Introduction to Krylov subspace methods - IMAGe

with R(x) = ‖b − Ax‖ 2This is Petrov-Galerkin aasubcase of of a ageneral Galerkin methodit is (b −K finite-element =: L, x 1 ∈ x 0 Axframework. + K, 1 v) = 0 ∀v ∈ AK =: L, xv ∈ AK = L.1 ∈ x 0 + K, v ∈ AK = L.b − Ax = b − A(x 0 + V m y) = r 0 − AV m y = βV 1 − V m+1 H m y = VThis is Petrov-Galerkin a subcase of a general Galerkin methodubcase of a general Galerkin method as seen in thefinite-element framework.GMRESFind y minimizingGMREShere r 0 = βV 1 , β = ‖r 0 ‖ 2 .Any vector in xAny vector in 0 + K m can be written as x = xcan be written as x = 0 +x 0 V+ m y where y ∈ R⇒ ?(y) GMRES = ‖V m+1 (βẽ 1 − H m y)‖ 2 = ‖βẽ 1 − H m y‖ V m2y where yAnyDef. ?(y) = ‖b − Ax‖Def. vector ?(y) in x‖b 0 + KAx‖ m can 2 = ‖b − A(x2 be ‖bwritten − A(x 0 + 0 asV+ x m y)‖ThusV= mxy)‖ 0 2 + 2V m y where y ∈ Rwritten as x = x 0 + V m y where y ∈ R mb −Using A(x 0 Def. ?(y) = ‖b − Ax‖+ Arnoldi:2 = ‖b − A(x 0 + V m y)‖ 2V m y)‖ 2b − Ax = b − A(x 0b − Ax = b − A(x 0 + V m y) = r 0+ V m y) = r 0 −x m AV=m yx= 0 +βVv m 1 −y mV m+1 H m y = V m+− AV m y = βV 1 − V m+1 H m y = Vb − Axwhere r 0 = b − A(x 0 + V= βV 1 , β = ‖r 0 m y) y m = = r 0 arg − AV min m y ‖βẽ = βV 11 − HV m+1 m y‖ H 2‖ 2 .y m y = V m+where ⇒ r 0 ?(y) = βV 1 ‖V , β = ‖r 0 } {{ }‖ 2 .Equivalent where r 0 = to βV 1 m+1 , β = (βẽ ‖r 1 0 ‖ − 2 . H m y)‖ 2 = ‖βẽ 1 − H m y‖ 2⇒ Thus?(y) = ‖V m+1 (βẽ 1 Solved y)‖ using ‖βẽ least-squares⇒ ?(y) = ‖V 1 − H m y‖ 2m+1 (βẽ 1 − H m y)‖ 2 = ‖βẽ 1 − H m y‖ 2Thus ThusSolve: H T x m = x 0 + v m ymH mm y = H T mβẽ 1 or (H m H T m)u = βẽ 1y m = arg x m min = x ‖βẽ 0 + vv y m m 1 −yH mm y‖x m = x 0 + v 2m y mythen y = H T mu } y m = arg min{{‖βẽ 1 1− HH m y‖ m} y‖ 2 2arg min ‖βẽ 1 − H m y‖ y2GMRES y (Saad and Schultz, }Solved [7]): using{{ {{ least-squares} }360 − AV m y = βV 1 − V m+1 H m y = V m+1 (βẽ 1 − H m y)y)‖ 2 = ‖βẽ 1 − H m y‖ 2