Introduction to Krylov subspace methods - IMAGe

More documents

Recommendations

Info

‖λ 0 ‖ 2We have from Arnoldi’s algo :βFOMV T m AV m y = H m r 0 = H m (βv 1 ) = βẽ 1⇒ x m = x 0 + V m (Mm −1 )βẽ 1Combing MGS/Arnoldi with the above leads to FOM :r 0 = b − Ax 0 , β = ‖r 0 ‖ 2 , v 1 = r 0 /βH m = {h ij } m i,j=1 : Set H m = 0for j = 1, 2, · · · , m doendw j = Av jfor i = 1, 2, · · · , j doendh ij = (w j , v i )w j = w j − h ij v ih j+1,j = ‖w j ‖ 2 , if h j+1,j = 0 set m = j exitv j+1 = w j /h j+1,jy m = H −1m (βẽ 1 ), x m = x 0 + V m y m 34
5 KRYLOV SUBSPACE METHODSRemarks :FOM• When A is symmetric, Arnoldi’s algorithm1) When A is symmetric, Arnoldi’s algorreduces to the Lanczos methods.•••H m becomes a tridiagonal matrixbecomes triadiagonal !Arnoldi becomes Lanczos algorithm...2) Noticing orthogonality of the residualsCG is an efficient implementation of FOMleads to CG as we desired it! (i.e. effiCG=FOM+Lanczos35
Page 1 and 2: Introduction to Krylovsubspace meth
Page 3 and 4: Why iterative methods?• Most disc
Page 5 and 6: et r n = f − Au n ,Let r n = f
Page 7 and 8: Gauss-Seidel A = L + D + U ; M = L
Page 11 and 12: For a SPD matrix A, considerg(x) :=
Page 13 and 14: SUBSPACE METHODSKRYLOV SUBSPACE MET
Page 15 and 16: Steepest descent (SD)• Homework:
Page 17 and 18: 〈Ap, p〉r k 〉 = 〈r k , r k
Page 19 and 20: ∴α i = − 〈di , e i 〉 A〈d
Page 21 and 22: 〈r , p〉 = 〈r , p〉 − α
Page 23 and 24: iFrom:Suppose u i = r i (cheap!)p i
Page 25 and 26: e A-conjugate GS,ThenConjugatee i =
Page 27 and 28: α i−1 〈Ap i−1 , p i−1 〉
Page 29 and 30: (√w understand that CG/SDesc./Ric
Page 31 and 32: Projection methods5 KRYLOV SUBSPACE
Page 33: TakingV T m AV m y = H m r 0 = H m
Page 37 and 38: GMRES5 KRYLOV SUBSPACE METHODS 36r
Page 39: Thank you!39

Introduction to Krylov subspace methods - IMAGe

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?