Polynomial (chaos) approximation of maximum eigenvalue functions

Abstract

This paper is concerned with polynomial approximations of the spectral abscissa function (defined by the supremum of the real parts of the eigenvalues) of a parameterized eigenvalue problem, which are closely related to polynomial chaos approximations if the parameters correspond to realizations of random variables. Unlike previous work, we highlight the major role of this function smoothness properties. Even if the eigenvalue problem matrices are analytic functions of the parameters, the spectral abscissa function may not be differentiable, and even non-Lipschitz continuous, due to multiple rightmost eigenvalues counted with multiplicity. This analysis demonstrates smoothness properties not only heavily affect the approximation errors of the Galerkin and collocation based polynomial approximations, but also the numerical errors in the evaluation of coefficients in the Galerkin approach with integration methods. A documentation of the experiments, conducted on the benchmark problems through the software Chebfun, is publicly available.

Appendix: Galerkin approach for the spectral abscissa functions in Example 1

The spectral abscissa functions for eigenvalue problems , , and are given by

$$ \begin{array}{lllllll} \alpha^{\text{I}}(\omega)=e^{\omega},\quad \alpha^{\text{II}}(\omega) = \left\{\begin{array}{lllllll} 0, & \text{ if }\omega\in[-1,0),\\ \omega, &\text{ if }\omega\in[0,1]; \end{array}\right.\quad \alpha^{\text{III}}(\omega)=\left\{\begin{array}{llllll} 0, &\text{ if }\omega\in[-1,0),\\ \sqrt{\omega}, &\text{ if }\omega\in[0,1]; \end{array}\right. \end{array} $$

(34)

respectively. In this Appendix, we analytically compute the quantities, which appear in the evaluation of the coefficients c_i in (6).

The ρ-norms of the Legendre polynomial are given by $\|p_{i}\|_{\rho }^{2}=\frac {1}{2i + 1}$ for all $i\in \mathbb {N}$ (see e.g. 22.2.10 in [1]).

To analytically express the coefficients c_i in (6), the evaluation of the ρ-inner product of spectral abscissae (34) and the i th Legendre polynomials is needed. We start analyzing α^I(ω) and we furnish the corresponding ρ-inner product iteratively, starting from:

$$\langle \alpha^{\text{I}},p_{0}\rangle_{\rho}=\frac{1}{2}\left( e-e^{-1}\right). $$

Set $i\in \mathbb {N}$ and i ≥ 1, by using integration by parts, and the following Legendre polynomial properties

$$p_{i}(\pm1) = (\pm1)^{i}, \quad\!\text{and } \frac{\mathrm{d} p_{i + 1}(\omega)}{\mathrm{d}\omega} = \sum\limits_{k = 0}^{\lfloor i/2\rfloor}\frac{p_{i-2k}(\omega)}{\|p_{i-2k}\|_{\rho}^{2}},\quad\! \text{ where } \left\lfloor\frac{i}{2}\right\rfloor = \max_{j\in\mathbb{N}}\left\{j\!\leq\! \frac{i}{2}\right\}, $$

we get

$$\langle\alpha^{\text{I}},p_{i + 1}\rangle_{\rho} = \frac{1}{2}\left( e+\frac{(-1)^{i}}{e} - 2\sum\limits_{k = 0}^{\lfloor i/2\rfloor}\frac{\langle\alpha^{\text{I}},p_{i-2k}\rangle_{\rho}}{\|p_{i-2k}\|_{\rho}^{2}}\right) = \frac{1}{2}\left( e+\frac{(-1)^{i}}{e} - 2\sum\limits_{k = 0}^{\lfloor i/2\rfloor} {c}_{i}^{\text{I}}\right), $$

where ${c}_{i}^{\text {I}}$ indicates the i th coefficients of polynomial approximation (5) of α^I(ω).

For all $i\in \mathbb {N}$, the ρ-inner product of the spectral abscissae α^II(ω) and α^III(ω) can be evaluated by using the relation 22.13.8 and 22.13.9 in [1]:

$$\begin{array}{@{}rcl@{}} \langle\alpha^{\text{II}},p_{i}\rangle_{\rho}&=&\left\{\begin{array}{llllll} \frac{(-1)^{j}{\Gamma}\left( j-\frac{1}{2}\right)}{4{\Gamma}\left( -\frac{1}{2}\right){\Gamma}\left( j + 2\right)}, &{\text{if }i = 2j},\\ \frac{1}{6}, &\text{if }i = 1,\\ 0, &\text{if }i = 2j + 1,\ i>1 \end{array}\right.\\ \langle\alpha^{\text{III}},p_{i}\rangle_{\rho}&=&\left\{\begin{array}{lllll} \frac{(-1)^{j}{\Gamma}\left( j-\frac{1}{4}\right){\Gamma}\left( \frac{3}{4}\right)}{4{\Gamma}\left( -\frac{1}{4}\right){\Gamma}\left( j+\frac{7}{4}\right)}, &\text{if }i = 2j,\\ \frac{(-1)^{j}{\Gamma}\left( j+\frac{1}{4}\right){\Gamma}\left( \frac{5}{4}\right)} {4{\Gamma}\left( \frac{1}{4}\right){\Gamma}\left( j+\frac{9}{4}\right)}, &\text{if }i = 2j + 1, \end{array}\right. \end{array} $$

where Γ(⋅) denotes the Gamma function. The two last inner product are, in fact, rational numbers which can be computed without any error via a symbolic software.