Lipschitz Continuous Research Articles

Triebel–Lizorkin regularity and bi-Lipschitz maps: Composition operator and inverse function regularity

We study the stability of Triebel–Lizorkin regularity of bounded functions and Lipschitz functions under bi-Lipschitz changes of variables and the regularity of the inverse function of a Triebel–Lizorkin bi-Lipschitz map in Lipschitz domains. To obtain the results we provide an equivalent norm for the Triebel–Lizorkin spaces with fractional smoothness in uniform domains in terms of the first-order difference of the last weak derivative available averaged on balls.

Duality arguments in the analysis of a viscoelastic contact problem

We consider a mathematical model which describes the quasistatic frictionless contact of a viscoelastic body with a rigid-plastic foundation. We describe the mechanical assumptions, list the hypotheses on the data and provide three different variational formulations of the model in which the unknowns are the displacement field, the stress field and the strain field, respectively. These formulations have a different structure. Nevertheless, we prove that they are pairwise dual of each other. Then, we deduce the unique weak solvability of the contact problem as well as the Lipschitz continuity of its weak solution with respect to the data. The proofs are based on recent results on history-dependent variational inequalities and inclusions. Finally, we present numerical simulations in the study of the contact problem, together with the corresponding mechanical interpretations.

A necessary and sufficient condition for the global existence of solutions to nonlinear reaction‐diffusion equations on the half‐spaces in ℝN

In this paper, we study the existence and nonexistence of the global solutions to nonlinear reaction‐diffusion equations where is the half‐space , is a nonnegative continuous function, and is a locally Lipschitz function with some additional properties. The purpose of this paper is to give a necessary and sufficient condition for the existence of global solutions as follows: There is no global solution for any nonnegative and nontrivial initial data if and only if for every . In fact, we introduce a very special curve in to obtain the lower bound of decay of the heat semigroup, which is essential to prove the main result.

Random periodic solutions of SDEs: Existence, uniqueness and numerical issues

The aim of this paper is to discuss existence and uniqueness of random periodic solutions to stochastic differential equations (SDEs) with multiplicative noise under a one-sided Lipschitz condition, as well as on their numerical approximation via two classes of stochastic θ-methods, i.e., θ-Maruyama methods with θ∈[1/2,1] and θ-Milstein ones with θ∈[0,1]. The existence of the random periodic solutions as the limit of the pull-back flows of the discretized SDEs and the strong convergence rate of the aforementioned methods are also investigated. Selected numerical experiments confirming the theoretical analysis are also given.

Leader-following cluster-delay consensus control for first-order nonlinear multi-agent systems based on event-triggered mechanism

In this paper, the event-triggered-based cluster-delay consensus control problem is investigated for leader-following nonlinear multi-agent systems (MASs). In control design, both state triggering in the agents’ network and constant time-delay in the leaders’ communication network are considered. Under the framework of Lyapunov function stability theory, the Lipschitz condition is used to overcome the influence caused by time-delay. In order to further effectively utilize data transmission resources, and reduce the communication load of the topology network between each agent, an event-triggered mechanism is established. Subsequently, a robust cluster consensus control strategy is proposed based on event-triggered mechanism, which can ensure all signals of the controlled system are bounded, and the tracking errors converge to zero. In addition, it can also effectively avoid the Zeno behavior. Finally, the effectiveness of the presented control method and theory is verified by a simulation example.

On the localization of fractal discontinuity lines from noisy data

We consider the ill-posed problem of localizing (finding the position of) the discontinuity lines of a function of two variables: the function is smooth outside the discontinuity lines, and at each point on the line it has a discontinuity of the first kind. We construct averaging procedures and study global discrete regularizing algorithms for approximating discontinuity lines. Lipschitz conditions are imposed on the discontinuity lines. A parametric family of fractal lines is constructed, for which all conditions can be checked analytically. A fractal is indicated that has a large fractal dimension, for which the efficiency of the constructed methods can be guaranteed.

Formula omitted] error estimates of unsymmetric RBF collocation for second order quasilinear elliptic equations

The paper proves the convergence of unsymmetric radial basis functions collocation for second order quasilinear elliptic equations. L2 error is obtained based on the kernel-based trial spaces generated by the compactly supported radial basis functions. For the unsymmetric collocation case, we obtain the convergence result when the testing discretization is finer than the trial discretization. The convergence rates depend on the regularity of the solution, the Lipschitz continuity of the Frechét derivative of quasilinear operator, the smoothness of the computing domain, and the approximation of scaled kernel-based spaces.

Numerical Analysis of Split-Step Backward Euler Method with Truncated Wiener Process for a Stochastic Susceptible-Infected-Susceptible Model.

This article deals with the numerical positivity, boundedness, convergence, and dynamical behaviors for stochastic susceptible-infected-susceptible (SIS) model. To guarantee the biological significance of the split-step backward Euler method applied to the stochastic SIS model, the numerical positivity and boundedness are investigated by the truncated Wiener process. Motivated by the almost sure boundedness of exact and numerical solutions, the convergence is discussed by the fundamental convergence theorem with a local Lipschitz condition. Moreover, the numerical extinction and persistence are initially obtained by an exponential presentation of the stochastic stability function and strong law of the large number for martingales, which reproduces the existing theoretical results. Finally, numerical examples are given to validate our numerical results for the stochastic SIS model.

Gradient-based optimisation of the conditional-value-at-risk using the multi-level Monte Carlo method

In this work, we tackle the problem of minimising the Conditional-Value-at-Risk (CVaR) of output quantities of complex differential models with random input data, using gradient-based approaches in combination with the Multi-Level Monte Carlo (MLMC) method. In particular, we consider the framework of multi-level Monte Carlo for parametric expectations introduced in [24] and propose modifications of the MLMC estimator, error estimation procedure, and adaptive MLMC parameter selection to ensure the estimation of the CVaR and sensitivities for a given design with a prescribed accuracy. We then propose combining the MLMC framework with an alternating inexact minimisation-gradient descent algorithm, for which we prove Q-linear convergence in the optimisation iterations under the assumptions of strong convexity and Lipschitz continuity of the gradient of the objective function. We demonstrate the performance of our approach on two numerical examples of practical relevance, which evidence the same optimal asymptotic cost-tolerance behaviour as standard MLMC methods for fixed design computations of output expectations.

An Analysis on the Existence of Mild Solution and Optimal Control for Semilinear Thermoelastic System

In this article, the main objective is the conversation about the optimal control problem of the semilinear thermoelastic system, in which the control term is placed solely in the thermal equation. We discuss the existence and uniqueness of mild solutions by applying the contraction mapping for the considered system. By assuming some conditions specified Lagrange’s problem acknowledges at least one optimal control pair. For proving the main results, we are assuming the Lipschitz condition on the nonlinear term.

Capacity of rings and mappings generate embeddings of Sobolev spaces

In this paper we give characterizations of mappings generate embeddings of Sobolev spaces in the terms of ring capacity inequalities. In addition we prove that such mappings are Lipschitz mappings in the sub-hyperbolic type capacitary metrics.

Energy dissipation-preserving GSAV-Fourier–Galerkin spectral schemes for space-fractional nonlinear wave equations in multiple dimensions

In this paper, we utilize the generalized scalar auxiliary variable (GSAV) approach proposed in recent paper [SIAM J. Numer. Anal., 60 (2022), 1905–1931] for space-fractional nonlinear wave equation to construct a novel class of linearly implicit energy dissipation-preserving finite difference/spectral scheme. The unconditional energy dissipation property and unique solvability of the fully discrete scheme are first established. Next, we apply the mathematical induction to discuss the convergence results of the proposed scheme in one- and two-dimensional spaces without the assumption of global Lipschitz condition for the nonlinear term which is necessary for the almost all previous works. Moreover, the convergence of one-dimensional space is unconditional but conditional for two-dimensional space, due to the fractional Sobolev inequalities of one-dimensional space are not equivalent to the high-dimensional versions. Subsequently, the efficient implementations of the proposed schemes are introduced in detail. Finally, extensive numerical comparisons are reported to confirm the effectiveness of the proposed schemes and the correctness of the theoretical analyses.

Recursive solution of initial value problems with temporal discretization

We construct a continuous domain, as a model of interval analysis, for temporal discretization of differential equations. By using this domain, and the domain of Lipschitz maps, we formulate a generalization of the Euler operator, which exhibits second-order convergence. We prove computability of the operator within the framework of effectively given domains. The operator only requires the vector field of the differential equation to be Lipschitz continuous, in contrast to the related operators in the literature which require the vector field to be at least continuously differentiable. Within the same framework, we also analyze temporal discretization and computability of another variant of the Euler operator formulated according to Runge-Kutta theory. We prove that, compared with this variant, the second-order operator that we formulate directly, not only imposes weaker assumptions on the vector field, but also exhibits superior convergence rate. We implement the first-order, second-order, and Runge-Kutta Euler operators using arbitrary-precision interval arithmetic, and report on some experiments. The experiments confirm our theoretical results. In particular, we observe the superior convergence rate of our second-order operator compared with the Runge-Kutta Euler and the common (first-order) Euler operators.

R2Net: Efficient and flexible diffeomorphic image registration using Lipschitz continuous residual networks.

Classical diffeomorphic image registration methods, while being accurate, face the challenges of high computational costs. Deep learning based approaches provide a fast alternative to address these issues; however, most existing deep solutions either lose the good property of diffeomorphism or have limited flexibility to capture large deformations, under the assumption that deformations are driven by stationary velocity fields (SVFs). Also, the adopted squaring and scaling technique for integrating SVFs is time- and memory-consuming, hindering deep methods from handling large image volumes. In this paper, we present an unsupervised diffeomorphic image registration framework, which uses deep residual networks (ResNets) as numerical approximations of the underlying continuous diffeomorphic setting governed by ordinary differential equations, which is parameterized by either SVFs or time-varying (non-stationary) velocity fields. This flexible parameterization in our Residual Registration Network (R2Net) not only provides the model's ability to capture large deformation but also reduces the time and memory cost when integrating velocity fields for deformation generation. Also, we introduce a Lipschitz continuity constraint into the ResNet block to help achieve diffeomorphic deformations. To enhance the ability of our model for handling images with large volume sizes, we employ a hierarchical extension with a multi-phase learning strategy to solve the image registration task in a coarse-to-fine fashion. We demonstrate our models on four 3D image registration tasks with a wide range of anatomies, including brain MRIs, cine cardiac MRIs, and lung CT scans. Compared to classical methods SyN and diffeomorphic VoxelMorph, our models achieve comparable or better registration accuracy with much smoother deformations. Our source code is available online at https://github.com/ankitajoshi15/R2Net.

Finite Sample Analysis for Structured Discrete System Identification

We consider a discrete-time dynamical system over a discrete state-space, which evolves according to a structured Markov model called Bernoulli Autoregressive (BAR) model. Our goal is to obtain sample complexity bounds for the problem of estimating the parameters of this model using an indirect Maximum Likelihood Estimator. Our sample complexity bounds exploit the structure of the BAR model and are established using concentration inequalities for random matrices and Lipschitz functions.

Convergence Rates for the Truncated Euler-Maruyama Method for Nonlinear Stochastic Differential Equations

In this paper, our main aim is to investigate the strong convergence rate of the truncated Euler-Maruyama approximations for stochastic differential equations with superlinearly growing drift coefficients. When the diffusion coefficient is polynomially growing or linearly growing, the strong convergence rate of arbitrarily close to one half is established at a single time T or over a time interval [0,T], respectively. In both situations, the common one-sided Lipschitz and polynomial growth conditions for the drift coefficients are not required. Two examples are provided to illustrate the theory.

Hybrid Nash Equilibrium Seeking Under Partial-Decision Information: An Adaptive Dynamic Event-Triggered Approach

This paper is concerned with the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hybrid</i> Nash equilibrium (NE) seeking problem over a network in a partial-decision information scenario. Each agent has access to both its own cost function and local decision information of its neighbors. First, an adaptive gradient-based algorithm is constructed in a fully distributed manner with the guaranteed convergence to the NE, where the network communication is required. Second, in order to save communication cost, a novel event-triggered scheme, namely, edge-based adaptive dynamic event-triggered (E-ADET) scheme, is proposed with on-line-tuned triggering parameter and threshold, and such a scheme is proven to be fully distributed and free of Zeno behavior. Then, a hybrid NE seeking algorithm, which is also fully distributed, is constructed under the E-ADET scheme. By means of the Lipschitz continuity and the strong monotonicity of the pseudo-gradient mapping, we show the convergence of the proposed algorithms to the NE. Compared with the existing distributed algorithms, our algorithms remove the requirement on global information, thereby exhibiting the merits of both flexibility and scalability. Finally, two examples are provided to validate the proposed NE seeking methods.

Dynamics of multi-valued retarded p-Laplace equations driven by nonlinear colored noise

This paper mainly considers the long-term behavior of p-Laplace equations with infinite delays driven by nonlinear colored noise. We firstly prove the existence of weak solutions to the equation, but the uniqueness of solutions cannot be guaranteed due to the lack of Lipschitz continuity conditions, and thus generate a multi-valued dynamical system. Moreover, the regularity of solutions is also proved. Then we prove the existence of a pullback attractor. Subsequently, the measurability of the pullback attractor and the multi-valued dynamical system are also proved.

On the Robustness of Belief-Rule-Based Expert Systems

Belief rule base (BRB) expert system has been widely used in complex system modeling. Robustness is crucial to the modeling performance and safety of BRB. For a better understanding and utility of BRB, there is thereby an urgent need to know what kind of influence each part of BRB may have when the disturbance occurs. Aiming at this, a more comprehensive analysis of BRB robustness is conducted in this article. First, the Lipschitz condition for BRB is defined. With the definitions, a new robustness analysis method of BRB is proposed, which is conducted from four aspects: 1) the input transformation; 2) the matching degree calculation; 3) the matching degree normalization; and 4) the rule aggregation. Moreover, five guidelines for BRB construction are proposed by analyzing its robustness, which can offer a practical guide for users to establish, adjust, and improve the BRB model for specific applications. The robustness analysis of the BRB expert system for the relay health-state evaluation is conducted to verify the effectiveness of the proposed method.

Adaptive dynamic output feedback FTC for nonlinear systems with unknown nonlinearities and faults

AbstractThe problem of adaptive dynamic output feedback fault tolerant control (FTC) for a class of nonlinear systems with unknown nonlinearities and faults is investigated, in which the nonlinearities do not need to satisfy the Lipschitz condition, so the system can represent a wider range of nonlinear system models. An adaptive dynamic output feedback FTC is designed, where the output and its derivatives are fully applied to compensate the actuator faults on systems. The designed control method can effectively offset the effect of faults on systems so as to stabilized the nonlinear system. Furthermore, based on the designed controller, the coupling term can be eliminated reasonably and less conservative stability conditions for the closed‐loop system have been obtained. The proposed control strategy pledges that the states of the closed‐loop system are uniformly ultimately bounded (UUB). Two numerical examples are presented to exhibit the effectiveness and merits of the proposed control scheme.