Experiments on polynomial (chaos) approximation of maximum eigenvalue functions: Tutorial

This tutorial reviews the numerical experiments contained in the article, Fenzi & Michiels (2018) "Polynomial (chaos) approximation of maximum eigenvalue functions: efficiency and limitations", providing a template that can be modified for explorations of your own. The tutorial explores the polynomial approximation of smooth, non-differentiable and not even Lipschitz continuous benchmark functions in the univariate and bivariate cases. The analyzed functions arise from parameter eigenvalue problems; more in details, they are the real part of the rightmost eigenvalue (the so-called spectral abscissa). The polynomial approximations are obtained by Galerkin and collocation approaches. In the Galerkin approach, the numerical approximation of the coefficients in the univariate case is achieved by extended (or composite) Trapezoidal and Simpson's rules or by Gauss and Clenshaw-Curtis quadrature rules. For the bivariate case, the coefficients are approximated by tensorial and non-tensorial Clenshaw-Curtis cubature rules, based on tensor-product Chebyshev grid and Padua points, respectively. The collocation approach interpolates the function on Chebyshev points in the univariate case, while for the bivariate case the interpolant nodes are given by tensor-product Chebyshev grid and Padua points.nrpages: 31status: publishe