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Sparse Approximation of Parametric Matrix Functions

by C. Schwab and M. Trautner and J. Westermann

(Report number 2026-38)

Abstract
For functions of parametric matrices on high-dimensional parameter spaces with analytic parameter dependence, we propose algorithms for sparse, parametric surrogates of such functions and analyze their approximation rates. Specifically, we consider parametric matrices \(\boldsymbol{A}_{\boldsymbol{y}}\) depending on a finite sequence of \(J\) parameters \(\boldsymbol{y} = (y_j)_{1\leq j\leq J}\). We analyze surrogates of certain holomorphic functions \(f(z)\) of such matrices, and of certain factorizations such as the \(\boldsymbol{L}\boldsymbol{D}\boldsymbol{L}^\top\), Cholesky- and \(\boldsymbol{Q}\boldsymbol{R}\) decompositions. We prove that domains of parametric holomorphy propagate to the functions and factorizations. Implications of the results are exponentially convergent sparse collocation approximations for surrogates of the matrix functions and the parametric factors. We illustrate the scope of the results using examples with discretized parametric partial differential equations, where we accelerate uncertainty quantification computations involving the finite element stiffness matrix and its Cholesky factor. This enables parsimonious surrogate construction of factored many-parametric matrices and holomorphic functions thereof, with error guarantees. Numerical examples illustrate applications including efficient multi-query Galerkin discretizations of parametric partial differential equations that arise in uncertainty quantification and optimization.

Keywords: Cholesky decomposition, Matrix factorization, Matrix function, Polynomial approximation, High dimensionality, Sparse grid, Polynomial interpolation, Holomorphy

@Techreport{STW26_1193,
  author = {C. Schwab and M. Trautner and J. Westermann},
  title = {Sparse Approximation of Parametric Matrix Functions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-38},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-38.pdf },
  year = {2026}
}

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