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13 Cards in this Set

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MPP Banded Sparse GE FLOPs

1D: 5N ~ 5h^-1


2D: O(N^4) ~ O(h^-4)


3D: O(N^7) ~ O(H^-7)

MPP Banded Sparse GE Storage

1D: 4N ~ 4h^-1


2D: 2N^3 ~ 2h^-3


3D: 2N^5 ~ 2H^-5

MPP residual computation FLOPs

1D: 6N ~ 6h^-1


2D: 10N^2 ~ 10h^-2


3D: 14N^3 ~ 14H^-3

MPP residual computation Storage

1D: 2N ~ 2h^-1


2D: 2N^2 ~ 2h^-2


3D: 2N^3 ~ 2H^-3

GE FLOPs

(2/3)N^3

Back solve FLOPs

N^2

Multiplying Ax

2N^2 - N

Solve Tridiag Ax = b

GE = 5N - 5


BS = 3N-2

J_0_(x)

(1/2)xTAx - xTb

FOR with MPP

rho(u^(n+1) - u^(n)) = f - Au^n


uij^(n+1) = (1/rho)(fij +ui+1j^n + ui-1j^n + uij+1^n + uij-1^n)

Steepest Descent Algorithm

A: SPD; guess x0, r0 = b-Ax0


For n = 1:Itmax


dn = rn


alphn = (dnTrn/dnTAdn)


xn+1 = xn + alphndn


rn+1 = {b - Axn+1, or rn - alphnAdn}


Test for convergence

Conjugate Gradient Method Algorithm

Given x0, r0 = b - Ax0, d0 = r0


alphn = (dnTrn/dnTAdn)


xn+1 = xn + alphndn


rn+1 = {b - Axn+1, or rn - alphnAdn}


Test for Convergence


In divergent


Bn+1 = rnTrn/rn+1Trn+1


dn+1 = rn+1 + Bn+1dn

<x,d>A

xTAd = bTd for symmetric matrices