二阶退化抛物型方程的系数反问题的理论和数值算法研究
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摘要
本文主要考虑具退化系数的二阶抛物型方程的系数反演问题,研究在适当的附加条件下解的唯一性,最优控制问题解的存在性、唯一性、稳定性,以及稳定的数值计算方法。此类问题在人口预测与控制,多孔介质流体力学,以及金融衍生产品定价等许多应用科学领域有重要意义。与普通的抛物型方程系数反问题不同,这里的方程在部分边界存在退化。方程的退化性一方面会导致在部分边界可能会缺少边界条件,另一方面会导致方程的解可能没有好的正则性。另外,由于反问题的不适定性,终端观测数据的微小扰动将会导致解的巨大变化。
第一章对本文的数学模型及其相关的研究背景作了简要介绍,
第二章中介绍了一些关于二阶线性偏微分方程的定理,引理和预备知识。
第三章讨论了一个利用终端观测值重构二阶退化抛物型方程的辐射系数的反问题。我们首先证明了原问题的解的唯一性,进而在最优控制理论框架下将原问题转化为一个优化问题,证明了最优解的存在性和所满足的必要条件。由于控制泛函非凸,一般来说没有唯一性。在假设T比较小的情况下,我们利用极小元所满足的必要条件结合正问题的一些先验估计结果,证明了极小元的唯一性和稳定性。
第四章主要从数值分析的角度讨论了一个利用区域内部给定的附加条件重构二阶退化抛物型方程的一阶项系数的反问题。该问题有明显的金融背景,在金融衍生品定价等研究领域具有重要意义。与前而类似,我们先证明了反问题的解的唯一性,这说明了问题的提法是正确的。接下来我们利用有限差分方法给出了正问题的一个隐式计算格式。对于反问题,我们采用预测-校验方法构造了一个迭代算法,并针对其中的数值微分问题提出了两种计算方法。最后,我们进行了数值实验,数值结果表明算法是稳定的,且收敛速度很快。
第五章讨论了一个二元泛函极小元的适定性问题,该问题来源于一个同时重构初值和辐射系数的逆热传导问题。与普通的一元控制问题不同,这里的控制泛函含有两个独立的未知函数和两个独立的正则化参数。我们先导出了控制泛函极小元所满足的必要条件,并采用分项估计的方法,在假设T比较小的前提下,证明了极小元的唯一性和稳定性。
In this paper, we mainly discuss some inverse coefficient problems for second-order degenerate parabolic equations. Under some appropriate additional condi-tions, we will study the uniqueness of the solution, the existence, uniqueness, sta-bility of the solution for the corresponding optimal control problem, and stable numerical computation methods for the solution of the inverse problem. Such kinds of problems have great significance in the fields of population prediction and control, porous media fluid mechanics, financial mathematics and other applied science. Be-ing different from ordinary inverse coefficient problems in parabolic equations, there exists degeneracy on a part of boundaries in the mathematical model. On one hand, the degeneracy may cause the corresponding boundary conditions missing; on the other hand, it can also make that the solution may has no good regularity. More-over, due to the ill-posedness for inverse problems, arbitrarily small changes in the final measurement data may lead to arbitrarily large changes in the solution.
In the first chapter, we introduce the mathematical models and their research background.
Some lemmas, theorems and preliminary knowledge regarding to the second-order linear partial differential equation are given in the second chapter.
In the third chapter, we discuss an inverse problem of identifying the radiation coefficient in a second-order degenerate parabolic equation using the final obser-vations. The uniqueness of the original problem is proved, and then the inverse problem is transformed into an optimization problem on the basis of optimal con-trol framework. The existence and necessary condition of the optimal solution are obtained. Since the control functional is non-convex, we can not guarantee the uniqueness of the optimal solution. Assuming that T is relatively small, we succeed in proving the uniqueness and stability of the minimizer by utilizing the necessary condition and some prior estimates of the direct problem.
In the fourth chapter, we mainly from the numerical analysis angle discuss an inverse problem of reconstructing the first-order term coefficient in a second-order degenerate parabolic equation using the extra condition imposed in the domain. Such a problem has obvious financial background, and is of great importance in the field of financial derivatives pricing. Similarly, we first prove the uniqueness of solution for the inverse problem, which illustrate the statement of the problem is correct. Then an implicit scheme on the basis of finite difference method is designed to obtain the numerical solution of the direct problem. For the inverse problem, we apply the predictor-corrector method to construct an iterative algorithm, and propose two numerical methods for the numerical differential problem arising in the iterative procedure. Finally, the numerical experiments are also done and the corresponding numerical results show that the algorithm is stable and it converges very quickly.
In the fifth chapter, we discuss the well-posedness of the minimizer of a bi-nary functional. This problem originates from an inverse problem of simultaneously reconstructing the initial value and radiation coefficient in a heat conduction equa-tion. Being different from other ordinary one-variable control problems, the cost functional constructed in the paper contains two independent unknown functions and two independent regularization parameters. We first derive the necessary con-dition which should be satisfied by the minimizer of the cost functional. Then we use the method of estimating by part to prove the uniqueness and stability of the minimizer on the condition of that the parameter T is assumed to be relatively small.
第一章对本文的数学模型及其相关的研究背景作了简要介绍,
第二章中介绍了一些关于二阶线性偏微分方程的定理,引理和预备知识。
第三章讨论了一个利用终端观测值重构二阶退化抛物型方程的辐射系数的反问题。我们首先证明了原问题的解的唯一性,进而在最优控制理论框架下将原问题转化为一个优化问题,证明了最优解的存在性和所满足的必要条件。由于控制泛函非凸,一般来说没有唯一性。在假设T比较小的情况下,我们利用极小元所满足的必要条件结合正问题的一些先验估计结果,证明了极小元的唯一性和稳定性。
第四章主要从数值分析的角度讨论了一个利用区域内部给定的附加条件重构二阶退化抛物型方程的一阶项系数的反问题。该问题有明显的金融背景,在金融衍生品定价等研究领域具有重要意义。与前而类似,我们先证明了反问题的解的唯一性,这说明了问题的提法是正确的。接下来我们利用有限差分方法给出了正问题的一个隐式计算格式。对于反问题,我们采用预测-校验方法构造了一个迭代算法,并针对其中的数值微分问题提出了两种计算方法。最后,我们进行了数值实验,数值结果表明算法是稳定的,且收敛速度很快。
第五章讨论了一个二元泛函极小元的适定性问题,该问题来源于一个同时重构初值和辐射系数的逆热传导问题。与普通的一元控制问题不同,这里的控制泛函含有两个独立的未知函数和两个独立的正则化参数。我们先导出了控制泛函极小元所满足的必要条件,并采用分项估计的方法,在假设T比较小的前提下,证明了极小元的唯一性和稳定性。
In this paper, we mainly discuss some inverse coefficient problems for second-order degenerate parabolic equations. Under some appropriate additional condi-tions, we will study the uniqueness of the solution, the existence, uniqueness, sta-bility of the solution for the corresponding optimal control problem, and stable numerical computation methods for the solution of the inverse problem. Such kinds of problems have great significance in the fields of population prediction and control, porous media fluid mechanics, financial mathematics and other applied science. Be-ing different from ordinary inverse coefficient problems in parabolic equations, there exists degeneracy on a part of boundaries in the mathematical model. On one hand, the degeneracy may cause the corresponding boundary conditions missing; on the other hand, it can also make that the solution may has no good regularity. More-over, due to the ill-posedness for inverse problems, arbitrarily small changes in the final measurement data may lead to arbitrarily large changes in the solution.
In the first chapter, we introduce the mathematical models and their research background.
Some lemmas, theorems and preliminary knowledge regarding to the second-order linear partial differential equation are given in the second chapter.
In the third chapter, we discuss an inverse problem of identifying the radiation coefficient in a second-order degenerate parabolic equation using the final obser-vations. The uniqueness of the original problem is proved, and then the inverse problem is transformed into an optimization problem on the basis of optimal con-trol framework. The existence and necessary condition of the optimal solution are obtained. Since the control functional is non-convex, we can not guarantee the uniqueness of the optimal solution. Assuming that T is relatively small, we succeed in proving the uniqueness and stability of the minimizer by utilizing the necessary condition and some prior estimates of the direct problem.
In the fourth chapter, we mainly from the numerical analysis angle discuss an inverse problem of reconstructing the first-order term coefficient in a second-order degenerate parabolic equation using the extra condition imposed in the domain. Such a problem has obvious financial background, and is of great importance in the field of financial derivatives pricing. Similarly, we first prove the uniqueness of solution for the inverse problem, which illustrate the statement of the problem is correct. Then an implicit scheme on the basis of finite difference method is designed to obtain the numerical solution of the direct problem. For the inverse problem, we apply the predictor-corrector method to construct an iterative algorithm, and propose two numerical methods for the numerical differential problem arising in the iterative procedure. Finally, the numerical experiments are also done and the corresponding numerical results show that the algorithm is stable and it converges very quickly.
In the fifth chapter, we discuss the well-posedness of the minimizer of a bi-nary functional. This problem originates from an inverse problem of simultaneously reconstructing the initial value and radiation coefficient in a heat conduction equa-tion. Being different from other ordinary one-variable control problems, the cost functional constructed in the paper contains two independent unknown functions and two independent regularization parameters. We first derive the necessary con-dition which should be satisfied by the minimizer of the cost functional. Then we use the method of estimating by part to prove the uniqueness and stability of the minimizer on the condition of that the parameter T is assumed to be relatively small.
引文
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