Asymptotics of quasi-stationary distributions of small noise stochastic dynamical systems in unbounded domains

We consider a collection of Markov chains that model the evolution of multitype biological populations. The state space of the chains is the positive orthant, and the boundary of the orthant is the absorbing state for the Markov chain and represents the extinction states of different population types. We are interested in the long-term behavior of the Markov chain away from extinction, under a small noise scaling. Under this scaling, the trajectory of the Markov process over any compact interval converges in distribution to the solution of an ordinary differential equation (ODE) evolving in the positive orthant. We study the asymptotic behavior of the quasi-stationary distributions (QSD) in this scaling regime. Our main result shows that, under conditions, the limit points of the QSD are supported on the union of interior attractors of the flow determined by the ODE. We also give lower bounds on expected extinction times which scale exponentially with the system size. Results of this type when the deterministic dynamical system obtained under the scaling limit is given by a discrete-time evolution equation and the dynamics are essentially in a compact space (namely, the one-step map is a bounded function) have been studied by Faure and Schreiber (2014). Our results extend these to a setting of an unbounded state space and continuous-time dynamics. The proofs rely on uniform large deviation results for small noise stochastic dynamical systems and methods from the theory of continuous-time dynamical systems. In general, QSD for Markov chains with absorbing states and unbounded state spaces may not exist. We study one basic family of binomial-Poisson models in the positive orthant where one can use Lyapunov function methods to establish existence of QSD and also to argue the tightness of the QSD of the scaled sequence of Markov chains. The results from the first part are then used to characterize the support of limit points of this sequence of QSD.

PhotoNs-GPU: A GPU accelerated cosmological simulation code

We present a GPU-accelerated cosmological simulation code, PhotoNs-GPU, based on an algorithm of Particle Mesh Fast Multipole Method (PM-FMM), and focus on the GPU utilization and optimization. A proper interpolated method for truncated gravity is introduced to speed up the special functions in kernels. We verify the GPU code in mixed precision and different levels of theinterpolated method on GPU. A run with single precision is roughly two times faster than double precision for current practical cosmological simulations. But it could induce an unbiased small noise in power spectrum. Compared with the CPU version of PhotoNs and Gadget-2, the efficiency of the new code is significantly improved. Activated all the optimizations on the memory access, kernel functions and concurrency management, the peak performance of our test runs achieves 48% of the theoretical speed and the average performance approaches to ∼35% on GPU.

Quasipotentials in the nonequilibrium stationary states or a method to get explicit solutions of Hamilton–Jacobi equations

We assume that a system at a mesoscopic scale is described by a field ϕ(x, t) that evolves by a Langevin equation with a white noise whose intensity is controlled by a parameter 1 / Ω . The system stationary state distribution in the small noise limit (Ω → ∞) is of the form P st [ϕ] ≃ exp(−ΩV 0[ϕ]), where V 0[ϕ] is called the quasipotential. V 0 is the unknown of a Hamilton–Jacobi equation. Therefore, V 0 can be written as an action computed along a path that is the solution from Hamilton’s equation that typically cannot be solved explicitly. This paper presents a theoretical scheme that builds a suitable canonical transformation that permits us to do such integration by deforming the original path into a straight line and including some weights along with it. We get the functional form of such weights through conditions on the existence and structure of the canonical transformation. We apply the scheme to get the quasipotential algebraically for several one-dimensional nonequilibrium models as the diffusive and reaction–diffusion systems.

Robust Group Synchronization via Cycle-Edge Message Passing

We propose a general framework for solving the group synchronization problem, where we focus on the setting of adversarial or uniform corruption and sufficiently small noise. Specifically, we apply a novel message passing procedure that uses cycle consistency information in order to estimate the corruption levels of group ratios and consequently solve the synchronization problem in our setting. We first explain why the group cycle consistency information is essential for effectively solving group synchronization problems. We then establish exact recovery and linear convergence guarantees for the proposed message passing procedure under a deterministic setting with adversarial corruption. These guarantees hold as long as the ratio of corrupted cycles per edge is bounded by a reasonable constant. We also establish the stability of the proposed procedure to sub-Gaussian noise. We further establish exact recovery with high probability under a common uniform corruption model.

Flocking dynamics of a coupled system in noisy environments

We investigate the effect of noise on the emergence of flocking in a coupled systems by constructing a second-order coupled dynamics model, in which the inter-subgroup interaction is affected by external interference and disturbances. Based on the principle of comparison, we derive sufficient conditions for flocking and non-flocking in terms of the noise intensity and system parameters. We report that the coupled population cannot aggregate if the noise is large, while under the premise of small noise, the flocking behavior will emerge. Moreover, we find form the simulation that appropriate noise can promote the generation of flocking.

Assessment of Quad-Frequency Long-Baseline Positioning with BeiDou-3 and Galileo Observations

Quad-frequency signals have thus far been available for all satellites of BeiDou-3 and Galileo systems. The major benefit of quad-frequency signals is that more extra-wide-lane (EWL) combinations can be formed with quad-frequency than with triple- or dual-frequency, of which the ambiguities can be fixed instantaneously in medium and long baselines. In this paper, the long-baseline positioning algorithm based on optimal triple-frequency EWL/wide-lane (WL) combinations of BeiDou-3 and Galileo is proposed. First, the theoretical precision of multi-frequency combinations of BeiDou-3 and Galileo is studied, and EWL/WL combinations with a small noise amplitude factor and a small ionospheric scalar factor are selected. Then, geometry-free methods are used to estimate the a priori precision of EWL/second EWL/WL signals for different combination schemes. Second, the double-differenced (DD) geometry-based function models of quad-frequency configurations and three different triple-frequency configurations are given, and the DD ionospheric delays are estimated as unknown parameters. In the end, the real BeiDou-3 and Galileo data are used to evaluate the positioning preference. The results show that, when using fixed EWL observations to constrain WL ambiguities, the proposed triple-frequency EWL/WL signals composed of (B1I,B3I,B2a) of BeiDou-3 and (E1,E5a,E6) of Galileo can achieve the same precision as the quad-frequency signals. Therefore, the method proposed in this article can realize long-baseline instantaneous decimeter-level positioning while reducing the dimension of matrix and improving calculation efficiency.