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arXiv 2201.12054math.NAcs.NA
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第一类积分方程超定系统的正则化最小范数解

Regularized minimal-norm solution of an overdetermined system of first kind integral equations

Patricia Díaz de Alba, Luisa Fermo, Federica Pes, Giuseppe Rodriguez

发表信息：
Numerical Algorithms 92, 471-502 (2023)
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AI总结：

针对第一类积分方程超定系统离散化后的不适定问题，提出基于再生核希尔伯特空间与奇异函数展开的正则化最小范数求解方法，经测试验证其在光滑解场景下效果优异，非光滑解场景也能提供有效信息。

摘要：

第一类积分方程的超定系统在诸多应用中均有出现。当右端项被离散化时，由此产生的有限数据问题是不适定的，且存在无穷多个解。我们提出了一种数值方法，用于在存在边界约束的情况下计算最小范数解。该算法源于Riesz表示定理，在再生核希尔伯特空间中运行。由于由此产生的线性系统病态性极强，我们构建了一种依赖于离散参数的正则化方法。该方法基于最小范数解按定义问题的积分算子的奇异函数展开的形式。我们测试了两种用于自动确定正则化参数的估计技术，即偏差原理和L-curve方法。针对两个人工测试问题的数值结果表明，所提方法性能优异。最后，我们研究了一个地球物理应用中典型的特定模型，该模型可复现频域电磁感应装置的读数。结果表明，当待求解的解是光滑的时，新方法极其有效；即便对于非光滑解，该方法也能提供关于解的重要信息。

英文摘要：

Overdetermined systems of first kind integral equations appear in many applications. When the right-hand side is discretized, the resulting finite-data problem is ill-posed and admits infinitely many solutions. We propose a numerical method to compute the minimal-norm solution in the presence of boundary constraints. The algorithm stems from the Riesz representation theorem and operates in a reproducing kernel Hilbert space. Since the resulting linear system is strongly ill-conditioned, we construct a regularization method depending on a discrete parameter. It is based on the expansion of the minimal-norm solution in terms of the singular functions of the integral operator defining the problem. Two estimation techniques are tested for the automatic determination of the regularization parameter, namely, the discrepancy principle and the L-curve method. Numerical results concerning two artificial test problems demonstrate the excellent performance of the proposed method. Finally, a particular model typical of geophysical applications, which reproduces the readings of a frequency domain electromagnetic induction device, is investigated. The results show that the new method is extremely effective when the sought solution is smooth, but gives significant information on the solution even for non-smooth solutions.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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