We address the following inverse problem in quantum statistical physics: does the quantum free energy (von Neumann entropy + kinetic energy) admit a unique minimizer among the density operators having a given local density n(x)? We give a positive answer to that question, in dimension one. This enables to define rigourously the notion of local quantum equilibrium, or quantum Maxwellian, which i...

In 1957 Abrikosov published his work on periodic solutions to the linearized Ginzburg-Landau equations. Abrikosov’s analysis assumes periodic boundary conditions, which are very different from the natural boundary conditions the minimizer of the Ginzburg-Landau energy functional should satisfy. In the present work we prove that the global minimizer of the fully non-linear functional can be appr...

We consider mass-constrained minimizers for a class of non-convex energy functionals involving a double-well potential. Based upon global quadratic lower bounds to the energy, we introduce a simple strategy to find sufficient conditions on a given critical point (metastable state) to be a global minimizer. We show that this strategy works well for the one exact and known metastable state: the c...

We consider the approximate minimization of a given polynomial on the standard simplex, obtained by taking the minimum value over all rational grid points with given denominator r ∈ N. It was shown in [De Klerk, E., Laurent, M., Sun, Z.: An error analysis for polynomial optimization over the simplex based on the multivariate hypergeometric distribution. SIAM J. Optim. 25(3) 1498–1514 (2015)] th...

We prove that the global minimizer of the Ginzburg-Landau functional of super-conductors in an external magnetic eld is, below the rst critical eld, the vortex-less solution found in S1].

In the first part of this contribution we prove the global existence and uniqueness of a trajectory that globally converges to the minimizer of the Gross-Pitaevskii energy functional for a large class of external potentials. Using the method of Sobolev gradients we can provide an explicit construction of this minimizing sequence. In the second part we numerically apply these results to a specif...

In this paper we present a simple algorithm for global optimization. This algorithm combines random searches with eecient local minimization algorithms. The proposed algorithm begins with an initial \local minimizer." In each iteration, a search direction is generated randomly, along which some points are chosen as the initial points for the local optimization algorithm and several \local minim...

We address estimation problems where the sought-after solution is defined as the minimizer of an objective function composed of a quadratic data-fidelity term and a regularization term. We especially focus on nonsmooth and/or nonconvex regularization terms because of their ability to yield good estimates. This work is dedicated to the stability of the minimizers of such nonsmooth and/or nonconv...

A popular class of algorithms to optimize the dual LP relaxation of the discrete energy minimization problem (a.k.a. MAP inference in graphical models or valued constraint satisfaction) are convergent message-passing algorithms, such as max-sum diffusion, TRW-S, MPLP and SRMP. These algorithms are successful in practice, despite the fact that they are a version of coordinate minimization applie...

Replacing linear diffusion by a degenerate diffusion of porous medium type is known to regularize the classical two-dimensional parabolic-elliptic Keller-Segel model [10]. The implications of nonlinear diffusion are that solutions exist globally and are uniformly bounded in time. We analyse the stationary case showing the existence of a unique, up to translation, global minimizer of the associa...