Introduction to Krylov subspace methods - IMAGe

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g(x) = x Ax − b x, x = py ]e D = p T Ap = diag[λ 1 λ 2 · · · λ ml column vectors andg(x) = x T Ax − b T x,ConjugateWe now,have :gradientλ ix = py(y) = y T p T Apy − b T py = y T Dy − ˜by• Start with one step of SD•,λ i∴ λ i y 2 i = ˜b i y i ⇒ y i = ˜b iof A.an A-conjugate basis cheaply?x k = x (k−1) + α (k−1) pr k = r (k−1) − α (k−1) Apα (k−1) : = 〈r(k−1) , p〉〈Ap, p〉with D : diagonalg(y) = y T p T Apy − b T py = y T Dy −How to ˜by find/generate an A-conjugate basis cheaply?es of A.∴ λ i y 2 i = ˜b i y i ⇒ y i = ˜b iate an A-conjugate basis cheaply?α (k−1) : = 〈r(k−1) , p〉〈Ap, p〉xFind a recurrence to compute A-conjugatek = x (k−1) + α (k−1) p1) Pick first vector of the basis as br basis...0 ≡ p 0 ≡ r 0 (first α is nonk = r (k−1) − α (k−1) Ap•2) Choose and recurrence to obtain the next A-conjugate ”p”α (k−1) : = How 〈r(k−1) , p〉short will the recurrence be?〈Ap, p〉(How short will the recurrence be?)of the basis as b 0 ≡ p 0 ≡ r 0 (first For all α isp,nonzero.) (→ SD)rrence to obtain the next A-conjugate ”p”the recurrence be?)〈r k , p〉 = 〈r k−1 , p〉 − α k−1 〈Ap, Thep〉new residual is orthogonal to the search direction.〈r k , p〉 = 0where λ i are eigenvalues of A.x k = x (k−1) + α (k−1) pr k = r (k−1) − α (k−1) ApNotice:〈r k , p〉 = 〈r k−1 , p〉 − α k−1 〈Ap, p〉〈r k , p〉 = 0the basis as b 0 ≡kp 0 ≡ rConjugate 0 (first α isGram-Schmidt nonzero.) (→ SD) Procedure 20∴r k ⊥ p
〈r , p〉 = 〈r , p〉 − α 〈Ap, p〉〈r k , p〉 = 0Conjugate gradient• Conjugate Gram-Schmidt:∴r k ⊥ pThe new residual is orthogonal to the search direction.Conjugate Gram-Schmidt ProcedureSuppose ”m” linearly independent vectors u 0 , u 1 , u 2 , · · · , u m−1 .To construct d i , take u i and subtract out any component that are not A-orthogonalto the previous ”d” vectors.5 KRYLOV SUBSPACE METHODS(1) d 0 = u 0(2) for i > 0,endd i = u i +∑i−1k=0β ik d kUsing A-orthogonality〈d i , Ad j 〉 = 〈u i +∑i−1k=0= u T i Ad j +β ik d k , Ad j 〉∑i−1k=0β ik (d k ) T Ad j= u T i Ad j + β ij (d j ) T Ad j = 0What is u? We need to keep all d’s∴β ij = − uT i Adj(d j ) T Ad jThis is not cheap, since we need to keep the vectors ”d” to21
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: ∴α i = − 〈di , e i 〉 A〈d
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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