Introduction to Krylov subspace methods - IMAGe

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Parallelism ?• Inner products in parallel (reduction-scatter)•Matrix-vec<strong>to</strong>r products: very efficient waysof doing this for FEM and high-orderelements• Avoid default partitioning schemes in matrixlibraries38
Thank you!39
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 and 34: TakingV T m AV m y = H m r 0 = H m
Page 35 and 36: 5 KRYLOV SUBSPACE METHODSRemarks :F
Page 37: GMRES5 KRYLOV SUBSPACE METHODS 36r

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