A linear-time approximate convex envelope algorithm using the double Legendre–Fenchel transform with application to phase separation

Contento, Lorenzo; Ern, Alexandre; Vermiglio, Rossana

doi:10.1007/s10589-014-9666-8

Cited by 9 publications

(7 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…convex conjugate) twice. Hence, one can use the existing linear-time algorithms for approximating Legendre-Fenchel transformation (e.g., [ 87 , 88 ]) for approximating

.…”

Section: Information Bottleneck and Privacy Funnel: Definitions Anmentioning

confidence: 99%

Bottleneck Problems: An Information and Estimation-Theoretic View

Asoodeh

Calmon

2020

Entropy

View full text Add to dashboard Cite

Information bottleneck (IB) and privacy funnel (PF) are two closely related optimization problems which have found applications in machine learning, design of privacy algorithms, capacity problems (e.g., Mrs. Gerber’s Lemma), and strong data processing inequalities, among others. In this work, we first investigate the functional properties of IB and PF through a unified theoretical framework. We then connect them to three information-theoretic coding problems, namely hypothesis testing against independence, noisy source coding, and dependence dilution. Leveraging these connections, we prove a new cardinality bound on the auxiliary variable in IB, making its computation more tractable for discrete random variables. In the second part, we introduce a general family of optimization problems, termed “bottleneck problems”, by replacing mutual information in IB and PF with other notions of mutual information, namely f-information and Arimoto’s mutual information. We then argue that, unlike IB and PF, these problems lead to easily interpretable guarantees in a variety of inference tasks with statistical constraints on accuracy and privacy. While the underlying optimization problems are non-convex, we develop a technique to evaluate bottleneck problems in closed form by equivalently expressing them in terms of lower convex or upper concave envelope of certain functions. By applying this technique to a binary case, we derive closed form expressions for several bottleneck problems.

show abstract

“…convex conjugate) twice. Hence, one can use the existing linear-time algorithms for approximating Legendre-Fenchel transformation (e.g., [ 87 , 88 ]) for approximating

.…”

Section: Information Bottleneck and Privacy Funnel: Definitions Anmentioning

confidence: 99%

Bottleneck Problems: An Information and Estimation-Theoretic View

Asoodeh

Calmon

2020

Entropy

View full text Add to dashboard Cite

show abstract

“…As is well-known, the MSA overpredicts the critical temperature, which is about T * crit = 0.05 according to Monte Carlo simulations [48]. For non-symmetric electrolytes, a study of the convexity of the bulk free energy functional can be found in [49].…”

Section: Methodsmentioning

confidence: 90%

“…The approximate subset K * * of the two-dimensional phase space then consists of two connected components (associated with the dilute and condensed phases) separated by the stripe {ρ ♭ tot ≤ρ + +ρ − ≤ρ ♯ tot } whose boundary are two parallel lines in the (ρ + , ρ − )plane. A comparison with the exact subset K * * determined numerically by computing the double Legendre-Fenchel transform of f (see [49]) shows relative differences in predicted ionic densities below 10 −4 away from the interface and 10 −2 near the interface.…”

Section: 1mentioning

confidence: 99%

Numerical study of density functional theory with mean spherical approximation for ionic condensation in highly charged confined electrolytes

et al. 2014

View full text Add to dashboard Cite

We investigate numerically a density functional theory (DFT) for strongly confined ionic solutions in the canonical ensemble by comparing predictions of ionic concentration profiles and pressure for the double-layer configuration to those obtained with Monte Carlo (MC) simulations and the simpler Poisson-Boltzmann (PB) approach. The DFT consists of a bulk (ion-ion) and an ion-solid part. The bulk part includes nonideal terms accounting for long-range electrostatic and short-range steric correlations between ions and is evaluated with the mean spherical approximation and the local density approximation. The ion-solid part treats the ion-solid interactions at the mean-field level through the solution of a Poisson problem. The main findings are that ionic concentration profiles are generally better described by PB than by DFT, although DFT captures the nonmonotone co-ion profile missed by PB. Instead, DFT yields more accurate pressure predictions than PB, showing in particular that nonideal effects are important to describe highly confined ionic solutions. Finally, we present a numerical methodology capable of handling nonconvex minimization problems so as to explore DFT predictions when the reduced temperature falls below the critical temperature.

show abstract

“…Note that the variable-switching formulation cannot do it either. Future works should tackle this question perhaps by combining the unified formulation with some judicious approaches advocated by [25,11].…”

Section: Alteration In the Three-root Regionmentioning

confidence: 99%

An analysis of the unified formulation for the equilibrium problem of compositional multiphase mixtures

et al. 2021

View full text Add to dashboard Cite

In this paper, we conduct a thorough mathematical analysis of the unified formulation advocated by Lauser et al. [Adv. Water Res. 34 (2011) 957–966] for compositional multiphase flows in porous media. The interest of this formulation lies in its potential to automatically handle the appearance and disappearance of phases. However, its practical implementation turned out to be not always robust for realistic fugacity laws associated with cubic equations of state, as shown by Ben Gharbia and Flauraud [Oil Gas Sci. Technol. 74 (2019) 43]. By focusing on the subproblem of phase equilibrium, we derive sufficient conditions for the existence of the corresponding system of equations. We trace back the difficulty of cubic laws to a deficiency of the Gibbs functions that comes into play due to the ``unifying’’ feature of the new formulation. We propose a partial remedy for this problem by extending the domain of definition of these functions in a natural way. Besides, we highlight the crucial but seemingly unknown fact that the unified formulation encapsulates all the properties known to physicists on phase equilibrium, such as the tangent plane criterion and the minimization of the Gibbs energy of the mixture.

show abstract

A linear-time approximate convex envelope algorithm using the double Legendre–Fenchel transform with application to phase separation

Cited by 9 publications

References 17 publications

Bottleneck Problems: An Information and Estimation-Theoretic View

Bottleneck Problems: An Information and Estimation-Theoretic View

Numerical study of density functional theory with mean spherical approximation for ionic condensation in highly charged confined electrolytes

An analysis of the unified formulation for the equilibrium problem of compositional multiphase mixtures

Contact Info

Product

Resources

About