arXiv subjects

Wei Gong

Publications and source records attributed to Wei Gong.

At least 55 records · Page 3Linked to original sources

arXiv2024

Analysis and approximation to parabolic optimal control problems with measure-valued controls in time

In this paper, we investigate an optimal control problem governed by parabolic equations with measure-valued controls over time. We establish the well-posedness of the optimal control problem and derive the first-order optimality condition using Clarke's subgradients, revealing a sparsity structure in time for the optimal control. Consequently, these optimal control problems represent a generalization of impulse control for evolution equations. To discretize the optimal control problem, we employ the space-time finite element method. Here, the state equation is approximated using piecewise linear and continuous finite elements in space, alongside a Petrov-Galerkin method utilizing piecewise constant trial functions and piecewise linear and continuous test functions in time. The control variable is discretized using the variational discretization concept. For error estimation, we initially derive a priori error estimates and stabilities for the finite element discretizations of the state and adjoint equations. Subsequently, we establish weak-* convergence for the control under the norm $\mathcal{M}(\bar I_c;L^2(ω))$, with a convergence order of $O(h^\frac{1}{2}+τ^\frac{1}{4})$ for the state.

math.OC↗

arXiv2023

A new error analysis for parabolic Dirichlet boundary control problems

In this paper, we consider the finite element approximation to a parabolic Dirichlet boundary control problem and establish new a priori error estimates. In the temporal semi-discretization we apply the DG(0) method for the state and the variational discretization for the control, and obtain the convergence rates $O(k^{\frac{1}{4}})$ and $O(k^{\frac{3}{4}-\varepsilon})$ $(\varepsilon>0)$ for the control for problems posed on polytopes with $y_0\in L^2(Ω)$, $y_d\in L^2(I;L^2(Ω))$ and smooth domains with $y_0\in H^{\frac{1}{2}}(Ω)$, $y_d\in L^2(I;H^1(Ω))\cap H^{\frac{1}{2}}(I;L^2(Ω))$, respectively. In the fully discretization of the optimal control problem posed on polytopal domains, we apply the DG(0)-CG(1) method for the state and the variational discretization approach for the control, and derive the convergence order $O(k^{\frac{1}{4}} +h^{\frac{1}{2}})$, which improves the known results by removing the mesh size condition $k=O(h^2)$ between the space mesh size $h$ and the time step $k$. As a byproduct, we obtain a priori error estimate $O(h+k^{1\over 2})$ for the fully discretization of parabolic equations with inhomogeneous Dirichlet data posed on polytopes, which also improves the known error estimate by removing the above mesh size condition.

math.NA↗

arXiv2023

A new finite element method for elliptic optimal control problems with pointwise state constraints in energy spaces

In this paper we propose a new finite element method for solving elliptic optimal control problems with pointwise state constraints, including the distributed controls and the Dirichlet or Neumann boundary controls. The main idea is to use energy space regularizations in the objective functional, while the equivalent representations of the energy space norms, i.e., the $H^{-1}(Ω)$-norm for the distributed control, the $H^{1/2}(Γ)$-norm for the Dirichlet control and the $H^{-1/2}(Γ)$-norm for the Neumann control, enable us to transform the optimal control problem into an elliptic variational inequality involving only the state variable. The elliptic variational inequalities are second order for the three cases, and include additional equality constraints for Dirichlet or Neumann boundary control problems. Standard $C^0$ finite elements can be used to solve the resulted variational inequality. We provide preliminary a priori error estimates for the new algorithm for solving distributed control problems. Extensive numerical experiments are carried out to validate the accuracy of the new algorithm.

math.NA↗

arXiv2023

Energy-Efficient WiFi Backscatter Communication for Green IoTs

The boom of the Internet of Things has revolutionized people's lives, but it has also resulted in massive resource consumption and environmental pollution. Recently, Green IoT (GIoT) has become a worldwide consensus to address this issue. In this paper, we propose EEWScatter, an energy-efficient WiFi backscatter communication system to pursue the goal of GIoT. Unlike previous backscatter systems that solely focus on tags, our approach offers a comprehensive system-wide view on energy conservation. Specifically, we reuse ambient signals as carriers and utilize an ultra-low-power and battery-free design for tag nodes by backscatter. Further, we design a new CRC-based algorithm that enables the demodulation of both ambient and tag data by only a single receiver while using ambient carriers. Such a design eliminates system reliance on redundant transceivers with high power consumption. Results demonstrate that EEWScatter achieves the lowest overall system power consumption and saves at least half of the energy. What's more, the power consumption of our tag is only 1/1000 of that of active radio. We believe that EEWScatter is a critical step towards a sustainable future.

cs.NI↗

arXiv2023

On the well-posedness of tracking Dirichlet data for Bernoulli free boundary problems

The aim of this paper is to study the shape optimization method for solving the Bernoulli free boundary problem, a well-known ill-posed problem that seeks the unknown free boundary through Cauchy data. Different formulations have been proposed in the literature that differ in the choice of the objective functional. Specifically, it was shown respectively in [14] and [16] that tracking Neumann data is well-posed but tracking Dirichlet data is not. In this paper we propose a new well-posed objective functional that tracks Dirichlet data at the free boundary. By calculating the Euler derivative and the shape Hessian of the objective functional we show that the new formulation is well-posed, i.e., the shape Hessian is coercive at the minimizers. The coercivity of the shape Hessian may ensure the existence of optimal solutions for the nonlinear Ritz-Galerkin approximation method and its convergence, thus is crucial for the formulation. As a summary, we conclude that tracking Dirichlet or Neumann data in its energy norm is not sufficient, but tracking it in a half an order higher norm will be well-posed. To support our theoretical results we carry out extensive numerical experiments.

math.NA↗

arXiv2023

CRC-based Reliable WiFi Backscatter Communication for Supply Chain Management

Supply chain management is aimed to keep going long-term performance of the supply chain and minimize the costs. Backscatter technology provides a more efficient way of being able to identify items and real-time monitoring. Among the backscatter systems, the ambient backscatter communication (AmBC) system provides a prospect of ultra-low energy consumption and does not require controlled excitation devices. In this paper, we introduce CRCScatter, a CRC reverse algorithm-based AmBC system using a single access point (AP). A CRC reverse decoder is applied to reverse the ambient data from CRC32 sequence in the backscatter packet and realize single-AP decoding. Based on the nature of DBPSK modulation in WiFi signal, the CRCScatter system obtains the tag data by XOR and Differential decoder. Our simulation results verify the effectiveness of our proposed system in the low SNR regime. The average decoding time of CRCScatter system is independent of the length of tag data. Furthermore, our system can append redundant bits in the tag data to improve the decoding accuracy while not increasing the decoding time.

cs.NI↗

arXiv2022

A New Global Divergence Free and Pressure-Robust HDG Method for Tangential Boundary Control of Stokes Equations

In [ESAIM: M2AN, 54(2020), 2229-2264], we proposed an HDG method to approximate the solution of a tangential boundary control problem for the Stokes equations and obtained an optimal convergence rate for the optimal control {that reflects its global regularity}. However, the error estimates depend on the pressure, and the velocity is not divergence free. The importance of pressure-robust numerical methods for fluids was addressed by John et al. [SIAM Review, 59(2017), 492-544]. In this work, we devise a new HDG method to approximate the solution of the Stokes tangential boundary control problem; the HDG method is also of independent interest for solving the Stokes equations. This scheme yields a $\mathbf{H}(\mathrm{div})$ conforming, globally divergence free, and pressure-robust solution. To the best of our knowledge, this is the first time such a numerical scheme has been obtained for an optimal boundary control problem for the Stokes equations. We also provide numerical experiments to show the performance of the new HDG method and the advantage over the non pressure-robust scheme.

math.NA↗

arXiv2022

Convergence analysis of the Schwarz alternating method for unconstrained elliptic optimal control problems

In this paper we analyze the Schwarz alternating method for unconstrained elliptic optimal control problems. We discuss the convergence properties of the method in the continuous case first and then apply the arguments to the finite difference discretization case. In both cases, we prove that the Schwarz alternating method is convergent if its counterpart for an elliptic equation is convergent. Meanwhile, the convergence rate of the method for the elliptic equation under the maximum norm also gives a uniform upper bound (with respect to the regularization parameter $α$) of the convergence rate of the method for the optimal control problem under the maximum norm of proper error merit functions in the continuous case or vectors in the discrete case. Our numerical results corroborate our theoretical results and show that with $α$ decreasing to zero, the method will converge faster. We also give some exposition of this phenomenon.

math.NA↗

arXiv2021

Homoscatter: Towards efficient connectivity for ZigBee backscatter system

Recent advances in backscatter open a promising direction for ultra-low power communication. However, the state-of-art ZigBee backscatter system, Interscatter, has several drawbacks to deploy. Its backscatter tag and exciting source, Bluetooth, can hardly decode packets from other ZigBee nodes, which left Interscatter one-way communication. Besides, it adopts instantaneous phase change to modulate information, producing obvious sidelobes and interfering devices working on neighboring channels severely. To address the problems mentioned above, we introduce Homoscatter, a novel ZigBee backscatter system that adopts specific ZigBee devices to generate a single tone and leverages continuous phase change to modulate information, which eliminates spectral leakage. It also does codeword translation on the packet header of exciting packets, improving the utilization of ambient signal. The prototype of Homoscatter consists of a microchip radio, a backscatter tag, and a commodity receiver. The evaluations show that the occupied bandwidth of Homoscatter achieves 3x smaller than Interscatter. When the channel capacity is 17.5 kbps, the continuous phase change modulation achieves 13 kbps with the codeword translation on the excitation header. Based on the widely spread IoT devices, Homoscatter is a practical way to build an efficient connection between IoT devices.

eess.SP↗

arXiv2021

Bayesian Convolutional Neural Networks for Seven Basic Facial Expression Classifications

The seven basic facial expression classifications are a basic way to express complex human emotions and are an important part of artificial intelligence research. Based on the traditional Bayesian neural network framework, the ResNet18_BNN network constructed in this paper has been improved in the following three aspects: (1) A new objective function is proposed, which is composed of the KL loss of uncertain parameters and the intersection of specific parameters. Entropy loss composition. (2) Aiming at a special objective function, a training scheme for alternately updating these two parameters is proposed. (3) Only model the parameters of the last convolution group. Through testing on the FER2013 test set, we achieved 71.5% and 73.1% accuracy in PublicTestSet and PrivateTestSet, respectively. Compared with traditional Bayesian neural networks, our method brings the highest classification accuracy gain.

cs.CV↗

arXiv2018

Resisting Selfish Mining Attacks in the Bicomp

Selfish mining, which is an attack on the integrity of the Bitcoin network, was first proposed by Cornell researchers Emin Gun Sirer and Ittay Eyal in 2013. Selfish mining attack also exists in most Nakamoto consensus protocols. Generally speaking, selfish mining strategy can comprise a Nakamoto consensus system with less than 25% mining power of the whole system. We have discussed how the Bicomp can resist selfish mining in our former paper "Bicomp: A Bilayer Scalable Nakamoto Consensus Protocol". In this technical report, we give a detailed derivation on the conditions a selfish attacker should meet to earn more revenues through selfish mining. And we also get a conclusion that through adjusting macroblock difficulties together with tenure lengths, the Bicomp protocol has high resistant towards selfish mining.

cs.NI↗

arXiv2016

Convergence of $L^2$-norm based adaptive finite element method for elliptic optimal control problems

This paper aims to study the convergence of adaptive finite element method for control constrained elliptic optimal control problems under $L^2$-norm. We prove the contraction property and quasi-optimal complexity for the $L^2$-norm errors of both the control, the state and adjoint state variables with $L^2$-norm based AFEM, this is in contrast to and improve our previous work [13] where convergence of AFEM based on energy norm had been studied and suboptimal convergence for the control variable was obtained and observed numerically. For the discretization we use variational discretization for the control and piecewise linear and continuous finite elements for the state and adjoint state. Under mild assumptions on the initial mesh and the mesh refinement algorithm to keep the adaptive meshes sufficiently mildly graded we prove the optimal convergence of AFEM for the control problems, numerical results are provided to support our theoretical findings.

math.NA↗

arXiv2015

Adaptive finite element method for elliptic optimal control problems: convergence and optimality

In this paper we consider the convergence analysis of adaptive finite element method for elliptic optimal control problems with pointwise control constraints. We use variational discretization concept to discretize the control variable and piecewise linear and continuous finite elements to approximate the state variable. Based on the well-established convergence theory of AFEM for elliptic boundary value problems, we rigorously prove the convergence and quasi-optimality of AFEM for optimal control problems with respect to the state and adjoint state variables, by using the so-called perturbation argument. Numerical experiments confirm our theoretical analysis.

math.NA↗

arXiv2014

A multilevel correction method for optimal controls of elliptic equation

We propose in this paper a multilevel correction method to solve optimal control problems constrained by elliptic equations with the finite element method. In this scheme, solving optimization problem on the finest finite element space is transformed to a series of solutions of linear boundary value problems by the multigrid method on multilevel meshes and a series of solutions of optimization problems on the coarsest finite element space. Our proposed scheme, instead of solving a large scale optimization problem in the finest finite element space, solves only a series of linear boundary value problems and the optimization problems in a very low dimensional finite element space, and thus can improve the overall efficiency for the solution of optimal control problems governed by PDEs.

math.NA↗

arXiv2014

A finite element method for Dirichlet boundary control problems governed by parabolic PDEs

Finite element approximations of Dirichlet boundary control problems governed by parabolic PDEs on convex polygonal domains are studied in this paper. The existence of a unique solution to optimal control problems is guaranteed based on very weak solution of the state equation and $L^2(0,T;L^2(Γ))$ as control space. For the numerical discretization of the state equation we use standard piecewise linear and continuous finite elements for the space discretization of the state, while a dG(0) scheme is used for time discretization. The Dirichlet boundary control is realized through a space-time $L^2$-projection. We consider both piecewise linear, continuous finite element approximation and variational discretization for the controls and derive a priori $L^2$-error bounds for controls and states. We finally present numerical examples to support our theoretical findings.

math.OC↗

arXiv2013

Data Aggregation without Secure Channel: How to Evaluate a Multivariate Polynomial Securely

Much research has been conducted to securely outsource multiple parties' data aggregation to an untrusted aggregator without disclosing each individual's data, or to enable multiple parties to jointly aggregate their data while preserving privacy. However, those works either assume to have a secure channel or suffer from high complexity. Here we consider how an external aggregator or multiple parties learn some algebraic statistics (e.g., summation, product) over participants' data while any individual's input data is kept secret to others (the aggregator and other participants). We assume channels in our construction are insecure. That is, all channels are subject to eavesdropping attacks, and all the communications throughout the aggregation are open to others. We successfully guarantee data confidentiality under this weak assumption while limiting both the communication and computation complexity to at most linear.

cs.CR↗

arXiv2013

Lattice Boltzmann simulations of apparent slip and contact angle in hydrophobic micro-channels

In this paper, we applied the Shan-Chen multiphase Lattice Boltzmann method to simulate two different parameters, contact angle (a static parameter) and slip length (a dynamic parameter), and we proposed a relationship between them by fitting those numerical simulation results. By changing the values of the strength of interaction between fluid particles (SIF) and the strength of interaction between fluid and solid surface (SIFS), we simulated a series of contact angles and slip lengths. Our numerical simulation results show that both SIF and SIFS have little effects on the relationship between contact angle and slip length. Using the proposed relationship between slip length and contact angle, we further derived an equation to determine the upper limit of nano-particles' diameter under which drag-reduction can be achieved when using nano-particles adsorbing method.

physics.flu-dyn↗

arXiv2008

Direct numerical integration of one-loop Feynman diagrams for N-photon amplitudes

One approach to the calculation of cross sections for infrared-safe observables in high energy collisions at next-to-leading order is to perform all of the integrations, including the virtual loop integration, by Monte Carlo numerical integration. In a previous paper, two of us have shown how one can perform such a virtual loop integration numerically after first introducing a Feynman parameter representation. In this paper, we perform the integration directly, without introducing Feynman parameters, after suitably deforming the integration contour. Our example is the N-photon scattering amplitude with a massless electron loop. We report results for N = 6 and N = 8.

hep-ph↗