12_IntroToFastMethods

12_IntroToFastMethods - Fast Methods Donald R Wilton Vikram...

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1 Donald R. Wilton Vikram Jandhyala F ast M ethods
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2 Matrix Memory Requirements for Direct Solvers 1.E- 0 6 1.E- 0 4 1.E- 0 2 1.E+0 0 1.E+0 2 1.E+0 4 1.E+0 6 1.E+00 1.E+03 1.E+06 1.E+09 Number of Unknowns Memory in [GB] Why Are Fast Methods Needed for Large MoM Problems? 2 N ⇒∝ 6 decades 3 decades slope = = 2
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3 Approximate Computation Times for Large Problems Computation Requirements of Direct Solver at 1nS per FLOP 1.E-09 1.E-06 1.E-03 1.E+00 1.E+03 1.E+06 1 10 100 1000 10000 100000 100000 0 Number of Unknowns Computation Time (Yrs. ) 3 N ⇒∝ 3 decades 1 decade slope = = 3 1GFLOP 1TFLOP Approx. limit of human “Cyber- patience”
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4 Main Features of Fast Methods We assume solution uses an iterative, not a direct method Use redundant information in Mom matrix and/or Green’s function to reduce storage requirements (“compress” the matrix) and speed up the solution process
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5 Iterative Methods 1 0 1 ,1 , 2 , nn n n n ε = =+ = −< " Instead of directly solving by, e.g. Gaussian elimination, we iterate on an equation of the form where is an initial guess and until we achieve convergence, say Ax b x Bx c x xx 12 < = , and/or . The process must usually be sped up by the system, i.e., premultiplying by a matrix and solving the modified system P PAx Pb preconditioning
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6 Iterating the Preconditioned System () 1 - The preconditioner should in some sense approximate the inverse of the system matrix, , or equivalently When this is the case, we may view the te PA P AI IP rm i A n th N 1 1 - - nn =+ ±²³²´ Ab 0 e identity as a "small" correction to the RHS, leading to the simple iterative procedu x I PA x Pb x I PA x re Pb
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12_IntroToFastMethods - Fast Methods Donald R Wilton Vikram...

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