Model Order Reduction for Parametric Hermitian Eigenvalue Problems: Local Acceleration with Taylor-Reduced Basis Method

1. Problem Statement

The paper addresses the computational challenge of approximating the eigenspaces and eigenvalues of large-scale parametric Hermitian matrices $A(\mu)$, where $\mu \in \mathbb{R}^d$ is a multi-dimensional parameter vector.

• Context: In fields like quantum physics and chemistry, solving the full eigenvalue problem (EVP) for every parameter instance is computationally prohibitive ($N \gg 1$).
• Specific Challenge: Unlike parametric source problems (linear systems), parametric EVPs suffer from:
  1. Lack of Galerkin orthogonality.
  2. Absence of reliable residual-based error estimators.
  3. Regularity issues: Individual eigenvalues and eigenvectors may not be analytic (or even continuous) in the multivariate case, especially near eigenvalue crossings.
• Goal: Develop a local Model Order Reduction (MOR) method that efficiently approximates the total eigenprojector $P(\mu)$ (associated with a cluster of eigenvalues) and the Ritz-value sum (trace of the projected matrix) in a neighborhood of a reference parameter $\mu_0$.

2. Methodology: Taylor-Reduced Basis Method (Taylor-RBM)

The authors propose the Taylor-RBM, a projection-based MOR technique that constructs a reduced basis space using the derivatives of the spectral projector rather than solution snapshots.

A. Theoretical Foundation

• Analyticity of Projectors: While individual eigenvectors may be discontinuous, the total eigenprojector $P(\mu)$ associated with an isolated cluster of eigenvalues is proven to be real-analytic in a neighborhood of $\mu_0$ (provided the cluster is separated from the rest of the spectrum by a gap).
• Taylor Expansion: The method relies on the power series expansion of $P(\mu)$ around $\mu_0$:
  $$P(\mu) = \sum_{\beta} (\mu - \mu_0)^\beta P^{(\beta)}$$
  where $\beta$ is a multi-index and $P^{(\beta)}$ are the coefficient matrices (Taylor coefficients).

B. Construction of the Reduced Space

Instead of solving the full problem at multiple points (Lagrange-RBM), the Taylor-RBM constructs the reduced space $V_n$ by spanning the ranges of the first $n$ orders of Taylor coefficients:
$$V_n = \text{span}\left\{ \text{ran}(P^{(\beta)}) : |\beta| \leq n \right\}$$
The reduced model is then obtained by projecting $A(\mu)$ onto $V_n$ and solving a small $r \times r$ eigenvalue problem ($r = \dim V_n \ll N$).

C. Efficient Assembly Algorithm (Key Innovation)

Directly computing $P^{(\beta)}$ involves dense $N \times N$ matrices, which is infeasible for large $N$. The authors derive a recursive algorithm (Algorithm 1) that avoids dense matrix operations:

1. Parametrization: They introduce a "wave operator" $K(\mu)$ and an effective Hamiltonian $B(\mu)$ such that the projector range is spanned by $U_0 + K(\mu)$, where $U_0$ is the initial eigenvector basis.
2. Recursion: They derive recursive formulas for the Taylor coefficients $K^{(\beta)}$ and $B^{(\beta)}$ using the reduced resolvent $S_0$ and the derivatives of the system matrix $A^{(\beta)}$.
3. Complexity: The algorithm only requires operations on thin matrices ($N \times m_0$) and solves linear systems involving the shifted matrix $(A_0 - \lambda_0 I)$, making it scalable for large systems.

3. Key Contributions

1. Rigorous Error Analysis

• Subspace Error Bound: Theorem 4.3 establishes an a priori error bound for the eigenprojector approximation:
  $$\|Q_{rb}(\mu) - P(\mu)\| \leq C \cdot \|\Pi_n^\perp P(\mu)\|$$
  This shows the error is controlled by how well the Taylor-RB space approximates the true projector, scaled by the spectral gap and residual.
• Squaring Effect: Theorem 4.4 proves that the error in the Ritz-value sum (eigenvalue approximation) converges at a quadratic rate ($O(|\mu-\mu_0|^{2n+2})$) relative to the subspace error ($O(|\mu-\mu_0|^{n+1})$). This is a crucial result for eigenvalue problems.

2. Comparison with Truncated Power Series

The paper analyzes the difference between the Taylor-RBM (projection onto the subspace) and the classical Truncated Power Series (TPS) approach (directly using the truncated series $P^{[n]}(\mu)$ as an approximation).

• Result: The Taylor-RBM provides a superior approximation because it performs a variational projection (Rayleigh-Ritz) onto the subspace, whereas TPS is a direct truncation that does not guarantee orthogonality or optimality.

3. Multivariate Generalization

The work extends perturbation theory and RBM from univariate ($d=1$) to multivariate ($d > 1$) parameter spaces, addressing the combinatorial complexity of multi-index derivatives and providing explicit recursive formulas for the coefficients.

4. Numerical Results

The authors validate the method using the XXZ quantum spin chain model (a standard benchmark in quantum many-body physics).

• Dimension Growth: The dimension of the reduced space grows polynomially with the order $n$ and the number of parameters $d$, but often significantly slower than the theoretical worst-case bound due to linear dependencies in the derivative ranges.
• Convergence Rates:
  • Numerical experiments confirm the theoretical convergence rates: $O(h^{n+1})$ for eigenspace error and $O(h^{2n+2})$ for eigenvalue error.
  • The method is robust even near eigenvalue crossings (where the spectral gap vanishes), provided the order $n$ is sufficiently high.
• Comparison with Classical Perturbation Theory (PT):
  • Efficiency: Taylor-RBM is significantly faster to assemble than classical PT because it avoids constructing dense $N \times N$ projector matrices.
  • Accuracy: Taylor-RBM outperforms the truncated power series approximation, particularly in regions slightly outside the strict radius of analytic convergence. The reduced basis captures the "expressivity" of the eigenspace better than the raw truncated series.

5. Significance and Impact

• Bridging Communities: The paper successfully bridges Model Order Reduction (RBM) and Mathematical Physics (Perturbation Theory), providing a unified framework for parametric eigenvalue problems.
• Computational Feasibility: By deriving a recursive, matrix-free assembly procedure, the authors make high-order local approximation of large-scale quantum systems computationally feasible.
• Theoretical Insight: The explicit error bounds and the demonstration of the "squaring effect" for eigenvalues provide a solid theoretical foundation for using Taylor-based methods in quantum chemistry and physics simulations.
• Future Potential: The method offers a pathway to accelerate the assembly of reduced bases for large parameter domains, potentially enabling real-time or many-query simulations of complex quantum systems.

In summary, this paper presents a mathematically rigorous and computationally efficient framework for approximating parametric eigenvalue problems using Taylor expansions of spectral projectors, overcoming the limitations of classical perturbation theory and standard snapshot-based RBM.