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Parameterizing Intersecting Surfaces via Invariants

Timon S. Gutleb, Rhyan Barrett, Julia Westermayr, Christoph Ortner

TL;DR

The paper tackles reconstructing multiple intersecting hypersurfaces from unordered, value-sorted data by projecting onto smooth invariants formed from elementary symmetric polynomials and related invariants. It analyzes three invariant-based reconstruction variants—Frobenius companion, Schmeisser, and Chebyshev colleague matrices—and demonstrates their convergence and stability properties, including root-conditioning near cusps and backward-stable eigenvalue solves for the Chebyshev approach. Key findings show that invariants enable near-spectral convergence on smooth data and robust cusp resolution under noise, with applications to graphene Dirac cones and $SO_2$ potential energy surfaces illustrating practical gains over direct interpolation. The work lays a foundation for efficient surrogate models of crossing surfaces in chemistry and materials science and suggests integrating these methods with modern machine-learning pipelines and invariant representations for improved reliability. The methods offer a pathway to accurate, scalable reconstructions of complex multi-surface landscapes essential for excited-state dynamics and band-structure analyses.

Abstract

We introduce and analyze numerical companion matrix methods for the reconstruction of hypersurfaces with crossings from smooth interpolants given unordered or, without loss of generality, value-sorted data. The problem is motivated by the desire to machine learn potential energy surfaces arising in molecular excited state computational chemistry applications. We present simplified models which reproduce the analytically predicted convergence and stability behaviors as well as two application-oriented numerical experiments: the electronic excited states of Graphene featuring Dirac conical cusps and energy surfaces corresponding to a sulfur dioxide ($SO_2$) molecule in different configurations.

Parameterizing Intersecting Surfaces via Invariants

TL;DR

The paper tackles reconstructing multiple intersecting hypersurfaces from unordered, value-sorted data by projecting onto smooth invariants formed from elementary symmetric polynomials and related invariants. It analyzes three invariant-based reconstruction variants—Frobenius companion, Schmeisser, and Chebyshev colleague matrices—and demonstrates their convergence and stability properties, including root-conditioning near cusps and backward-stable eigenvalue solves for the Chebyshev approach. Key findings show that invariants enable near-spectral convergence on smooth data and robust cusp resolution under noise, with applications to graphene Dirac cones and potential energy surfaces illustrating practical gains over direct interpolation. The work lays a foundation for efficient surrogate models of crossing surfaces in chemistry and materials science and suggests integrating these methods with modern machine-learning pipelines and invariant representations for improved reliability. The methods offer a pathway to accurate, scalable reconstructions of complex multi-surface landscapes essential for excited-state dynamics and band-structure analyses.

Abstract

We introduce and analyze numerical companion matrix methods for the reconstruction of hypersurfaces with crossings from smooth interpolants given unordered or, without loss of generality, value-sorted data. The problem is motivated by the desire to machine learn potential energy surfaces arising in molecular excited state computational chemistry applications. We present simplified models which reproduce the analytically predicted convergence and stability behaviors as well as two application-oriented numerical experiments: the electronic excited states of Graphene featuring Dirac conical cusps and energy surfaces corresponding to a sulfur dioxide () molecule in different configurations.
Paper Structure (23 sections, 13 theorems, 49 equations, 12 figures, 4 algorithms)

This paper contains 23 sections, 13 theorems, 49 equations, 12 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Elementary symmetric polynomials
  4. Companion matrices
  5. Colleague matrices and Chebyshev polynomials
  6. Reconstruction of Multi-Surfaces
  7. Problem statement and preliminaries
  8. Reconstruction from invariants, Frobenius variant
  9. Symmetric tridiagonal Schmeisser variant
  10. Chebyshev Colleague matrix variant
  11. Sensitivity analysis
  12. Errors in the Chebyshev colleague matrix method
  13. Errors in the Schmeisser companion matrix method
  14. Numerical experiments
  15. Toy problems
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let $p(x) = x^n + \sum_{j=0}^{n-1} a_j x^j$ be a degree $n$ monic polynomial with coefficients $a_j \in \mathbb{R}$ and (possibly complex and repeated) roots $r_j, j=1, \dots, n$, then

Figures (12)

  • Figure 1: Intersecting, smooth sinusoidal curves in (a), sorted by value in (b), revealing non-smooth cusps in the "$j$-th entry by value" function.
  • Figure 2: (a-b) Semi-logarithmic maximum absolute error and gap weighted error on all $1000$ random gridpoints (sampled uniformly) for the problem in Eqs. (\ref{['eq:toy1start']}--\ref{['eq:toy1end']}) solved via Frobenius, Schmeisser companion, Chebyshev colleague or direct Chebyshev interpolation approaches. (c-d) show analogous error plots with added perturbations of indicated magnitude on the surface or ESP evaluations.
  • Figure 3: Endpoint plot of Figure \ref{['fig:1dtoyerrors']}(c) including some additional data points, showing that the converged (in degree) error for using the Chebyshev colleague approach for the problem in (\ref{['eq:toy1start']}--\ref{['eq:toy1end']}) scales like the square root of the perturbations, consistent with Theorem \ref{['thm:rootconditioning']} in light of the multiplicity $2$ roots observed in this problem, cf. Figure \ref{['fig:firsttoyexample']}.
  • Figure 4: (a) shows the surfaces $\bar{\mathbf{z}}$ in (\ref{['eq:2dsinusoidaltoyeq']}--\ref{['eq:2dsinusoidaltoyeqend']}), (b) shows a corresponding value-ordered reconstruction with cusps.
  • Figure 5: (a) shows a semi-logarithmic plot of the max. abs. error for the problem in (\ref{['eq:2dsinusoidaltoyeq']}--\ref{['eq:2dsinusoidaltoyeqend']}) using the Chebyshev colleague approach, (b) shows an analogous plot for reconstructing the conical cusp problem in (\ref{['eq:toyconicalsinh']}--\ref{['eq:toyconicalsinhend']}).
  • ...and 7 more figures

Theorems & Definitions (22)

  • Definition 2.1
  • Theorem 2.1: Viète's formula
  • Theorem 2.2
  • Theorem 2.3: Schmeisser companion matrix, schmeisser1993real
  • Theorem 2.4: Colleague matrix, good1961colleaguetrefethen2019approximation
  • Remark 2.1
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3: fox1968chebyshev
  • Theorem 4.1
  • ...and 12 more