A Study on the Regularization Methods of Inverse Problems

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Project title

Code/計畫編號

NSC89-2211-E019-023

Translated Name/計畫中文名

反算問題之正規化研究

Project Coordinator/計畫主持人

Shyh-Rong Kuo

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Harbor and River Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=581412

Year

2000

Start date/計畫起

01-08-2000

Expected Completion/計畫迄

01-07-2001

Bugetid/研究經費

305千元

ResearchField/研究領域

土木水利工程

本計畫主要是處理過定邊界條件(Overspecified boundary condition)的反算問題，已知的資訊包含：分析領域的範圍、控制方程式及材料性質，另外有一部份邊界已知過定的邊界條件，而另外一部份的邊界條件則完全未知。在這種反算問題中，已知的邊界條件往往包含著人為或儀器的量測誤差，一旦系統太過敏感，則些微的量測誤差，會使求得的解和真實相距甚遠。當遭遇此等劣性問題時，一般文獻上會尋求一相近系統求得原系統的近似解，此方法即為正規化。一般文獻上常見的Tikhonov 正規化方法，有單位不一致的問題，會因單位改變而影響正規化效果。為了避免上述缺點，本計畫擬以能量最小值的觀念，提出一改良的無因次正規化方法。在多自由度的反算問題中，若已知邊界自由度總數小於未知邊界自由度總數時，本計畫擬將一輔助系統加於原結構上，並由應變能最小值的觀點，提出一個輔助系統正規方法。另外當已知邊界自由度總數大於未知邊界自由度總數時，為避免奇異現象，本計畫擬將一能量型式的模量(Norm)加於原結構的應變能，利用能量泛涵最小值的觀念提出另一種改良的正規化方法，此二種正規化參數皆是無因次，可克服單位不一致的問題。 This research studies the inverse problems with overspecified boundary condition. Tikhonov Regularization Method is often used to deal with this kind of ill-posed problems. But the results obtain from the regularization method would be influenced by different measureing units. To improve this fault, we propose a new method called Stiffness Matrix Method (S.M.M.). It takes use of the minimum of energy functional and its regularization parameter is nondimensional. S.M.M. can be expected to be used in different units. From the results of SDOF examples, we know that the Tikhonov Regularization Method is not suitable for the problem whose major frequence(or wave number) range concentrates at low frequency (or wave number). Oppositely, when the major frequence (or wave number) range is wide and the Gaussian window is not suitable. It is expected that the S.M.M. should be applicated for any case. In MDOF inverse problems, except adopting S.M.M., we propose another regularization method, Assistant System Method (A.S.M.). To avoid possible singular phenomenon, we adopt S.M.M. when the number of degree of freedom on known boundary is great than on unknown boundary. Conversely, when the numbers of degree of freedom on known boundary is less than on unknown boundary, we suggest to use A.S.M.

Keyword(s)

反算問題
正規化
邊界條件
無因次分析
Inverse problem
Regularization
Boundary condition
Dimensionless analysis

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A Study on the Regularization Methods of Inverse Problems

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