非线性不适定问题的动力系统方法研究
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摘要
动力系统方法,也可称其为连续正则化方法,是求解非线性不适定问题的一个行之有效的方法。它克服了原有迭代法收敛定理中对算子较强的限制条件,证明了所研究系统的稳定性。众所周知,动力系统理论及其稳定性一直都是引人关注的问题。而Lyapunov稳定性理论作为分析系统稳定性的重要方法在本文中得到了进一步的研究、完善。本文将从以下几个方面用动力系统方法的思想及Lyapunov稳定性理论研究非线性不适定问题。
动力系统可分为离散动力系统和连续动力系统两种。本文先从离散的动力系统入手,基于连续的Landweber法,构造了一个求解非线性不适定问题的Runge–Kutta(简称R–K)型Landweber方法,并研究了该方法的收敛性,给出了存在扰动误差情况下的收敛率。与Landweber迭代法的数值比较表明,R–K型方法的收敛速度更快、更稳定。
当算子无界时,去掉算子的Fr′echet可微性及算子的一些非线性条件,本文引入了一个无导数的特殊结构,并提出了一个参数识别问题。在较之以前更弱的限制条件下,基于正问题的可解性,从理论上证明了所提参数识别问题的收敛性及稳定性。
针对非线性不适定算子方程的优化问题,在原有的Lyapunov稳定定理的基础上,给出了一个新的Lyapunov稳定性引理,并用此引理证明了该优化问题的收敛性。这一新的Lyapunov稳定性引理的限制条件要稍弱于Xu提出的稳定性定理的条件。所以,本文中的稳定性引理是原有稳定性定理的拓展,是本文的一个创新之处。
由于经典的求解非线性不适定问题的迭代法都是局部收敛的,借鉴同伦方法的优点,本文构造了一个鲁棒的、大范围收敛的同伦正则化方法用以识别参数,利用Lyapunov稳定性理论证明了该正则化方法的收敛性。并通过分析具体的数值算例,更加肯定了当扰动误差相同时,与Landweber迭代法相比,该同伦正则化方法更稳定,收敛范围更大。
基于Sobolev空间中偏微分方程的可解性及稳定性,本文利用类似于动力系统的证明方法,提出了用水平集方法识别非线性抛物分布式参数系统。并在偏差原则作为终止法则的前提下,验证了水平集方法的正则性。
Dynamical system methods, which can also be called continuous regularizationmethod, are very effective methods for solving nonlinear ill-posed problems. Thesemethods overcome the stronger restrictive conditions on the operator in the conver-gence theorems known for the corresponding iterative methods, and prove the stabilityof the systems which we investigated. It is well known that the theory of dynamicalsystems and their stability have been remarkable issue. The Lyapunov theory, whichare important techniques for analyzing the stability of the systems, obtains furtherdevelopment, and becomes to be perfect in this paper. We will investigate the nonlin-ear ill-posed problems by means of the idea of dynamical systems and the Lyapunovtheory in some aspects.
Dynamical systems can be divided into two kinds of systems, that is, one is adiscrete system, and the other is continuous. Firstly, we set about to study the dis-crete one. Based on the continuous Landweber method, we construct a Runge–Kutta(simply as R–K) type Landweber method for solving nonlinear ill-posed problems,and investigate the convergent property of this method. Furthermore, we obtain theconvergence rate of this method when the perturbed data with noise exists. Comparedwith the numerical performance of Landweber method, the convergence rate of R–Ktype method is higher, and the method is more stable.
When the operators are unbounded we get ride of the Fr′echet differentiability ofthe parameter–to–output map as well as conditions restricting its nonlinearity, intro-duce a derivative-free and special structure, and put forward a parameter identification.Convergence and stability to the parameter identification have been proved in theoryunder the more weak restrictive conditions associated with the solvability of the directproblem.
Aiming at a minimization problem for solving nonlinear ill-posed problems, wegive a new lemma of Lyapunov stability based on the original Lyapunov stability.Moreover, we use this lemma to prove the convergence of the minimization problem.The restrictive conditions on this new lemma of stability are weaker than those Xuproposed. Therefore, the lemma of stability is a continuation of the original one, and a remarkable innovation in this paper.
Since the classical iterative methods for nonlinear ill-posed problems are all lo-cally convergent, we construct a robust and widely convergent homotopy regulariza-tion to identify the parameter in view of the properties of homotopy method, andproved this method to be convergent in the light of the theory of Lyapunov stability.Compared with Landweber iteration, a concrete numerical example proved this ho-motopy regularization to be more stable and widely convergent with the same noise.
Based on the solvability and stability of the partial differential equation inSobolev space, we use the similar idea to the dynamical systems and put forwarda level set method for the identification problem of the nonlinear parabolic distributedparameter systems. Moreover, we validate the level set approach to be a regularizationif the discrepancy principle is used as a stopping rule.
动力系统可分为离散动力系统和连续动力系统两种。本文先从离散的动力系统入手,基于连续的Landweber法,构造了一个求解非线性不适定问题的Runge–Kutta(简称R–K)型Landweber方法,并研究了该方法的收敛性,给出了存在扰动误差情况下的收敛率。与Landweber迭代法的数值比较表明,R–K型方法的收敛速度更快、更稳定。
当算子无界时,去掉算子的Fr′echet可微性及算子的一些非线性条件,本文引入了一个无导数的特殊结构,并提出了一个参数识别问题。在较之以前更弱的限制条件下,基于正问题的可解性,从理论上证明了所提参数识别问题的收敛性及稳定性。
针对非线性不适定算子方程的优化问题,在原有的Lyapunov稳定定理的基础上,给出了一个新的Lyapunov稳定性引理,并用此引理证明了该优化问题的收敛性。这一新的Lyapunov稳定性引理的限制条件要稍弱于Xu提出的稳定性定理的条件。所以,本文中的稳定性引理是原有稳定性定理的拓展,是本文的一个创新之处。
由于经典的求解非线性不适定问题的迭代法都是局部收敛的,借鉴同伦方法的优点,本文构造了一个鲁棒的、大范围收敛的同伦正则化方法用以识别参数,利用Lyapunov稳定性理论证明了该正则化方法的收敛性。并通过分析具体的数值算例,更加肯定了当扰动误差相同时,与Landweber迭代法相比,该同伦正则化方法更稳定,收敛范围更大。
基于Sobolev空间中偏微分方程的可解性及稳定性,本文利用类似于动力系统的证明方法,提出了用水平集方法识别非线性抛物分布式参数系统。并在偏差原则作为终止法则的前提下,验证了水平集方法的正则性。
Dynamical system methods, which can also be called continuous regularizationmethod, are very effective methods for solving nonlinear ill-posed problems. Thesemethods overcome the stronger restrictive conditions on the operator in the conver-gence theorems known for the corresponding iterative methods, and prove the stabilityof the systems which we investigated. It is well known that the theory of dynamicalsystems and their stability have been remarkable issue. The Lyapunov theory, whichare important techniques for analyzing the stability of the systems, obtains furtherdevelopment, and becomes to be perfect in this paper. We will investigate the nonlin-ear ill-posed problems by means of the idea of dynamical systems and the Lyapunovtheory in some aspects.
Dynamical systems can be divided into two kinds of systems, that is, one is adiscrete system, and the other is continuous. Firstly, we set about to study the dis-crete one. Based on the continuous Landweber method, we construct a Runge–Kutta(simply as R–K) type Landweber method for solving nonlinear ill-posed problems,and investigate the convergent property of this method. Furthermore, we obtain theconvergence rate of this method when the perturbed data with noise exists. Comparedwith the numerical performance of Landweber method, the convergence rate of R–Ktype method is higher, and the method is more stable.
When the operators are unbounded we get ride of the Fr′echet differentiability ofthe parameter–to–output map as well as conditions restricting its nonlinearity, intro-duce a derivative-free and special structure, and put forward a parameter identification.Convergence and stability to the parameter identification have been proved in theoryunder the more weak restrictive conditions associated with the solvability of the directproblem.
Aiming at a minimization problem for solving nonlinear ill-posed problems, wegive a new lemma of Lyapunov stability based on the original Lyapunov stability.Moreover, we use this lemma to prove the convergence of the minimization problem.The restrictive conditions on this new lemma of stability are weaker than those Xuproposed. Therefore, the lemma of stability is a continuation of the original one, and a remarkable innovation in this paper.
Since the classical iterative methods for nonlinear ill-posed problems are all lo-cally convergent, we construct a robust and widely convergent homotopy regulariza-tion to identify the parameter in view of the properties of homotopy method, andproved this method to be convergent in the light of the theory of Lyapunov stability.Compared with Landweber iteration, a concrete numerical example proved this ho-motopy regularization to be more stable and widely convergent with the same noise.
Based on the solvability and stability of the partial differential equation inSobolev space, we use the similar idea to the dynamical systems and put forwarda level set method for the identification problem of the nonlinear parabolic distributedparameter systems. Moreover, we validate the level set approach to be a regularizationif the discrepancy principle is used as a stopping rule.
引文
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