不适定问题高效算法研究
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摘要
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ι1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止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ι1,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ι1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。
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 de-viation of the solution. However, in practical applications, the data are always noisy and uncertain due to corruption by inherent measurement errors. Mean-while, 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 l1 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 fun-damental 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 nu- merical 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 equation. With assump-tion on the prior belief about the form of the parameter and an assignment of normal error in sensor measurements, we derive the solution to the statistical in-verse problem analytically. A hierarchical Bayesian model is adopted for selecting the regularization parameter. The posterior probability density depends implicitly on the parameter through the forward model, and is exploring using the Markov chain Monte Carlo for obtaining relevant statistics. Then an augmented Tikhonov regularization(a-TR) method that could determine the regularization parameter and noise level is using to reconstruct the unknown heat source. Numerical results for several benchmark test problems are indicate that the Bayesian inference ap-proach and a-TR method are accurate and flexible methods for inverse problems.
Part 3 consider a novel approach for quantifying the parametric uncertainty associated with a stochastic problem output. The new approach differs from the standard stochastic collocation methods in that it is based on ideas directly linked to the recently developed compressed sensing theory. We provide theoretical anal-ysis on the validity of the approach. Numerical tests are provided to examine the performance of the method and validate the theoretical finding. This opens an avenue for constructing stochastic surrogate models to accelerate the Bayesian inference approach for parameter estimation problems associated with partial dif-ferential equations.
第一部分讨论基本解方法结合确定性止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ι1,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ι1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。
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 de-viation of the solution. However, in practical applications, the data are always noisy and uncertain due to corruption by inherent measurement errors. Mean-while, 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 l1 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 fun-damental 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 nu- merical 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 equation. With assump-tion on the prior belief about the form of the parameter and an assignment of normal error in sensor measurements, we derive the solution to the statistical in-verse problem analytically. A hierarchical Bayesian model is adopted for selecting the regularization parameter. The posterior probability density depends implicitly on the parameter through the forward model, and is exploring using the Markov chain Monte Carlo for obtaining relevant statistics. Then an augmented Tikhonov regularization(a-TR) method that could determine the regularization parameter and noise level is using to reconstruct the unknown heat source. Numerical results for several benchmark test problems are indicate that the Bayesian inference ap-proach and a-TR method are accurate and flexible methods for inverse problems.
Part 3 consider a novel approach for quantifying the parametric uncertainty associated with a stochastic problem output. The new approach differs from the standard stochastic collocation methods in that it is based on ideas directly linked to the recently developed compressed sensing theory. We provide theoretical anal-ysis on the validity of the approach. Numerical tests are provided to examine the performance of the method and validate the theoretical finding. This opens an avenue for constructing stochastic surrogate models to accelerate the Bayesian inference approach for parameter estimation problems associated with partial dif-ferential equations.
引文
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