Determination of the time-dependent reaction coefficient and the heat flux in a nonlinear inverse heat conduction problem

ABSTRACT

Diffusion processes with reaction generated by a nonlinear source are commonly encountered in practical applications related to ignition, pyrolysis and polymerization. In such processes, determining the intensity of reaction in time is of crucial importance for control and monitoring purposes. Therefore, this paper is devoted to such an identification problem of determining the time-dependent coefficient of a nonlinear heat source together with the unknown heat flux at an inaccessible boundary of a one-dimensional slab from temperature measurements at two sensor locations in the context of nonlinear transient heat conduction. Local existence and uniqueness results for the inverse coefficient problem are proved when the first three derivatives of the nonlinear source term are Lipschitz continuous functions. Furthermore, the conjugate gradient method (CGM) for separately reconstructing the reaction coefficient and the heat flux is developed. The ill-posedness is overcome by using the discrepancy principle to stop the iteration procedure of CGM when the input data is contaminated with noise. Numerical results show that the inverse solutions are accurate and stable.     

区间粒子算法与线源反问题求解

目前国内外对线源反问题数值求解尚没有一种成熟有效的算法。本文在研究区间搜索算法基础上,提出了一种新的求解算法—区间粒子算法(Range Particle Algorithm)来求解线源反问题。首先简要介绍了线源反问题的求解特点,并根据线源方程建立了反问题求解的目标函数;其次基于该目标函数,设计了区间粒子算法来求解,探讨了算法实现的基本步骤和参数调整问题;最后通过模拟数据和实测数据分别检验了该算法求解的效果,结果表明区间粒子算法求解精度高、收敛速度快和计算稳定,在线源反问题数值求解中是适用的。     

Numerical solution of time-dependent component with sparse structure of source term for a time fractional diffusion equation

We consider an inverse time-dependent component of source term with sparse structure for the time fractional diffusion equation in the present paper. We prove the uniqueness of the inverse problem with nonlocal observation data by Laplace transform technique. Concerning the sparsity of the source term, we transform the inverse source problem into an elastic-net regularization optimization problem. The semi-smooth Newton method is adopted to solve the optimization problem and the superconvergence of the semi-smooth Newton algorithm is proven. Several numerical examples are tested to verify the efficiency of the algorithm.     

Solvability of the Nonlocal Inverse Parabolic Problem and Numerical Results

In this paper, we consider the unique solvability of the inverse problem of determining the right-hand side of a parabolic equation whose leading coefficient depends on time variable under nonlocal integral overdetermination condition. We obtain sufficient conditions for the unique solvability of the inverse problem. The existence and uniqueness of the solution of the inverse parabolic problem upon the data are established using the fixed point theorem. This inverse problem appears extensively in the modelling of various phenomena in engineering and physics. For example, seismology, medicine, fusion welding, continuous casting, metallurgy, aircraft, oil and gas production during drilling and operation of wells. In addition, the numerical solution of the inverse problem is studied by using the Crank-Nicolson finite difference method together with the Tikhonov regularization to find a stable and accurate approximate solution of finite differences. The resulting nonlinear system of parabolic equation is solved computationally using the MATLAB subroutine lsqnonlin. Both analytical and numerically simulated noisy input data are inverted. The root mean square error values for various noise levels for both continuous and discontinuous time-dependent heat source term are compared. Numerical results presented for two examples show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data. Furthermore, the choice of the regularization parameter is also discussed based on the trial and error technique.     

基于自适应差异进化法的电力系统无功优化

差异进化算法（DE）是一种新的进化算法,近年来的研究和应用已经展示出很大的应用潜力,但其中的某些参数需通过试验确定,影响了实用性。提出一种自适应差异进化算法（FADE）,能使算法的控制参数粮据求解问题的不同在优化过程中自适应发生改变,并应用于无功优化问题。通过IEEE-30节点算例系统的仿真结果证明,与DE和GA算法相比,模糊差异进化算法具有很强的自适应性及通用性。     

On Resolving Inverse Nonstationary Scattering Problems in a Two-Dimensional Homogeneous Layered Medium by the τ–<Emphasis Type="Italic">p</Emphasis> Radon Transform

We consider a two-dimensional nonstationary inverse scattering problem in a layered homogeneous acoustic medium. The data consist of a scattered wavefield from a surface point source registered on the boundary of the half-plane. We prove the uniqueness of the recovery of an acoustic impedance and velocity in a medium from the scattering data. An algorithm for solving an inverse twodimensional scattering problem as a one-dimensional problem with the parameter based on the τ–p Radon transformation is constructed. Also, the numerical modeling results for the direct scattering problem and solutions of a pair of inverse scattering problems in a layered homogeneous acoustic medium are presented. The proposed algorithm is applicable to data processing in geophysical prospecting both for surface seismics and vertical seismic profiling.     

自适应调整控制参数的差异演化算法

控制参数选取是包括差异演化在内的演化算法设计时所面临的一个重要问题,对算法的性能有着重大影响.针对差异演化算法参数选取问题,提出一种利用个体适应度作为参数调整决策依据,并结合一定的调整概率对F和CR进行自适应调整的方法,解决了手工设置控制参数的不便.同时利用交叉操作生成双子代个体与父代个体竞争形成新一代种群,加快了算法的收敛.对标准测试函数的仿真实验结果表明,该算法无论在最优解质量和收敛速度上都优于相关算法,尤其对于高维函数而言.     

热连轧精轧负荷分配优化的多目标反向迭代法

在求解热精轧负荷分配优化的过程中,针对目标函数中加权系数难以确定的问题,提出多目标反向迭代法(multi-objective inversion iteration algorithm,MOII),该法通过数值迭代的方式实现了热精轧机组负荷分配的多目标优化.首先,在综合考虑板形、板厚精度和质量的基础上,设计负荷分配优化的多目标函数;然后,基于轧制力能参数与轧机入口和出口板带厚度的单调性关系,设计MOII算法,并对MOII算法的合理性及异常情况进行分析,给出其计算流程;最后,基于实际生产数据进行仿真,结果表明,MOII算法不仅能够实现负荷分配的多目标优化,而且求解速度快、精度高,体现出良好的在线应用前景;与其他方法得到的解相比,MOII算法求得的解有更好的性能.     

Finite‐time stabilization of stochastic nonlinear systems with SiISS inverse dynamics

This paper investigates the finite‐time control problem for a class of stochastic nonlinear systems with stochastic integral input‐to‐state stablility (SiISS) inverse dynamics. Motivated by finite‐time stochastic input‐to‐state stability and the concept of SiISS using Lyapunov functions, a novel finite‐time SiISS using Lyapunov functions is introduced firstly. Then, by adopting this novel finite‐time SiISS small‐gain arguments, using the backstepping technique and stochastic finite‐time stability theory, a systematic design and analysis algorithm is proposed. Given the control laws that guarantee global stability in probability or asymptotic stability in probability, our design algorithm presents a state‐feedback controller that can ensure the solution of the closed‐loop system to be finite‐time stable in probability. Finally, a simulation example is given to demonstrate the effectiveness of the proposed control scheme. Copyright © 2017 John Wiley & Sons, Ltd.     

用边界粒子法求解柯西反问题的数值计算软件包

为推动无网格方法在反问题中的应用及相关计算软件的发展,介绍基于MATLAB开发的用边界粒子法(Boundary Particle Method,BPM)数值模拟柯西反问题的计算软件包;描述数值算法的理论基础,并结合标准算例介绍软件包的基本功能和使用方法.该软件包可求解含不同形式源项(多项式函数、三角函数、指数函数及其耦合函数等)的椭圆型偏微分方程,并且可用于处理任意几何区域问题.本软件包具有简单易用的图形用户界面,在数值模拟柯西反问题时仅需边界测量数据,且效率高、精度高.     

Numerical method for the simultaneous determination of unknown parameters in a parabolic equation

Numerical procedures for the solution of an inverse problem of simultaneously determining unknown parameters in a linear parabolic equation are considered. The approach proposed is to approximate unknown functions using Chebyshev polynomials, which are determined consecutively from the solutions of the minimization problems based on overspecified data. Finally, the results of a numerical experiment are displayed.     

Numerical procedure for the determination of an unknown parameter in a parabolic equation

A numerical procedure for an inverse problem of determining unknown source parameter of one-dimensional parabolic equation subject to the specification of the solution at internal point along with the usual initial boundary conditions is considered. By using some transformation the problem is reformulated to a nonlocal parabolic problem. Some numerical examples using the proposed numerical procedure are presented.     

一种双域Montgomery求逆算法与硬件实现

有限域上的求逆运算是椭圆曲线密码算法的关键运算之一。分别对GF（p）和GF（2ⁿ）域上的Montgomery模逆算法进行分析,并将GF（2ⁿ）域上的Montgomery模逆算法中对变量阶数的比较进行了改进,这样不仅利于GF（p）和GF（2ⁿ）域上的模逆运算在统一的硬件结构上实现,也解决了数据位数较大时进行阶数比较延迟较大的问题,在此基础上提出一种基于GF（p）和GF（2ⁿ）双域上统一的模逆算法,并根据算法,采用双域可伸缩运算单元,实现了一种可扩展的统一Montgomery模逆硬件结构。设计采用Verilog-HDL语言进行硬件描述,并基于0.18 μm工艺标准单元库进行了综合,结果表明该设计与其他设计相比具有灵活性好、性能高的特点。     

基于核估计的超定混合共轭盲信号分离方法

当混合信号的个数多于源信号时,盲源分离模型中的混合矩阵被描述为一个超定矩阵,因此不能直接通过估计逆矩阵的方法来得到分离矩阵。针对该线性超定混合情况提出了一种基于共轭梯度的盲源分离方法。该方法基于最小互信息准则,通过对行满秩分离矩阵的奇异值分解而引入了超定盲源分离的代价函数。利用共轭梯度优化算法推导出了迭代计算分离矩阵的更新公式。在每次迭代计算中,利用随机变量概率密度估计的核函数法在线估计分离信号的评价函数。避免了诸多传统盲分离算法中只能凭经验选取特定的非线性函数来代替评价函数的问题。仿真结果验证了所提算法的有效性。     

Unsupervised selection of a finite Dirichlet mixture model: an MML-based approach

This paper proposes an unsupervised algorithm for learning a finite Dirichlet mixture model. An important part of the unsupervised learning problem is determining the number of clusters which best describe the data. We extend the minimum message length (MML) principle to determine the number of clusters in the case of Dirichlet mixtures. Parameter estimation is done by the expectation-maximization algorithm. The resulting method is validated for one-dimensional and multidimensional data. For the one-dimensional data, the experiments concern artificial and real SAP image histograms. The validation for multidimensional data involves synthetic data and two real applications: shadow detection in images and summarization of texture image databases for efficient retrieval. A comparison with results obtained for other selection criteria is provided.     

Coupling of logistic regression analysis and local search methods for characterization of water distribution system contaminant source

Accidental or intentional drinking water contamination has long been and remains a major threat to water security throughout the world. An inverse problem can be constructed, given sensor measurements in a water distribution system (WDS), to identify the contaminant source characteristics by integrating a WDS simulation model with an optimization method. However, this approach requires numerous compute-intensive simulation runs to evaluate potential solutions; thus, determining the best source characteristic within a reasonable computational time is challenging. In this paper, we describe the development of a WDS contamination characterization algorithm by coupling a statistical model with a heuristic search method. The statistical model is used to identify potential source locations of contamination and a local search aims at further refining contaminant source characteristics. Application of the proposed approach to two illustrative example water distribution networks demonstrates its capability of adaptively discovering contaminant source characteristics as well as evaluating the degree of non-uniqueness of solutions. The results also showed that the local search as an optimizer has better performance than a standard evolutionary algorithm (EA).     

A meshless method to the numerical solution of an inverse reaction–diffusion–convection problem

In this paper, the determination of the source term in a reaction–diffusion convection problem is investigated. First with suitable transformations, the problem is reduced, then a new meshless method based on the use of the heat polynomials as basis functions is proposed to solve the inverse problem. Due to the ill-posed inverse problem, the Tikhonov regularization method with a generalized cross-validation criterion is employed to obtain a numerical stable solution. Finally, some numerical examples are presented to show the accuracy and effectiveness of the algorithm.     

利用有限差分法计算真实头模型脑电正问题

脑电研究领域的两个关键问题是脑电正问题和脑电逆问题,脑电正问题是脑电逆问题的基础.由于复杂、非规则真实头模型中的脑电正问题不存在解析解,因此脑电源分析依赖于正问题数值算法的精度和效率.文章首先详细推导了有限差分算法求解三维各向同性脑电正问题的数学模型,然后在三层同心球模型上通过与解析解比较验证了该算法的精度和效率,最后将该算法应用于真实头模型.仿真结果表明,有限差分法可以有效地处理任意形状几何体的电位场分布问题,是模拟计算真实头模型中脑电正问题的有力工具.     

Direct and inverse problems on nonlinear rods

In this paper a class of models on nonlinear rods, which includes spatial inhomogeneities, varying cross-sectional area and arbitrary memory functions, is considered. The wave splitting technique is applied to provide a formulation suitable for numerical computation of direct and inverse problems. Due to the nonlinearity of the material, there are no well defined characteristics other than the leading edge, so the method of characteristics, highly successful in the computation of linear wave splitting problems, is abandoned. A standard finite difference method is employed for the direct problem, and a shooting method is introduced for the inverse problem. The feasibility of the inverse algorithm is presented in various numerical examples.     

Construction of substantially different solutions of an inverse problem for a toroidal plasma equilibrium equation

The paper is devoted to the reconstruction of the current density in toroidal plasma, using experimentally measured data. The toroidal current density is characterized by two functions in the right-hand side of the Grad-Shafranov equation which, together with the poloidal flux, is to be deter-mined. The question about the uniqueness of the solution of the inverse problem was usually not addressed in the numerical methods developed over the past few decades. However, the theoretical study of this question for simplified models showed the possibility of the existence of substantially different solutions. For a correct understanding of the physical properties of a pulse it is necessary to analyze all possible solutions of the inverse problem in its physically correct formulation. This formulation is presented in the paper. A new numerical method for determining all substantially different solutions of the inverse problem is proposed. Examples of the existence of such solutions are constructed for plasma parameters which are close to experimental.