Anouncements. Multigrid Solvers. Multigrid Solvers. Multigrid Solvers. Multigrid Solvers. Multigrid Solvers
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1 Anouncements ultgrd Solvers The readng semnar starts ths week: o Usuall t wll e held n NEB 37 o Ths week t wll e n arland 3 chael Kazhdan (6657 ultgrd Solvers Recall: To compute the soluton to the osson equaton A usng a Jaco solver we: We start wth an ntal guess, We generate a sequence of mproved guesses {,,,,} convergng to the soluton: lm A ultgrd Solvers Recall: To do ths, we decompose the matr A as the sum A: o s the dagonal part of A o s everthng else And defne the update rule as: ( ( whch has the fed-pont propert that f s the soluton A, then: ( ultgrd Solvers Goal: To evaluate the convergence propertes of the update rule: Gven a matr A, a vector, and an ntal guess, how quckl wll the seres {,,, } converge to the correct answer? ( ( ultgrd Solvers Goal: Equvalentl, we can thnk of the queston n terms of the error If we have the vector such that A we can ask: Gven a matr A, and an ntal guess -, how quckl wll the seres {,,, } converge to zero?
2 ultgrd Solvers roof of Equvalence: To show the equvalence, we need to show that f s the soluton to the equaton A, then f: o {,,, } s the Jaco sequence generated solvng A wth ntal guess, and o {,,, } s the Jaco sequence generated solvng A wth ntal guess - Then: ultgrd Solvers roof ( nducton: (: Clearl true, defnton of ultgrd Solvers ultgrd Solvers roof ( nducton: Assume true for : ( ( ( ( ( ( A ( ( ( roof ( nducton: Assume true for : ( ( ( ( ( ( A ( ( ( ultgrd Solvers ultgrd Solvers roof ( nducton: Assume true for : ( ( ( ( ( ( A ( ( ( roof ( nducton: Assume true for : ( ( ( ( ( ( A ( ( (
3 3 ultgrd Solvers roof ( nducton: Assume true for : A ultgrd Solvers roof ( nducton: Assume true for : A ultgrd Solvers roof ( nducton: Assume true for : A ultgrd Solvers Convergence: The queston now can e formulated as follows: Gven a matr A, and an ntal guess, how quckl wll the seres {,,, } converge to zero? ultgrd Solvers In the case, the aplacan matr can e epressed as: A ultgrd Solvers In the case, the aplacan matr can e epressed as: How does the operator: act on a vector? A
4 ultgrd Solvers In the case, the aplacan matr can e epressed as: A How does the operator: act It defnes on a vector a vector? whose k-th coeffcent s the average of the (k--th and (k-th coeffcents of ultgrd Solvers Eample : When the ntal error s hgh-frequenc: ultgrd Solvers Eample : When the ntal error s hgh-frequenc: ultgrd Solvers Eample : When the ntal error s lower-frequenc: the convergence s ver fast! ultgrd Solvers Eample : When the ntal error s lower-frequenc: ultgrd Solvers Eample : When the ntal error s lower-frequenc: the convergence slows down In a gven teraton, the change n the k-th coeffcent s determned how much the coeffcent dffers the convergence from the slows average down of ts neghors 4
5 ultgrd Solvers Eample : When the ntal error s lower-frequenc: ultgrd Solvers Ke Idea: Transform error so low-frequences ecome hghfrequences: In a gven teraton, the change n the k-th coeffcent s determned how much the coeffcent dffers the convergence from the slows average down of ts neghors So the smoother the error, the slower the convergence ultgrd Solvers Ke Idea: Transform error so low-frequences ecome hghfrequences: ultgrd Solvers Gven the equaton A: Restrcton: Compute the low-resoluton equaton: A ~ ~ ~ Restrct to lower resoluton ultgrd Solvers Gven the equaton A: Restrcton: Compute the low-resoluton equaton: A ~ ~ ~ ow-res Solve: Solve for the low-resoluton soluton ~ ultgrd Solvers Gven the equaton A: Restrcton: Compute the low-resoluton equaton: A ~ ~ ~ ow-res Solve: Solve for the low-resoluton soluton ~ 3 rojecton: Instantate the hgh-resoluton soluton usng the low-resoluton soluton 5
6 ultgrd Solvers Gven the equaton A: Restrcton: Compute the low-resoluton equaton: A ~ ~ ~ ow-res Solve: Solve for the low-resoluton soluton ~ 3 rojecton: Instantate the hgh-resoluton soluton usng the low-resoluton soluton 4 Hgh-Res Solve: Solve for the hgh-resoluton soluton ultgrd Solvers Gven the equaton A: Restrcton: Compute the low-resoluton equaton: A ~ ~ ~ ow-res Solve: Solve for the low-resoluton soluton ~ 3 rojecton: Instantate the hgh-resoluton soluton usng the low-resoluton soluton 4 Hgh-Res Solve: Solve for the hgh-resoluton soluton Solves for the low-res part of ultgrd Solvers Gven the equaton A: Restrcton: Compute the low-resoluton equaton: A ~ ~ ~ ow-res Solve: Solve for the low-resoluton soluton ~ 3 rojecton: Instantate the hgh-resoluton soluton usng the low-resoluton soluton 4 Hgh-Res Solve: Solve for the hgh-resoluton soluton Solves for the low-res part of Solves for the hgh-res part of 6
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