Deterministic and Stochastic Error Bounds in Numerical Analysis

By Erich Novak. 1988

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Citations in the corpus

Convergence guarantees for kernel-based quadrature rules in misspecified settings (2016, neutral): “, c = 1/2) is given by n−b with b = r/d [15].”
On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions (2015, background): “ third problem is of interest (and outside of the scope of this paper), namely the problem of ﬁnding the minimum of a function given (potentially noisy ) function evaluations. For noiseless problems, Novak (1988, page 26) shows that the approximation and optimization problems have the same worst-cas”
Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance (2017, unknown): “from μ with uniform weights αi := n−1 for 1 ≤ i ≤ n is asymptotically suboptimal [39].”
Control Functionals for Quasi-Monte Carlo Integration (2015, background): “It is known that this rate is best-possible [19] and explicit algorithms to generate design points that attain this rate are now available for many (but not all) values of α [6].”
Set membership identification of nonlinear systems (2000, background): “Analgorithm � g is called globally optimal if Eg(� g )= inf � Eg(� ). This is the optimality concept usually investigated inthe approximationtheory literature ( Traub et al., 1988; Novak, 1988; Wasilkowski & Wo> zniakowski, 2001).”
Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands (2010, unknown): “The upper bound is best possible (apart from the power of the logN factor), since there is also a lower bound on the standard deviation; see [13].”
Bayesian calibration of computer models (2001, unknown): “See Novak (1988), for example.”
Quadrature rules and distribution of points on manifolds (2010, background): “ence (1) and (2) imply (3), and by (4) this latter is optimal. When the manifold is a torus or a sphere these results are essentially known, and indeed there is a huge literature on this subject. See [24] for deterministic and stochastic error bounds in numerical analysis. In particular, (3) and (4)”
Is There a Curse of Dimensionality for Contraction Fixed Points in the Worst Case (2002, background): “It is known that the exponent p∗ that appears in Theorem 5 is 1; see Novak (1988).”
Discrepancy Theory and Quasi-Monte Carlo Integration (2014, unknown): “For more results in this direction we refer to Sloan andWoźniakowski [108] or to Novak and Woźniakowski [85, Chapter 16].”
Tractability of Multivariate Problems (2008, background): “4 is mainly based on results of Wasilkowski [324], see also [206].”
On the Equivalence between Quadrature Rules and Random Features (2015, background): “(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) , and thus a degradation compared to the non-squared L 2 -loss of n 1 / 2 (plus additional logarithmic terms), which corresponds to the (non-improvable) result of Novak (1988, page 36).”
A lower bound for the worst-case cubature error on spheres of arbitrary dimension (2006, methodology): “The method of proof is analogous to the one in the special case of S2 in [6] which was inspired by the ideas with which lower bounds for the worst-case cubature error in certain spaces of continuous functions on the unit cube [0, 1]r are derived (see [1,8]).”