论文标题
在二维奇异扰动的非线性抛物线PDE中，渐近膨胀正规化用于逆源问题
Asymptotic expansion regularization for inverse source problems in two-dimensional singularly perturbed nonlinear parabolic PDEs
论文作者
论文摘要
在本文中，我们开发了一种渐近扩张 - 调查方法（AER）方法，用于二维非线性和非组织奇异扰动的部分微分方程（PDES）中的逆源问题。这种方法的关键思想是使用渐近膨胀理论，这使我们能够确定溶液对具有锐利过渡层的给定PDE的存在和唯一性的条件。作为副产品，我们在可测量数量的源函数和一阶渐近近似之间得出了一个更简单的链路方程，并且基于该方程式，我们提出了一种有效的反转算法AER，对于反源问题。我们证明，这种简化不会降低反转结果的准确性，尤其是对于嘈杂数据的反问题。提供了各种数值示例，以证明我们新方法的效率。
In this paper, we develop an asymptotic expansion-regularization (AER) method for inverse source problems in two-dimensional nonlinear and nonstationary singularly perturbed partial differential equations (PDEs). The key idea of this approach is the use of the asymptotic-expansion theory, which allows us to determine the conditions for the existence and uniqueness of a solution to a given PDE with a sharp transition layer. As a by-product, we derive a simpler link equation between the source function and first-order asymptotic approximation of the measurable quantities, and based on that equation we propose an efficient inversion algorithm, AER, for inverse source problems. We prove that this simplification will not decrease the accuracy of the inversion result, especially for inverse problems with noisy data. Various numerical examples are provided to demonstrate the efficiency of our new approach.
