Kernelized approaches to streaming compression of scientific data

Benjamin P. Russo; Richard Archibald

doi:10.3934/ammc.2024017

Article Contents

2024, Volume 2, Issue 3: 322-347. Doi: 10.3934/ammc.2024017

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Kernelized approaches to streaming compression of scientific data

Benjamin P. Russo^1, , and
Richard Archibald^2,

1.
Riverside Research, New York, NY 10038, USA
2.
Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

^*Corresponding author: Benjamin P. Russo
^*Corresponding author: Benjamin P. Russo

Received: May 2024

Revised: August 2024

Early access: September 2024

Published: September 2024

This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the work for publication, acknowledges that the US government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the submitted manuscript version of this work, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

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Abstract

In this paper three algorithms are developed for the streaming compression of scientific data. The algorithms presented are reliant on the theory of vector-valued reproducing kernel Hilbert spaces and operator valued kernels. The scientific data is modeled as a snapshot of time dependent vector-field $ F(x, t) $ over a manifold $ M $ and the recovery of the data is framed as a learning problem. These processes are then appropriately modified and analyzed for the streaming scenario in which data is generated without the ability to revisit past entries.

Keywords:

Streaming data,
online compression,
vector valued reproducing kernel Hilbert spaces.

Mathematics Subject Classification: 41A05, 46E22, 94A08.

Citation:

FullText(HTML)

Figure 1. $ E_{[0, T]}(t) $ for the surrogate model generated by Algorithm 1 on both data sets

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Figure 2. The $ L^2 $ percent error over time, $ E_{\text{basis}}(t) $, due to the choice of basis

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Figure 3. A sample reconstruction of $ \alpha_j(t) $ for a randomly chosen wavelet function. The dashed line represents the kernel interpolation and the dots represent the Halton sequence of centers

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Figure 4. $ E_{M}(x) $ for Algorithm 1 applied to data sets A (left) and data set B (right). All figures displayed on a log scale

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Figure 5. Sample reconstruction by Algorithm 1 of a single image from data set B

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Figure 6. 5000 Halton sequence points over the domain $ M $ overlayed on a snapshot of the data

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Figure 7. Sample reconstruction by Algorithm 2 using Gaussian kernels of a single image from data set A

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Figure 8. Sample reconstruction by Algorithm 2 using thin-plate spline kernels of a single image from data set B

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Figure 9. The generated coefficient functions for selected few basis functions $ \phi_j(t) = \sqrt{2}\{\cos( \pi\cdot j \cdot t)\}_{j = 1}^{199}\cup\{1\} $ for data set A

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Figure 10. The generated coefficient functions for selected few basis functions $ \phi_j(t) = \sqrt{2} \{\cos( \pi\cdot j \cdot t)\}_{j = 1}^{199}\cup\{1\} $ for data set B

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Figure 11. $ E_{[0, T]}(t) $ for Algorithm 2 applied to data sets A (top) and data set B (bottom)

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Figure 12. $ E_{M}(x) $ for Algorithm 2 applied to data sets A (left) and data set B (right). All figures displayed on a log scale

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Figure 13. $ F(x_i, t) $ for $ t\in [0, T] $ at two different interpolation points in the domain $ M $ taken from data set A with additive noise

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Figure 14. In this figure we show an example frame of a reconstruction produced by Algorithm 3 applied to data set A and data set A with additive noise

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Figure 15. $ E_M(x) $ for Algorithm 3 applied to data set A and data set A with additive noise. Displayed on a log scale

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Figure 16. $ E_{[0, T]}(t) $ for Algorithm 3 applied to data set A (blue) and data set A with additive noise (orange)

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Figure 17. Coefficient functions produced by Algorithm 3 applied to a noisy signal corresponding to a few kernels over the time domain $ k(t, c) $ where $ c $ is a center belonging to the Halton sequence. Compare with Figure 18 which are the same functions derived from a non-noisy signal

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Figure 18. Coefficient functions produced by Algorithm 3 applied to a noise-free signal corresponding to a few kernels over the time domain $ k(t, c) $ where $ c $ is a center belonging to the Halton sequence. Compare to Figure 17

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Table 1. Description of fluid flow data sets

data set	Description
total simulation	$ 100, 000 $ time-steps using the Lattice-Boltzmann method with $ \Delta x = \Delta t = 1 $.
data set A	$ 1, 000 $ equally spaced time-steps between snapshots $ 32, 000 $ and $ 42, 000 $.
data set B	$ 1, 000 $ equally spaced time-steps between snapshots $ 22, 000 $ and $ 32, 000 $.

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Table 2. Summary of the performance of Algorithm 1–3 with chosen parameters and details given in Sections 5.1–5.3. Here, A$ ^* $ represents data set A with noise

data	alg.	basis	kernel	start size	compressed	$\mathit{\boldsymbol{ E}}_{\text{overall}} $
A	1	Haar	Gaussian	$ 16, 000 \times 1000 $	$ 4000 \times 202 $	$ 24.36 \% $
	2	Fourier	Gaussian	$ 16, 000 \times 1000 $	$ 5000 \times 200 $	$ 17.90 \% $
	3	N/A	Gaussian/spline	$ 16, 000 \times 1000 $	$ 3200 \times 202 $	$ 20.40\% $
A$ ^* $	3	N/A	Gaussian/spline	$ 16, 000 \times 1000 $	$ 3200 \times 202 $	$ 23.41\% $
B	1	Haar	Gaussian	$ 16, 000 \times 1000 $	$ 4000 \times 202 $	$ 23.44 \% $
B	2	Fourier	thin-plate spline	$ 16, 000 \times 1000 $	$ 5000 \times 200 $	$ 14.74 \% $

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