Distributed Adaptive Gradient Optimization Algorithm

In this paper, a distributed optimization problem with general differentiable convex objective functions is studied for single-integrator and double-integrator multi-agent systems. Two distributed adaptive optimization algorithm is introduced which uses the relative information to construct the gain of the interaction term. The analysis is performed based on the Lyapunov functions, the analysis of the system solution and the convexity of the local objective functions. It is shown that if the gradients of the convex objective functions are continuous, the team convex objective function can be minimized as time evolves for both single-integrator and double-integrator multi-agent systems. Numerical examples are included to show the obtained theoretical results.Comment: 12 pages, 3 figure

BFDA: A Matlab Toolbox for Bayesian Functional Data Analysis

We provide a MATLAB toolbox, BFDA, that implements a Bayesian hierarchical model to smooth multiple functional data with the assumptions of the same underlying Gaussian process distribution, a Gaussian process prior for the mean function, and an Inverse-Wishart process prior for the covariance function. This model-based approach can borrow strength from all functional data to increase the smoothing accuracy, as well as estimate the mean-covariance functions simultaneously. An option of approximating the Bayesian inference process using cubic B-spline basis functions is integrated in BFDA, which allows for efficiently dealing with high-dimensional functional data. Examples of using BFDA in various scenarios and conducting follow-up functional regression are provided. The advantages of BFDA include: (1) Simultaneously smooths multiple functional data and estimates the mean-covariance functions in a nonparametric way; (2) flexibly deals with sparse and high-dimensional functional data with stationary and nonstationary covariance functions, and without the requirement of common observation grids; (3) provides accurately smoothed functional data for follow-up analysis.Comment: A tool paper submitted to the Journal of Statistical Softwar

Control of Spin-Exchange Interaction between Alkali-Earth Atoms via Confinement-Induced Resonances in a Quasi 1+0 Dimensional System

A nuclear-spin exchange interaction exists between two ultracold fermionic alkali-earth (like) atoms in the electronic $^{1}{\rm S}_{0}$ state ($g$-state) and $^{3}{\rm P}_{0}$ state ($e$-state), and is an essential ingredient for the quantum simulation of Kondo effect. We study the control of this spin-exchange interaction for two atoms simultaneously confined in a quasi-one-dimensional (quasi-1D) tube, where the $g$-atom is freely moving in the axial direction while the $e$-atom is further localized by an additional axial trap and behaves as a quasi-zero-dimensional (quasi-0D) impurity. In this system, the two atoms experience effective-1D spin-exchange interactions in both even and odd partial wave channels, whose intensities can be controlled by the characteristic lengths of the confinements via the confinement-induced-resonances (CIRs). In current work, we go beyond that pure-1D approximation. We model the transverse and axial confinements by harmonic traps with finite characteristic lengths $a_\perp$ and $a_z$, respectively, and exactly solve the "quasi-1D + quasi-0D" scattering problem between these two atoms. Using the solutions we derive the effective 1D spin-exchange interaction and investigate the locations and widths of the even/odd wave CIRs for our system. It is found that when the ratio $a_z/a_\perp$ is larger, the CIRs can be induced by weaker confinements, which are easier to be realized experimentally. The comparison between our results and the recent experiment shows that the two experimentally observed resonance branches of the spin-exchange effect are due to an even-wave CIR and an odd-wave CIR, respectively. Our results are advantageous for the control and description of either the effective spin-exchange interaction or other types of interactions between ultracold atoms in quasi 1+0 dimensional systems.Comment: 14 pages, 3 figures. Compare to previous version, we did a major revision in current versio

Distributed Subgradient-based Multi-agent Optimization with More General Step Sizes

A wider selection of step sizes is explored for the distributed subgradient algorithm for multi-agent optimization problems, for both time-invariant and time-varying communication topologies. The square summable requirement of the step sizes commonly adopted in the literature is removed. The step sizes are only required to be positive, vanishing and non-summable. It is proved that in both unconstrained and constrained optimization problems, the agents' estimates reach consensus and converge to the optimal solution with the more general choice of step sizes. The idea is to show that a weighted average of the agents' estimates approaches the optimal solution, but with different approaches. In the unconstrained case, the optimal convergence of the weighted average of the agents' estimates is proved by analyzing the distance change from the weighted average to the optimal solution and showing that the weighted average is arbitrarily close to the optimal solution. In the constrained case, this is achieved by analyzing the distance change from the agents' estimates to the optimal solution and utilizing the boundedness of the constraints. Then the optimal convergence of the agents' estimates follows because consensus is reached in both cases. These results are valid for both a strongly connected time-invariant graph and time-varying balanced graphs that are jointly strongly connected

A Novel Carrier Waveform Inter-Displacement Modulation Method in Underwater Communication Channel

As the main way of underwater wireless communication, underwater acoustic communication is one of the focuses of ocean research. Compared with the free space wireless communication channel, the underwater acoustic channel suffers from more severe multipath effect, the less available bandwidth and the even complex noise. The underwater acoustic channel is one of the most complicated wireless communication channels. To achieve a reliable underwater acoustic communication, Phase Shift Keying (PSK) modulation and Passive Time Reversal Mirror (PTRM) equalization are considered to be a suitable scheme. However, due to the serious distortion of the received signal caused by the channel, this scheme suffers from a high Bit Error Rate (BER) under the condition of the low Signal to Noise Ratio (SNR). To solve this problem, we proposes a Carrier Waveform Inter-Displacement (CWID) modulation method based on the Linear Frequency Modulation (LFM) PSK and PTRM scheme. The new communication scheme reduces BER by increasing the difference from the carrier waveform for different symbols. Simulation results show the effectiveness and superiority of the proposed method.Comment: 8 pages, 11 figure

Quantum Defect Theory for Orbital Feshbach Resonance

In the ultracold gases of alkali-earth (like) atoms, a new type of Feshbach resonance, i.e., the orbital Feshbach resonance (OFR), has been proposed and experimentally observed in ultracold $^{173}$Yb atoms. When the OFR of the $^{173}$Yb atoms occurs, the energy gap between the open and closed channels is smaller by two orders of magnitudes than the van der Waals energy. As a result, quantitative accurate results for the low-energy two-body problems can be obtained via multi-channel quantum defect theory (MQDT), which is based on the exact solution of the Schr$\ddot{{\rm o}}$dinger equation with the van der Waals potential. In this paper we use the MQDT to calculate the two-atom scattering length, effective range, and the binding energy of two-body bound states for the systems with OFR. With these results we further study the clock-transition spectrum for the two-body bound states, which can be used to experimentally measure the binding energy. Our results are helpful for the quantitative theoretical and experimental researches for the ultracold gases of alkali-earth (like) atoms with OFR.Comment: 11 pages, 6 figuer

Realized volatility and parametric estimation of Heston SDEs

We present a detailed analysis of \emph{observable} moments based parameter estimators for the Heston SDEs jointly driving the rate of returns $R_t$ and the squared volatilities $V_t$. Since volatilities are not directly observable, our parameter estimators are constructed from empirical moments of realized volatilities $Y_t$, which are of course observable. Realized volatilities are computed over sliding windows of size $\varepsilon$, partitioned into $J(\varepsilon)$ intervals. We establish criteria for the joint selection of $J(\varepsilon)$ and of the sub-sampling frequency of return rates data. We obtain explicit bounds for the $L^q$ speed of convergence of realized volatilities to true volatilities as $\varepsilon \to 0$. In turn, these bounds provide also $L^q$ speeds of convergence of our observable estimators for the parameters of the Heston volatility SDE. Our theoretical analysis is supplemented by extensive numerical simulations of joint Heston SDEs to investigate the actual performances of our moments based parameter estimators. Our results provide practical guidelines for adequately fitting Heston SDEs parameters to observed stock prices series

Parametric Estimation from Approximate Data: Non-Gaussian Diffusions

We study the problem of parameters estimation in Indirect Observability contexts, where $X_t \in R^r$ is an unobservable stationary process parametrized by a vector of unknown parameters and all observable data are generated by an approximating process $Y^{\varepsilon}_t$ which is close to $X_t$ in $L^4$ norm. We construct consistent parameter estimators which are smooth functions of the sub-sampled empirical mean and empirical lagged covariance matrices computed from the observable data. We derive explicit optimal sub-sampling schemes specifying the best paired choices of sub-sampling time-step and number of observations. We show that these choices ensure that our parameter estimators reach optimized asymptotic $L^2$-convergence rates, which are constant multiples of the $L^4$ norm $|| Y^{\varepsilon}_t - X_t ||$

Orbital Feshbach Resonance with Small Energy Gap between Open and Closed Channels

Recently a new type of Feshbach resonance, i.e., orbital Feshbach resonance (OFR) was proposed for the ultracold alkali-earth (like) atoms, and experimentally observed in the ultracold gases of $^{{\rm 173}}$Yb atoms. Unlike most of the magnetic Feshbach resonances of ultracold alkali atoms, when the OFR of $^{{\rm 173}}$Yb atoms appears, the energy gap between the thresholds of the open channel (OC) and the closed channel (CC) is much smaller than the characteristic energy of the inter-atomic interaction, i.e., the van der Waals energy. In this paper we study the OFR in the systems with small CC-OC threshold gap. We show that in these systems the OFR can be induced by the coupling between the OC and either an isolated bound state of the CC or the scattering states of the CC. Moreover, we also show that in each case the two-channel Huang-Yang pesudopoential is always applicable for the approximate calculation of the low-energy scattering amplitude. Our results imply that in the theoretical calculations for these systems it is appropriate to take into account the contributions from the scattering states of the CC

Enhancing Kondo Coupling in Alkaline-Earth Atomic Gases with Confinement-induced Resonances in Mixed Dimensions

The Kondo effect describes the spin-exchanging interaction between localized impurity and the itinerant fermions. The ultracold alkaline-earth atomic gas provides a natural platform for quantum simulation of the Kondo model, utilizing its long-lived clock state and the nuclear-spin exchanging interaction between the clock state and the ground state. One of the key issue now is whether the Kondo temperature can be high enough to be reached in current experiment, for which we have proposed using a transverse confinement to confine atoms into a one-dimensional tube and to utilize the confinement-induced resonance to enhance the Kondo coupling. In this work, we further consider the $1+0$ dimensional scattering problem when the clock state is further confined by an axial harmonic confinement. We show that this axial confinement for the clock state atoms not only plays a role for localizing them, but also can act as an additional control knob to reach the confinement-induced resonance. We show that by combining both the transverse and the axial confinements, the confinement-induced resonance can be reached in the practical conditions and the Kondo effect can be attainable in this system.Comment: 6 pages, 5 figure