Abstract
A general concept for parallelizing quasi-Monte Carlo methods is introduced. By considering the distribution of computing jobs across a multiprocessor as an additional problem dimension, the straightforward application of quasi-Monte Carlo methods implies parallelization. The approach in fact partitions a single low-discrepancy sequence into multiple low-discrepancy sequences. This allows for adaptive parallel processing without synchronization, i.e. communication is required only once for the final reduction of the partial results. Independent of the number of processors, the resulting algorithms are deterministic, and generalize and improve upon previous approaches.
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- 1.
Actually, any quadrature rule could be chosen.
- 2.
The partitions can also be scaled to fill the (s + 1)-dimensional unit cube again. In other words, one could reuse the component chosen for selecting samples for each job, which is more efficient since one component less must be generated. Reformulating Eq. 2 accordingly, requires only the generation of s-dimensional samples:
$${S}_{j} \approx \frac{1} {n}{\sum \nolimits }_{i=0}^{n-1}{\chi }_{ j}(N \cdot {x}_{i,c}) \cdot f({x}_{i,1},\ldots ,{x}_{i,c-1},N \cdot {x}_{i,c} - j,{x}_{i,c+1},\ldots ,{x}_{i,s})$$However, this variant is not recommended, because the resulting ensemble of samples may not be well-stratified in the dimension c.
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Acknowledgements
This work has been dedicated to Stefan Heinrich’s 60th birthday. The authors thank Matthias Raab for discussion.
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Keller, A., Grünschloß, L. (2012). Parallel Quasi-Monte Carlo Integration by Partitioning Low Discrepancy Sequences. In: Plaskota, L., Woźniakowski, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2010. Springer Proceedings in Mathematics & Statistics, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27440-4_27
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