Uniform weak tractability

Siedlecki, Paweł

doi:10.1016/j.jco.2013.04.006

Cited by 55 publications

(28 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A problem is called uniformly weakly tractable [23] if it is (α,β)-weakly tractable for all α,β > 0. From Lemma 7.3 we can conclude that the approximation problem Id : H s,p (T d ) → L 2 (T d ) is (α, β)-weakly tractable for α > p/s and all β > 0 (which has also been observed in [24,Thm 4.1]).…”

Section: Tractability Analysismentioning

confidence: 99%

Counting Via Entropy: New Preasymptotics for the Approximation Numbers of Sobolev Embeddings

Kühn¹,

Mayer²,

Ullrich

2016

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

We study the optimal linear L 2 -approximation by operators of finite rank (i.e., approximation numbers) for the isotropic periodic Sobolev space H s (T d ) of fractional smoothness on the d-torus. For a family of weighted norms, which penalize Fourier coefficientsf (k) by a weight w s,p (k) = (1 + k p p ) s/p , 0 < p ≤ ∞, we prove that the n-th approximation number of the embedding Id :is characterized by the n-th (non-dyadic) entropy number of the embedding id : ℓ d p → ℓ d ∞ raised to the power s. From the known behavior of these entropy numbers we gain a complete understanding of the approximation numbers in terms of n and d for all n ∈ N and d ∈ N.

show abstract

Section: Tractability Analysismentioning

confidence: 99%

Counting Via Entropy: New Preasymptotics for the Approximation Numbers of Sobolev Embeddings

Kühn¹,

Mayer²,

Ullrich

2016

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

show abstract

“…The concept of weak tractability was recently strengthened in [12] by introducing the notion of uniform weak tractability, which holds for multivariate integration defined over the class C k d (L) iff lim d+ε −1 →∞ ln n(ε, C k d (L)) d α + ε −α = 0 for all α ∈ (0, 1).…”

Section: The Case Of L Jd =mentioning

confidence: 99%

The curse of dimensionality for numerical integration of smooth functions II

Hinrichs

Novak

Ullrich

et al. 2014

Journal of Complexity

View full text Add to dashboard Cite

We prove the curse of dimensionality in the worst case setting for numerical integration for a number of classes of smooth d-variate functions. Roughly speaking, we consider different bounds for the directional or partial derivatives of f ∈ C k (D d ) and ask whether the curse of dimensionality holds for the respective classes of functions. We always assume that D d ⊂ R d has volume one and we often assume additionally that D d is either convex or that its radius is proportional to √ d. In particular, D d can be the unit cube. We consider various values of k including the case k = ∞ which corresponds to infinitely differentiable functions. We obtain necessary and sufficient conditions, and in some cases a full characterization for the curse of dimensionality. For infinitely differentiable functions we prove the curse if the bounds on the successive derivatives are appropriately large. The proof technique is based on a volume estimate of a neighborhood of the convex hull of n points which decays exponentially fast in d. For k = ∞, we also study conditions for quasi-polynomial, weak and uniform weak tractability. In particular, weak tractability holds if all directional derivatives are bounded by one. It is still an open problem if weak tractability holds if all partial derivatives are bounded by one.

show abstract

“…then the approximation problem is called (s, t)-weakly tractable. Clearly, (1, 1)weak tractability is just weak tractability, whereas the approximation problem is uniformly weakly tractable if it is (s, t)-weakly tractable for all positive s and t (see [19]). Also, if the approximation problem suffers from the curse of dimensionality, then for any s > 0, 0 < t ≤ 1, it is not (s, t)-weakly tractable.…”

Section: Tractability Analysismentioning

confidence: 99%

Approximation numbers of Sobolev and Gevrey type embeddings on the sphere and on the ball—Preasymptotics, asymptotics, and tractability

Chen

Wang

2019

Journal of Complexity

View full text Add to dashboard Cite

In this paper, we investigate optimal linear approximations (napproximation numbers ) of the embeddings from the Sobolev spaces H r (r > 0) for various equivalent norms and the Gevrey type spaces G α,β (α, β > 0) on the sphere S d and on the ball B d , where the approximation error is measured in the L 2 -norm. We obtain preasymptotics, asymptotics, and strong equivalences of the above approximation numbers as a function in n and the dimension d. We emphasis that all equivalence constants in the above preasymptotics and asymptotics are independent of the dimension d and n. As a consequence we obtain that for the absolute error criterion the approximation problems I d : H r → L 2 are weakly tractable if and only if r > 1, not uniformly weakly tractable, and do not suffer from the curse of dimensionality. We also prove that for any α, β > 0, the approximation problems I d : G α,β → L 2 are uniformly weakly tractable, not polynomially tractable, and quasi-polynomially tractable if and only if α ≥ 1. r/d exist, having the same value for various norms. We also prove that for 0 < α < 1,, lim n→∞ e βγn α/d a n (I d : G α,β (S d ) → L 2 (S d )) = 1.

show abstract

Uniform weak tractability

Cited by 55 publications

References 3 publications

Counting Via Entropy: New Preasymptotics for the Approximation Numbers of Sobolev Embeddings

Counting Via Entropy: New Preasymptotics for the Approximation Numbers of Sobolev Embeddings

The curse of dimensionality for numerical integration of smooth functions II

Approximation numbers of Sobolev and Gevrey type embeddings on the sphere and on the ball—Preasymptotics, asymptotics, and tractability

Contact Info

Product

Resources

About