We study a 1D geometry of a plasma confined between two conducting floating walls with applications to laboratory plasmas. These plasmas are characterized by a quasi-neutral bulk that is joined to the wall by a thin boundary layer called sheath that is positively charged. Although analytical solutions are available in the sheath and the pre-sheath, joining the two areas by one analytical solution is still an open problem which requires the numerical resolution of the fluid equations coupled to Poisson equation. Current numerical schemes use high-order discretizations to correctly capture the electron current in the sheath, presenting unsatisfactory results in the boundary layer and they are not adapted to all the possible collisional regimes. In this work, we identify the main numerical challenges that arise when attempting the simulations of such configuration and we propose explanations for the observed phenomena via numerical analysis. We propose a numerical scheme with controlled diffusion as well as new discrete boundary conditions that address the identified issues.
相關內容
Compact finite-difference (FD) schemes specify derivative approximations implicitly, thus to achieve parallelism with domain-decomposition suitable partitioning of linear systems is required. Consistent order of accuracy, dispersion, and dissipation is crucial to maintain in wave propagation problems such that deformation of the associated spectra of the discretized problems is not too severe. In this work we consider numerically tuning spectral error, at fixed formal order of accuracy to automatically devise new compact FD schemes. Grid convergence tests indicate error reduction of at least an order of magnitude over standard FD. A proposed hybrid matching-communication strategy maintains the aforementioned properties under domain-decomposition. Under evolution of linear wave-propagation problems utilizing exponential integration or explicit Runge-Kutta methods improvement is found to remain robust. A first demonstration that compact FD methods may be applied to the Z4c formulation of numerical relativity is provided where we couple our header-only, templated C++ implementation to the highly performant GR-Athena++ code. Evolving Z4c on test-bed problems shows at least an order in magnitude reduction in phase error compared to FD for propagated metric components. Stable binary-black-hole evolution utilizing compact FD together with improved convergence is also demonstrated.
In many developing nations, a lack of poverty data prevents critical humanitarian organizations from responding to large-scale crises. Currently, socioeconomic surveys are the only method implemented on a large scale for organizations and researchers to measure and track poverty. However, the inability to collect survey data efficiently and inexpensively leads to significant temporal gaps in poverty data; these gaps severely limit the ability of organizational entities to address poverty at its root cause. We propose a transfer learning model based on surface temperature change and remote sensing data to extract features useful for predicting poverty rates. Machine learning, supported by data sources of poverty indicators, has the potential to estimate poverty rates accurately and within strict time constraints. Higher temperatures, as a result of climate change, have caused numerous agricultural obstacles, socioeconomic issues, and environmental disruptions, trapping families in developing countries in cycles of poverty. To find patterns of poverty relating to temperature that have the highest influence on spatial poverty rates, we use remote sensing data. The two-step transfer model predicts the temperature delta from high resolution satellite imagery and then extracts image features useful for predicting poverty. The resulting model achieved 80% accuracy on temperature prediction. This method takes advantage of abundant satellite and temperature data to measure poverty in a manner comparable to the existing survey methods and exceeds similar models of poverty prediction.
Achieving and maintaining cooperation between agents to accomplish a common objective is one of the central goals of Multi-Agent Reinforcement Learning (MARL). Nevertheless in many real-world scenarios, separately trained and specialized agents are deployed into a shared environment, or the environment requires multiple objectives to be achieved by different coexisting parties. These variations among specialties and objectives are likely to cause mixed motives that eventually result in a social dilemma where all the parties are at a loss. In order to resolve this issue, we propose the Incentive Q-Flow (IQ-Flow) algorithm, which modifies the system's reward setup with an incentive regulator agent such that the cooperative policy also corresponds to the self-interested policy for the agents. Unlike the existing methods that learn to incentivize self-interested agents, IQ-Flow does not make any assumptions about agents' policies or learning algorithms, which enables the generalization of the developed framework to a wider array of applications. IQ-Flow performs an offline evaluation of the optimality of the learned policies using the data provided by other agents to determine cooperative and self-interested policies. Next, IQ-Flow uses meta-gradient learning to estimate how policy evaluation changes according to given incentives and modifies the incentive such that the greedy policy for cooperative objective and self-interested objective yield the same actions. We present the operational characteristics of IQ-Flow in Iterated Matrix Games. We demonstrate that IQ-Flow outperforms the state-of-the-art incentive design algorithm in Escape Room and 2-Player Cleanup environments. We further demonstrate that the pretrained IQ-Flow mechanism significantly outperforms the performance of the shared reward setup in the 2-Player Cleanup environment.
We consider the problem of tracking an unknown time varying parameter that characterizes the probabilistic evolution of a sequence of independent observations. To this aim, we propose a stochastic gradient descent-based recursive scheme in which the log-likelihood of the observations acts as time varying gain function. We prove convergence in mean-square error in a suitable neighbourhood of the unknown time varying parameter and illustrate the details of our findings in the case where data are generated from distributions belonging to the exponential family.
We consider a time-varying first-order autoregressive model with irregular innovations, where we assume that the coefficient function is H\"{o}lder continuous. To estimate this function, we use a quasi-maximum likelihood based approach. A precise control of this method demands a delicate analysis of extremes of certain weakly dependent processes, our main result being a concentration inequality for such quantities. Based on our analysis, upper and matching minimax lower bounds are derived, showing the optimality of our estimators. Unlike the regular case, the information theoretic complexity depends both on the smoothness and an additional shape parameter, characterizing the irregularity of the underlying distribution. The results and ideas for the proofs are very different from classical and more recent methods in connection with statistics and inference for locally stationary processes.
We consider an auction design problem where a seller sells multiple homogeneous items to a set of connected buyers. Each buyer only knows the buyers she directly connects with and has a diminishing marginal utility valuation for the items. The seller initially only connects to some buyers who can be directly invited to the sale by the seller. Our goal is to design an auction to incentivize the buyers who are aware of the auction to further invite their neighbors to join the auction. This is challenging because the buyers are competing for the items and they would not invite each other by default. Thus, rewards need to be given to buyers who diffuse information, but the rewards should be carefully designed to guarantee both invitation incentives and the seller's revenue. Solutions have been proposed recently for the settings where each buyer requires at most one unit and demonstrated the difficulties of the design. We move this forward to propose the very first diffusion auction for the multi-unit demand settings to improve both the social welfare and the seller's revenue.
Kinetic equations model the position-velocity distribution of particles subject to transport and collision effects. Under a diffusive scaling, these combined effects converge to a diffusion equation for the position density in the limit of an infinite collision rate. Despite this well-defined limit, numerical simulation is expensive when the collision rate is high but finite, as small time steps are then required. In this work, we present an asymptotic-preserving multilevel Monte Carlo particle scheme that makes use of this diffusive limit to accelerate computations. In this scheme, we first sample the diffusive limiting model to compute a biased initial estimate of a Quantity of Interest, using large time steps. We then perform a limited number of finer simulations with transport and collision dynamics to correct the bias. The efficiency of the multilevel method depends on being able to perform correlated simulations of particles on a hierarchy of discretization levels. We present a method for correlating particle trajectories and present both an analysis and numerical experiments. We demonstrate that our approach significantly reduces the cost of particle simulations in high-collisional regimes, compared with prior work, indicating significant potential for adopting these schemes in various areas of active research.
Time-dependent Maxwell's equations govern electromagnetics. Under certain conditions, we can rewrite these equations into a partial differential equation of second order, which in this case is the vectorial wave equation. For the vectorial wave, we investigate the numerical application and the challenges in the implementation. For this purpose, we consider a space-time variational setting, i.e. time is just another spatial dimension. More specifically, we apply integration by parts in time as well as in space, leading to a space-time variational formulation with different trial and test spaces. Conforming discretizations of tensor-product type result in a Galerkin--Petrov finite element method that requires a CFL condition for stability. For this Galerkin--Petrov variational formulation, we study the CFL condition and its sharpness. To overcome the CFL condition, we use a Hilbert-type transformation that leads to a variational formulation with equal trial and test spaces. Conforming space-time discretizations result in a new Galerkin--Bubnov finite element method that is unconditionally stable. In numerical examples, we demonstrate the effectiveness of this Galerkin--Bubnov finite element method. Furthermore, we investigate different projections of the right-hand side and their influence on the convergence rates. This paper is the first step towards a more stable computation and a better understanding of vectorial wave equations in a conforming space-time approach.
Consider the problem of solving systems of linear algebraic equations $Ax=b$ with a real symmetric positive definite matrix $A$ using the conjugate gradient (CG) method. To stop the algorithm at the appropriate moment, it is important to monitor the quality of the approximate solution. One of the most relevant quantities for measuring the quality of the approximate solution is the $A$-norm of the error. This quantity cannot be easily computed, however, it can be estimated. In this paper we discuss and analyze the behaviour of the Gauss-Radau upper bound on the $A$-norm of the error, based on viewing CG as a procedure for approximating a certain Riemann-Stieltjes integral. This upper bound depends on a prescribed underestimate $\mu$ to the smallest eigenvalue of $A$. We concentrate on explaining a phenomenon observed during computations showing that, in later CG iterations, the upper bound loses its accuracy, and is almost independent of $\mu$. We construct a model problem that is used to demonstrate and study the behaviour of the upper bound in dependence of $\mu$, and developed formulas that are helpful in understanding this behavior. We show that the above mentioned phenomenon is closely related to the convergence of the smallest Ritz value to the smallest eigenvalue of $A$. It occurs when the smallest Ritz value is a better approximation to the smallest eigenvalue than the prescribed underestimate $\mu$. We also suggest an adaptive strategy for improving the accuracy of the upper bounds in the previous iterations.
In this paper, an oscillation-free spectral volume (OFSV) method is proposed and studied for the hyperbolic conservation laws. The numerical scheme is designed by introducing a damping term in the standard spectral volume method for the purpose of controlling spurious oscillations near discontinuities. Based on the construction of control volumes (CVs), two classes of OFSV schemes are presented. A mathematical proof is provided to show that the proposed OFSV is stable and has optimal convergence rate and some desired superconvergence properties when applied to the linear scalar equations. Both analysis and numerical experiments indicate that the damping term would not destroy the order of accuracy of the original SV scheme and can control the oscillations discontinuities effectively. Numerical experiments are presented to demonstrate the accuracy and robustness of our scheme.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191