorithm has access only to random bits or to random variables with finite range.…”

Section: Introductionmentioning

confidence: 99%

Lower bounds for the number of random bits in Monte Carlo algorithms

Heinrich¹

2020

Preprint

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We continue the study of restricted Monte Carlo algorithms in a general setting. Here we show a lower bound for minimal errors in the setting with finite restriction in terms of deterministic minimal errors. This generalizes a result of [11] to the adaptive setting. As a consequence, the lower bounds on the number of random bits from [11] also hold in this setting. We also derive a lower bound on the number of needed bits for integration of Lipschitz functions over the Wiener space, complementing a result of [5].

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“…Proof of Theorem 1. We begin with the decomposition of the cube D as in [8,9]. Let n 0 be sufficiently large integer such that n 0…”

Section: Lemma 6 Let (ψ I ) L I=1 Be a Collection Of Functions In B(h R P (D))mentioning

confidence: 99%