Introduction to Krylov subspace methods - IMAGe

on-stationary approaches :son,hes :u n+1 =ρ = κ(M u n + −1 τ n A) M −1 −(f 1Iterative methodsκ(M −1 A) + 1 .− Au n )κ(M −1 A) + 1 .(Multi-step)k∑Non-stationary approaches u n+1 : = u n +u n+1 = u n i=0k∑k∑ + τ n M −1 (f − Au n )Let’s callu n+1 à =uM = n+1 −1 Au n and+ = u ˜f = n M+−1 f.n∑(Multi-step)i=0 u i=0k∑n+1 = u n + (k∑˜f − Ãun )i=0τ n+1 M −1 (f − Au n )u n+1 = u n u+n+1 = u n + ˜r ni=0i=0Let’s call à = M −1 Auand n+1 ˜f = M˜ru n n+1 −1 + f.= ˜r( ˜f n − ØrnÃun )˜r n+1 = (I −u n+1 u n+1 = u= n u+ n + ˜r nÃ)˜rn( ˜f Ãun )u ˜r n+1 = (Iu n − Ã)n+1˜r + n0u n+1 ∴= u˜r n = + (I ˜r n − Ã)n˜r 0˜r n+1 = ˜r n − Ørn ˜r n+1 ˜r n = P˜r n (Ã)˜r0 − Ørn , P n (0) = I˜r n+1 = (I − Ã)˜rnρ = κ(M −1 A) − 1(Multi-step)u n+1 = u n + τ n M −1 (f − Au n )ulti-step)Non-stationary approach:τMulti-steps: u n+1 = u n n+1 M −1 (f − Au n )+ τ ni M −1 (f − Au n )n+1 = u n + τ n M −1 (f − Au n )1 = u n +et’s call à = M −1 A and ˜f = M −1 f.u n+1 = u n + τ n M −1 (f − ALet’s call à = M −1 A and ˜f = M −1 f.nd ˜f = M −1 f.u n+1 = u n + ( ˜f − Ãun )u n+1 = u n + ˜r nτ n+1 M −1 (f − Au n )˜r n+1 = ˜r n − A˜r nτ n+1 M −1 (f − Au n )τ n+1 M −1 (f −u n+1 = u n + ( ˜f − Ãun )˜r n+1 = ˜r n − Ørnwhere P n : monic polynomial of degree n. 8