几类抛物型方程反边值问题的数值求解

发布时间：2018-12-13 16:02

【摘要】：扩散过程是一类重要的自然现象,从数学研究上来看,扩散过程在一定的假设条件下可以用带有定解条件的抛物型方程模型来描述.在很多场合,扩散过程的物理机制是明确的(对应于扩散方程是已知的),但描述扩散过程的边界状态信息是未知的,此时需要通过其他可测量的间接信息来反演边界状态信息,进而确定整个扩散过程.这类问题就是扩散方程模型的反边值问题.边界状态是抛物型方程描述扩散系统的重要参数,包括边界形状和边界条件参数.边界条件主要包括Dirichlet边界条件、Neumann边界条件和更一般的Robin(阻尼)边界条件.Robin边界条件中的Robin系数表征了边界界面对整个扩散过程的阻尼作用,在工程应用中有重要的意义.关于前两种边界条件的反问题,已经有很多研究工作,而关于Robin系数反演的研究工作较少,尤其是不连续Robin系数的反演问题,已有工作更少.数学上的难点在于,Robin系数的重建是一·类非线性不适定问题.这显然增加了反问题理论分析和数值实现的难度.因此,本文围绕Robin系数反演及相关问题展开研究,主要工作包括以下三个方面.首先,研究利用边界上关于时间方向的积分型数据,反演依赖于空间变量的Robin系数.在很多物理情形下,由于客观条件的限制,扩散物浓度的逐点数据很难测量,而扩散物浓度的某种(时间或空间)平均值相对易于测量,因此研究这类非局部积分型测量数据更具有实际意义。近年来,利用该类数据反演相关参数的研究也得到了广泛的重视,本部分反问题的测量数据正是此类非局部数据.对此反问题,本文建立了唯一性,并且首次建立了条件稳定性.证明思路是,基于解的位势表达式,将非线性反问题模型转换为关于待定密度函数和未知Robin系数的耦合的非线性积分方程组.在此基础上,构造了双参数正则化泛函,并从理论上严格分析了优化方案的适定性和收敛性.在数值实现方面,与已有的共轭梯度法相比,我们提出的交替迭代法计算更快,反演效果更好,对于误差较大的噪音数据,这种优势更明显.其次,研究利用末时刻的温度(浓度)分布同时反演边界Robin系数和初始状态的反问题.初始状态的反演是经典的线性反问题,在工业中有广泛的应用背景,但是在边界Robin系数也未知的情形下,这类反问题除了未知成份更多以外,更重要的是,反问题变为非线性的了,从而使得初始状态和Robin系数的同时反演难度更大,也更有意义.现有的研究基本还是空白.针对该问题,本文利用最值原理和特征函数展开理论,证明了 Robin系数在允许集内是唯一确定的.进一步,基于数据磨光化和结合初值的拟逆正则化思想,提出了一种同时重建两个参数的正则化方案,并且给出了正则化参数选取策略和误差分析.本文的重建方案分两步进行.第一步重建Robin系数,第二步基于重建的Robin系数重建初值,显然第一步的重建对整个重建方案至关重要,并且非线性地影响着整个重建方案.本文细致分析了两步之间的误差传递,并给出了正则化参数的选取策略,这是本文的重要创新之处.在此基础上,基于位势理论提出了有效的数值实现方案,数值结果与理论分析的结果一致.最后,研究一维扩散模型中,利用边界一端的Dirichlet数据反演边界另一端不连续的Robin系数(依赖于时间变量的函数).在实际工程中,边界Robin系数不连续是很重要的情形,可能预示着系统内部出现某类故障,那么Robin系数就可以作为检测系统是否出现故障的重要指标.此时,不连续Robin系数的检测尤其是不连续位置的检测就具有特别重要的意义.对不连续的Robin系数,本文基于Fourier变换,在L2空间的允许集内建立了反问题的唯一性.我们的结果不同于连续函数空间的唯一性,是本文另一理论创新点.此外,本文将原问题转化为求解带全变差罚项的优化问题,并对该正则化方案进行了理论分析.数值实现方面,提出了一种有效的双循环迭代算法.该算法的基本思想是将非线性的数据匹配项和近似全变差罚项交替迭代,实现极小化.这种交替迭代的策略一方面可以降低原问题的非线性,无需考虑正则化参数的选取,另一方面可以提高计算速度.同时,这种根据非线性项的不同性质交替迭代处理的思想,还可以推广到包含多个非线性项的数值实现中,对其它相关问题的数值求解具有重要的借鉴意义.
[Abstract]:The diffusion process is an important natural phenomenon. From the mathematical research, the diffusion process can be described by a parabolic equation model with a fixed solution under certain assumptions. In many cases, the physical mechanism of the diffusion process is clear (corresponding to the diffusion equation is known), but the boundary state information describing the diffusion process is unknown, at which time the boundary state information needs to be inverted by other measurable indirect information, and thus the entire diffusion process is determined. This kind of problem is the inverse boundary value problem of the diffusion equation model. The boundary condition is an important parameter of the parabolic equation describing the diffusion system, including boundary shape and boundary condition parameters. Boundary conditions mainly include Dirichlet boundary conditions, Neumann boundary conditions and more general Robin (damping) boundary conditions. The Robin coefficient in the Robin boundary condition indicates the damping effect of the boundary interface on the whole diffusion process, which is of great significance in the engineering application. There are many researches on the inverse problem of the first two boundary conditions, and the research work on the inversion of the Robin coefficient is less, especially the problem of the inversion of the non-continuous Robin coefficient, and has less work. The difficulty of the mathematics is that the reconstruction of the Robin coefficient is a class-class non-linear discomfort problem. This obviously increases the difficulty of theoretical analysis and numerical implementation of the inverse problem. Therefore, this paper studies the inversion of the Robin coefficient and the related problems, and the main work includes the following three aspects. First, the integration type data on the time direction on the boundary is used to retrieve the Robin coefficient which is dependent on the spatial variable. In many physical circumstances, the point-by-point data of the diffusion concentration is difficult to measure due to the limitation of objective conditions, and some (time or space) average value of the diffusion concentration is relatively easy to measure, so it is more practical to study the non-local integral type measurement data. In recent years, the data of this kind of data are used to retrieve the relevant parameters, and the measurement data of this partial inverse problem is this kind of non-local data. In this paper, the uniqueness is set up in this paper, and the condition stability is established for the first time. It is proved that the non-linear inverse problem model is transformed into a non-linear integral equation system with respect to the coupling of the undetermined density function and the unknown Robin coefficient based on the potential expression of the solution. On this basis, the double-parameter regularization function is constructed, and the appropriate and the convergence of the optimization scheme are analyzed strictly from the theory. In the aspect of numerical realization, the alternative iteration method proposed by us is faster, the inversion effect is better, and the advantage is more obvious for the noise data with large error compared with the existing common-current gradient method. Next, the inverse problem of the boundary Robin coefficient and the initial state is studied by using the temperature (concentration) distribution at the end time. the inversion of the initial state is a classical linear inverse problem, which has a wide application background in the industry, but in the case where the boundary Robin coefficient is also unknown, the inverse problem is more important than the unknown component, and more importantly, the inverse problem becomes non-linear, so that the inversion difficulty of the initial state and the Robin coefficient is more difficult and more meaningful. The existing research is essentially blank. In order to solve this problem, this paper uses the principle of the maximum value and the characteristic function to expand the theory, and proves that the Robin coefficient is uniquely determined within the permission set. Further, based on the quasi-inverse regularized idea of data burnishing and combining initial values, a regularized scheme for reconstructing two parameters at the same time is proposed, and the regularization parameter selection strategy and error analysis are given. The reconstruction scheme of this paper is carried out in two steps. The first step rebuilds the Robin coefficient, and the second step reconstructs the initial value based on the reconstructed Robin coefficient, and it is clear that the reconstruction of the first step is critical to the overall reconstruction scheme and that the entire reconstruction scheme is non-linear. In this paper, the error transfer between two steps is analyzed in detail, and the selection strategy of regularized parameters is given, which is the important innovation of this paper. On this basis, the effective numerical solution is proposed based on the potential theory, and the numerical results are consistent with the results of the theoretical analysis. Finally, in the one-dimensional diffusion model, the non-continuous Robin coefficient at the other end of the boundary is inverted by the Dirichlet data at one end of the boundary (as a function of the time variable). In practical engineering, the non-continuous boundary Robin coefficient is very important, which may indicate a certain type of fault inside the system, then the Robin coefficient can be used as an important index for detecting the failure of the system. In this case, the detection of the non-continuous Robin coefficient, in particular the detection of the discontinuous position, is of particular importance. In this paper, the uniqueness of the inverse problem is set up in the permission set of the L2 space based on the Fourier transform. Our results are different from the uniqueness of the continuous function space, which is another theoretical innovation point of this paper. In addition, this paper is to transform the original problem into the optimization problem with the total variogram, and the regularized scheme is analyzed theoretically. In this paper, an effective two-cycle iterative algorithm is proposed. The basic idea of the algorithm is to iterate the non-linear data matching term and the approximate full-variation penalty term to realize the minimization. The strategy of the alternative iteration can reduce the non-linearity of the original problem on the one hand, without taking into account the selection of the regularized parameters, and on the other hand, the calculation speed can be improved. At the same time, the idea of alternating iterative processing according to different properties of non-linear term can also be extended to the numerical solution containing multiple non-linear terms, which has important reference significance to the numerical solution of other related problems.
【学位授予单位】：东南大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O241.82

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