一般而言，大部分工程問題的解析上，可依其輸入(Input source)、系統模式(System module)和輸出(Output response)三者之間的關係分成兩大類。第一類的問題是探討不同輸入值對於已知之系統模式所造成的輸出變化為何。第二類的問題則是藉由已知的系統模式和輸出來推算出其輸入值；或是藉由已知的輸入及輸出，來探求其系統模式。我們統稱第一類的問題為正算問題(Direct problem)，第二類的問題為反算問題或稱逆向問題(Inverse problem)。在本文第一章中，主要是利用反算問題中的共軛梯度法(Conjugate Gradient Method)來進行反算分析工作。問題的物理模型主要是在探討藉由蓄水層中某一位置的濃度分佈之量測，來預測地下蓄水層邊界上面的污染源強度。在這個質傳問題的反算中，我們可以利用數值實驗來模擬實際的濃度分佈且在考慮量測誤差的情況下，檢驗反算分析的正確性。結果顯示在初始猜值為未知任意猜值的情況下，我們可以成功的運用共軛梯度法進行反算分析得到為時間及位置函數的污染源強度之預測值。在第二章中，我們主要是探討最佳化控制理論之共軛梯度法應用於半導體摻雜擴散製程時，於要求半導體所需的無差排濃度下，成功地預測出最適當的摻雜擴散濃度。在這個最佳化控制的問題中，我們將進行數值實驗來加以驗證此反算分析的可靠度。本文與之前的研究不同的是能在相同的物理模型下利用共軛梯度法反算出更正確的結果，或是利用最佳化控制理論以單一摻雜濃度控制函數預測出良好的無差排濃度。由以上兩章的結果顯示，共軛梯度法應用在質傳問題中的污染源強度反算預測及摻雜濃度最佳化控制中，均可以得到很好的研究成果。

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