Introduction to Krylov subspace methods - IMAGe

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v w.r.t. A is the endConsequence lowest degree monic polynomial ⎡e grade always equal or smaller than m.R m×m of a vector v, v ∈ m w.r.t. A is the lowest degree hmonic polynomial)v = 0. For ADef. ∈ RThe m×m , we grade , vhave ∈of Ra m that vector , we have the v w.r.t. grade thatA the 11 hisof the grade v 12 hlowest is 21 · · · hh 22of degree v is monic polysmaller p(≠ Consequence1) than dim(K m.Projection0) s.t. m ]1j· ·(A; p(A)v v)) = 0. = For min(m, A ∈ Rmethodsm×mgrade(v)), h 21 ∈ R m h, 22 we have that the gradeor smaller than m.alwaysAv equal j =1) dim(K m or smaller[v 1 than v 2 m. · · · v j+1 0· · · hh 32j · ·]nce Av Consequence j =[v 1 (A; vv)) 2 = · · min(m, · v j+1 grade(v))0 h 32 · · · h 0 2j ·· ; v))2)= min(m,(∗)grade(v))0 ⎢ · · ·rade(v)) 2) 1) (∗) dim(K m (A; v)) = min(m, grade(v))⎢ ⎣⎣h j+1,j ·rade of a vector v w.r.t. A is the lowest degree monic polynomialp(A)v = 0. For A ∈ R m×m , v ∈ R m , we have that the grade of v ism (A; v)) = min(m, grade(v))• From line 4 and 6 we have:Notice that v j+1 h j+1,j = Av j − ∑ ji=1 h ijv i . Then Av j =2) (∗)Notice that v j+1 h j+1,j = Av j − ∑ ji=1 h ijv i . Then Av j = ∑ j+1i=1vh ∑Notice that v j+1 h j+1,jj ⎡i=1 h ijv then i . Then Av j = ∑ = Av j − ∑ jj+1i=1 h i=1 h ijv i . Then Av j = ∑ j+1j+1 h j+1,j = Av j − ∑ ji=1 h ijv i . Then Av j j+1j+1,j = Av j − ∑ ji=1 h ijv i . Then Av j = ∑ 0LOV SUBSPACE METHODSj+1i=1 h i=1 ijv i :⎡ ⎡i=1 h ijv i⎡:⎡ ⎡ijv i : ⎡ ⎤h 11h ∗ ∗ 11 h 12h · · · 12 · · · ⎤· · ∗·hh 1j11 h 11 h 12 h 12 · · · · h 1j h 11· · ·h 12 · · · h 1j ·⎡in matrix form:5 KRYLOV SUBSPACE⎡⎤h 21 h 22 · h h 21 h h 2jh 11 h 12 · · · 1j · · · ]22 · · · h 2j · · ·]21 h 21 h 22 h 22 · · · h 2j · · ·h METHODS 0 ∗ · ·21 h ⎤⎡ ]⎤]]j =[v 1 v 2 j+1 0 h 32 · h 2jAv h 21 h 22 j · · vh 2 v 2j · j+1 0 h 32 · · · h 2j · · ·Av j =1 v 2 · · v j+1 0 h 32 · 2j =[v 1 v 2 · · · v j+1 0 h 32 · · h 2j · · ·]]∗ ∗ · 22 · · · · 0·∗1 ]· · · 0A[v 1 v 2 · · · v m =.[v 1 v 2 · · · v m+1 Av j =0 · ·[v 0 ·· · ·]0 v 2 · · · · · · j+1 0 ] ..0 ∗ h· 32· · · · · ∗A[v 1 v 2 · · · v1 m =[v 10v 2 · · ⎢· ⎢ v m+1 V.m = Vm ⎥T V m+1 H m =. ⎢v ⎣h j+1j+1 0 h 32 ·⎢· · h.2j · · h·⎣h j+1,j · · ⎦j+1,j j+1,j · · ·⎥⎦⎣ ⎡ 0 .. · · ∗·⎣ .. ⎥0 .. ∗0⎦1 ⎢· · ·⎢ . 0 · · ·0⎣.. ⎥⎣ 01 0⎢⎡ ⎥⎡1 ∗ ∗⎦⎤ 0: AV m ⎡⎣= V m+1 H m hVj+1,jm T AV ∗ m ∗ · = ⎦V· T m×(m+1)m V m+1 ⎤ H∗ m = ∗ · · · ∗∗ ∗ · · · ∗ ⎢ 0 ⎡ . ∗0 ∗ · · · ∗ ∗· · ·] ]0 ∗ · ⎣ .. 0]0 ∗ · · · ∗]. : VAV A[v 1 v 0 · · ·[ ⎡2 · v m =[v 1 v . 2 · · · v] [ ⎤m+1 0 ..⎡m T AV 2 · · ·] mv[ m = VH m = m+1 m where H mH [v 1 v 2 · · v] m := {H m deleted the last row}m+1 0 .. . ⎢· · v = v v · · · v ⎢ 0 .. ∗ ∗ 1ark : In practice, a modified Gram-Schmidt ∗(MGS) ⎥ ⎢ is used ] ⎢ on the ⎥ Ho 32⎢.
TakingV T m AV m y = H m r 0 = H m (βv 1 ) = βẽ 1FOM⎢V ⇒ x m = x 0 + V m (Mm −1 )βẽm T AV m = Vm T V m+1 H m = ⎢ . 1Combing MGS/Arnoldi with the above ⎣ leads ..to FOM 0:Generater 0 = b − Ax 0 , β = ‖r 0 ‖ 2 , v 1 = r 0 /βH m = {h ij } m i,j=1 : Set H m = 0w j = Av jfor i = 1, 2, · · · , j doAVh ij = (w j , v i )m y m = r 0w j = w j − h ij v iFULL endV ORTHOGONALIZATION METHOD (FOMm T AVh m y m = Vj+1,j = ‖w j ‖ m T r 0 = V2 , if h j+1,j = m T (βv0 set m 1 ) = βẽ= j exit 1endv j+1 = w j /h j+1,j⎡⎤1 · · · 0⎢1 0⎥⎦y m = H −1m (βẽ 1 ), x m = x 0 + V m y m1 0m×(m+1)⎡∗ ∗0 ∗0⎢⎣Vfor j m T AV= 1, 2, m = H· · · , m m where Hdom := {H m deleted the lastRemark : In practice, a modified Gram-Schmidt (MGS) is uSolution isx m = x 0 + V m y mholder version of Arnoldi’s algo.y m = Hm −1 (βẽ 1 )L = K = K m (A; r 0 ) ⊕ b − Ax m ⊥ K m (A; r 0 )Galerkin condi33
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: Projection methods5 KRYLOV SUBSPACE
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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