# Uniform Design on Manifold

Chapter

## Abstract

Uniform design aims to scatter points as evenly as possible on certain domain. Although in real applications, the experimental domain is often quite arbitrary, the discrepancies frequently used to measure the uniformity of experimental designs are often defined on the unit cube. In this paper, we will introduce a unified framework to measure the uniformity of an experimental design on manifold. We will give some examples to illustrate the construction of uniform designs on some specific manifolds and provide a stochastic algorithm to construct uniform designs on the unit semi-spherical surface and on the unit spherical surface. Numerical results show that the algorithm performs well.

## Notes

### Acknowledgements

The author would like to thank the referees for their valuable and helpful comments. This research is supported by Natural Science Foundation of China (11671290) and Jiangsu Provincial Key Subject on Statistics (GD10700118).

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