Low discrepancy sequences, which are based on radical inversion, expose an intrinsic stratification. New algorithms are presented to efficiently enumerate the points of the Halton and (t, s)-sequences per stratum. This allows for consistent and adaptive integro-approximation as for example in image synthesis.
Summary.Many experiments in computer graphics imply that the average quality of quasi-Monte Carlo integro-approximation is improved as the minimal distance of the point set grows. While the definition of (t, m, s)-nets in base b guarantees extensive stratification properties, which are best for t = 0, sampling points can still lie arbitrarily close together. We remove this degree of freedom, report results of two computer searches for (0, m, 2)-nets in base 2 with maximized minimum distance, and present an inferred construction for general m. The findings are especially useful in computer graphics and, unexpectedly, some (0, m, 2)-nets with the best minimum distance properties cannot be generated in the classical way using generator matrices.
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 quasiMonte 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.
The quality parameter t of (t, m, s)-nets controls extensive stratification properties of the generated sample points. However, the definition allows for points that are arbitrarily close across strata boundaries. We continue the investigation of (t, m, s)-nets under the constraint of maximizing the mutual distance of the points on the unit torus and present two new constructions along with algorithms. The first approach is based on the fact that reordering (t, s)-sequences can result in (t, m, s + 1)-nets with varying toroidal distance, while the second algorithm generates points by permutations instead of matrices.
The simulation of light transport often involves specular and transmissive surfaces, which are modeled by functions that are not square integrable. However, in many practical cases unbiased Monte Carlo methods are not able to handle such functions efficiently and consistent Monte Carlo methods are applied. Based on quasi-Monte Carlo integration, a deterministic alternative to the stochastic approaches is introduced. The new method for deterministic consistent functional approximation uses deterministic consistent density estimation.
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