Certified Model Order Reduction for parametric Hermitian eigenproblems

TL;DR

The article addresses efficient, certified approximation of the smallest eigenvalue $\lambda_1(\bm{\mu})$ and its eigenspace $\mathcal{W}_1(\bm{\mu})$ for large parametric Hermitian matrices $\mathbf{A}(\bm{\mu})$. It develops a projection-based MOR framework with offline/online phases, connecting reduced basis ideas to eigenproblem MOR and introducing a novel eigenspace error bound that depends on the spectral gap $\gamma(\bm{\mu})$. A practical two-stage greedy strategy first certifies the spectral gap and then constructs a reduced space for the ground-state eigenspace, with computable upper/lower bounds and a dimension-recovery theorem to ensure exact eigenspace dimension matching. Numerical experiments on randomly generated matrices and parametric quantum spin systems validate the method, showing substantial speed-ups and reliable error control even for challenging, highly degenerate problems. This work offers a rigorous, scalable path to ground-state identification in parametric quantum models and other parametric PDE contexts where spectral properties are central.

Abstract

This article deals with the efficient and certified numerical approximation of the smallest eigenvalue and the associated eigenspace of a large-scale parametric Hermitian matrix. For this aim, we rely on projection-based model order reduction (MOR), i.e., we approximate the large-scale problem by projecting it onto a suitable subspace and reducing it to one of a much smaller dimension. Such a subspace is constructed by means of weak greedy-type strategies. After detailing the connections with the reduced basis method for source problems, we introduce a novel error estimate for the approximation error related to the eigenspace associated with the smallest eigenvalue. Since the difference between the second smallest and the smallest eigenvalue, the so-called spectral gap, is crucial for the reliability of the error estimate, we propose efficiently computable upper and lower bounds for higher eigenvalues and for the spectral gap, which enable the assembly of a subspace for the MOR approximation of the spectral gap. Based on that, a second subspace is then generated for the MOR approximation of the eigenspace associated with the smallest eigenvalue. We also provide efficiently computable conditions to ensure that the multiplicity of the smallest eigenvalue is fully captured in the reduced space. This work is motivated by a specific application: the repeated identifications of the states with minimal energy, the so-called ground states, of parametric quantum spin system models.

Figures (7)

• Figure 1: Approximation of the spectral gap of a dense parametric Hermitian matrix $\mathbf{A}(\mu)\in\mathbb{R}^{N\times N}$ as in \ref{['eqn:ex2:aff']} with $N=2\cdot 10^3$; the resulting reduced space $\mathcal{V}_{\gamma}$ for the gap approximation has dimension $r={62}$.
• Figure 2: Approximation of the eigenspace associated to the smallest eigenvalue of a dense parametric Hermitian matrix $\mathbf{A}(\mu)\in\mathbb{R}^{N\times N}$ as in \ref{['eqn:ex2:aff']} with $N=2\cdot 10^3$; the resulting reduced space $\mathcal{V}_{\mathcal{W}}$ for the eigenspace approximation has dimension $r=39$.
• Figure 3: QSS example, xxz chain model: Approximation of the spectral gap over a discrete domain consisting of $35\times 35$ Chebyschev-nodes for $\mathbf{A}(\bm \mu)\in\mathbb{R}^{N\times N}$ sparse matrix as in \ref{['eqn:QSS:xxz:aff']} with $N=2^{14}=16384$. The resulting reduced space $\mathcal{V}_{\gamma}$ has dimension $r={265}$.
• Figure 4: QSS example, xxz chain model: Approximation of the eigenspace over a discrete domain consisting of $35\times 35$ Chebyschev-nodes for $\mathbf{A}(\bm \mu)\in\mathbb{R}^{N\times N}$ sparse matrix as in \ref{['eqn:QSS:xxz:aff']} with $N=2^{14}= 16384$. The resulting reduced space $\mathcal{V}_{\mathcal{W}}$ has dimension $r={134}$.
• Figure 5: QSS example, bblq chain model: Approximation of the spectral gap for $\mathbf{A}(\bm \mu)\in\mathbb{R}^{N\times N}$ sparse matrix as in \ref{['eqn:QSS:bblq:aff']} with $N=59049$; $\log_{10}$ scale is used. The resulting reduced space $\mathcal{V}_{\gamma}$ has dimension $r={859}$. Red crosses in both the plots indicate the selected interpolation points by \ref{['alg:gap']}.

Theorems & Definitions (24)

• Theorem 1.1
• Example 2.1
• Example 2.2
• Remark 2.4
• Lemma 2.5
• proof : Proof of \ref{['sub-lemma:EigVal by EigVec']} and \ref{['sub-lemma:EigVec by EigVal']}
• Theorem 2.6: Upper bound for the eigenspace projection error
• proof
• Theorem 3.1: Subspace lower bound for eigenvalues
• Remark 3.2