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On the convergence of generalized polynomial chaos expansions

Oliver G. Ernst, Antje Mugler, Hans-Jörg Starkloff, Elisabeth Ullmann (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

A number of approaches for discretizing partial differential equations with random data are based on generalized polynomial chaos expansions of random variables. These constitute generalizations of the polynomial chaos expansions introduced by Norbert Wiener to expansions in polynomials orthogonal with respect to non-Gaussian probability measures. We present conditions on such measures which imply mean-square convergence of generalized polynomial chaos expansions to the correct limit and complement...

On the convergence of generalized polynomial chaos expansions

Oliver G. Ernst, Antje Mugler, Hans-Jörg Starkloff, Elisabeth Ullmann (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

A number of approaches for discretizing partial differential equations with random data are based on generalized polynomial chaos expansions of random variables. These constitute generalizations of the polynomial chaos expansions introduced by Norbert Wiener to expansions in polynomials orthogonal with respect to non-Gaussian probability measures. We present conditions on such measures which imply mean-square convergence of generalized polynomial...

On the convergence of Newton's method under ω*-conditioned second derivative

Ioannis K. Argyros, Saïd Hilout (2011)

Applicationes Mathematicae

We provide a new semilocal result for the quadratic convergence of Newton's method under ω*-conditioned second Fréchet derivative on a Banach space. This way we can handle equations where the usual Lipschitz-type conditions are not verifiable. An application involving nonlinear integral equations and two boundary value problems is provided. It turns out that a similar result using ω-conditioned hypotheses can provide usable error estimates indicating only linear convergence for Newton's method.

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