  This paper proposes a new approach to construct high quality space-filling
sample designs. First, we propose a novel technique to quantify the
space-filling property and optimally trade-off uniformity and randomness in
sample designs in arbitrary dimensions. Second, we connect the proposed metric
(defined in the spatial domain) to the objective measure of the design
performance (defined in the spectral domain). This connection serves as an
analytic framework for evaluating the qualitative properties of space-filling
designs in general. Using the theoretical insights provided by this
spatial-spectral analysis, we derive the notion of optimal space-filling
designs, which we refer to as space-filling spectral designs. Third, we propose
an efficient estimator to evaluate the space-filling properties of sample
designs in arbitrary dimensions and use it to develop an optimization framework
to generate high quality space-filling designs. Finally, we carry out a
detailed performance comparison on two different applications in 2 to 6
dimensions: a) image reconstruction and b) surrogate modeling on several
benchmark optimization functions and an inertial confinement fusion (ICF)
simulation code. We demonstrate that the propose spectral designs significantly
outperform existing approaches especially in high dimensions.
