Instantaneous control of interacting particle systems in the mean-field limit

Controlling large particle systems in collective dynamics by a few agents is a subject of high practical importance, e.g., in evacuation dynamics. In this paper we study an instantaneous control approach to steer an interacting particle system into a certain spatial region by repulsive forces from a few external agents, which might be interpreted as shepherd dogs leading sheep to their home. We introduce an appropriate mathematical model and the corresponding optimization problem. In particular, we are interested in the interaction of numerous particles, which can be approximated by a mean-field equation. Due to the high-dimensional phase space this will require a tailored optimization strategy. The arising control problems are solved using adjoint information to compute the descent directions. Numerical results on the microscopic and the macroscopic level indicate the convergence of optimal controls and optimal states in the mean-field limit,i.e., for an increasing number of particles.Comment: arXiv admin note: substantial text overlap with arXiv:1610.0132

Simulations in statistical physics and biology: some applications

One of the most active areas of physics in the last decades has been that of critical phenomena, and Monte Carlo simulations have played an important role as a guide for the validation and prediction of system properties close to the critical points. The kind of phase transitions occurring for the Betts lattice (lattice constructed removing 1/7 of the sites from the triangular lattice) have been studied before with the Potts model for the values q=3, ferromagnetic and antiferromagnetic regime. Here, we add up to this research line the ferromagnetic case for q=4 and 5. In the first case, the critical exponents are estimated for the second order transition, whereas for the latter case the histogram method is applied for the occurring first order transition. Additionally, Domany's Monte Carlo based clustering technique mainly used to group genes similar in their expression levels is reviewed. Finally, a control theory tool --an adaptive observer-- is applied to estimate the exponent parameter involved in the well-known Gompertz curve. By treating all these subjects our aim is to stress the importance of cooperation between distinct disciplines in addressing the complex problems arising in biology. Contents: Chapter 1 - Monte Carlo simulations in stat. physics; Chapter 2: MC simulations in biology; Chapter 3: Gompertz equationComment: 82 pages, 33 figures, 4 tables, somewhat reduced version of the M.Sc. thesis defended in Jan. 2006 at IPICyT, San Luis Potosi, Mx. (Supervisers: Drs. R. Lopez-Sandoval and H.C. Rosu). Last sections 3.3 and 3.4 can be found at http://lanl.arxiv.org/abs/physics/041108

Statistical Mechanics of maximal independent sets

The graph theoretic concept of maximal independent set arises in several practical problems in computer science as well as in game theory. A maximal independent set is defined by the set of occupied nodes that satisfy some packing and covering constraints. It is known that finding minimum and maximum-density maximal independent sets are hard optimization problems. In this paper, we use cavity method of statistical physics and Monte Carlo simulations to study the corresponding constraint satisfaction problem on random graphs. We obtain the entropy of maximal independent sets within the replica symmetric and one-step replica symmetry breaking frameworks, shedding light on the metric structure of the landscape of solutions and suggesting a class of possible algorithms. This is of particular relevance for the application to the study of strategic interactions in social and economic networks, where maximal independent sets correspond to pure Nash equilibria of a graphical game of public goods allocation

Automatic Construction of Predictive Neuron Models through Large Scale Assimilation of Electrophysiological Data.

We report on the construction of neuron models by assimilating electrophysiological data with large-scale constrained nonlinear optimization. The method implements interior point line parameter search to determine parameters from the responses to intracellular current injections of zebra finch HVC neurons. We incorporated these parameters into a nine ionic channel conductance model to obtain completed models which we then use to predict the state of the neuron under arbitrary current stimulation. Each model was validated by successfully predicting the dynamics of the membrane potential induced by 20-50 different current protocols. The dispersion of parameters extracted from different assimilation windows was studied. Differences in constraints from current protocols, stochastic variability in neuron output, and noise behave as a residual temperature which broadens the global minimum of the objective function to an ellipsoid domain whose principal axes follow an exponentially decaying distribution. The maximum likelihood expectation of extracted parameters was found to provide an excellent approximation of the global minimum and yields highly consistent kinetics for both neurons studied. Large scale assimilation absorbs the intrinsic variability of electrophysiological data over wide assimilation windows. It builds models in an automatic manner treating all data as equal quantities and requiring minimal additional insight

Distributed coordination of self-organizing mechanisms in communication networks

The fast development of the Self-Organizing Network (SON) technology in mobile networks renders the problem of coordinating SON functionalities operating simultaneously critical. SON functionalities can be viewed as control loops that may need to be coordinated to guarantee conflict free operation, to enforce stability of the network and to achieve performance gain. This paper proposes a distributed solution for coordinating SON functionalities. It uses Rosen's concave games framework in conjunction with convex optimization. The SON functionalities are modeled as linear Ordinary Differential Equation (ODE)s. The stability of the system is first evaluated using a basic control theory approach. The coordination solution consists in finding a linear map (called coordination matrix) that stabilizes the system of SON functionalities. It is proven that the solution remains valid in a noisy environment using Stochastic Approximation. A practical example involving three different SON functionalities deployed in Base Stations (BSs) of a Long Term Evolution (LTE) network demonstrates the usefulness of the proposed method.Comment: submitted to IEEE TCNS. arXiv admin note: substantial text overlap with arXiv:1209.123