求解大型二次連續型最佳化控制問題之數值方法

Abstract

以有限元素法模擬的大型震動結構通常可以被離散化成二階動態系統的矩陣方程式。這些動態系統的解可以用公式表示成大型稀疏漢米爾頓矩陣特徵問題的解。傳統上採用的方法-舒而分解-所面臨的問題就是必須將整個矩陣存入電腦中，這種方法在電腦的記憶體不足時是無法運作的。而子空間逼近法通常會運用到矩陣的異動值和反矩陣來增加收斂速度，後者的運算量是相當貴的。並且實際上的困難是如何決定出這些異動值。 在這篇論文裡我們提議運用同倫分列式的方法，配合分割與合成的策略來計算異動值。並藉由主要奇子空間疊代法來逼近連續時間李卡提等式的解。論文中還列出了詳細的演算法和實驗的數據和經驗。最後我們針對實驗結果提出可能潛在的問題並討論之。

Large vibrating structures modeled by finite element methods are usually discretrized into second order dynamical systems of matrix equations. Solutions of such systems can be formulated as solving large sparse Hamiltonian eigenproblem. QR-type or Schur method suffer from fill-in's and are not practical for large problem. Subspace approaches usually require shift and invert to accelerate the convergence rate. It is, however, difficult in practice to determine the shift values. In this thesis we propose to use the divide-and-conquer homotopy-determinant algorithm for computing shift values, and dominant singular subspace updating method for approximating the solution of the underlined continuous-time Riccati equation. The detailed algorithms is presented, experiment results and observed difficulties is discussed.