A Sharp Bound on the ss-Energy and Its Applications to Averaging Systems

The {\em $s$-energy} is a generating function of wide applicability in network-based dynamics. We derive an (essentially) optimal bound of $(3/\rho s)^{n-1}$ on the $s$-energy of an $n$-agent symmetric averaging system, for any positive real $s\leq 1$, where~$\rho$ is a lower bound on the nonzero weights. This is done by introducing the new dynamics of {\em twist systems}. We show how to use the new bound on the $s$-energy to tighten the convergence rate of systems in opinion dynamics, flocking, and synchronization

Geographic Gossip: Efficient Averaging for Sensor Networks

Gossip algorithms for distributed computation are attractive due to their simplicity, distributed nature, and robustness in noisy and uncertain environments. However, using standard gossip algorithms can lead to a significant waste in energy by repeatedly recirculating redundant information. For realistic sensor network model topologies like grids and random geometric graphs, the inefficiency of gossip schemes is related to the slow mixing times of random walks on the communication graph. We propose and analyze an alternative gossiping scheme that exploits geographic information. By utilizing geographic routing combined with a simple resampling method, we demonstrate substantial gains over previously proposed gossip protocols. For regular graphs such as the ring or grid, our algorithm improves standard gossip by factors of $n$ and $\sqrt{n}$ respectively. For the more challenging case of random geometric graphs, our algorithm computes the true average to accuracy $\epsilon$ using $O(\frac{n^{1.5}}{\sqrt{\log n}} \log \epsilon^{-1})$ radio transmissions, which yields a $\sqrt{\frac{n}{\log n}}$ factor improvement over standard gossip algorithms. We illustrate these theoretical results with experimental comparisons between our algorithm and standard methods as applied to various classes of random fields.Comment: To appear, IEEE Transactions on Signal Processin

Stirring up trouble: Multi-scale mixing measures for steady scalar sources

The mixing efficiency of a flow advecting a passive scalar sustained by steady sources and sinks is naturally defined in terms of the suppression of bulk scalar variance in the presence of stirring, relative to the variance in the absence of stirring. These variances can be weighted at various spatial scales, leading to a family of multi-scale mixing measures and efficiencies. We derive a priori estimates on these efficiencies from the advection--diffusion partial differential equation, focusing on a broad class of statistically homogeneous and isotropic incompressible flows. The analysis produces bounds on the mixing efficiencies in terms of the Peclet number, a measure the strength of the stirring relative to molecular diffusion. We show by example that the estimates are sharp for particular source, sink and flow combinations. In general the high-Peclet number behavior of the bounds (scaling exponents as well as prefactors) depends on the structure and smoothness properties of, and length scales in, the scalar source and sink distribution. The fundamental model of the stirring of a monochromatic source/sink combination by the random sine flow is investigated in detail via direct numerical simulation and analysis. The large-scale mixing efficiency follows the upper bound scaling (within a logarithm) at high Peclet number but the intermediate and small-scale efficiencies are qualitatively less than optimal. The Peclet number scaling exponents of the efficiencies observed in the simulations are deduced theoretically from the asymptotic solution of an internal layer problem arising in a quasi-static model.Comment: 37 pages, 7 figures. Latex with RevTeX4. Corrigendum to published version added as appendix

On the convergence of mirror descent beyond stochastic convex programming

In this paper, we examine the convergence of mirror descent in a class of stochastic optimization problems that are not necessarily convex (or even quasi-convex), and which we call variationally coherent. Since the standard technique of "ergodic averaging" offers no tangible benefits beyond convex programming, we focus directly on the algorithm's last generated sample (its "last iterate"), and we show that it converges with probabiility $1$ if the underlying problem is coherent. We further consider a localized version of variational coherence which ensures local convergence of stochastic mirror descent (SMD) with high probability. These results contribute to the landscape of non-convex stochastic optimization by showing that (quasi-)convexity is not essential for convergence to a global minimum: rather, variational coherence, a much weaker requirement, suffices. Finally, building on the above, we reveal an interesting insight regarding the convergence speed of SMD: in problems with sharp minima (such as generic linear programs or concave minimization problems), SMD reaches a minimum point in a finite number of steps (a.s.), even in the presence of persistent gradient noise. This result is to be contrasted with existing black-box convergence rate estimates that are only asymptotic.Comment: 30 pages, 5 figure

Decaying Turbulence in Generalised Burgers Equation

We consider the generalised Burgers equation $$\frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2}=0,\ t \geq 0,\ x \in S^1,$$ where $f$ is strongly convex and $\nu$ is small and positive. We obtain sharp estimates for Sobolev norms of $u$ (upper and lower bounds differ only by a multiplicative constant). Then, we obtain sharp estimates for small-scale quantities which characterise the decaying Burgers turbulence, i.e. the dissipation length scale, the structure functions and the energy spectrum. The proof uses a quantitative version of an argument by Aurell, Frisch, Lutsko and Vergassola \cite{AFLV92}. Note that we are dealing with \textit{decaying}, as opposed to stationary turbulence. Thus, our estimates are not uniform in time. However, they hold on a time interval $[T_1, T_2]$, where $T_1$ and $T_2$ depend only on $f$ and the initial condition, and do not depend on the viscosity. These results give a rigorous explanation of the one-dimensional Burgers turbulence in the spirit of Kolmogorov's 1941 theory. In particular, we obtain two results which hold in the inertial range. On one hand, we explain the bifractal behaviour of the moments of increments, or structure functions. On the other hand, we obtain an energy spectrum of the form $k^{-2}$. These results remain valid in the inviscid limit.Comment: arXiv admin note: substantial text overlap with arXiv:1201.5567, arXiv:1107.486

Binding of molecules to DNA and other semiflexible polymers

A theory is presented for the binding of small molecules such as surfactants to semiflexible polymers. The persistence length is assumed to be large compared to the monomer size but much smaller than the total chain length. Such polymers (e.g. DNA) represent an intermediate case between flexible polymers and stiff, rod-like ones, whose association with small molecules was previously studied. The chains are not flexible enough to actively participate in the self-assembly, yet their fluctuations induce long-range attractive interactions between bound molecules. In cases where the binding significantly affects the local chain stiffness, those interactions lead to a very sharp, cooperative association. This scenario is of relevance to the association of DNA with surfactants and compact proteins such as RecA. External tension exerted on the chain is found to significantly modify the binding by suppressing the fluctuation-induced interaction.Comment: 15 pages, 7 figures, RevTex, the published versio

Multidimensional potential Burgers turbulence

We consider the multidimensional generalised stochastic Burgers equation in the space-periodic setting: $\partial \mathbf{u}/\partial t+$ $(\nabla f(\mathbf{u}) \cdot \nabla)$ $\mathbf{u} -\nu \Delta \mathbf{u}=$ $\nabla \eta,\quad t \geq 0,\ \mathbf{x} \in \mathbb{T}^d=(\mathbb{R}/\mathbb{Z})^d,$ under the assumption that $\mathbf{u}$ is a gradient. Here $f$ is strongly convex and satisfies a growth condition, $\nu$ is small and positive, while $\eta$ is a random forcing term, smooth in space and white in time. For solutions $\mathbf{u}$ of this equation, we study Sobolev norms of $\mathbf{u}$ averaged in time and in ensemble: each of these norms behaves as a given negative power of $\nu$. These results yield sharp upper and lower bounds for natural analogues of quantities characterising the hydrodynamical turbulence, namely the averages of the increments and of the energy spectrum. These quantities behave as a power of the norm of the relevant parameter, which is respectively the separation $\mathbf{l}$ in the physical space and the wavenumber $\mathbf{k}$ in the Fourier space. Our bounds do not depend on the initial condition and hold uniformly in $\nu$. We generalise the results obtained for the one-dimensional case in \cite{BorW}, confirming the physical predictions in \cite{BK07,GMN10}. Note that the form of the estimates does not depend on the dimension: the powers of $\nu, |\mathbf{k}|, \mathbf{l}$ are the same in the one- and the multi-dimensional setting.Comment: arXiv admin note: substantial text overlap with arXiv:1201.556