A numerical comparison of solvers for large-scale, continuous-time algebraic Riccati equations and LQR problems

In this paper, we discuss numerical methods for solving large-scale continuous-time algebraic Riccati equations. These methods have been the focus of intensive research in recent years, and significant progress has been made in both the theoretical understanding and efficient implementation of various competing algorithms. There are several goals of this manuscript: first, to gather in one place an overview of different approaches for solving large-scale Riccati equations, and to point to the recent advances in each of them. Second, to analyze and compare the main computational ingredients of these algorithms, to detect their strong points and their potential bottlenecks. And finally, to compare the effective implementations of all methods on a set of relevant benchmark examples, giving an indication of their relative performance

Simple Approximations of Semialgebraic Sets and their Applications to Control

Many uncertainty sets encountered in control systems analysis and design can be expressed in terms of semialgebraic sets, that is as the intersection of sets described by means of polynomial inequalities. Important examples are for instance the solution set of linear matrix inequalities or the Schur/Hurwitz stability domains. These sets often have very complicated shapes (non-convex, and even non-connected), which renders very difficult their manipulation. It is therefore of considerable importance to find simple-enough approximations of these sets, able to capture their main characteristics while maintaining a low level of complexity. For these reasons, in the past years several convex approximations, based for instance on hyperrect-angles, polytopes, or ellipsoids have been proposed. In this work, we move a step further, and propose possibly non-convex approximations , based on a small volume polynomial superlevel set of a single positive polynomial of given degree. We show how these sets can be easily approximated by minimizing the L1 norm of the polynomial over the semialgebraic set, subject to positivity constraints. Intuitively, this corresponds to the trace minimization heuristic commonly encounter in minimum volume ellipsoid problems. From a computational viewpoint, we design a hierarchy of linear matrix inequality problems to generate these approximations, and we provide theoretically rigorous convergence results, in the sense that the hierarchy of outer approximations converges in volume (or, equivalently, almost everywhere and almost uniformly) to the original set. Two main applications of the proposed approach are considered. The first one aims at reconstruction/approximation of sets from a finite number of samples. In the second one, we show how the concept of polynomial superlevel set can be used to generate samples uniformly distributed on a given semialgebraic set. The efficiency of the proposed approach is demonstrated by different numerical examples

Effectively Stable Dark Matter

We study dark matter (DM) which is cosmologically long-lived because of standard model (SM) symmetries. In these models an approximate stabilizing symmetry emerges accidentally, in analogy with baryon and lepton number in the renormalizable SM. Adopting an effective theory approach, we classify DM models according to representations of $SU(3)_C\times SU(2)_L\times U(1)_Y \times U(1)_B\times U(1)_L$, allowing for all operators permitted by symmetry, with weak scale DM and a cutoff at or below the Planck scale. We identify representations containing a neutral long-lived state, thus excluding dimension four and five operators that mediate dangerously prompt DM decay into SM particles. The DM relic abundance is obtained via thermal freeze-out or, since effectively stable DM often carries baryon or lepton number, asymmetry sharing through the very operators that induce eventual DM decay. We also incorporate baryon and lepton number violation with a spurion that parameterizes hard breaking by arbitrary units. However, since proton stability precludes certain spurions, a residual symmetry persists, maintaining the cosmological stability of certain DM representations. Finally, we survey the phenomenology of effectively stable DM as manifested in probes of direct detection, indirect detection, and proton decay.Comment: 7 pages, 1 figure, 4 table

Cosmology and Hierarchy in Stabilized Warped Brane Models

We examine the cosmology and hierarchy of scales in models with branes immersed in a five-dimensional curved spacetime subject to radion stabilization. When the radion field is time-independent and the inter-brane spacing is stabilized, the universe can naturally find itself in the radiation-dominated epoch. This feature is independent of the form of the stabilizing potential. We recover the standard Friedmann equations without assuming a specific form for the bulk energy-momentum tensor. In the models considered, if the observable brane has positive tension, a solution to the hierarchy problem requires the presence of a negative tension brane somewhere in the bulk. We find that the string scale can be as low as the electroweak scale. In the situation of self-tuning branes where the bulk cosmological constant is set to zero, the brane tensions have hierarchical values. In the case of a polynomial stabilizing potential no new hierarchy is created.Comment: Version to appear in PL

Exploring the Levinthal limit in protein folding

According to the thermodynamic hypothesis, the native state of proteins is uniquely defined by their amino acid sequence. On the other hand, according to Levinthal, the native state is just a local minimum of the free energy and a given amino acid sequence, in the same thermodynamic conditions, can assume many, very different structures that are as thermodynamically stable as the native state. This is the Levinthal limit explored in this work. Using computer simulations, we compare the interactions that stabilize the native state of four different proteins with those that stabilize three non-native states of each protein and find that the nature of the interactions is very similar for all such 16 conformers. Furthermore, an enhancement of the degree of fluctuation of the non-native conformers can be explained by an insufficient relaxation to their local free energy minimum. These results favor Levinthal's hypothesis that protein folding is a kinetic non-equilibrium process.FCT - Foundation for Science and Technology, Portugal [UID/Multi/04326/2013]; Fundacao de Amparo a Pesquisa do Estado de Sao Paulo (FAPESP); Conselho Nacional de Desenvolvimento Cientia co e Tecnologico (CNPq

A Turbulent Model for the Interstellar Medium. II. Magnetic Fields and Rotation

We present results from two-dimensional numerical simulations of a supersonic turbulent flow in the plane of the galactic disk, incorporating shear, thresholded and discrete star formation (SF), self-gravity, rotation and magnetic fields. A test of the model in the linear regime supports the results of the linear theory of Elmegreen (1991). In the fully nonlinear turbulent regime, while some results of the linear theory persist, new effects also emerge. Some exclusively nonlinear effects are: a) Even though there is no dynamo in 2D, the simulations are able to maintain or increase their net magnetic energy in the presence of a seed uniform azimuthal component. b) A well-defined power-law magnetic spectrum and an inverse magnetic cascade are observed in the simulations, indicating full MHD turbulence. Thus, magnetic field energy is generated in regions of SF and cascades up to the largest scales. c) The field has a slight but noticeable tendency to be aligned with density features. d) The magnetic field prevents HII regions from expanding freely, as in the recent results of Slavin \& Cox (1993). e) A tendency to exhibit {\it less} filamentary structures at stronger values of the uniform component of the magnetic field is present in several magnetic runs. f) For fiducial values of the parameters, the flow in general appears to be in rough equipartition between magnetic and kinetic energy. There is no clear domination of either the magnetic or the inertial forces. g) A median value of the magnetic field strength within clouds is $\sim 12\mu$G, while for the intercloud medium a value of $\sim 3\mu$G is found. Maximum contrasts of up to a factor of $\sim 10$ are observed.Comment: Plain TeX file, 25 pages. Gzipped, tarred set of Tex file plus 17 figures and 3 tables (Postscript) available at ftp://kepler.astroscu.unam.mx/incoming/enro/papers/mhdgturb.tar.g