A very fast C++ implementation of the Halton sequence that supports both Faure-permutations and random digit permutations. Together with the component (i / N) this will yield a Hammersley point set. Comes with a Python script to generate optimized code for arbitrary dimensions. It also includes code to enumerate samples in a 2D domain, for example pixels.

A C++ implementation of the small linear feedback shift registers (LFSR) for use in a
Markov chain quasi-Monte Carlo context, as described in *S. Chen, M. Matsumoto, T.
Nishimura, and A. Owen:* New inputs and methods for Markov chain quasi-Monte
Carlo.

A C++ implementation of a simple and fast random-access Sobol’-sequence generator,
based on the direction numbers published in *S. Joe and F. Y. Kuo:* Constructing Sobol sequences with better
two-dimensional projections.

with S. Premoze, A. Keller, and M. Raab.

with M. Raab and A. Keller.

In H. Wozniakowski and L. Plaskota (eds.), Monte Carlo and Quasi-Monte Carlo Methods 2010, Springer-Verlag, Berlin, 2012.

*Abstract:* 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.

with A. Keller.

In H. Wozniakowski and L. Plaskota (eds.), Monte Carlo and Quasi-Monte Carlo Methods 2010, Springer-Verlag, Berlin, 2012.

*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.

with A. Keller and M. Droske.

In H. Wozniakowski and L. Plaskota (eds.), Monte Carlo and Quasi-Monte Carlo Methods 2010, Springer-Verlag, Berlin, 2012.

*Abstract:* 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.

with M. Stich, S. Nawaz, and A. Keller.

In Proceedings of the Conference on High Performance Graphics (HPG ‘11).

*Abstract:* When a bounding volume hierarchy is used for accelerating the intersection
*of
rays and scene geometry, one common way to incorporate motion blur is to interpolate
node bounding volumes according to the time of the ray. However, such hierarchies
typically exhibit large overlap between bounding volumes, which results in an
inefficient traversal. This work builds upon the concept of spatially partitioning nodes
during tree construction in order to reduce overlap in the presence of moving objects.
The resulting hierarchies are often significantly cheaper to traverse than those
generated by classic approaches.

with A. Keller.

In P. L’Ecuyer and A. Owen (eds.), Monte Carlo and Quasi-Monte Carlo Methods 2008, Springer-Verlag, Berlin, 2009.

*Abstract:* 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.

Diploma thesis at Ulm University, Germany, 2008.

Supervisors: A. Keller and M. Weber.

with J. Hanika , R. Schwede, and A. Keller.

In A. Keller, S. Heinrich, and H. Niederreiter (eds.), Monte Carlo and Quasi-Monte Carlo Methods 2006, Springer-Verlag, Berlin, 2008.

*Abstract:* 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.

with M. Raab, J. Hanika, M. Finckh, and A. Keller.