Slepians.jl

Motivation

An elegant solution to the Slepian, Landau, and Pollak problem was needed to generate 3D-ΔPDF maps (also look here). However, current methods, especially the naive "punch and fill" algorithm, introduced unintended ripples and noise due to sharp edges during interpolation and was not scalable to higher dimensions. Since the algorithm was sequential, it was a huge bottleneck in our applications of X-Ray analysis. Processing multiple gigabytes of data every minute for our applications was impractical.

Significance

To mitigate the noise and the runtime issues, I proposed leveraging the inherent sparsity of the data to our advantage. I used functions localized both in their Fourier transforms and the real space to minimize the leakage artifacts in the 3D-ΔPDF maps. Using a sparse representation for the diffuse scattering in the real space, I created interpretable PDFs with minimal noise. I used the Nystrom method and DPSS to solve the Fredholm equations

Solving Fredholm equations using the above methods also restructured the solution to be embarrassingly parallel. This solved the crucial problem relating to runtime. The benchmark for these is available here

These improvements allowed my team to perform enhanced material characterization and improved data quality with broader applicability and optimized computational efficiency.

Installation

Slepians.jl is unregistered and relies on unregistered packages. To avoid difficulties in which Julia does not know the relevant URLS, I have created a registry bbkt-reg.jl which will tell your installation where to find this package and its dependencies. Begin with adding the registry using

pkg> registry add https://github.com/lootie/bbkt-reg.jl

then one can simply add the Slepians.jl package as

pkg> add Slepians

Examples

The jupyter notebooks in the Examples/ directory illustrate the functionality of this package.

Funding

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Materials Science and Engineering.

References

Please see the below papers

``` @article{bronez1988spectral, title={Spectral estimation of irregularly sampled multidimensional processes by generalized prolate spheroidal sequences}, author={Bronez, Thomas P}, journal={IEEE Transactions on Acoustics, Speech, and Signal Processing}, volume={36}, number={12}, pages={1862--1873}, year={1988}, publisher={IEEE} }

@article{slepian1978prolate, title={Prolate spheroidal wave functions, Fourier analysis, and uncertainty—V: The discrete case}, author={Slepian, David}, journal={Bell System Technical Journal}, volume={57}, number={5}, pages={1371--1430}, year={1978}, publisher={Wiley Online Library} }

@article{simons2011spatiospectral, title={Spatiospectral concentration in the Cartesian plane}, author={Simons, Frederik J and Wang, Dong V}, journal={GEM-International Journal on Geomathematics}, volume={2}, number={1}, pages={1--36}, year={2011}, publisher={Springer} } ```

NOTE:

Some of these files have been forked from my mentor Dr. Charlotte Haley @lootie, who at the time of editing this is affiliated with Argonne National Laboratory. She had been working on this problem before I joined ANL as an intern. I thank her immensely for her mentorship and guidance and for introducing me to the beautiful world of spectral analysis. The files linked to in this modified README.md are either solely or majorly my contributions to the AXMAS group.