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When Are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals?
It is proved that the minimalworst case error of quasi-Monte Carlo algorithms does not depend on the dimensiondiff the sum of the weights is finite, and the minimal number of function values in the worst case setting needed to reduce the initial error by ? is bounded byC??p, where the exponentp? 1, 2], andCdepends exponentially on thesum of weights.
Tractability of Multivariate Problems
The main purpose of this book is to study weighted spaces and to obtain conditions on the weights that are necessary and sufficient to achieve various notions of tractability, depending on how to measure the lack of exponential dependence.
Explicit Cost Bounds of Algorithms for Multivariate Tensor Product Problems
An upper bound is obtained, which is independent of d, for the number, n(?, d), of points for which discrepancy is at most ?, n (?, d) ? 7.26??2.454, ?d, ? ? 1.
Information-based complexity seeks to develop general results about the intrinsic difficulty of solving problems where available information is partial or approximate and to apply these results to…
The inverse of the star-discrepancy depends linearly on the dimension
We study bounds on the classical ∗-discrepancy and on its inverse. Let n∞(d, e) be the inverse of the ∗-discrepancy, i.e., the minimal number of points in dimension d with the ∗-discrepancy at most…
Tractability of Multivariate Problems Volume II: Standard Information for Functionals
A general theory of optimal algorithms
This work is a general theory of optimal algorithms, it will really give you the good idea to be successful.
Estimating the Largest Eigenvalue by the Power and Lanczos Algorithms with a Random Start
This paper addresses the problem of computing an approximation to the largest eigenvalue of an $n \times n$ large symmetric positive definite matrix with relative error at most $\varepsilon $ with sharp bounds on the average relative error and on the probabilistic relative failure.
Convergence and Complexity of Newton Iteration for Operator Equations
An optimal convergence condition for Newton iteration in a Banach space is established and which stronger condition must be imposed to also assure good complexity.
Tractability of Multivariate Integration for Weighted Korobov Classes
We study the worst-case error of multivariate integration in weighted Korobov classes of periodic functions of d coordinates. This class is defined in terms of weights ?j which moderate the behavior…