论文标题
为椭圆方程的库奇问题构建差异的准可逆性方法
Constructing a variational quasi-reversibility method for a Cauchy problem for elliptic equations
论文作者
论文摘要
在针对逆问题和不良问题的正则化理论的最新发展中，已设计了一种变异的准可逆性（QR）方法来解决一类时期的准线性抛物线寄生虫问题。该方法被称为基于PDE的方法，依靠将合适的扰动操作员添加到原始问题中，因此依靠获得相应的稳定运算符，这使我们遇到了类似前向的问题。在这项工作中，我们为此类操作员建立了新的条件估计，以解决椭圆方程的原型库奇问题。此问题基于反热传导问题的固定情况，在给定部分边界数据的情况下，人们希望在某个介质中识别热量分布。使用新的QR方法，我们获得了波型方程的二阶初始值问题，该方程可以使用先验估计和紧凑性参数来推导其弱的可溶性。当我们研究所提出的方案的Hölder收敛速率时，通过类似于Carleman的函数加权，在变化环境中探索了一种新型的能量估计。此外，分析了该方案的线性化版本。提供数值示例以证实我们的理论分析。
In the recent developments of regularization theory for inverse and ill-posed problems, a variational quasi-reversibility (QR) method has been designed to solve a class of time-reversed quasi-linear parabolic problems. Known as a PDE-based approach, this method relies on adding a suitable perturbing operator to the original problem and consequently, on gaining the corresponding fine stabilized operator, which leads us to a forward-like problem. In this work, we establish new conditional estimates for such operators to solve a prototypical Cauchy problem for elliptic equations. This problem is based on the stationary case of the inverse heat conduction problem, where one wants to identify the heat distribution in a certain medium, given the partial boundary data. Using the new QR method, we obtain a second-order initial value problem for a wave-type equation, whose weak solvability can be deduced using a priori estimates and compactness arguments. Weighted by a Carleman-like function, a new type of energy estimates is explored in a variational setting when we investigate the Hölder convergence rate of the proposed scheme. Besides, a linearized version of this scheme is analyzed. Numerical examples are provided to corroborate our theoretical analysis.
