Introduction to Krylov subspace methods - IMAGe

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adient Method Methodat is p ?x k+1 = x k + 〈rk , r k 〉What is p ?〈Ar k , r k 〉 rkSteepest descent (SD)Gradient MethodAx k+1 = Ax k + 〈rk , r k 〉〈Ar k , r k 〉 Arkown as(AKA) ”Steepest Descent(SD)”b − Axp = p −∇g = −∇g = −(Ax = −(Ax k+1 = b − Ax− b) − b) = Ax b k − 〈rk , r k 〉2 Gradient Method= 〈Ar Ax rk , = r k 〉 rArkp = −∇g = −(Ax − b) = b − Ax = r.4), Take: taking p = r 0 p r,0 = , r 0 r k+1 = r k − 〈rk , r k 〉〈Ar,k , r k 〉 Arkg(xx k+1 x k+1 = x k + 〈rk , r k 〈r〉k+1 x, 0 r k 〉 = 〈r k , r k 〉 − 〈rk , r k 〉 1 , x 2 ) = Notice:= x + 〈rk , r k 〉x k+1 〈Ar k , r= k 〉x k rk+ 〈rk , r k 〉 x 0 〈Ar k , r k 〉〈Ark , r k 〉∴〈Ar k 〈r k+1 〈Ar , r k , k r k 〉, rk 〉 =r k 0〉 rk : orthogonalAx k+1 = Ax k + 〈rk , r k 〉〈Ar k , r k 〉 Ark∴ 〈r k+1 , r k ∴ 〉 = 〈r 0 k+1 : orthogona , k 〉 = 0own as(AKA) as(AKA) ”Steepest ”Steepest Descent(SD)” Descent(SD)” r k+1 = r k − 〈rAx k+1 = Ax k + 〈rk , r k 〉Ax k+1 = Ax〈Ar k + 〈rk , r k 〉〈Ar k , r k 〉 Ark k 〈Ar , r k k 〉, Arkr k x〉 Ark 0b − Ax k+1 = b − Ax k − 〈rk , r k 〉r k+1 = r k − 〈rk , r k 〉b − Ax k+1 = b − Ax k − 〈rk , r k 〉b − Ax k+1 〈Ar k , r= k 〉 Arkb − Ax〈Ar k − 〈rk g(x, 1 ,r k x〉2 ) = kk 〈Ar , r k k 〉, Arkr k 〉 Ark〈r k+1 , r k 〉 = 〈r k , r k 〉 − 〈rk , r k 〉〈Ar k , r k 〉〈Ark , r k 〉r k+1 = r k − 〈rk , r k 〉r k+1 = r〈Ar k − 〈rk , r k 〉k , r k k〉 Arkk Ark∴ 〈r k+1 , r k 〉 = 0 : orthogonal2D example (2x2 matrix!)so known as(AKA) ”Steepest Descent(SD)om (5.4), taking p = r 0 ,b − Ax= b − Ax〈Ar k , r〈A〈r k+1 , r k 〉 = 〈r k , r k 〉∴ 〈r k+1 , r k 〉 = 0 :p = −∇g = −(Ax − bg(x 1x 0 14x k+1 = x k + 〈rk , rg
Steepest descent (SD)• Homework:• Show:5 KRYLOV SUBSPACE METHODSShow that SD converges as :‖e n ‖ A ≤( κ(A) − 1κ(A) + 1) n‖e 0 ‖ A , where ‖v‖ 2 A = v T Av with A : SPDe.g. Orthogonal basis of R 2 : d 0 , d 1 (d 1 ⊥ d 0 )e 0 = α 0 d 0 + α 1 d 1 ↑α 1 d 115
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: SUBSPACE METHODSKRYLOV SUBSPACE MET
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

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