Simplistically put, using a random, Monte Carlo style algorithm, but deterministically, by sampling at well-chosen points.

Key words: *low discrepancy set/sequence*.

Some of the series of points used are nice for parallelised algorithms, by the way, in the same way that randomised algorithms are.

Sobol nets, others? Do Gray codes, fit in here? If we aren’t doing this incrementally we can pre-generate a point set rather than a sequence.

See

- Quasi Monte Carlo introduction by Sasha Nikolov
- algorithmic complexity
- Practical example: Generating Hammersley point sets for image synthesis.
- Dirk Nuyens’ magic point shop contains “QMC point generators and generating vectors” (MATLAB/octave/C++)
- When Random Numbers Are Too Random: Low Discrepancy Sequences « The blog at the bottom of the sea

## References

Beck, József. 1996. “Discrepancy Theory.” In

*Handbook of Combinatorics*, edited by Vera T Sós. Vol. 2. MIT Press, Cambridge, MA.Chazelle, Bernard. 2001.

*The discrepancy method: randomness and complexity*. 1st paperback ed. Cambridge: Cambridge Univ. Press.Dick, Josef, Frances Y. Kuo, and Ian H. Sloan. 2013. “High-Dimensional Integration: The Quasi-Monte Carlo Way.”

*Acta Numerica*22 (May): 133–288.Kuipers, L., and H. Niederreiter. 2012.

*Uniform Distribution of Sequences*. Dover Publications.Matousek, Jiri. 2010. “Geometric Discrepancy: An Illustrated Guide.”

*Algorithms and Combinatorics*18.Panneton, François, and Pierre L’Ecuyer. 2006. “Infinite-Dimensional Highly-Uniform Point Sets Defined via Linear Recurrences in \(\mathbb{F}_{2^w }\).” In

*Monte Carlo and Quasi-Monte Carlo Methods 2004*, edited by Harald Niederreiter and Denis Talay, 419–29. Springer Berlin Heidelberg.Parlett, Beresford N. 1992. “Some Basic Information on Information-Based Complexity Theory.”

*Bulletin of the American Mathematical Society*26 (1): 3–27.Schwab, C., and A. M. Stuart. 2012. “Sparse Deterministic Approximation of Bayesian Inverse Problems.”

*Inverse Problems*28 (4): 045003.Sobol’, Il’ya Meerovich. 1966. “Distribution of Points in a Cube and Integration Nets.”

*Uspekhi Matematicheskikh Nauk*21 (5): 271–72.Srinivasan, Aravind. 2000. “Low-Discrepancy Sets for High-Dimensional Rectangles: A Survey.”

*Bulletin of the EATCS*70: 67–76.Stuart, Andrew M. 2010. “Inverse Problems: A Bayesian Perspective.”

*Acta Numerica*19: 451–559.Traub, J. F. 1988. “Information-Based Complexity.”

Wang, Xiaoqun, and Ian H. Sloan. 2008. “Low Discrepancy Sequences in High Dimensions: How Well Are Their Projections Distributed?”

*Journal of Computational and Applied Mathematics*213 (2): 366–86.Yang, Jiyan, Vikas Sindhwani, Haim Avron, and Michael Mahoney. 2014. “Quasi-Monte Carlo Feature Maps for Shift-Invariant Kernels.”

*arXiv:1412.8293 [Cs, Math, Stat]*, December.
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