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Simple iteration procedure

residue. use of pre-conditionner. correction. residue. use of pre-conditionner. Simple iteration procedure. Solve. Known approximate solution. Preconditionning:. Jacobi. Gauss-Seidel. Lower triangle. Spectral radius (magnitude of largest eigenvalue ). Convergence.

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Simple iteration procedure

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  1. residue use of pre-conditionner correction residue use of pre-conditionner Simple iteration procedure Solve Known approximate solution Preconditionning: Jacobi Gauss-Seidel Lower triangle

  2. Spectral radius (magnitude of largest eigenvalue) Convergence Convergence rate (from now on, we replace M-1A by A and M-1b by y): ( ) Field of values:

  3. = Orthomin(1) and orthomin(2) Krylov subspace Residue orthogonal to subspace Projection on Orthomin(2): Project on larger subspace Orthodir(n) GMRES Beyond that, project on all previous values of A ri

  4. Key elements of iterative solvers • Good pre-conditionner (often based on nearest interactions) • Fast matrix-vector multiplication (e.g. based on Fast Multipoles) (get far below N2 complexity, e.g. N log2 N) • Iterative technique with good convergence/stability properties (get far below N iterations) Get far below O(N3) complexity (D N log2 N) Number of iterations

  5. Modified Gram-Schmidt (MGS) algorithm

  6. MGS defines a QR decompositon For overdetermined problems, QR solution = least-squares solution Normal equations:

  7. Arnoldi algorithm: MGS on Krylov subspace normalize

  8. 2 Primitive GMRES Principle: the solution belongs to the (orthogonalized) Krylov subspace… …with coefficients obtained in the least-square sense …for instance with the help of a QR solution ! Then, the residue is: A qk-1 i.e. it is « minimized over subspace »: A qk

  9. Arnoldi algorithm: Upper Hessenberg Matrix H33 h43 0 0 H43

  10. -1 Arnoldi recursion Recursion formula: nxn nxk nx(k+1) (k+1)xk Proof: (j=1, i=1,2 loop) (j=k, i=1..k loop) (previous iteration)

  11. || || k+1 comp. of QR minimization = All 0’s, except entry 1 as proven earlier QR decomposition of Succession of orthogonal transformations: FFH=I

  12. Recursive orthogonal transformations for H Orthogonalisation realised at previous iteration: Same operations on new Hessenberg matrix: Find new operation to null that entry from same operations applied to new column of H

  13. Givens rotations if d≠0 if d=0 Expression of residual 2 2 2 2 Norm of k+1 comp. of

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