scholarly journals Variational Approximation of Functionals Defined on 1-dimensional Connected Sets: The Planar Case

2018 ◽  
Vol 50 (6) ◽  
pp. 6307-6332 ◽  
Author(s):  
Mauro Bonafini ◽  
Giandomenico Orlandi ◽  
Édouard Oudet
Keyword(s):  
Variational Approximation ◽  
Planar Case ◽  
Connected Sets

Variational approximation of functionals defined on 1-dimensional connected sets in ℝn

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mauro Bonafini ◽  
Giandomenico Orlandi ◽  
Édouard Oudet
Keyword(s):  
Variational Problems ◽  
Convergence Result ◽  
Steiner Tree Problem ◽  
Variational Approximation ◽  
Planar Case ◽  
Ginzburg Landau ◽  
Connected Sets ◽  
Steiner Problems ◽  
Γ Convergence ◽  
Euclidean Steiner Tree Problem

AbstractIn this paper we consider the Euclidean Steiner tree problem and, more generally, (single sink) Gilbert–Steiner problems as prototypical examples of variational problems involving 1-dimensional connected sets in {\mathbb{R}^{n}}. Following the analysis for the planar case presented in [M. Bonafini, G. Orlandi and E. Oudet, Variational approximation of functionals defined on 1-dimensional connected sets: The planar case, SIAM J. Math. Anal. 50 2018, 6, 6307–6332], we provide a variational approximation through Ginzburg–Landau type energies proving a Γ-convergence result for {n\geq 3}.

scholarly journals Verification Studies for the Noh Problem Using Nonideal Equations of State and Finite Strength Shocks

2018 ◽  
Vol 3 (2) ◽  
Author(s):  
Sarah C. Burnett ◽  
Kevin G. Honnell ◽  
Scott D. Ramsey ◽  
Robert L. Singleton
Keyword(s):  
Equations Of State ◽  
National Laboratory ◽  
Jump Conditions ◽  
Self Similarity ◽  
Planar Case ◽  
Los Alamos National Laboratory ◽  
Flow Equations ◽  
Arbitrary Strength ◽  
Verification Test ◽  
Gamma Law

The Noh verification test problem is extended beyond the commonly studied ideal gamma-law gas to more realistic equations of state (EOSs) including the stiff gas, the Noble-Abel gas, and the Carnahan–Starling EOS for hard-sphere fluids. Self-similarity methods are used to solve the Euler compressible flow equations, which, in combination with the Rankine–Hugoniot jump conditions, provide a tractable general solution. This solution can be applied to fluids with EOSs that meet criterion such as it being a convex function and having a corresponding bulk modulus. For the planar case, the solution can be applied to shocks of arbitrary strength, but for the cylindrical and spherical geometries, it is required that the analysis be restricted to strong shocks. The exact solutions are used to perform a variety of quantitative code verification studies of the Los Alamos National Laboratory Lagrangian hydrocode free Lagrangian (FLAG).

Identifying graph clusters using variational inference and links to covariance parametrization

2009 ◽  
Vol 367 (1906) ◽  
pp. 4407-4426
Author(s):  
David Barber
Keyword(s):  
Mean Field ◽  
Matrix Decomposition ◽  
Difficult Problem ◽  
Positive Definite ◽  
Variational Inference ◽  
Field Theories ◽  
Variational Approximation ◽  
Positive Definite Matrices ◽  
The Matrix ◽  
Non Gaussian

Finding clusters of well-connected nodes in a graph is a problem common to many domains, including social networks, the Internet and bioinformatics. From a computational viewpoint, finding these clusters or graph communities is a difficult problem. We use a clique matrix decomposition based on a statistical description that encourages clusters to be well connected and few in number. The formal intractability of inferring the clusters is addressed using a variational approximation inspired by mean-field theories in statistical mechanics. Clique matrices also play a natural role in parametrizing positive definite matrices under zero constraints on elements of the matrix. We show that clique matrices can parametrize all positive definite matrices restricted according to a decomposable graph and form a structured factor analysis approximation in the non-decomposable case. Extensions to conjugate Bayesian covariance priors and more general non-Gaussian independence models are briefly discussed.

Maximal connected sets—application to microcell CDMA

1994 ◽  
Vol 7 (1) ◽  
pp. 29-32 ◽  
Author(s):  
A. Z. Tirkel ◽  
C. F. Osborne ◽  
N. Mee ◽  
G. A. Rankin ◽  
A. McAndrew
Keyword(s):  
Connected Sets
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