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Critical Point Extraction from Multivariate Functional Approximation

Guanqun Ma, David Lenz, Tom Peterka, Hanqi Guo, Bei Wang

TL;DR

This work addresses the challenge of performing topological data analysis directly on continuous implicit representations by introducing CPE-MFA, the first critical-point extraction framework for Multivariate Functional Approximation (MFA). The method combines span filtration based on the strong convex hull of B-spline control points, Newton-based critical-point search within candidate spans, and spatial hashing to remove duplicates, all optimized for parallel execution. Across Schwefel, CESM, S3D, QMC, and RTI datasets, CPE-MFA demonstrates scalable extraction of isolated, non-degenerate critical points from MFA representations and shows that upsampling the associated PL reconstructions improves alignment with PL-based references, validating the approach as a bridge between continuous MFA and discrete topological descriptors. The results highlight the potential to enable robust topological data analysis directly on continuous implicit models at scale, with clear avenues for extending to topological descriptors and higher dimensions.

Abstract

Advances in high-performance computing require new ways to represent large-scale scientific data to support data storage, data transfers, and data analysis within scientific workflows. Multivariate functional approximation (MFA) has recently emerged as a new continuous meshless representation that approximates raw discrete data with a set of piecewise smooth functions. An MFA model of data thus offers a compact representation and supports high-order evaluation of values and derivatives anywhere in the domain. In this paper, we present CPE-MFA, the first critical point extraction framework designed for MFA models of large-scale, high-dimensional data. CPE-MFA extracts critical points directly from an MFA model without the need for discretization or resampling. This is the first step toward enabling continuous implicit models such as MFA to support topological data analysis at scale.

Critical Point Extraction from Multivariate Functional Approximation

TL;DR

This work addresses the challenge of performing topological data analysis directly on continuous implicit representations by introducing CPE-MFA, the first critical-point extraction framework for Multivariate Functional Approximation (MFA). The method combines span filtration based on the strong convex hull of B-spline control points, Newton-based critical-point search within candidate spans, and spatial hashing to remove duplicates, all optimized for parallel execution. Across Schwefel, CESM, S3D, QMC, and RTI datasets, CPE-MFA demonstrates scalable extraction of isolated, non-degenerate critical points from MFA representations and shows that upsampling the associated PL reconstructions improves alignment with PL-based references, validating the approach as a bridge between continuous MFA and discrete topological descriptors. The results highlight the potential to enable robust topological data analysis directly on continuous implicit models at scale, with clear avenues for extending to topological descriptors and higher dimensions.

Abstract

Advances in high-performance computing require new ways to represent large-scale scientific data to support data storage, data transfers, and data analysis within scientific workflows. Multivariate functional approximation (MFA) has recently emerged as a new continuous meshless representation that approximates raw discrete data with a set of piecewise smooth functions. An MFA model of data thus offers a compact representation and supports high-order evaluation of values and derivatives anywhere in the domain. In this paper, we present CPE-MFA, the first critical point extraction framework designed for MFA models of large-scale, high-dimensional data. CPE-MFA extracts critical points directly from an MFA model without the need for discretization or resampling. This is the first step toward enabling continuous implicit models such as MFA to support topological data analysis at scale.
Paper Structure (21 sections, 1 theorem, 11 equations, 8 figures, 5 tables, 2 algorithms)

This paper contains 21 sections, 1 theorem, 11 equations, 8 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Technical Background
  4. Multivariate Functional Approximation
  5. Critical Point Extraction
  6. Method
  7. Span Filtration
  8. Critical Point Extraction in a Single Span
  9. Duplication Removal
  10. Time Complexity
  11. Experimental Results
  12. Schwefel Dataset
  13. CESM Dataset
  14. Turbulent Combustion Dataset
  15. Quantum Monte Carlo Dataset
  16. ...and 6 more sections

Key Result

Theorem 1

If $(u_1,\cdots, u_d)$ is a point in the span $[t^{(1)}_{j_1},t^{(1)}_{j_1+1}]\times\cdots\times[t^{(d)}_{j_d},t^{(d)}_{j_d+1}]$, then $F(u_1,\cdots, u_d)$ lies in the convex hull defined by control points $P_{h_1,\cdots, h_d}$, where $j_l \leq h_l \leq j_l + p$, for $1\leq l \leq d$.

Figures (8)

  • Figure 1: Left: a 1-dimensional B-spline curve (top), and a 2-dimensional B-spline surface (bottom). $P_i$ are control points, control meshes are in black, approximated curve/surface are in green. Right: an overall MFA model pipeline reproduced from peterka2022multivariate.
  • Figure 2: Scientific datasets: CESM (top), S3D (middle left), QMC (middle right), and RTI (bottom).
  • Figure 3: The Schwefel function with red critical points identified by CPE-MFA: (left) top view (right) side view.
  • Figure 4: CESM dataset with critical points identified by CPE-MFA and TTK-MFA. Top: critical points from blocks A, B, and C, respectively. Middle: critical points from upsampled blocks A, B, and C (labeled as A$^*$, B$^*$, and C$^*$), respectively, with a ratio of $10^2$. Bottom: zoomed-in views of regions in the domain with and without upsampling (at a ratio of $10^2$).
  • Figure 5: S3D dataset with critical points identified by CPE-MFA and TTK-MFA. Top: critical points from blocks A, B, and C, respectively. Middle: critical points from upsampled blocks A, B, and C (labeled as A$^*$, B$^*$, and C$^*$), respectively, with a ratio of $10^2$. Bottom: ($1, 1^*, 2, 2^*, 3,3^*$): zoomed-in views of regions in the domain with and without upsampling at a ratio of $10^2$; (a$^{**}$, b$^{**}$): better critical point alignments (in regions a$^{*}$, b$^{*}$, respectively) after upsampling at a ratio of $10^4$.
  • ...and 3 more figures

Theorems & Definitions (1)

  • Theorem 1: Strong Convex Hull Property