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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 SCF algorithms for the Hartree-Fock equations

Eric Cancès, Claude Le Bris (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The present work is a mathematical analysis of two algorithms, namely the Roothaan and the level-shifting algorithms, commonly used in practice to solve the Hartree-Fock equations. The level-shifting algorithm is proved to be well-posed and to converge provided the shift parameter is large enough. On the contrary, cases when the Roothaan algorithm is not well defined or fails in converging are exhibited. These mathematical results are confronted to numerical experiments performed by chemists.

On the Convergence of the Approximate Free Boundary for the Parabolic Obstacle Problem

Paola Pietra, Claudio Verdi (1985)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Si discretizza il problema dell'ostacolo parabolico con differenze all'indietro nel tempo ed elementi finiti lineari nello spazio e si dimostrano stime dell'errore per la frontiera libera discreta.

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