以再生核局部化徑向基函數配置法求解反算問題

Reproducing Kernel Enhanced Local Radial Basis Collocation Method for Solving Inverse Problems

陳原嘉(Yuan-Chia Chen)

指導教授 ： 楊子儀

國立交通大學/工學院/土木工程系所/碩士(2019年)

摘要

反算問題具有不完整之邊界條件，一直以來如何有效求解為計算力學研究領域中待克服之難題。雖然以徑向基函數配置法做數值近似具有指數收斂率，然其所建立之離散系統為滿矩陣，往往導致病態矩陣系統之問題；反觀再生核函數配置法，雖僅具有代數收斂率，然其系統矩陣相對穩定。有鑑於此，本研究使用局部化徑向基函數配置法求解反算問題，透過局部化後之徑向基函數，解決病態矩陣系統之問題。其中，本論文研究透過多個不同類型之反算問題驗證局部化徑向基函數配置法對於反算問題數值求解之精度與運算效率。

關鍵字

反算問題 ； 局部化徑向基函數 ； 強形式配置法 ； 多重連通域

並列摘要

As inverse problems have been known for the incomplete boundary conditions, how to solve it effectively remains a challenging task in the field of computational mechanics. Although the radial basis collocation method has exponential convergence rate, the resulting discrete systems are full matrices and thus have ill-conditioned systems. In contrast, the reproducing kernel collocation method has algebraic convergence rate, but the resulting systems are more stable compared to the ones obtained by the global approximation. As such, this work introduces the localized radial basis collocation method to solve inverse problems in order to get rid of ill-conditioned systems. In particular, different types of inverse problems are provided to demonstrate the accuracy of approximation and efficiency of calculation by using the localized radial basis collocation method.

並列關鍵字

inverse problem ； localized radial basis function ； strong form collocation method ； multiply connected domain

參考文獻

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Chakib, A. and Nachaoui, A. [2006] “Convergence analysis for finite element approximation to an inverse Cauchy problem,” Inverse Probl. 22, 1191-1206.

Chen, J.S., Hu, H. Y. and Hu, W. [2008] “Reproducing kernel enhanced local radial basis collocation method,” Int. J. Numer. Meth. Engng 2008; 75:600-627

Chi, S. W., Chen, J. S., Hu, H. Y. and Yang, J. P. [2013] “A Gradient Reproducing Kernel Collocation Method for Boundary Value Problems,” International Journal for Numerical Methods in Engineering 93, 1381-1402