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• ????????
• ???
• ??????????????
• ??????
• yhzhao_at_sccas.cn

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??

• ?????????
• ??????????????
• ????????????
• ?/???(single and multiple vector)???
• Lanczos ?Arnoldi??
• Jacobi-Davidson??
• ???????????CG(Conjugate-Gradient)?
• ?????????
• ?????????

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???????

• ???????????????,(?????????????????????),?????
  ??????????
• Ax?Mx
• ????????????A?M??????????????
• ???????????????????????????????,?????????????
  ?????????????????

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• ?????
• ??????????????Maxwell????????????????
  ??????????
• ??????????????????????????????,????????????????
  ???????
• ??????
• ??????????????????????????????
• ?? ?????(????), ????????????

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• Nano/Material-Science
• 
  
• Accelerator Modeling

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?????????? Lanczos?Arnoldi??

• ?/???(single and multiple vector)???
• ?(Power)???????????Rayleigh???(RQI)??
• Lanczos?Arnoldi??
• Lanczos?Arnoldi??????Krylov???????????
• ??????????????Ritz??????????????????,???????
  ???-????
• ?????????????????????????????Lanczos?Arnoldi??
• ?????????????????????Lanczos?Arnoldi??
• ??????????????Lanczos?Arnoldi????????????,???????
  ??????-??????,??????????,??? ???
• ????????????????,???????????

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??????????Jacobi-Davidson??

• ??????Lanczos?Arnoldi???band Lanczos?
• Jacobi-Davidson??
• Jacobi-Davidson????????JOCC(Jacobi
  orthogonal Component Correction)?Davidson??????G.
  L.G.Sleijpen??1996????
• ??Lanczos?Arnoldi????,??????????,????
  ????????????????????,????A?????????,??????????????
  ???????Davidson?Jacobi-Davidson?????Arnoldi??????
  
• ?Lanczos?Arnoldi????,Jacobi-Davidson
  ???????????,????????????????,??????????????,??????
  ????????Rayleigh???????Jacobi-Davidson
  ???????????Krylov????? ?

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??????????Jacobi-Davidson??

• ????
• ?????????Vu1,u2,,uk,????????
• 1.??Bs?s ?????(?j, sj),????A?Ritz??ujVsj
• 2.???????? ,??
• 3.?????
• ???v?
• 4. ?v????????????,?????Ritz??
• ????,?????????Jacobi-Davidson???

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??????????Jacobi-Davidson??

• ???????????
• ??????????????????(?GMRES?BCGSTAB)?????????
  ????? ????????????
• ????????,??????????? ??
• ???????????
• ????
• ????????,? ??z

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??????????Jacobi-Davidson??

• ???
• JD?????????????????,???????????????,?
  ???????
• ???????????Dynamic thick ??????
• ??
• ????????????????,???????????????Ritz??u
  1,u2,,uk ,?????????
• ??Q( u1,u2,,uk )

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??????????Jacobi-Davidson??

• ?Jacobi-Davidison??
• JD????????k??????????????,??????????k?
  ??????????k?????????,????????Ritz????
• JDCG??
• Y.Notay2002??????????????,??JD???????
  ???(CG)?,??JDCG???
• JDQR???JDQZ??
• ???
• JD????????????????????????????????????
  ?????,???????????????????????????????????????????-
  ???????????????
• ???PRIMME
• ????????????????

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?????????????CG??

• ???CG??
• CG???Chebyshev?????,????????????????
  CG????,????LOBPCG?DACG???
• LOBPCG(Locally Optimal Block ,Preconditioned
  Conjugate Gradient)
• ???????????????????(A.VKayazev,2001)?
  ?????????????????,?????Lanczos?Davidson?????
  ????,????????
• ?????????,??????????,??????????????
  ?,????????????????
• LOBPCG?????????,???????????????????
  ??????,?????????Rayleigh-Ritz???
• LOBPCG?? ???????BLOPEX
  ,MATLAB

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?????????????CG??

• DACG(Deflation-Accelerated Conjugate
  Gradient )
• ???????????CG?????Rayleigh??CG???,???
  ??????????
• ??????????,DACG??????ARPACK??????????
  ?????, DACG???????????Jacobi-Davidson???
• ???????????Jacobi????????FSAI?AINV,?
  ????Jacobi-DACG,FSAI-DACG?AINA-DACG???
• ?????????????????????????????
• ??????PDACG

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????????

• ????????????????-???,???-??????????
  ??????????????????????????
• ?????????
• ???(?)????(RSCompress Row storage)
• ????(?)????(BCRSBlock Compress Row storage)
  
• ????????(CDSCompressed Diagonal Storage )
• ????????(JDSJagged Diagonal Sstorage)

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????????

• ??????
• val????????????????
• col_ind??val??????
• row_ptr??????????val????
• Val10,-2,3,9,3,7,8,7,3,8,7,5,8,9,9,13,4,2,-
  1
• Col_ind1,5,1,2,6,2,3,4,1,3,4,5,2,4,5,6,2,5,
  6
• Row_ptr1,3,6,9,13,17,20

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????????

• ????????BCRS(Block compress row storage)
• ??????????????????????
• ?????????????????????
• ?CRS??????,BCRS?????????????
  val11,12,0,22,31,32,41,42,33,0,43,44,55,56,0,66,
  0,0,67,0,77,78,87,88
• AN1,5,9,13,17,21 AJ1,1,2,3,4,4
  AI1,2,4,6,7

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????????

• ??????(the compressed diagonal storageCDS
  )
• ??????????????????,?????????????????
  ???????????????????
• ????Val(6,-11)???????

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????????

• ?????

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?????????

• ??????
• ????????????????????????,??????????????
  ?????????,??????????????????
• ??????????????????????????
• ???????????????,???????
• ????
• ???????????????????SPMD??,??MPI????????
  ???????MPI-????,????-????
• ??????????OpenMP???????
  ?????????JAVA???????

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?????????

• ????
• C,C?FORTRAN77/90
• ??????????
• ?????????,????????
• ???
• ????????,??????????????????????????????
  ????,??LAPACK,BLAS, PETSC,MATLAB??

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• ??!