Bayesian random persistence diagram generation: An application to material microstructure analysis

Farzana Nasrin; Theodore Papamarkou; Austin Lawson; Na Gong; Orlando Rios; Vasileios Maroulas

doi:10.3934/fods.2024018

Article Contents

2024, Volume 6, Issue 3: 361-378. Doi: 10.3934/fods.2024018

This issue Previous Article Eikonal depth: An optimal control approach to statistical depths Next Article Combinatorial persistent homology transform

Bayesian random persistence diagram generation: An application to material microstructure analysis

1.
Department of Mathematics, University of Hawai'i at Manoa, Hawai'i, US
2.
Department of Mathematics, The University of Manchester, Manchester, UK
3.
Department of Mathematics, University of Tennessee, Knoxville, TN, USA
4.
Pellissippi State Community College, Knoxville, TN, USA
5.
Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN, USA

^*Corresponding author: Farzana Nasrin
^*Corresponding author: Farzana Nasrin

Received: July 2023

Revised: March 2024

Early access: April 22, 2024

Published online: April 22, 2024

The work has been partially supported by the ARO W911NF-21-1-0094 (VM).

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Abstract

Data analysis helps identify changes in the microstructure of materials, but is often hindered by the cost and time requirements of experimental data generation. Data augmentation provides an in silico alternative. A recent data augmentation algorithm, known as the random persistence diagram generator (RPDG), samples a sequence of synthetic topological summaries from a possibly limited amount of data. RPDG relies on a parametric model for persistence diagrams, namely a pairwise interacting point process (PIPP). Herein, we develop a Bayesian approach to infer the PIPP parameters and call the resulting pipeline the Bayesian RPDG (BRPDG). We showcase that BRPDG exhibits higher discriminative power than RPDG in the identification of materials structural changes.

Keywords:

Mathematics Subject Classification: 62F15, 60G55, 62R40.

Citation:

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Figure 1. Construction of a PD from the empirical density function of the Schmid factor at $ 950^{\circ} C $ by using the sub-level set filtration. For more details about the computation of the empirical density function of the Schmid factor, see Section 4

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Figure 2. Schmid factors of NGAuSS annealed at (a) and (f) $ 950^\circ{C} $, (b) and (g) $ 850^\circ{C} $, (c) and (h) $ 800^\circ{C} $, (d) and (i) $ 750^\circ{C} $, (e) and (j) $ 700^\circ{C} $. The top row displays Schmid factors, with light and dark gray regions representing lower and higher Schmid factor values, respectively. The bottom row shows the empirical densities (histograms) of Schmid factors for each temperature

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Figure 3. Each plot shows the running averages of the parameters for an annealing temperature using the formula in equation (9).

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Figure 4. The trace plots of all parameters relevant to the five annealing temperatures we consider.

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Figure 5. The $ p- $values obtained from the same KS test discussed in Section 4.3 to show how sensitive the model is with respect to the number of mesh grid points in the Voronoi tessellation model

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Table 1. $ p $-values of KS tests based on BRPDG for all annealing temperature pairs $ (T_j, T_k),\; j\neq k $

	$ T_2 $	$ T_3 $	$ T_4 $	$ T_5 $
$ T_1 $	$ 0.03 $	$ 1.51\times 10^{-5} $	$ 1.51\times 10^{-5} $	$ 7.08\times 10^{-7} $
$ T_2 $		$ 5.93\times 10^{-5} $	$ 1.51\times 10^{-5} $	$ 3.47\times 10^{-6} $
$ T_3 $			$ 2.0\times 10^{-4} $	$ 3.05\times 10^{-10} $
$ T_4 $				$ 1.51\times 10^{-5} $

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Table 2. $ p $-values of KS tests based on RPDG for all annealing temperature pairs $ (T_j, T_k),\; j\neq k $

	$ T_2 $	$ T_3 $	$ T_4 $	$ T_5 $
$ T_1 $	$ 0.063 $	$ 0.002 $	$ 0.123 $	$ 0.002 $
$ T_2 $		$ 0.572 $	$ 0.791 $	$ 5.932\times 10^{-5} $
$ T_3 $			$ 0.123 $	$ 3.471\times 10^{-6} $
$ T_4 $				$ 2.099\times 10^{-4} $

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Table 3. Specification of the proposal densities of Equation (10) and of jump points $ \mathbf{r}^{(T_j)} $

Temperature	$ \mu_{i}^{(T_j)} $	$ \sigma_{i}^{(T_j)} $	$ w_{i}^{(T_j)} $	$ \mathbb{W}^{(T_j)} $	$ \mathbf{r}^{(T_j)} $
$ T_1 $	$ (0.2,15) $ $ (5.2,2.5) $ $ (7,8) $	0.2 0.5 0.5	0.2 0.4 0.4	$ [0,8]\times[0,16] $	$ (0,4,6,8) $
$ T_2 $	$ (0.5,12.3) $ $ (1.5,0.5) $ $ (7.8,1) $	0.1 0.5 0.1	0.4 0.4 0.2	$ [-1,9]\times[-1,13] $	$ (0,3,6,9) $
$ T_3 $	$ (0.4,11.7) $ $ (1,0.5) $ $ (4,0.5) $	0.25 0.25 0.25	1/3 1/3 1/3	$ [-2,11]\times[-2,15] $	$ (0,2,4,6) $
$ T_4 $	$ (0,10) $ $ (1.9,0.1) $ $ (4.8,0.3) $	0.1 0.5 0.2	0.2 0.4 0.4	$ [-1,6]\times[0,11] $	$ (0,2,4,6) $
$ T_5 $	$ (0.1,12) $ $ (3.8,1) $	0.1 0.3	1/3 2/3	$ [-0.5,6]\times[0,13] $	$ (0,3) $

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