半導體元件模擬中各種預調共軛梯度法向量計算之性能

Abstract

四種預調共軛梯度法：預調共軛梯度(PCG)、修正預調共軛梯度(MPCG)、 共軛梯度平方(CGS)及雙共軛梯度穩定(BICGST)，被用來求解線性化的半 導體元件方程式系統。係數矩陣被分解成三種不完全喬可利斯基型式：標 準型、無根型及劃一型，並且當做預調矩陣。在矩陣倒反的向量運算上， 吳人採用截斷紐曼展開式。原始及各種截斷矩陣倒反的計算性能係用康衛 氏3840電腦上的向量及純量最佳化來檢查。 在康衛氏3840電腦上，向量最佳化比純量最佳化可得到3倍的提升速率。 標準型及無根型分解，仍保持原矩陣的對稱性並可用在所有共軛梯度法上 。劃一型分解則成非對稱性僅可用在共軛梯度平方及雙共軛梯度穩定法上 。無根型及劃一型分解可用在截斷矩陣倒反上。使用無根型分解的預調共 軛梯度法可得到最省時的運算,使用無根型分解的修正預調共軛法則得到 最平坦的收斂行為。使用劃一型分解的雙共軛梯度穩定法在計算花費及收 斂行為上都比共軛梯度平方法好。在康衛氏3840電腦上執行向量最佳化運 算, 原始方法比截斷技巧有利。 Four preconditioned CG-type methods, PCG, MPCG, CGS and BICGST, have beenemployed to solve the linearized system of semiconductor device equations. The coefficient matrix can be factorized into three incomplete Choleski forms : the standard (L1+D1)1/D1(D1+U1), the root-free (L2+I)D2(I+U2) andthe scaled (L3+I)(I+U3), which can be used as the preconditioning matrix. A truncated Neumann expansion is adopted for vector computation in the matrix inversion. The computation performance of the original and various truncated inversions has been examined using the vector and scalar optimization of a CONVEX 3840 computer. The vector optimization gives about a speedup factor of 3 over the scalar optimization on the CONVEX 3840 computer. The original and the root-free factorizations remain symmetric as the original matrix and can be used in all CG methods. The scaled factorization is nonsymmetric and can be used only for CGS and BICGST methods. The root-free and the scaled factorizations can be used in the truncated inversion. The PCG method using the root-free factorization gives the best performance in terms of the CPU time. The MPCG method using the root-free factorization gives smoother convergence behavior.The BICGST method using the scaled factorization is better than CGS in the computational cost and convergence behavior. The original method compares favourably with the truncated implementation, when the computations were performed in vector optimization on the CONVEX 3840 computer.