arXiv subjects

Jinde Cao

Publications and source records attributed to Jinde Cao.

At least 37 records · Page 2Linked to original sources

arXiv2024

Mixed Finite Element Method for Multi-layer Elastic Contact Systems

With the development of multi-layer elastic systems in the field of engineering mechanics, the corresponding variational inequality theory and algorithm design have received more attention and research. In this study, a class of equivalent saddle point problems with interlayer Tresca friction conditions and the mixed finite element method are proposed and analyzed. Then, the convergence of the numerical solution of the mixed finite element method is theoretically proven, and the corresponding algebraic dual algorithm is given. Finally, through numerical experiments, the mixed finite element method is not only compared with the layer decomposition method, but also its convergence relationship with respect to the spatial discretization parameter $H$ is verified.

math.NA↗

arXiv2024

A layer decomposition method for multi-layer elastic contact systems with interlayer Tresca friction

With the increasing demand for the accuracy of numerical simulation of pavement mechanics, the variational inequality model and its induced finite element method which can simulate the interlayer contact state becomes a potential solution. In this paper, a layer decomposition algorithm for solving variational inequality models of multi-layer elastic contact systems with interlayer Tresca friction conditions is studied. Continuous and discrete versions of the algorithm and their convergence theorems have been proposed and proved successively. Then, the algebraic form of the executable optimization algorithm and the numerical experimental results verify the practicability of the variational inequality model and its algorithm in the pavement mechanics modeling.

math.NA↗

arXiv2024

Variational inequalities of multilayer elastic systems with interlayer friction: existence and uniqueness of solution and convergence of numerical solution

Based on the mathematical-physical model of pavement mechanics, a multilayer elastic system with interlayer friction conditions is constructed. Given the complex boundary conditions, the corresponding variational inequalities of the partial differential equations are derived, so that the problem can be analyzed under the variational framework. First, the existence and uniqueness of the solution of the variational inequality is proved; then the approximation error of the numerical solution based on the finite element method is analyzed, and when the finite element space satisfies certain approximation conditions, the convergence of the numerical solution is proved; finally, in the trivial finite element space, the convergence order of the numerical solution is derived. The above conclusions provide basic theoretical support for solving the displacement-strain problem of multilayer elastic systems under the framework of variational inequalities.

math.NA↗

arXiv2023

Universal bifurcation scenarios in delay-differential equations with one delay

We show that delay-differential equations (DDE) exhibit universal bifurcation scenarios, which are observed in large classes of DDEs with a single delay. Each such universality class has the same sequence of stabilizing or destabilizing Hopf bifurcations. These bifurcation sequences and universality classes can be explicitly described by using the asymptotic continuous spectrum for DDEs with large delays. Here, we mainly study linear DDEs, provide a general transversality result for the delay-induced bifurcations, and consider three most common universality classes. For each of them, we explicitly describe the sequence of stabilizing and destabilizing bifurcations. We also illustrate the implications for a nonlinear Stuart-Landau oscillator with time-delayed feedback.

math.DS↗

arXiv2023

Fixed-Time Gradient Flows for Solving Constrained Optimization: A Unified Approach

The accelerated method in solving optimization problems has always been an absorbing topic. Based on the fixed-time (FxT) stability of nonlinear dynamical systems, we provide a unified approach for designing FxT gradient flows (FxTGFs). First, a general class of nonlinear functions in designing FxTGFs is provided. A unified method for designing first-order FxTGFs is shown under PolyakL jasiewicz inequality assumption, a weaker condition than strong convexity. When there exist both bounded and vanishing disturbances in the gradient flow, a specific class of nonsmooth robust FxTGFs with disturbance rejection is presented. Under the strict convexity assumption, Newton-based FxTGFs is given and further extended to solve time-varying optimization. Besides, the proposed FxTGFs are further used for solving equation-constrained optimization. Moreover, an FxT proximal gradient flow with a wide range of parameters is provided for solving nonsmooth composite optimization. To show the effectiveness of various FxTGFs, the static regret analysis for several typical FxTGFs are also provided in detail. Finally, the proposed FxTGFs are applied to solve two network problems, i.e., the network consensus problem and solving a system linear equations, respectively, from the respective of optimization. Particularly, by choosing component-wisely sign-preserving functions, these problems can be solved in a distributed way, which extends the existing results. The accelerated convergence and robustness of the proposed FxTGFs are validated in several numerical examples stemming from practical applications.

math.OC↗

arXiv2023

PID-inspired Continuous-time Distributed Optimization

This paper proposes two novel distributed continuous-time algorithms inspired by PID control to solve distributed optimization problems. The algorithms are referred to as first-order and second-order, respectively, depend on the intrinsic dynamics of the agents in the network. Sufficient conditions are derived so that both algorithms converge exponentially over undirected connected graphs. Finally, numerical simulations illustrate the effectiveness and efficiency of the proposed algorithms.

math.OC↗

arXiv2023

Multi/Single-stage structured zero-gradient-sum approach for prescribed-time optimization

Prescribed-time convergence mechanism has become a prominent research focus in the current field of optimization and control due to its ability to precisely control the target completion time. The recently arisen prescribed-time algorithms for distributed optimization, currently necessitate multi-stage structures to achieve global convergence. This paper introduces two modified zero-gradient-sum algorithms, each based on a multi-stage and a single-stage structural frameworks established in this work. These algorithms are designed to achieve prescribed-time convergence and relax two common yet stringent conditions. This work also bridges the gap in current research on single-stage structured PTDO algorithm. The excellent convergence performance of the proposed algorithms is validated through a case study.

math.OC↗

arXiv2023

Resilient Controller Synthesis Against DoS Attacks for Vehicular Platooning in Spatial Domain

This paper proposes a vehicular platoon control approach under Denial-of-Service (DoS) attacks and external disturbances. DoS attacks increase the service time on the communication network and cause additional transmission delays, which consequently increase the risk of rear-end collisions of vehicles in the platoon. To counter DoS attacks, we propose a resilient control scheme that exploits polytopic overapproximations of the closed-loop dynamics under DoS attacks. This scheme allows synthesizing robust controllers that guarantee tracking of both the desired spacing policy and spatially varying reference velocity for all space-varying DoS attacks satisfying a hard upper bound on the attack duration. In addition, L2 string stability conditions are derived to ensure that external perturbations do not grow as they propagate through the platoon, thus ensuring the string stability. Numerical simulations illustrate the effectiveness of the proposed control method.

eess.SY↗

arXiv2023

A data-driven rutting depth short-time prediction model with metaheuristic optimization for asphalt pavements based on RIOHTrack

Rutting of asphalt pavements is a crucial design criterion in various pavement design guides. A good road transportation base can provide security for the transportation of oil and gas in road transportation. This study attempts to develop a robust artificial intelligence model to estimate different asphalt pavements' rutting depth clips, temperature, and load axes as primary characteristics. The experiment data were obtained from 19 asphalt pavements with different crude oil sources on a 2.038 km long full-scale field accelerated pavement test track (RIOHTrack, Road Track Institute) in Tongzhou, Beijing. In addition, this paper also proposes to build complex networks with different pavement rutting depths through complex network methods and the Louvain algorithm for community detection. The most critical structural elements can be selected from different asphalt pavement rutting data, and similar structural elements can be found. An extreme learning machine algorithm with residual correction (RELM) is designed and optimized using an independent adaptive particle swarm algorithm. The experimental results of the proposed method are compared with several classical machine learning algorithms, with predictions of Average Root Mean Squared Error, Average Mean Absolute Error, and Average Mean Absolute Percentage Error for 19 asphalt pavements reaching 1.742, 1.363, and 1.94\% respectively. The experiments demonstrate that the RELM algorithm has an advantage over classical machine learning methods in dealing with non-linear problems in road engineering. Notably, the method ensures the adaptation of the simulated environment to different levels of abstraction through the cognitive analysis of the production environment parameters.

cs.AI↗

arXiv2023

DISA: A Dual Inexact Splitting Algorithm for Distributed Convex Composite Optimization

In this paper, we propose a novel Dual Inexact Splitting Algorithm (DISA) for distributed convex composite optimization problems, where the local loss function consists of a smooth term and a possibly nonsmooth term composed with a linear mapping. DISA, for the first time, eliminates the dependence of the convergent step-size range on the Euclidean norm of the linear mapping, while inheriting the advantages of the classic Primal-Dual Proximal Splitting Algorithm (PD-PSA): simple structure and easy implementation. This indicates that DISA can be executed without prior knowledge of the norm, and tiny step-sizes can be avoided when the norm is large. Additionally, we prove sublinear and linear convergence rates of DISA under general convexity and metric subregularity, respectively. Moreover, we provide a variant of DISA with approximate proximal mapping and prove its global convergence and sublinear convergence rate. Numerical experiments corroborate our theoretical analyses and demonstrate a significant acceleration of DISA compared to existing PD-PSAs.

math.OC↗

arXiv2023

LP-BFGS attack: An adversarial attack based on the Hessian with limited pixels

Deep neural networks are vulnerable to adversarial attacks. Most $L_{0}$-norm based white-box attacks craft perturbations by the gradient of models to the input. Since the computation cost and memory limitation of calculating the Hessian matrix, the application of Hessian or approximate Hessian in white-box attacks is gradually shelved. In this work, we note that the sparsity requirement on perturbations naturally lends itself to the usage of Hessian information. We study the attack performance and computation cost of the attack method based on the Hessian with a limited number of perturbation pixels. Specifically, we propose the Limited Pixel BFGS (LP-BFGS) attack method by incorporating the perturbation pixel selection strategy and the BFGS algorithm. Pixels with top-k attribution scores calculated by the Integrated Gradient method are regarded as optimization variables of the LP-BFGS attack. Experimental results across different networks and datasets demonstrate that our approach has comparable attack ability with reasonable computation in different numbers of perturbation pixels compared with existing solutions.

cs.CR↗

arXiv2023

Decentralized Inexact Proximal Gradient Method With Network-Independent Stepsizes for Convex Composite Optimization

This paper proposes a novel CTA (Combine-Then-Adapt)-based decentralized algorithm for solving convex composite optimization problems over undirected and connected networks. The local loss function in these problems contains both smooth and nonsmooth terms. The proposed algorithm uses uncoordinated network-independent constant stepsizes and only needs to approximately solve a sequence of proximal mappings, which is advantageous for solving decentralized composite optimization problems where the proximal mappings of the nonsmooth loss functions may not have analytical solutions. For the general convex case, we prove an O(1/k) convergence rate of the proposed algorithm, which can be improved to o(1/k) if the proximal mappings are solved exactly. Furthermore, with metric subregularity, we establish a linear convergence rate for the proposed algorithm. Numerical experiments demonstrate the efficiency of the algorithm.

math.OC↗

arXiv2023

On some analytic properties of nabla tempered fractional calculus

Despite many applications regarding fractional calculus have been reported in literature, it is still unknown how to model some practical process. One major challenge in solving such a problem is that, the nonlocal property is needed while the infinite memory is undesired. Under this context, a new kind nabla fractional calculus accompanied by a tempered function is formulated. However, many properties of such fractional calculus needed to be discovered. From this, this paper gives particular emphasis to the topic. Some remarkable properties like the equivalence relation, the nabla Taylor formula, and the nabla Laplace transform for such nabla fractional calculus are developed and analyzed. It is believed that this work greatly enriches the mathematical theory of nabla tempered fractional calculus and provides high value and huge potential for further applications.

math.GM↗

arXiv2022

Some useful inequalities for nabla tempered fractional calculus

This paper gives particular emphasis to the nabla tempered fractional calculus, which involves the multiplication of the rising function kernel with tempered functions, and provides a more flexible alternative with considerable promise for practical applications. Some remarkable inequalities for such nabla fractional calculus are developed and analyzed, which greatly enrich the mathematical theory of nabla tempered fractional calculus. Numerical results confirm the validity of the developed properties once again, which also reveals that the introduction of tempered function provides high value and huge potential.

math.CA↗

arXiv2022

BALPA: A Balanced Primal-Dual Algorithm for Nonsmooth Optimization with Application to Distributed Optimization

In this paper, we propose a novel primal-dual proximal splitting algorithm (PD-PSA), named BALPA, for the composite optimization problem with equality constraints, where the loss function consists of a smooth term and a nonsmooth term composed with a linear mapping. In BALPA, the dual update is designed as a proximal point for a time-varying quadratic function, which balances the implementation of primal and dual update and retains the proximity-induced feature of classic PD-PSAs. In addition, by this balance, BALPA eliminates the inefficiency of classic PD-PSAs for composite optimization problems in which the Euclidean norm of the linear mapping or the equality constraint mapping is large. Therefore, BALPA not only inherits the advantages of simple structure and easy implementation of classic PD-PSAs but also ensures a fast convergence when these norms are large. Moreover, we propose a stochastic version of BALPA (S-BALPA) and apply the developed BALPA to distributed optimization to devise a new distributed optimization algorithm. Furthermore, a comprehensive convergence analysis for BALPA and S-BALPA is conducted, respectively. Finally, numerical experiments demonstrate the efficiency of the proposed algorithms.

math.OC↗

arXiv2022

Exploring Adversarial Examples and Adversarial Robustness of Convolutional Neural Networks by Mutual Information

A counter-intuitive property of convolutional neural networks (CNNs) is their inherent susceptibility to adversarial examples, which severely hinders the application of CNNs in security-critical fields. Adversarial examples are similar to original examples but contain malicious perturbations. Adversarial training is a simple and effective defense method to improve the robustness of CNNs to adversarial examples. The mechanisms behind adversarial examples and adversarial training are worth exploring. Therefore, this work investigates similarities and differences between normally trained CNNs (NT-CNNs) and adversarially trained CNNs (AT-CNNs) in information extraction from the mutual information perspective. We show that 1) whether NT-CNNs or AT-CNNs, for original and adversarial examples, the trends towards mutual information are almost similar throughout training; 2) compared with normal training, adversarial training is more difficult and the amount of information that AT-CNNs extract from the input is less; 3) the CNNs trained with different methods have different preferences for certain types of information; NT-CNNs tend to extract texture-based information from the input, while AT-CNNs prefer to shape-based information. The reason why adversarial examples mislead CNNs may be that they contain more texture-based information about other classes. Furthermore, we also analyze the mutual information estimators used in this work and find that they outline the geometric properties of the middle layer's output.

cs.LG↗

arXiv2022

Spectral Analysis and its applications for a class of scale-free network based on the weighted m-clique annex operation

The spectrum of network is an important tool to study the function and dynamic properties of network, and graph operation and product is an effective mechanism to construct a specific local and global topological structure. In this study, a class of weighted $m-$clique annex operation $τ_m^r(\cdot)$ controlled by scale factor $m$ and weight factor $r$ is defined, through which an iterative weighted network model $G_t$ with small-world and scale-free properties is constructed. In particular, when the number of iterations $t$ tends to infinity, the network has transfinite fractal property. Then, through the iterative features of the network structure, the iterative relationship of the eigenvalues of the normalized Laplacian matrix corresponding to the network is studied. Accordingly, some applications of the spectrum of the network, including the Kenemy constant, Multiplicative Degree-Kirchhoff index and the number of weighted spanning trees, are further given. In addition, we also study the effect of the two factors controlling network operation on the structure and function of the iterative weighted network $G_t$, so that the network operation can better simulate the real network and have more application potential in the field of artificial network.

math.DS↗

arXiv2022

Distributed Pinning Set Stabilization of Large-Scale Boolean Networks

In this article, we design the distributed pinning controllers to globally stabilize a Boolean network (BN), specially a sparsely connected large-scale one, towards a preassigned subset of state space through the node-to-node message exchange. Given an appointed state set, system nodes are partitioned into two disjoint parts, which respectively gather the nodes whose states are fixed or arbitrary with respect to the given state set. With such node division, three parts of pinned nodes are selected and the state feedback controllers are accordingly designed such that the resulting BN satisfies three conditions: the states of the other nodes cannot affect the nodal dynamics of fixed-state nodes, the subgraph of network structure induced by the fixed-state nodes is acyclic, and the steady state of the subnetwork induced by the fixed-state nodes lies in the state set given beforehand. If the BN after control is acyclic, the stabilizing time is revealed to be no more than the length of the longest path in the current network structure plus one. This enables us to further design the pinning controllers with the constraint of stabilizing time. Noting that the overall procedure runs in an exponentially increasing time with respect to the largest number of functional variables in the dynamics of pinned nodes, the sparsely-connected large-scale BNs can be well addressed in a reasonable amount of time. Finally, we demonstrate the applications of our theoretical results in a T-LGL survival signal network with $29$ nodes and T-cell receptor signaling network with $90$ nodes.

eess.SY↗