【摘要】：擴散過程是一類重要的自然現(xiàn)象,從數(shù)學(xué)研究上來看,擴散過程在一定的假設(shè)條件下可以用帶有定解條件的拋物型方程模型來描述.在很多場合,擴散過程的物理機制是明確的(對應(yīng)于擴散方程是已知的),但描述擴散過程的邊界狀態(tài)信息是未知的,此時需要通過其他可測量的間接信息來反演邊界狀態(tài)信息,進而確定整個擴散過程.這類問題就是擴散方程模型的反邊值問題.邊界狀態(tài)是拋物型方程描述擴散系統(tǒng)的重要參數(shù),包括邊界形狀和邊界條件參數(shù).邊界條件主要包括Dirichlet邊界條件、Neumann邊界條件和更一般的Robin(阻尼)邊界條件.Robin邊界條件中的Robin系數(shù)表征了邊界界面對整個擴散過程的阻尼作用,在工程應(yīng)用中有重要的意義.關(guān)于前兩種邊界條件的反問題,已經(jīng)有很多研究工作,而關(guān)于Robin系數(shù)反演的研究工作較少,尤其是不連續(xù)Robin系數(shù)的反演問題,已有工作更少.數(shù)學(xué)上的難點在于,Robin系數(shù)的重建是一·類非線性不適定問題.這顯然增加了反問題理論分析和數(shù)值實現(xiàn)的難度.因此,本文圍繞Robin系數(shù)反演及相關(guān)問題展開研究,主要工作包括以下三個方面.首先,研究利用邊界上關(guān)于時間方向的積分型數(shù)據(jù),反演依賴于空間變量的Robin系數(shù).在很多物理情形下,由于客觀條件的限制,擴散物濃度的逐點數(shù)據(jù)很難測量,而擴散物濃度的某種(時間或空間)平均值相對易于測量,因此研究這類非局部積分型測量數(shù)據(jù)更具有實際意義。近年來,利用該類數(shù)據(jù)反演相關(guān)參數(shù)的研究也得到了廣泛的重視,本部分反問題的測量數(shù)據(jù)正是此類非局部數(shù)據(jù).對此反問題,本文建立了唯一性,并且首次建立了條件穩(wěn)定性.證明思路是,基于解的位勢表達式,將非線性反問題模型轉(zhuǎn)換為關(guān)于待定密度函數(shù)和未知Robin系數(shù)的耦合的非線性積分方程組.在此基礎(chǔ)上,構(gòu)造了雙參數(shù)正則化泛函,并從理論上嚴格分析了優(yōu)化方案的適定性和收斂性.在數(shù)值實現(xiàn)方面,與已有的共軛梯度法相比,我們提出的交替迭代法計算更快,反演效果更好,對于誤差較大的噪音數(shù)據(jù),這種優(yōu)勢更明顯.其次,研究利用末時刻的溫度(濃度)分布同時反演邊界Robin系數(shù)和初始狀態(tài)的反問題.初始狀態(tài)的反演是經(jīng)典的線性反問題,在工業(yè)中有廣泛的應(yīng)用背景,但是在邊界Robin系數(shù)也未知的情形下,這類反問題除了未知成份更多以外,更重要的是,反問題變?yōu)榉蔷�性的了,從而使得初始狀態(tài)和Robin系數(shù)的同時反演難度更大,也更有意義.現(xiàn)有的研究基本還是空白.針對該問題,本文利用最值原理和特征函數(shù)展開理論,證明了 Robin系數(shù)在允許集內(nèi)是唯一確定的.進一步,基于數(shù)據(jù)磨光化和結(jié)合初值的擬逆正則化思想,提出了一種同時重建兩個參數(shù)的正則化方案,并且給出了正則化參數(shù)選取策略和誤差分析.本文的重建方案分兩步進行.第一步重建Robin系數(shù),第二步基于重建的Robin系數(shù)重建初值,顯然第一步的重建對整個重建方案至關(guān)重要,并且非線性地影響著整個重建方案.本文細致分析了兩步之間的誤差傳遞,并給出了正則化參數(shù)的選取策略,這是本文的重要創(chuàng)新之處.在此基礎(chǔ)上,基于位勢理論提出了有效的數(shù)值實現(xiàn)方案,數(shù)值結(jié)果與理論分析的結(jié)果一致.最后,研究一維擴散模型中,利用邊界一端的Dirichlet數(shù)據(jù)反演邊界另一端不連續(xù)的Robin系數(shù)(依賴于時間變量的函數(shù)).在實際工程中,邊界Robin系數(shù)不連續(xù)是很重要的情形,可能預(yù)示著系統(tǒng)內(nèi)部出現(xiàn)某類故障,那么Robin系數(shù)就可以作為檢測系統(tǒng)是否出現(xiàn)故障的重要指標(biāo).此時,不連續(xù)Robin系數(shù)的檢測尤其是不連續(xù)位置的檢測就具有特別重要的意義.對不連續(xù)的Robin系數(shù),本文基于Fourier變換,在L2空間的允許集內(nèi)建立了反問題的唯一性.我們的結(jié)果不同于連續(xù)函數(shù)空間的唯一性,是本文另一理論創(chuàng)新點.此外,本文將原問題轉(zhuǎn)化為求解帶全變差罰項的優(yōu)化問題,并對該正則化方案進行了理論分析.數(shù)值實現(xiàn)方面,提出了一種有效的雙循環(huán)迭代算法.該算法的基本思想是將非線性的數(shù)據(jù)匹配項和近似全變差罰項交替迭代,實現(xiàn)極小化.這種交替迭代的策略一方面可以降低原問題的非線性,無需考慮正則化參數(shù)的選取,另一方面可以提高計算速度.同時,這種根據(jù)非線性項的不同性質(zhì)交替迭代處理的思想,還可以推廣到包含多個非線性項的數(shù)值實現(xiàn)中,對其它相關(guān)問題的數(shù)值求解具有重要的借鑒意義.
[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.
【學(xué)位授予單位】：東南大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O241.82

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