Absence of nematic quasi-long-range order in two-dimensional liquid crystals with three director components

The Lebwohl-Lasher model describes the isotropic-nematic transition in liquid crystals. In two dimensions, where its continuous symmetry cannot break spontaneously, it is investigated numerically since decades to verify, in particular, the conjecture of a topological transition leading to a nematic phase with quasi-long-range order. We use scale invariant scattering theory to exactly determine the renormalization group fixed points in the general case of N director components (RPN 121 model), which yields the Lebwohl-Lasher model for N = 3. For N > 2 we show the absence of quasi-long-range order and the presence of a zero temperature critical point in the universality class of the O(N(N + 1)/2 12 1) model. For N = 2 the fixed point equations yield the Berezinskii-Kosterlitz-Thouless transition required by the correspondence RP1 3c O(2)

Globally convergent evolution strategies for constrained optimization

International audienceIn this paper we propose, analyze, and test algorithms for constrained optimization when no use of derivatives of the objective function is made. The proposed methodology is built upon the globally convergent evolution strategies previously introduced by the authors for unconstrained optimization. Two approaches are encompassed to handle the constraints. In a first approach, feasibility is first enforced by a barrier function and the objective function is then evaluated directly at the feasible generated points. A second approach projects first all the generated points onto the feasible domain before evaluating the objective function.The resulting algorithms enjoy favorable global convergence properties (convergence to stationarity from arbitrary starting points), regardless of the linearity of the constraints.The algorithmic implementation (i) includes a step where previously evaluated points are used to accelerate the search (by minimizing quadratic models) and (ii) addresses the particular cases of bounds on the variables and linear constraints. Our solver is compared to others, and the numerical results confirm its competitiveness in terms of efficiency and robustness

Critical points in the CP N-1model

We use scale invariant scattering theory to obtain the exact equations determining the renormalization group fixed points of the two-dimensional CP ( N-1) model, for N real. Also due to special degeneracies at N = 2 and 3, the space of solutions for N > 2 reduces to that of the O(N (2) - 1) model, and accounts for a zero temperature critical point. For N < 2 the space of solutions becomes larger than that of the O(N (2) - 1) model, with the appearance of new branches of fixed points relevant for criticality in gases of intersecting loops

A mixed-categorical correlation kernel for Gaussian process

Recently, there has been a growing interest for mixed-categorical meta-models based on Gaussian process (GP) surrogates. In this setting, several existing approaches use different strategies either by using continuous kernels (e.g., continuous relaxation and Gower distance based GP) or by using a direct estimation of the correlation matrix. In this paper, we present a kernel-based approach that extends continuous exponential kernels to handle mixed-categorical variables. The proposed kernel leads to a new GP surrogate that generalizes both the continuous relaxation and the Gower distance based GP models. We demonstrate, on both analytical and engineering problems, that our proposed GP model gives a higher likelihood and a smaller residual error than the other kernel-based state-of-the-art models. Our method is available in the open-source software SMT.Comment: version

A Parallel Evolution Strategy for an Earth Imaging Problem in Geophysics

International audienceIn this paper we propose a new way to compute a rough approximation solution, to be later used as a warm starting point in a more refined optimization process, for a challenging global optimization problem related to Earth imaging in geophysics. The warm start con- sists of a velocity model that approximately solves a full-waveform inverse problem at low frequency. Our motivation arises from the availability of massively parallel computing plat- forms and the natural parallelization of evolution strategies as global optimization methods for continuous variables.Our first contribution consists of developing a new and efficient parametrization of the velocity models to significantly reduce the dimension of the original optimization space. Our second contribution is to adapt a class of evolution strategies to the specificity of the physical problem at hands where the objective function evaluation is known to be the most expen- sive computational part. A third contribution is the development of a parallel evolution strategy solver, taking advantage of a recently proposed modification of these class of evolu- tionary methods that ensures convergence and promotes better performance under moderate budgets.The numerical results presented demonstrate the effectiveness of the algorithm on a realistic 3D full-waveform inverse problem in geophysics. The developed numerical approach allows us to successfully solve an acoustic full-waveform inversion problem at low frequencies on a reasonable number of cores of a distributed memory computer