Introduction to Krylov subspace methods - IMAGe

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Overview• Basic iterative methods:• Matrix splitting• Minimization• Gradient and conjugate gradient• Projection methods 2
Why iterative methods?• Most discretizations of PDE’s lead to largelinear systems• LU or factorizations require too muchmemory: fill in is a problem...• Why:• Preserve the structure of the matrix• Based only on matrix-vector products 3
Page 1: Introduction to Krylovsubspace meth
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 and 38: GMRES5 KRYLOV SUBSPACE METHODS 36r
Page 39: Thank you!39

Introduction to Krylov subspace methods - IMAGe

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