| 不适定问题高效算法研究 |
| Alternative Title | Efficient Numerical Methods for Inverse Problems
|
| 闫亮 |
| Thesis Advisor | 傅初黎
|
| 2011-06-01
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| Degree Grantor | 兰州大学
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| Place of Conferral | 兰州
|
| Degree Name | 博士
|
| Keyword | 无网格方法
贝叶斯推断
增广Tikhonov正则化
不确定量化
随机替代模型
压缩感知
谱随机方法
热源识别问题
抛物型Robin系数识别问题
|
| Abstract | 反问题在数学上往往是不适定的，对于数据很小的扰动将使解产生巨大的变化，因此利用数值求解非常困难。通常利用正则化算法可以得到稳定的数值解。
从算法上讲，处理不适定问题的正则化算法可以分为确定性方法以及随机方法。确定性方法理论相对完整，
随机方法着重讨论数据以及模型的不确定性对
问题的影响。本文试图针对抛物型方程热源识别问题以及Robin系数识别问题设计高效算法，特别对解的不确定性进行量化。全文分为三个部分，分别研究
求解不适定问题的确定性方法和随机方法以及处理随机偏微分方程的基于$l_1$优化的随机配点方法。
第一部分讨论基本解方法结合确定性正则化理论处理热源项分别为时间相关及空间相关的热源识别问题。基本解方法是一种真正的无网格方法，其基本思想是将
问题的解写成微分算子基本解线性组合形式，避免了对区域的离散。为了能够直接利用基本解方法，首先通过变换将原问题转化成齐次多边值问题，
通过该变换可以看出热源项仅为时间相关以及空间相关的热源问题的不适定程度与数值微分相当。由于所得到的线性方程组是严重病态的，本文采用离散Tikhonov
正则化方法并利用GCV策略选取正则化参数对病态方程组进行处理。
第二部分考虑贝叶斯推断方法在不适定问题中的应用。首先考虑在不同先验分布假设
下，贝叶斯方法和经典正则化方法的关系，以及贝叶斯方法在选取正则化参数上的灵活性。然后给出不同的抽样方法对后验状态空间进行求解，并讨论
抽样方法在贝叶斯方法解决不适定问题中的优缺点，以及可能采取的解决办法。接着，分析利用分层
贝叶斯模型得到的增广Tikhonov方法处理一般线性问题的框架。最后将所讨论的方法具体应用到Robin系数识别以及热源识别问题中。
第三部分提出结合压缩感知理论的随机配点方法。首先考虑贝叶斯随机替代模型与随机偏微分方程的关系。然后细致研究
基于$l_1$优化的随机配点方法，并给出该算法的收敛性结果。数值结果说明利用基于$l_1$优化的随机配点方法可以大大降低计算成本，
为设计快速贝叶斯方法提供了新思路。 |
| Other Abstract | Inverse problems are often ill-posed in the sense that the
solution may not exist and be unique, and more importantly, it fails
to depend continuously on the data such that a small perturbation in
the data may case an enormous deviation of the solution. However, in
practical applications, the data are always noisy and uncertain due
to corruption by inherent measurement errors. Meanwhile, the
forward model may be imperfect and imprecise due to the presence of
unmodeled physics. Therefore, the numerical solution of inverse
problems is very challenging. Regularization methods are the
standard approach for inverse problems. Algorithmically speaking,
existing techniques roughly divide into two categories:
deterministic and stochastic. There exist numerous mathematically
elegant theoretical results and computationally efficient numerical
algorithms for deterministic inverse techniques. However, they yield
only a point estimate of the solution, without quantifying the
associated uncertainties or rigorously considering the stochastic
nature of data noise. Stochastic approaches are necessary for
inverse problems under model uncertainties and for probabilistic
calibration. This thesis attempts to design efficient numerical
methods for the inverse heat source problems and the inverse Robin
problems associated with the parabolic problem. It consist of three
parts: Part 1 considers deterministic methods for the inverse heat
source problem; Part 2 discusses Bayesian inference approach for
inverse problems and Part 3 studies the stochastic collocation
method via $l_1$ minimization for stochastic partial differential
equations.
Part 1 discussed the use of the methods of fundamental solution
(MFS) for reconstructing the unknown heat source in parabolic
problems. The main idea of MFS is to approximate the unknown
solution by a linear combination of fundamental solutions whose
singularities are located outsider the solution domain. Since the
matrix arising from the MFS discretization is severely
ill-conditioned, a regularization solution is obtained by employing
the Tikhonov regularization, while the regularization parameter is
determined by the GCV criterion. For numerical verification, several
examples for solving inverse heat source problems with smooth and
non-smooth geometries in two- and three dimensional space are given.
Part 2 studies the Bayesian inference approach for uncertainly
quantification of inverse Robin problems associated with the
parabolic equ... |
| URL | 查看原文
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| Language | 中文
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| Document Type | 学位论文
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| Identifier | https://ir.lzu.edu.cn/handle/262010/225598
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| Collection | 数学与统计学院
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Recommended Citation
GB/T 7714 |
闫亮. 不适定问题高效算法研究[D]. 兰州. 兰州大学,2011.
|
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