6 research outputs found
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Weighted Geometric Discrepancies and Numerical Integration on Reproducing Kernel Hilbert Spaces
We extend the notion of L2-B-discrepancy introduced in [E. Novak, H. Wozniakowski, L2 discrepancy and multivariate integration, in: Analytic number theory. Essays in honour of Klaus Roth. W. W. L. Chen, W. T. Gowers, H. Halberstam, W. M. Schmidt, and R. C. Vaughan (Eds.), Cambridge University Press, Cambridge, 2009, 359"“388] to what we want to call weighted geometric L2-discrepancy. This extended notion allows us to consider weights to moderate the importance of different groups of variables, and additionally volume measures different from the Lebesgue measure as well as classes of test sets different from measurable subsets of Euclidean spaces. We relate the weighted geometric L2-discrepancy to numerical integration defined over weighted reproducing kernel Hilbert spaces and settle in this way an open problem posed by Novak and Wozniakowski. Furthermore, we prove an upper bound for the numerical integration error for cubature formulas that use admissible sample points. The set of admissible sample points may actually be a subset of the integration domain of measure zero. We illustrate that particularly in infinite dimensional numerical integration it is crucial to distinguish between the whole integration domain and the set of those sample points that actually can be used by algorithms
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Entropy, Randomization, Derandomization, and Discrepancy
The star discrepancy is a measure of how uniformly distributed a finite point set is in the d-dimensional unit cube. It is related to high-dimensional numerical integration of certain function classes as expressed by the Koksma-Hlawka inequality. A sharp version of this inequality states that the worst-case error of approximating the integral of functions from the unit ball of some Sobolev space by an equal-weight cubature is exactly the star discrepancy of the set of sample points. In many applications, as, e.g., in physics, quantum chemistry or finance, it is essential to approximate high-dimensional integrals. Thus with regard to the Koksma- Hlawka inequality the following three questions are very important: (i) What are good bounds with explicitly given dependence on the dimension d for the smallest possible discrepancy of any n-point set for moderate n? (ii) How can we construct point sets efficiently that satisfy such bounds? (iii) How can we calculate the discrepancy of given point sets efficiently? We want to discuss these questions and survey and explain some approaches to tackle them relying on metric entropy, randomization, and derandomization
A New Randomized Algorithm to Approximate the Star Discrepancy Based on Threshold Accepting
We present a new algorithm for estimating the star discrepancy of arbitrary point sets. Similar to the algorithm for discrepancy approximation of Winker and Fang [SIAM J. Numer. Anal. 34 (1997), 2028{2042] it is based on the optimization algorithm threshold accepting. Our improvements include, amongst others, a non-uniform sampling strategy which is more suited for higher-dimensional inputs and additionally takes into account the topological characteristics of given point sets, and rounding steps which transform axis-parallel boxes, on which the discrepancy is to be tested, into critical test boxes. These critical test boxes provably yield higher discrepancy values, and contain the box that exhibits the maximum value of the local discrepancy. We provide comprehensive experiments to test the new algorithm. Our randomized algorithm computes the exact discrepancy frequently in all cases where this can be checked (i.e., where the exact discrepancy of the point set can be computed in feasible time). Most importantly, in higher dimension the new method behaves clearly better than all previously known methods