Lipschitz Continuity Condition Research Articles

Explicit exponential Runge–Kutta methods for semilinear time-fractional integro-differential equations

In this work, we consider and analyze explicit exponential Runge–Kutta methods for solving semilinear time-fractional integro-differential equation, which involves two nonlocal terms in time. Firstly, the temporal Runge–Kutta discretizations follow the idea of exponential integrators. Subsequently, we utilize the spectral Galerkin method to introduce a fully discrete scheme. Then, we mainly focus on discussing the one-stage and two-stage methods for solving the proposed semilinear problem. Based on special abstract settings, we perform the convergence analysis for the proposed two different stage methods. In this process, we heavily use estimates about the operator family {S̃(t)}, and in combination with Lipschitz continuous condition. Finally, some numerical experiments confirm theoretical results. Meanwhile, applying this scheme to the related linear problem yields high-order convergence, highlighting the advantages of explicit exponential Runge–Kutta methods.

SBL-LCGL: sparse Bayesian learning based on Laplace distribution for robust cone-beam x-ray luminescence computed tomography

Objective. To address the quality and accuracy issues in the distribution of nanophosphors (NPs) using Cone-beam x-ray luminescence computed tomography (CB-XLCT) by proposing a novel reconstruction strategy.Approach. This paper introduces a sparse Bayesian learning reconstruction method termed SBL-LCGL, which is grounded in the Lipschitz continuous gradient condition and the Laplace prior to overcome the ill-posed inverse problem inherent in CB-XLCT.Main results. The SBL-LCGL method has demonstrated its effectiveness in capturing the sparse features of NPs and mitigating the computational complexity associated with matrix inversion. Both numerical simulation andin vivoexperiments confirm that the method yields satisfactory imaging results regarding the position and shape of the targets.Significance. The advancements presented in this work are expected to enhance the clinical applicability of CB-XLCT, contributing to its broader adoption in medical imaging and diagnostics.

An Improved Three-Term Conjugate Gradient Algorithm for Constrained Nonlinear Equations under Non-Lipschitz Conditions and Its Applications

This paper proposes an improved three-term conjugate gradient algorithm designed to solve nonlinear equations with convex constraints. The key features of the proposed algorithm are as follows: (i) It only requires that nonlinear equations have continuous and monotone properties; (ii) The designed search direction inherently ensures sufficient descent and trust-region properties, eliminating the need for line search formulas; (iii) Global convergence is established without the necessity of the Lipschitz continuity condition. Benchmark problem numerical results illustrate the proposed algorithm’s effectiveness and competitiveness relative to other three-term algorithms. Additionally, the algorithm is extended to effectively address the image denoising problem.

General inferential limits under differential and Pufferfish privacy

Differential privacy (DP) is a class of mathematical standards for assessing the privacy provided by a data-release mechanism. This work concerns two important flavors of DP that are related yet conceptually distinct: pure ε-differential privacy (ε-DP) and Pufferfish privacy. We restate ε-DP and Pufferfish privacy as Lipschitz continuity conditions and provide their formulations in terms of an object from the imprecise probability literature: the interval of measures. We use these formulations to derive limits on key quantities in frequentist hypothesis testing and in Bayesian inference using data that are sanitised according to either of these two privacy standards. Under very mild conditions, the results in this work are valid for arbitrary parameters, priors and data generating models. These bounds are weaker than those attainable when analysing specific data generating models or data-release mechanisms. However, they provide generally applicable limits on the ability to learn from differentially private data – even when the analyst's knowledge of the model or mechanism is limited. They also shed light on the semantic interpretations of the two DP flavors under examination, a subject of contention in the current literature.1

On the effect of clock offsets and quantization on learning-based adversarial games

In this work, we consider systems whose components suffer from clock offsets and quantization and study the effect of those on a reinforcement learning (RL) algorithm. Specifically, we consider an off-policy iterative RL algorithm for continuous-time systems, which uses input and state data to approximate the Nash-equilibrium of a zero-sum game. However, the data used by this algorithm are not consistent with one another, in that each of them originates from a slightly different time instant of the past, hence putting the convergence of the algorithm in question. We prove that, given that these timing inconsistencies remain below a certain threshold, the iterative off-policy RL algorithm will still converge epsilon-closely to the desired Nash policy. However, this result is conditional to a certain Lipschitz continuity and differentiability condition on the input-state data collected, which is indispensable in the presence of clock offsets. A similar result is also derived when quantization of the measured state is considered. Finally, unlike prior work, we provide a sufficiently rich data condition for the execution of the iterative RL algorithm, which can be verified a priori across all iteration indices. Simulations are performed, which verify and clarify theoretical findings.

A Modified Three-Term Conjugate Descent Derivative-Free Method for Constrained Nonlinear Monotone Equations and Signal Reconstruction Problems

Iterative methods for solving constraint nonlinear monotone equations have been developed and improved by many researchers. The aim of this research is to present a modified three-term conjugate descent (TTCD) derivative-free method for constrained nonlinear monotone equations. The proposed algorithm requires low storage memory; therefore, it has the capability to solve large-scale nonlinear equations. The algorithm generates a descent and bounded search direction dk at every iteration independent of the line search. The method is shown to be globally convergent under monotonicity and Lipschitz continuity conditions. Numerical results show that the suggested method can serve as an alternative to find the approximate solutions of nonlinear monotone equations. Furthermore, the method is promising for the reconstruction of sparse signal problems.

Unconditionally optimal error estimates of linearized Crank-Nicolson virtual element methods for quasilinear parabolic problems on general polygonal meshes

In this paper, we construct, analyze, and numerically validate a linearized Crank-Nicolson virtual element method (VEM) for solving quasilinear parabolic problems on general polygonal meshes. In particular, we consider the more general nonlinear term a(x, u), which does not require Lipschitz continuity or uniform ellipticity conditions. To ensure that the fully discrete solution remains bounded in L∞-norm, we construct two novel elliptic projections and apply a new error splitting technique. With the help of the boundedness of numerical solution and delicate analysis of the nonlinear term, we derive the optimal error estimates for any k-order VEMs without any time-step restrictions. Numerical experiments on various polygonal meshes validate the accuracy of theoretical analysis and the unconditional convergence of the proposed scheme.

Exponential stability of non-conformable fractional-order systems

Abstract Recently, the authors Guzman et al. (2018) introduced a new simple well-behaved definition of the fractional derivative called non-conformable fractional derivative. In this paper we study the exponential stability of non-conformable fractional-order systems by using the Lyapunov function and Gronwall inequality. These inequalities can be used as handy tools to research stability problems of nonlinear systems. Sufficient conditions for exponential stability are given using the Lyapunov theory. Further, deals with the state feedback stabilization problems for a family of nonlinear systems satisfying a Lipschitz continuity condition. The stability of the controller is proved by means of the new Lyapunov stability theorem given in this paper. A numerical example is given to illustrate the efficiency of the obtained result.

Generic Stability Implication From Full Information Estimation to Moving-Horizon Estimation

Optimization-based state estimation is useful for handling of constrained linear or nonlinear dynamical systems. It has an ideal form, known as full information estimation (FIE) which uses all past measurements to perform state estimation, and also a practical counterpart, known as moving-horizon estimation (MHE) which uses most recent measurements of a limited length to perform the estimation. This work reveals a generic link from robust stability of FIE to that of MHE, showing that the former implies at least a weaker robust stability of MHE which implements a long enough horizon. The implication strengthens to strict robust stability of MHE if the corresponding FIE satisfies a mild Lipschitz continuity condition. The revealed implications are then applied to derive new sufficient conditions for robust stability of MHE, which further reveals an intrinsic relation between the existence of a robustly stable FIE/MHE and the system being incrementally input/output-to-state stable.

Forward-backward splitting algorithm with self-adaptive method for finite family of split minimization and fixed point problems in Hilbert spaces

In this paper, we introduce an inertial forward-backward splitting method together with a Halpern iterative algorithm for approximating a common solution of a finite family of split minimization problem involving two proper, lower semicontinuous and convex functions and fixed point problem of a nonexpansive mapping in real Hilbert spaces. Under suitable conditions, we proved that the sequence generated by our algorithm converges strongly to a solution of the aforementioned problems. The stepsizes studied in this paper are designed in such a way that they do not require the Lipschitz continuity condition on the gradient and prior knowledge of operator norm. Finally, we illustrate a numerical experiment to show the performance of the proposed method. The result discussed in this paper extends and complements many related results in literature.

Lipschitz Continuity for Harmonic Functions and Solutions of the α¯-Poisson Equation

In this paper we investigate the solutions of the so-called α¯-Poisson equation in the complex plane. In particular, we will give sufficient conditions for Lipschitz continuity of such solutions. We also review some recently obtained results. As a corollary, we can restate results for harmonic and (p,q)-harmonic functions.

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.

Output feedback distributed optimization algorithms of second‐order Lipschitz nonlinear multi‐agent systems

AbstractThis paper introduces output feedback distributed optimization algorithms designed specifically for second‐order nonlinear multi‐agent systems. The agents are allowed to have heterogeneous dynamics, characterized by distinct nonlinearities, as long as they satisfy the Lipschitz continuity condition. For the case with unknown states, nonlinear state observers are designed first for each agent to reconstruct agents' unknown states. It is proven that the agents' unknown states are estimated accurately by the developed state observers. Then, based on the agents' state estimates and the gradient of each agent local cost function, a kind of output feedback distributed optimization algorithms are proposed for the considered multi‐agent systems. Under the proposed distributed optimization algorithms, all the agents' outputs asymptotically approach the minimizer of the global cost function which is the sum of all the local cost functions. By using Lyapunov stability theory, convex analysis, and input‐to‐state stability theory, the asymptotical convergence of the output feedback distributed optimization closed‐loop system is proven. Simulations are conducted to validate the efficacy of the proposed algorithms.

A new approach to decentralized constrained nonlinear estimation over noisy communication links

This paper investigates constrained nonlinear estimation over noisy communication links. The estimation is solved by a set of nodes in a fully distributed fashion. The unknown parameter of interest is assumed to be belonging to a closed convex set. The measurement of each node is nonlinearly related to the unknown parameter. The communication among adjacent nodes is corrupted by the additive communication noises. We propose a decentralized projection consensus+innovation algorithm with communication noises to solve the nonlinear estimation problem and develop a novel approach to analyze its convergence. For the case of fixed graph, by introducing an auxiliary matrix and combination of the graph Laplacian and the auxiliary matrix, we prove that the algorithm converges in mean square and almost surely if the combined persistence of excitation (CPE) condition holds and the measurement function satisfies the Lipschitz continuity and monotonicity conditions. Furthermore, for the case of the time-varying graphs, we establish the jointly combined persistence of excitation (JCPE) condition guaranteeing convergence in mean square. Both the CPE and JCPE conditions are proposed for the first time and do not require that the graph is balanced. The JCPE condition holds even when the graph is disconnected at infinitely many time instants. A simulation example is presented to demonstrate our theoretical results.

Nonlinear Inverse Problems for Equations with Dzhrbashyan–Nersesyan Derivatives

The unique solvability in the sense of classical solutions for nonlinear inverse problems to differential equations, solved for the oldest Dzhrbashyan–Nersesyan fractional derivative, is studied. The linear part of the equation contains a bounded operator, a continuous nonlinear operator that depends on lower-order Dzhrbashyan–Nersesyan derivatives, and an unknown element. The inverse problem is given by an equation, special initial value conditions for lower Dzhrbashyan–Nersesyan derivatives, and an overdetermination condition, which is defined by a linear continuous operator. Applying the fixed-point method for contraction mapping a theorem on the existence of a local unique solution is proved under the condition of local Lipschitz continuity of the nonlinear mapping. Analogous nonlocal results were obtained for the case of the nonlocally Lipschitz continuous nonlinear operator in the equation. The obtained results for the problem in arbitrary Banach spaces were used for the research of nonlinear inverse problems with time-dependent unknown coefficients at lower-order Dzhrbashyan–Nersesyan time-fractional derivatives for integro-differential equations and for a linearized system of dynamics of fractional Kelvin–Voigt viscoelastic media.

A Class of Quasilinear Equations with Distributed Gerasimov–Caputo Derivatives

Quasilinear equations in Banach spaces with distributed Gerasimov–Caputo fractional derivatives, which are defined by the Riemann–Stieltjes integrals, and with a linear closed operator A, are studied. The issues of unique solvability of the Cauchy problem to such equations are considered. Under the Lipschitz continuity condition in phase variables and two types of continuity over all variables of a nonlinear operator in the equation, we obtain two versions on a theorem on the nonlocal existence of a unique solution. Two similar versions of local unique solvability of the Cauchy problem are proved under the local Lipschitz continuity condition for the nonlinear operator. The general results are used for the study of an initial boundary value problem for a generalization of the nonlinear phase field system of equations with distributed derivatives with respect to time.

Quasilinear Fractional Order Equations and Fractional Powers of Sectorial Operators

The fractional powers of generators for analytic operator semigroups are used for the proof of the existence and uniqueness of a solution of the Cauchy problem to a first order semilinear equation in a Banach space. Here, we use an analogous construction of fractional powers Aγ for an operator A such that −A generates analytic resolving families of operators for a fractional order equation. Under the condition of local Lipschitz continuity with respect to the graph norm of Aγ for some γ∈(0,1) of a nonlinear operator, we prove the local unique solvability of the Cauchy problem to a fractional order quasilinear equation in a Banach space with several Gerasimov–Caputo fractional derivatives in the nonlinear part. An analogous nonlocal Lipschitz condition is used to obtain a theorem of the nonlocal unique solvability of the Cauchy problem. Abstract results are applied to study an initial-boundary value problem for a time-fractional order nonlinear diffusion equation.

A self-adaptive inertial extragradient method for a class of split pseudomonotone variational inequality problems

Abstract In this article, we study a class of pseudomonotone split variational inequality problems (VIPs) with non-Lipschitz operator. We propose a new inertial extragradient method with self-adaptive step sizes for finding the solution to the aforementioned problem in the framework of Hilbert spaces. Moreover, we prove a strong convergence result for the proposed algorithm without prior knowledge of the operator norm and under mild conditions on the control parameters. The main advantages of our algorithm are: the strong convergence result obtained without prior knowledge of the operator norm and without the Lipschitz continuity condition often assumed by authors; the minimized number of projections per iteration compared to related results in the literature; the inertial technique employed, which speeds up the rate of convergence; and unlike several of the existing results in the literature on VIPs with non-Lipschitz operators, our method does not require any linesearch technique for its implementation. Finally, we present several numerical examples to illustrate the usefulness and applicability of our algorithm.

Convergence analysis for iterative learning control of fractional-order nonlinear differential inclusion system

This paper considers a class of finite-time tracking problems of fractional-order uncertain system governed by nonlinear differential inclusion. We introduce a Lipschitz continuous condition into the set-valued mapping described by the closure of the convex hull of a set, and design PDγ-type and PIαDγ-type update laws with initial learning mechanisms in open loop and closed loop, respectively. Then, the Steiner-type selector and mathematical analysis tools are used to complete the convergence analysis. Finally, numerical examples are used to verify the validity of the theoretical results, and the tracking performance of system is compared for different values of γ,α and learning laws.

Local Convergence of Parameter Based Derivative Free Continuation Method for the Solution of Non-linear Equations

This paper's major purpose is to evaluate the local convergence of the parameter-based sixth- and seventh-order continuation iterative approach for solving nonlinear equations in R. This analysis assumes that the Fréchet derivative of the first order satisfies the Lipschitz continuity condition. Under these circumstances, we explore convergence analysis in order to investigate the existence and uniqueness region for the solution of our proposed strategies. Thus, we also offered the theoretical concept of the radii of convergence balls for the proposed approach. By determining the radii of the convergence balls and solving many numerical problems, we can verify the significance of our convergence study.