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Nonlinear reduced basis using mixture Wasserstein barycenters: application to an eigenvalue problem inspired from quantum chemistry

Maxime Dalery, Genevieve Dusson, Virginie Ehrlacher, Alexei Lozinski

TL;DR

This work tackles parametric eigenvalue problems modeling electronic-structure phenomena by introducing a nonlinear reduced-basis method based on mixture Wasserstein barycenters. It demonstrates that linear $L^2$-widths decay only algebraically, while Wasserstein- and mixture-Wasserstein-based widths decay much faster for 1D transport-dominated solutions, enabling compact nonlinear reduced bases. The method comprises an offline greedy algorithm to select snapshots and an online energy-minimization stage over barycenters, with numerical results showing rapid offline projection-error decay and robust online performance, including extrapolation beyond training data. The approach holds promise for scalable ROMs in higher-dimensional electronic-structure calculations and other transport-dominated regimes.

Abstract

The aim of this article is to propose a new reduced-order modelling approach for parametric eigenvalue problems arising in electronic structure calculations. Namely, we develop nonlinear reduced basis techniques for the approximation of parametric eigenvalue problems inspired from quantum chemistry applications. More precisely, we consider here a one-dimensional model which is a toy model for the computation of the electronic ground state wavefunction of a system of electrons within a molecule, solution to the many-body electronic Schrödinger equation, where the varying parameters are the positions of the nuclei in the molecule. We estimate the decay rate of the Kolmogorov n-width of the set of solutions for this parametric problem in several settings, including the standard L2-norm as well as with distances based on optimal transport. The fact that the latter decays much faster than in the traditional L2-norm setting motivates us to propose a practical nonlinear reduced basis method, which is based on an offline greedy algorithm, and an efficient stochastic energy minimization in the online phase. We finally provide numerical results illustrating the capabilities of the method and good approximation properties, both in the offline and the online phase.

Nonlinear reduced basis using mixture Wasserstein barycenters: application to an eigenvalue problem inspired from quantum chemistry

TL;DR

This work tackles parametric eigenvalue problems modeling electronic-structure phenomena by introducing a nonlinear reduced-basis method based on mixture Wasserstein barycenters. It demonstrates that linear -widths decay only algebraically, while Wasserstein- and mixture-Wasserstein-based widths decay much faster for 1D transport-dominated solutions, enabling compact nonlinear reduced bases. The method comprises an offline greedy algorithm to select snapshots and an online energy-minimization stage over barycenters, with numerical results showing rapid offline projection-error decay and robust online performance, including extrapolation beyond training data. The approach holds promise for scalable ROMs in higher-dimensional electronic-structure calculations and other transport-dominated regimes.

Abstract

The aim of this article is to propose a new reduced-order modelling approach for parametric eigenvalue problems arising in electronic structure calculations. Namely, we develop nonlinear reduced basis techniques for the approximation of parametric eigenvalue problems inspired from quantum chemistry applications. More precisely, we consider here a one-dimensional model which is a toy model for the computation of the electronic ground state wavefunction of a system of electrons within a molecule, solution to the many-body electronic Schrödinger equation, where the varying parameters are the positions of the nuclei in the molecule. We estimate the decay rate of the Kolmogorov n-width of the set of solutions for this parametric problem in several settings, including the standard L2-norm as well as with distances based on optimal transport. The fact that the latter decays much faster than in the traditional L2-norm setting motivates us to propose a practical nonlinear reduced basis method, which is based on an offline greedy algorithm, and an efficient stochastic energy minimization in the online phase. We finally provide numerical results illustrating the capabilities of the method and good approximation properties, both in the offline and the online phase.
Paper Structure (35 sections, 9 theorems, 151 equations, 11 figures, 3 algorithms)

This paper contains 35 sections, 9 theorems, 151 equations, 11 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. An eigenvalue problem inspired from quantum chemistry
  4. Kolmogorov widths in metric spaces
  5. Wasserstein spaces
  6. Wasserstein metric
  7. Slater mixture Wasserstein metric
  8. Theoretical estimations of the decay of Kolmogorov n-widths
  9. On linear approximations: case H = L2(R)
  10. On the Wasserstein transport metric: case M = P2(R) with d = W2
  11. Preliminary discussion
  12. Main results
  13. Transport approximation for $r\leq R$
  14. Transport approximation for $r \geq R$
  15. Transport approximation for $r \in \mathbb{R}^+$
  16. ...and 20 more sections

Key Result

Theorem 3.1

\newlabelthrm_linear_comp0 There exist positive constants $c_R$, $C_R$, $\tilde{c}_R$ and $\tilde{C}_R$ depending on $R$ such that for all $n\in\mathbb{N}^*$, and

Figures (11)

  • Figure 1: Three example solutions in $\mathcal{M}_{tr}$.
  • Figure 2: First seven elements selected in the reduced basis in the offline phase.
  • Figure 3: Decay of the projection error in the offline phase.
  • Figure 4: Decay of the projection error in $W_{2}$-distance in the offline phase for $\mathrm{MW}_{2}$-distance and $W_{2}$-distance greedy selection.
  • Figure 5: Example of projections on bases with 2,3,5, and 8 elements for $r = 1.266$.
  • ...and 6 more figures

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 3.1
  • Corollary 3.2
  • Proof 1
  • Remark 3.3
  • Proposition 3.4
  • Proof 2
  • Theorem 3.5
  • ...and 16 more