| 几类时间分数阶扩散波方程反问题的唯一性及算法研究 | |
| Alternative Title | Some studies on the uniqueness and algorithms of inverse problems for the time-fractional diffusion-wave equations |
| 张云 | |
| Subtype | 博士 |
| Thesis Advisor | 魏婷 |
| 2021-05-22 | |
| Degree Grantor | 兰州大学 |
| Place of Conferral | 兰州 |
| Degree Name | 理学博士 |
| Degree Discipline | 计算数学 |
| Keyword | 时间分数阶扩散波方程 反初值问题 反源项问题 分数阶阶数 零阶项系数 对流项系数 唯一性 Tikhonov正则化 迭代Tikhonov正则化方法 最速下降法 Nesterov策略 迭代正则化集合Kalman方法 交替方向乘子法 收敛性 |
| Abstract | 本文主要考虑时间分数阶扩散波方程(TFDWE)中的以下几类反问题：同时反演初值的问题，同时反演阶数与零阶项系数的问题，同时反演阶数与对流项系数的问题，最后给出了一种求解线性反问题的预处理交替方向乘子法(ADMM)，并成功应用到了源项的反演问题中。 在第一部分中，我们研究了TFDWE中利用Cauchy数据同时反演两个初值的反问题。通过Laplace变换和解析延拓的方法，我们证明了该反问题的唯一性，然后利用非稳态迭代Tikhonov正则化方法对反问题进行了求解，通过有限维逼近的方法得到了对初值的近似解，最后通过一维和二维情形下的数值算例来说明所给出的算法是有效的。 在第二部分中，我们从唯一性和数值方法两个方面，研究了一类TFDWE中利用边界测量数据来同时反演阶数和零阶项系数的问题。在证明了反问题的唯一性之后，我们又给出了正演算子的局部Lipschitz连续性的结果。然后利用Tikhonov正则化方法将该反问题转化成了一个变分问题，根据正演算子的连续性，我们证明了在先验选取正则化参数的前提下，Tikhonov正则化泛函的极小元是存在并且收敛到精确解的，最后使用了最速下降法和Nestrov加速策略来对变分问题进行求解，通过三个数值算例来说明所给算法的有效性和合理性。 在第三部分中，考虑了一维的带有对流项的TFDWE中，同时反演阶数和对流项系数的反问题。通过对原方程作变换，可以将对流项系数转移到零阶项系数上，消去原方程的对流项，然后我们可以证明阶数和新的零阶项系数是可以由两个边界点的观测数据唯一确定的，再利用一阶常微分方程的基本理论可以得到对流项系数的唯一性，最后我们在Bayes的观点下对未知的阶数和对流项系数进行了数值重构，利用迭代正则化集合Kalman方法进行了数值求解，进而给出了几个数值算例来说明求解方法的有效性。 在第四部分中，我们提出来一种求解线性反问题的残差方法，为了能够将该问题在数值上进行实现，我们又给出了一种预处理的交替方向乘子法(ADMM)进行求解。由于观测数据总是会不可避免的受到噪声的扰动，通过引入一个新的变量来代替扰动数据，然后再对这个新的变量作额外的约束，从而可以将要求解的反问题完全转换成一个带约束的优化问题，不需要假设拉格朗日乘子的存在，我们就可以给出算法的收敛性分析，最后我们将这种算法应用到TFDWE的反源项问题中，通过几个数值算例，可以看出我们所提出的算法的有效性。 |
| Other Abstract | In this thesis, we consider several inverse problems in time fractional diffusion-wave equations (TFDWE), i.e. inverse initial value problem, fractional order and zeroth-order coefficient inverse problem, fractional order and advection coefficient inverse problem, and a preconditioned alternating direction method of multipliers is proposed with application to the inverse source problem. In Part 1, we study the problem of recovering two initial values for a time-fractional diffusion-wave equation from boundary Cauchy data. We provide the uniqueness result for recovering two initial values simultaneously by the method of Laplace transformation and analytic continuation. And then we use a non-stationary iterative Tikhonov regularization method to solve the inverse problem and propose a finite dimensional approximation algorithm to find a good approximation to the initial values. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method. In Part 2, we investigate an inverse problem of recovering the zeroth-order coefficient and fractional order simultaneously in a time-fractional reaction-diffusion-wave equation by using boundary measurement data from both of uniqueness and numerical method. We prove the uniqueness of the considered inverse problem and the locally Lipschitz continuity of the forward operator. Then the inverse problem is formulated into a variational problem by the Tikhonov type regularization. Based on the continuity of the forward operator, we prove that the minimizer of the Tikhonov type functional exists and converges to the exact solution under an a priori choice rule of regularization parameter. The steepest descent method combined with Nesterov acceleration is adopted to solve the variational problem. Three numerical examples are presented to support the efficiency and rationality of our proposed method. In Part 3, the work is concerned with inverse problem of recovering the space-dependent advection coefficient and the fractional order in a one-dimensional time-fractional reaction-advection-diffusion-wave equation. Based on a transformation, the original equation can be changed into a new form without an advection term. Then we can show the uniqueness of recovering the fractional order and the zeroth order coefficient which contains the information of the "original"advection coefficient by the observation data at two end points. Under the theory of first-order ordinary differential equations, we obtain the uniqueness result of the advection coefficient. Lastly, we solve the inverse problem numerically from Bayes perspective by using iterative regularizing ensemble Kalman method, and numerical examples are presented to show the effectiveness of the proposed method. In Part 4, we propose a residual method for solving linear inverse problems. To implement this method, a preconditioned alternating direction method of multipliers (ADMM) is given. Since the observed data is always noise-contaminated, we introduce a new variable to replace the noisy data and then impose additional constraint on the new variable. Thus the inverse problem is totally converted into an optimization problem. Without taking account of the existence of the Lagrange multiplier, we provide the convergence result of the proposed method. Finally we apply this method to the inverse source problem in TFDWE. Numerical examples are presented to show the efficiency of the proposed method. |
| Pages | 164 |
| URL | 查看原文 |
| Language | 中文 |
| Document Type | 学位论文 |
| Identifier | https://ir.lzu.edu.cn/handle/262010/459373 |
| Collection | 数学与统计学院 |
| Affiliation | 数学与统计学院 |
| First Author Affilication | School of Mathematics and Statistics |
| Recommended Citation GB/T 7714 | 张云. 几类时间分数阶扩散波方程反问题的唯一性及算法研究[D]. 兰州. 兰州大学,2021. |
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