Bi-fidelity Boosted Quadrature Sampling

This repo contains code for our paper

N. Cheng, O. A. Malik, Y. Xu, S. Becker, A. Doostan, A. Narayan. Quadrature Sampling of Parametric Models with Bi-fidelity Boosting. SIAM/ASA Journal on Uncertainty Quantification, to appear. arXiv preprint arXiv:2209.05705, 2022.

The paper is available at arXiv.

Referencing this code

If you use this code in any of your own work, please reference our paper:

@article{cheng2022quadrature,
      title={Quadrature Sampling of Parametric Models with Bi-fidelity Boosting}, 
      author={Nuojin Cheng and Osman Asif Malik and Yiming Xu and Stephen Becker and Alireza Doostan and Akil Narayan},
      journal={SIAM/ASA Journal on Uncertainty Quantification},
      year={2024},
      eprint={2209.05705},
      archivePrefix={arXiv},
      primaryClass={math.NA}
}

Description of code

Main files

• cavity_flow_bifidelity_TD/cavity_flow_bifidelity_HC.m: Codes solving boosted quadrature sampling results for cavity flow data in total degree/hyperbolic cross space;

• beam_bifidelity_TD/beam_bifidelity_HC: Codes solving boosted quadrature sampling results for beam data in total degree/hyperbolic space;

Sampling functions

• hyperbolic_cross_sampling.m: This function samples and solves the least square problem with many different sampling options. The function utilizes the tensor structure and solves sampled least square problems with less time. The sampling methods provided are uniform sampling and leverage score sampling.

• total_degree_sampling.m: Same with above, but in the total degree space;

• det_rejection_sampling.m: This function allows sampling according to leveraged volume sampling. This is done via the determinantal rejection sampling algorith in [Derezinski et al., arXiv:1802.06749, 2018]. Moreover, it is implemented to allow both boosted and unboosted sampling for both unstructured and Kronecker structured matrices.

Help functions

• RegVol.m: This algorithm does volume sampling. It was presented in [Derezinski and Warmuth, JMLR 19, 2018].

• FastRegVol.m: This algorithm is a faster version of RegVol.m, and uses RegVol.m as a substep. It was also presented in [Derezinski and Warmuth, JMLR 19, 2018].

• data_loader.m: This function is used to load the various datasets.

• sub_tp_idx_set.m: This function computes the sub-index sets corresponding to tensor product, total degree, and hyperbolic cross polynomial spaces. It can be expanded to handle other polynomial spaces if necessary.

• my_legendre_1d.m: This function generates 1d Gauss-Legendre node between -1 and 1;

• mu.m: This function computes optimality coefficient of the given sketching and QoI vector b.

Plot functions

• synthetic_experiment_fig1.py: Codes generating Figure 1;
• synthetic_experiment_fig2.py: Codes generating Figure 2;
• cor_plot.py: Codes generating Figure 4 and Figure 8;
• plot.py: Codes generating Figure 5 and Figure 9.

Data

• cavity_flow_dataset: Cavity flow data for Section 4.2.1;
• X.mat/Y.mat: Composite beam data for Section 4.2.2.

Instructions for creating plots

For Figure 1 and Figure 2, simply run the codes in synthetic_experiment_fig1/fig2.py. For Figures 4,5,8,9, we should run codes in cavity_flow_bifidelity_TD, cavity_flow_bifidelity_HC.m, beam_bifidelity_TD, and beam_bifidelity_HC to generate data, and then run plotting functions in cor_plot.py/plot.py to attain figures. Note that the algorithms are random, so the figure of code result won't be exactly same with what the paper presents.