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| HAL: inria-00535806, version 1 |
| arXiv: 1011.3107 |
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| A probabilistic algorithm approximating solutions of a singular PDE of porous media type |
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| Nadia Belaribi 1, 2François Cuvelier 2 |
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| (2010-11-12) |
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| The object of this paper is a one-dimensional generalized porous media equation (PDE) with possibly discontinuous coefficient $\beta$, which is well-posed as an evolution problem in $L^1(\mathbb{R})$. In some recent papers of Blanchard et alia and Barbu et alia, the solution was represented by the solution of a non-linear stochastic differential equation in law if the initial condition is a bounded integrable function. We first extend this result, at least when $\beta$ is continuous and the initial condition is only integrable with some supplementary technical assumption. The main purpose of the article consists in introducing and implementing a stochastic particle algorithm to approach the solution to (PDE) which also fits in the case when $\beta$ is possibly irregular, to predict some long-time behavior of the solution and in comparing with some recent numerical deterministic techniques. |
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| 1: | École Nationale Supérieure de Techniques Avancées (ENSTA ParisTech) |
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ENSTA ParisTech |
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| 2: | Laboratoire d'Analyse, Géométrie et Applications (LAGA) |
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CNRS : UMR7539 – Université Paris XIII - Paris Nord – Université Paris VIII - Vincennes Saint-Denis |
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| 3: | MATHFI (INRIA Rocquencourt) |
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INRIA – Ecole des Ponts ParisTech – Université Paris XII - Paris Est Créteil Val-de-Marne |
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| Unité de Mathématiques Appliquées, ENSTA ParisTech |
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| Domain | : | Mathematics/Probability Mathematics/Numerical Analysis Statistics/Applications |
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| Stochastic particle algortithm – porous media equation – monotonicity – stochastic differential equations – non-parametric density estimation – kernel estimator |
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| inria-00535806, version 1 | |
| http://hal.inria.fr/inria-00535806 | |
| oai:hal.inria.fr:inria-00535806 | |
| From: Francesco Russo | |
| Submitted on: Friday, 12 November 2010 18:59:05 | |
| Updated on: Friday, 10 December 2010 11:31:54 | |
