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Collevecchio A.,University of Venice | Konig W.,Weierstrass Institute Berlin | Konig W.,TU Berlin | Morters P.,University of Bath | Sidorova N.,University College London
Communications in Mathematical Physics | Year: 2010

We consider a classical dilute particle system in a large box with pair-interaction given by a Lennard-Jones-type potential. The inverse temperature is picked proportionally to the logarithm of the particle density. We identify the free energy per particle in terms of a variational formula and show that this formula exhibits a cascade of phase transitions as the temperature parameter ranges from zero to infinity. Loosely speaking, the particle system separates into spatially distant components in such a way that within each phase all components are of the same size, which is the larger the lower the temperature. The main tool in our proof is a new large deviation principle for sparse point configurations. © 2010 Springer-Verlag.

Van Ackooij W.,Electricite de France | Van Ackooij W.,Ecole Centrale Paris | Henrion R.,Weierstrass Institute Berlin
SIAM Journal on Optimization | Year: 2014

Probabilistic constraints represent a major model of stochastic optimization. A possible approach for solving probabilistically constrained optimization problems consists in applying nonlinear programming methods. To do so, one has to provide sufficiently precise approximations for values and gradients of probability functions. For linear probabilistic constraints under Gaussian distribution this can be done successfully by analytically reducing these values and gradients to values of Gaussian distribution functions and computing the latter, for instance, by Genz's code. For nonlinear models one may fall back on the spherical-radial decomposition of Gaussian random vectors and apply, for instance, Deák's sampling scheme for the uniform distribution on the sphere in order to compute values of corresponding probability functions. The present paper demonstrates how the same sampling scheme can be used to simultaneously compute gradients of these probability functions. More precisely, we prove a formula representing these gradients in the Gaussian case as a certain integral over the sphere again. The result is also extended to alternative distributions with an emphasis on the multivariate Student's (or t-) distribution. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

Jansen S.,Weierstrass Institute Berlin | Konig W.,Weierstrass Institute Berlin | Konig W.,TU Berlin
Journal of Statistical Physics | Year: 2012

We consider an interacting particle system in continuous configuration space. The pair interaction has an attractive part. We show that, at low density, the system behaves approximately like an ideal mixture of clusters (droplets): we prove rigorous bounds (a) for the constrained free energy associated with a given cluster size distribution, considered as an order parameter, (b) for the free energy, obtained by minimising over the order parameter, and (c) for the minimising cluster size distributions. It is known that, under suitable assumptions, the ideal mixture has a transition from a gas phase to a condensed phase as the density is varied; our bounds hold both in the gas phase and in the coexistence region of the ideal mixture. The present paper improves our earlier results by taking into account the mixing entropy. © 2012 Springer Science+Business Media, LLC.

Eigel M.,Weierstrass Institute Berlin | Merdon C.,Weierstrass Institute Berlin
Journal of Scientific Computing | Year: 2015

We study a posteriori error estimates for convection–diffusion–reaction problems with possibly dominating convection or reaction and inhomogeneous boundary conditions. For the conforming FEM discretisation with streamline diffusion stabilisation, we derive reliable and efficient error estimators based on the reconstruction of equilibrated fluxes in an admissible discrete subspace of (Formula presented.). Error estimators of this type have become popular recently since they provide guaranteed error bounds without further unknown constants. The estimators can be improved significantly by some postprocessing and divergence correction technique. For an extension of the energy norm by a dual norm of the convection part of the differential operator, robustness of the error estimator with respect to the coefficients of the problem is achieved. Numerical benchmarks illustrate the good performance of the error estimators for singularly perturbed problems, in particular with dominating convection. © 2015 Springer Science+Business Media New York

van Ackooij W.,Electricite de France | Henrion R.,Weierstrass Institute Berlin | Moller A.,Weierstrass Institute Berlin | Zorgati R.,Electricite de France
Optimization and Engineering | Year: 2014

In this paper, we deal with a cascaded reservoir optimization problem with uncertainty on inflows in a joint chance constrained programming setting. In particular, we will consider inflows with a persistency effect, following a causal time series model with Gaussian innovations. We present an iterative algorithm for solving similarly structured joint chance constrained programming problems that requires a Slater point and the computation of gradients. Several alternatives to the joint chance constraint problem are presented. In particular, we present an individual chance constraint problem and a robust model. We illustrate the interest of joint chance constrained programming by comparing results obtained on a realistic hydro valley with those obtained from the alternative models. Despite the fact that the alternative models often require less hypothesis on the law of the inflows, we show that they yield conservative and costly solutions. The simpler models, such as the individual chance constraint one, are shown to yield insufficient robustness and are therefore not useful. We therefore conclude that Joint Chance Constrained programming appears as an approach offering a good trade-off between cost and robustness and can be tractable for complex realistic models. © 2013 Springer Science+Business Media New York.

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