Chebyshev interpolation for nonlinear eigenvalue problems

Abstract

This work is concerned with numerical methods for matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. In particular, we focus on eigenvalue problems for which the evaluation of the matrix-valued function is computationally expensive. Such problems arise, e.g., from boundary integral formulations of elliptic PDE eigenvalue problems and typically exclude the use of established nonlinear eigenvalue solvers. Instead, we propose the use of polynomial approximation combined with non-monomial linearizations. Our approach is intended for situations where the eigenvalues of interest are located on the real line or, more generally, on a pre-specified curve in the complex plane. A first-order perturbation analysis for nonlinear eigenvalue problems is performed. Combined with an approximation result for Chebyshev interpolation, this shows exponential convergence of the obtained eigenvalue approximations with respect to the degree of the approximating polynomial. Preliminary numerical experiments demonstrate the viability of the approach in the context of boundary element methods.

Polynomial Eigenvalue Problems: Theory, Computation, and Structure

“…Although there are precedents for doing this for scalar polynomials in [10], and even earlier in [33], the first serious effort in this direction for matrix polynomials was [5] and the earlier [17], where concrete templates for producing strong linearizations were provided, one for each of several classical polynomial bases, including Chebyshev, Newton, Lagrange, and Bernstein bases. This work has been used in [26], as part of a Chebyshev interpolation method for solving non-polynomial nonlinear eigenproblems. Additional examples for the Hermite and Lagrange bases have been developed and used in [81,82].…”

“…The linearization strategy has been so effective for polynomial eigenproblems that researchers have started to consider ways to extend this strategy to other nonlinear eigenproblems, especially to rational eigenproblems P(λ )x = 0, where the scalar φ i (λ ) functions in P(λ ) as in (26) are now rational functions of λ rather than just polynomials. Significant advances in this direction have been made in [77], and more recently in [35].…”

Memory-efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis

“…see also [9]. Invariant pairs generalize the notion of eigenpairs in the following sense: If (X, Λ) is minimal, i.e., V has full column rank, then the eigenvalues of Λ are eigenvalues of (4) and span(X) contains the corresponding (generalized) eigenvectors [16].…”

Constructing Strong Linearizations of Matrix Polynomials Expressed in Chebyshev Bases

“…Chebyshev polynomials are widely used in many areas of numerical analysis, especially in approximation theory [12,32,26,31]. A common approach to solving the nonlinear eigenvalue problem T (λ)x = 0, y * T (λ) = 0, for a holomorphic matrix-valued function T : Ω → C n×n , is to construct a matrix polynomial approximation P (λ) of the function T (λ), replacing the nonlinear eigenvalue problem with the polynomial eigenvalue problem P (λ)x = 0, y * P (λ) = 0 via Chebyshev interpolation [12]. One of the most common approaches to solve polynomial eigenvalue problems is to linearize the matrix polynomial.…”