Sparse-Grid Implementation of Fixed-Point Fast Sweeping WENO Schemes for Eikonal Equations

Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of fast sweeping schemes,fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order.The resulting iterative schemes have a fast convergence rate to steady-state solutions.Moreover,an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve the inverse operation of any nonlinear local system.Hence,they are robust and flexible,and have been combined with high-order accurate weighted essentially non-oscillatory(WENO)schemes to solve various hyperbolic PDEs in the literature.For multidimensional nonlinear problems,high-order fixed-point fast sweeping WENO methods still require quite a large amount of computational costs.In this technical note,we apply sparse-grid techniques,an effective approximation tool for multidimensional problems,to fixed-point fast sweeping WENO methods for reducing their computational costs.Here,we focus on fixed-point fast sweeping WENO schemes with third-order accuracy(Zhang et al.2006[41]),for solving Eikonal equations,an important class of static Hamilton-Jacobi(H-J)equations.Numerical experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse-grid computations of the fixed-point fast sweeping WENO schemes achieve large savings of CPU times on refined meshes,and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.

A Secure and Effective Energy-Aware Fixed-Point Quantization Scheme for Asynchronous Federated Learning

Asynchronous federated learning(AsynFL)can effectivelymitigate the impact of heterogeneity of edge nodes on joint training while satisfying participant user privacy protection and data security.However,the frequent exchange of massive data can lead to excess communication overhead between edge and central nodes regardless of whether the federated learning(FL)algorithm uses synchronous or asynchronous aggregation.Therefore,there is an urgent need for a method that can simultaneously take into account device heterogeneity and edge node energy consumption reduction.This paper proposes a novel Fixed-point Asynchronous Federated Learning(FixedAsynFL)algorithm,which could mitigate the resource consumption caused by frequent data communication while alleviating the effect of device heterogeneity.FixedAsynFL uses fixed-point quantization to compress the local and global models in AsynFL.In order to balance energy consumption and learning accuracy,this paper proposed a quantization scale selection mechanism.This paper examines the mathematical relationship between the quantization scale and energy consumption of the computation/communication process in the FixedAsynFL.Based on considering the upper bound of quantization noise,this paper optimizes the quantization scale by minimizing communication and computation consumption.This paper performs pertinent experiments on the MNIST dataset with several edge nodes of different computing efficiency.The results show that the FixedAsynFL algorithm with an 8-bit quantization can significantly reduce the communication data size by 81.3%and save the computation energy in the training phase by 74.9%without significant loss of accuracy.According to the experimental results,we can see that the proposed AsynFixedFL algorithm can effectively solve the problem of device heterogeneity and energy consumption limitation of edge nodes.

A Fixed-Point Iterative Method for Discrete Tomography Reconstruction Based on Intelligent Optimization

Discrete Tomography(DT)is a technology that uses image projection to reconstruct images.Its reconstruction problem,especially the binary image(0–1matrix)has attracted strong attention.In this study,a fixed point iterative method of integer programming based on intelligent optimization is proposed to optimize the reconstructedmodel.The solution process can be divided into two procedures.First,the DT problem is reformulated into a polyhedron judgment problembased on lattice basis reduction.Second,the fixed-point iterativemethod of Dang and Ye is used to judge whether an integer point exists in the polyhedron of the previous program.All the programs involved in this study are written in MATLAB.The final experimental data show that this method is obviously better than the branch and bound method in terms of computational efficiency,especially in the case of high dimension.The branch and bound method requires more branch operations and takes a long time.It also needs to store a large number of leaf node boundaries and the corresponding consumptionmatrix,which occupies a largememory space.

A Fixed-Point Fast Sweeping WENO Method with Inverse Lax-Wendroff Boundary Treatment for Steady State of Hyperbolic Conservation Laws

Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations(PDEs).The Gauss-Seidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady-state solutions.A nice property of fixed-point fast sweeping WENO methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems.Hence,they are easy to be applied to a general hyperbolic system.To deal with the difficulties associated with numerical boundary treatment when high-order finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains,inverse Lax-Wendroff(ILW)procedures were developed as a very effective approach in the literature.In this paper,we combine a fifthorder fixed-point fast sweeping WENO method with an ILW procedure to solve steadystate solution of hyperbolic conservation laws on complex computing regions.Numerical experiments are performed to test the method in solving various problems including the cases with the physical boundary not aligned with the grids.Numerical results show highorder accuracy and good performance of the method.Furthermore,the method is compared with the popular third-order total variation diminishing Runge-Kutta(TVD-RK3)time-marching method for steady-state computations.Numerical examples show that for most of examples,the fixed-point fast sweeping method saves more than half CPU time costs than TVD-RK3 to converge to steady-state solutions.

一种适于实现Datapath电路的FPGA逻辑模块

提出一种适于实现数据通路的FPGA逻辑模块。每个模块包括4个基于全加器的逻辑单元。逻辑单元分为组合和时序两部分：组合部分以1位全加器为基础，有两个输出端，这两个输出端在必要时可以合并在一起以实现功能更复杂的单输出函数；时序部分基于可配置的D型触发器。逻辑单元在结构上保证了电路的高速性。工艺映射的实验结果显示，在实现数据通路中的常用电路时，新逻辑单元比基于LUT的FPGA单元平均大约节省75%的MOS管数。

The Criteria for Reducing Centrally Restricted Three-Body Problem to Two-Body Problem

Our Solar System contains eight planets and their respective natural satellites excepting the inner two planets Mercury and Venus. A satellite hosted by a given Planet is well protected by the gravitational pertubation of much heavier planets such as Jupiter and Saturn if the natural satellite lies deep inside the respective host Planet Hill sphere. Each planet has a Hill radius aH and planet mean radius RP and the ratio R1=RP/aH. Under very low R1 (less than 0.006) the approximation of CRTBP (centrally restricted three-body problem) to two-body problem is valid and planet has spacious Hill lobe to capture a satellite and retain it. This ensures a high probability of capture of natural satellite by the given planet and Sun’s perturbation on Planet-Satellite binary can be neglected. This is the case with Earth, Mars, Jupiter, Saturn, Neptune and Uranus. But Mercury and Venus has R1=RP/aH =0.01 and 5.9862 × 10-3 respectively hence they have no satellites. There is a limit to the dimension of the captured body. It must be a much smaller body both dimensionally as well masswise. The qantitative limit is a subject of an independent study.

高性能16位定点DSP的Datapath设计及FPGA验证

随着数字化产品的不断普及,以高速、实时为特点的数字信号处理器(DigitalSignalProcessors,DSPs)越来越多地应用到国防和消费等各个领域。文章介绍了一款嵌入式高性能16位定点DSPs的Datapath(数据通路)设计,以Datapath中各功能单元为线索,叙述其电路实现及FPGA验证。

Fixed-Point Iteration Method for Solving the Convex Quadratic Programming with Mixed Constraints

The present paper is devoted to a novel smoothing function method for convex quadratic programming problem with mixed constrains, which has important application in mechanics and engineering science. The problem is reformulated as a system of non-smooth equations, and then a smoothing function for the system of non-smooth equations is proposed. The condition of convergences of this iteration algorithm is given. Theory analysis and primary numerical results illustrate that this method is feasible and effective.

POWER OPTIMIZATION FOR THE DATAPATH OF A 32-BIT RECONFIGURABLE PIPELINED DSP PROCESSOR

With the continuous increasing of circuit scale, the problem of power consumption is paid much more attention than before, especially in large designs. In this paper, an experience of optimizing the power consumption of the 16-bit datapath in a 32-bit reconfigurable pipelined Digital Signal Processor (DSP) is introduced. By keeping the old input values and preventing the useless switching of the logic blocks on the datapath, the power consumption is much lowered. At the same time, by relocating some logic blocks between different pipeline stages and employing some data forward logics, a better balanced pipeline is achieved to lower the power consumption for conditional computation instructions at very low timing and area costs. The effectivity of these power optimization technologies are proved by the experimental results. Finally, some ideas about how to reduce the power consumption of circuits are proposed, which are very effective and useful in practice designs, especially in pipelined ones.

Datapath电路的布局概述

集成电路在飞速发展,相关的设计能力却远远跟不上集成电路发展的速度,使得现有的设计模式也不再适用。Datapath在电路当中所占比例越来越高,传统的设计模式因没有考虑到datapath电路本身的特殊性使整个集成电路的性能受到影响,因此,datapath电路的设计变得尤为重要,而布局又是电路设计过程中至关重要的一部分,本文介绍了datapath的设计现状、datapath电路的结构特点和布局要求及其布局的基本方法。

Attractive fixed-point solution study of shell model for homogeneous isotropic turbulence

The attractive fixed-point solution of a nonlinear cascade model is stud- ied for the homogeneous isotropic turbulence containing a parameter C, introduced by Desnyansky and Novikov. With a traditional constant positive external force added on the first shell equation, it can be found that the attractive fixed-point solution of the model depends on both the parameter C and the external force. Thus, an explicit force is introduced to remove the effects of the external force on the attractive fixed-point solu- tion. F^arthermore, two groups of attractive fixed-point solutions are derived theoretically and studied numerically. One of the groups has the same scaling behavior of the velocity in the whole inertial range and agrees well with those observed by Bell and Nelkin for the nonnegative parameters. The other is found to have different scaling behaviors of the velocity at the odd and even number shells for the negative parameters. This special characteristic may be used to study the anomalous scaling behavior of the turbulence.

Quantum search for unknown number of target items by hybridizing fixed-point method with trail-and-error method

For the unsorted database quantum search with the unknown fraction λ of target items, there are mainly two kinds of methods, i.e., fixed-point and trail-and-error.(i) In terms of the fixed-point method, Yoder et al. [Phys. Rev. Lett.113 210501(2014)] claimed that the quadratic speedup over classical algorithms has been achieved. However, in this paper, we point out that this is not the case, because the query complexity of Yoder’s algorithm is actually in O(1/λ01/2)rather than O(1/λ1/2), where λ0 is a known lower bound of λ.(ii) In terms of the trail-and-error method, currently the algorithm without randomness has to take more than 1 times queries or iterations than the algorithm with randomly selected parameters. For the above problems, we provide the first hybrid quantum search algorithm based on the fixed-point and trail-and-error methods, where the matched multiphase Grover operations are trialed multiple times and the number of iterations increases exponentially along with the number of trials. The upper bound of expected queries as well as the optimal parameters are derived. Compared with Yoder’s algorithm, the query complexity of our algorithm indeed achieves the optimal scaling in λ for quantum search, which reconfirms the practicality of the fixed-point method. In addition, our algorithm also does not contain randomness, and compared with the existing deterministic algorithm, the query complexity can be reduced by about 1/3. Our work provides a new idea for the research on fixed-point and trial-and-error quantum search.

Low power implementation of datapath using regularity

Datapath accounts for a considerable part of power consumption in VLSI circuit design. This paper presents a method for physical implementation of datapath to achieve low power consumption. Regularity is a characteristic of datapath and the key of the proposed method, where synthesis is tightly combined with placement to make full use of regularity, so that low power consumption is achieved. In This paper, a new concept of Synthesis In Relative Placement (SIRP) is given to deal with the semi-regularity in some datapath. Experimental results of a sample circuit validated the proposed method.

Investigation on dynamic characteristics of a rod fastening rotor-bearing coupling system with fixed-point rubbing

The aim of this paper is to gain insight into the nonlinear vibration feature of a dynamic model of a gas turbine.First,a rod fastening rotor-bearing coupling model with fixed-point rubbing is proposed,where the fractal theory and the finite element method are utilized.For contact analysis,a novel contact force model is introduced in this paper.Meanwhile,the Coulomb model is adopted to expound the friction characteristics.Second,the governing equations of motion of the rotor system are numerically solved,and the nonlinear dynamic characteristics are analyzed in terms of the bifurcation diagram,Poincarémap,and time history.Third,the potential effects provided by contact degree of joint interface,distribution position,and amount of contact layer are discussed in detail.Finally,the contrast analysis between the integral rotor and the rod fastening rotor is conducted under the condition of fixed-point rubbing.

Modified Homotopy Method for a Class of Brouwer Fixed-point Problems

In this paper, we modify the homotopy method （proposed by Yu and Lin, Appl. Math. Comput., 74（1996）, 65） and hence make the modified method be able to solve Brouwer fixed-point problems in a broader class of nonconvex subsets in Rn. In addition, a simple example is given to show the effectiveness of the modified method.

FIXED-POINT BLIND ADAPTIVE MULTIUSER DETECTION ALGORITHM FOR IR UWB SYSTEMS IN MULTIPATH CHANNEL

Using the hypothesis that data transmitted by different users are statistically independent of each other,this paper proposes a fixed-point blind adaptive multiuser detection algorithm for Time-Hopping (TH) Impulse Radio (IR) Ultra Wide Band (UWB) systems in multipath channel,which is based on Independent Component Analysis (ICA) idea. The proposed algorithm employs maximizing negentropy criterion to separate the data packets of different users. Then the user characteristic se-quences are utilized to identify the data packet order of the desired user. This algorithm only needs the desired user’s characteristic sequence instead of channel information,power information and time-hoping code of any user. Due to using hypothesis of statistical independence among users,the proposed algorithm has the outstanding Bit Error Rate (BER) performance and the excellent ability of near-far resistance. Simulation results demonstrate that this algorithm has the performance close to that of Maximum-Likelihood (ML) algorithm and is a suboptimum blind adaptive multiuser detection algorithm of excellent near-far resistance and low complexity.

Traversable Wormholes and the Brouwer Fixed-Point Theorem

The Brouwer fixed-point theorem in topology states that for any continuous mapping f on a compact convex set into itself admits a fixed point, i.e., a point x0 such that f(x0) = x0. Under suitable conditions, this fixed point corresponds to the throat of a traversable wormhole, i.e., b(r0) = r0 for the shape function b = b(r). The possible existence of wormholes can therefore be deduced from purely mathematical considerations without going beyond the existing physical requirements.

Passive Attack Detection for a Class of Stealthy Intermittent Integrity Attacks

This paper proposes a passive methodology for detecting a class of stealthy intermittent integrity attacks in cyberphysical systems subject to process disturbances and measurement noise.A stealthy intermittent integrity attack strategy is first proposed by modifying a zero-dynamics attack model.The stealthiness of the generated attacks is rigorously investigated under the condition that the adversary does not know precisely the system state values.In order to help detect such attacks,a backward-in-time detection residual is proposed based on an equivalent quantity of the system state change,due to the attack,at a time prior to the attack occurrence time.A key characteristic of this residual is that its magnitude increases every time a new attack occurs.To estimate this unknown residual,an optimal fixed-point smoother is proposed by minimizing a piece-wise linear quadratic cost function with a set of specifically designed weighting matrices.The smoother design guarantees robustness with respect to process disturbances and measurement noise,and is also able to maintain sensitivity as time progresses to intermittent integrity attack by resetting the covariance matrix based on the weighting matrices.The adaptive threshold is designed based on the estimated backward-in-time residual,and the attack detectability analysis is rigorously investigated to characterize quantitatively the class of attacks that can be detected by the proposed methodology.Finally,a simulation example is used to demonstrate the effectiveness of the developed methodology.

Maximum Correntropy Kalman Filtering for Non-Gaussian Systems With State Saturations and Stochastic Nonlinearities

This paper tackles the maximum correntropy Kalman filtering problem for discrete time-varying non-Gaussian systems subject to state saturations and stochastic nonlinearities. The stochastic nonlinearities, which take the form of statemultiplicative noises, are introduced in systems to describe the phenomenon of nonlinear disturbances. To resist non-Gaussian noises, we consider a new performance index called maximum correntropy criterion(MCC) which describes the similarity between two stochastic variables. To enhance the “robustness” of the kernel parameter selection on the resultant filtering performance, the Cauchy kernel function is adopted to calculate the corresponding correntropy. The goal of this paper is to design a Kalman-type filter for the underlying systems via maximizing the correntropy between the system state and its estimate. By taking advantage of an upper bound on the one-step prediction error covariance, a modified MCC-based performance index is constructed. Subsequently, with the assistance of a fixed-point theorem, the filter gain is obtained by maximizing the proposed cost function. In addition, a sufficient condition is deduced to ensure the uniqueness of the fixed point. Finally, the validity of the filtering method is tested by simulating a numerical example.

THE EXISTENCE AND MULTIPLICITY OF k-CONVEX SOLUTIONS FOR A COUPLED k-HESSIAN SYSTEM

In this paper,we focus on the following coupled system of k-Hessian equations:{S_(k)(λ(D^(2)u))=f_(1)(|x|,-v)in B,S_(k)(λ(D^(2)v))=f2(|x|,-u)in B,u=v=0 on■B.Here B is a unit ball with center 0 and fi(i=1,2)are continuous and nonnegative functions.By introducing some new growth conditions on the nonlinearities f_(1) and f_(2),which are more flexible than the existing conditions for the k-Hessian systems(equations),several new existence and multiplicity results for k-convex solutions for this kind of problem are obtained.