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Uniform Approximation of Eigenproblems of a Large-Scale Parameter-Dependent Hermitian Matrix

Mattia Manucci, Emre Mengi, Nicola Guglielmi

TL;DR

The paper develops a projection-based framework for uniformly approximating the smallest eigenvalue of large parameter-dependent Hermitian matrices by maximizing a surrogate error over the full parameter domain rather than a discrete grid. It proves uniform convergence of the projected eigenvalue function and extends the approach to non-Hermitian cases by directly interpolating the smallest singular value, avoiding squaring the matrix. The method is demonstrated on a suite of parametric PDE-inspired problems, including a five-parameter Heston model, a thermal-block example, and Black-Scholes discretizations, showing accurate uniform approximation with modest reduced dimensions and highlighting benefits of continuum optimization and hybrid discrete-continuum strategies. The appendices establish the necessary Lipschitz continuity of eigenvalues, eigenvectors, and related bounds to support the global convergence theory and provide a rigorous foundation for the interpolation-based singular-value approach.

Abstract

We consider the uniform approximation of the smallest eigenvalue of a large parameter-dependent Hermitian matrix by that of a smaller counterpart obtained through projections. The projection subspaces are constructed iteratively by means of a greedy strategy; at each iteration the parameter where a surrogate error is maximal is computed and the eigenvectors associated with the smallest eigenvalues at the maximizing parameter value are added to the subspace. Unlike the classical approaches, such as the successive constraint method, that maximize such surrogate errors over a discrete and finite set, we maximize the surrogate error over the continuum of all permissible parameter values globally. We formally prove that the projected eigenvalue function converges to the actual eigenvalue function uniformly. In the second part, we focus on the uniform approximation of the smallest singular value of a large parameter-dependent matrix, in case it is non-Hermitian. The proposed frameworks on numerical examples, including those arising from discretizations of parametric PDEs, reduce the size of the large matrix-valued function drastically, while retaining a high accuracy over all permissible parameter values.

Uniform Approximation of Eigenproblems of a Large-Scale Parameter-Dependent Hermitian Matrix

TL;DR

The paper develops a projection-based framework for uniformly approximating the smallest eigenvalue of large parameter-dependent Hermitian matrices by maximizing a surrogate error over the full parameter domain rather than a discrete grid. It proves uniform convergence of the projected eigenvalue function and extends the approach to non-Hermitian cases by directly interpolating the smallest singular value, avoiding squaring the matrix. The method is demonstrated on a suite of parametric PDE-inspired problems, including a five-parameter Heston model, a thermal-block example, and Black-Scholes discretizations, showing accurate uniform approximation with modest reduced dimensions and highlighting benefits of continuum optimization and hybrid discrete-continuum strategies. The appendices establish the necessary Lipschitz continuity of eigenvalues, eigenvectors, and related bounds to support the global convergence theory and provide a rigorous foundation for the interpolation-based singular-value approach.

Abstract

We consider the uniform approximation of the smallest eigenvalue of a large parameter-dependent Hermitian matrix by that of a smaller counterpart obtained through projections. The projection subspaces are constructed iteratively by means of a greedy strategy; at each iteration the parameter where a surrogate error is maximal is computed and the eigenvectors associated with the smallest eigenvalues at the maximizing parameter value are added to the subspace. Unlike the classical approaches, such as the successive constraint method, that maximize such surrogate errors over a discrete and finite set, we maximize the surrogate error over the continuum of all permissible parameter values globally. We formally prove that the projected eigenvalue function converges to the actual eigenvalue function uniformly. In the second part, we focus on the uniform approximation of the smallest singular value of a large parameter-dependent matrix, in case it is non-Hermitian. The proposed frameworks on numerical examples, including those arising from discretizations of parametric PDEs, reduce the size of the large matrix-valued function drastically, while retaining a high accuracy over all permissible parameter values.
Paper Structure (11 sections, 9 theorems, 24 equations, 7 figures, 2 tables)

This paper contains 11 sections, 9 theorems, 24 equations, 7 figures, 2 tables.

Table of Contents

  1. Example 1 (continued)
  2. Example 2
  3. Test cases from parametrized PDEs
  4. Heston model
  5. Black-Scholes test problem
  6. Approximation of the Pseudospectra
  7. Conclusion and outlook
  8. Lipschitz continuity of TEXT
  9. Lipschitz continuity of eigenspaces (eigenvectors)
  10. Uniform Lipschitz continuity of TEXT
  11. Lipschitz continuity of NOTEXT

Key Result

Lemma 3

(Lipschitz Continuity of $\lambda^{\mathcal{V}}_{j}(\mu)$) Let $j$ be a positive integer. There exists a positive real scalar $\gamma_\lambda$ such that for every subspace $\mathcal{V}$ of $\ell^2(\mathbb{N})$ such that $\text{dim} \, \mathcal{V} \geq j$, we have

Figures (7)

  • Figure 1: (Concerns Example 1)$\: A(\mu)\in\mathbb{R}^{\mathsf{n}\times \mathsf{n}}$ full matrix as in \ref{['eqn:ex1:aff']}, $\mathsf{n}=10^2$ and $\: A^{V}(\mu)\in\mathbb{R}^{\mathsf{d} \times \mathsf{d}}$ with $\mathsf{d}=32$.
  • Figure 2: (Concerns Example 1)$\: A(\mu)\in\mathbb{R}^{\mathsf{n}\times \mathsf{n}}$ in \ref{['eqn:ex1:aff']}, $\mathsf{n}=10^2$.
  • Figure 3: (Concerns Example 2) Dense matrix $A(\mu)\in\mathbb{R}^{\mathsf{n}\times \mathsf{n}}$ as in \ref{['eqn:ex2:aff']}, $\mathsf{n} = 2000$, with projected matrix $A^V(\mu)\in\mathbb{R}^{\mathsf{d} \times \mathsf{d}}$, $\mathsf{d} = 32$.
  • Figure 4: (Concerns Example 2) Computational time (CT, in seconds) for \ref{['alg:sf']} and its key components.
  • Figure 5: (Heston model) Computational time (CT, in seconds) for \ref{['alg:sv']} with SSCM and its key components.
  • ...and 2 more figures

Theorems & Definitions (11)

  • Lemma 3
  • Definition 1: Invariant subspace
  • Definition 2: Simple invariant subspace
  • Theorem 6: Lipschitz continuity of simple invariant subspaces
  • Theorem 7
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Theorem 8
  • ...and 1 more