Conjugate Gradients: Idea
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- Steven Hood
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1 Overview Steepest Descent often takes steps in the same direction as earlier steps Wouldn t it be better every time we take a step to get it exactly right the first time? Again, in general we choose a new point Additionally, we could demand that the remaining error (NOT the residual) is orthogonal to our direction Unfortunately, we don t know the error (if we knew we would be done) Conjugate Gradients: Idea 81
2 Overview Instead, we demand the search direction to be A-orthogonal to the remaining error A-orthogonal vectors a,b satisfy Some A-orthogonal vectors: Conjugate Gradients: Idea 82
3 Overview Instead, we demand the search direction to be A-orthogonal to the remaining error A-orthogonal vectors a,b satisfy The remaining error shall be A-orthogonal to search direction: This is equivalent to finding the minimum point along the direction So far, everything is equivalent to Steepest Descent (except that p is not necessarily r) How to construct the A-orthogonal search directions p? Conjugate Gradients: Idea 83
4 Overview The process of making vectors orthogonal or A-orthogonal is called Gram-Schmidt orthogonalization Suppose n linear independent vectors are given. Then, one can find a new vector which is orthogonal to all previous vectors by Factors are found from the A-orthogonality condition: We can now create a vector that is A-orthogonal to all previous vectors. We can now put everything together Conjugate Gradients: Idea 84
5 We now choose the search directions to be the residual (as in Steepest Descent) but additionally make them A-orthogonal to the previous search direction. Doing so, the new search direction is A-orthogonal to ALL previous directions. Conjugate Gradients: Formulation 85
6 Conjugate Gradients: Preconditioning 86
7 J.R. Shewchuk: An Introduction to the Conjugate Gradient Method without the Agonizing Pain, Carnegiee Mellon Univ., (free on the web) Conjugate Gradients: Convergence 87
8 Conjugate Gradients: Preconditioning 88
9 Conjugate Gradients: Preconditioning 89
10 Symmetric Gauss Seidel Preconditioning 90
11 Symmetric Gauss Seidel Preconditioning 91
12 Symmetric Gauss Seidel Preconditioning 92
13 Symmetric Gauss Seidel Preconditioning 93
14 Review: Sym. Mult. Schwarz is Sym. Block Gauss Seidel 94
15 Multiplicative Schwarz DD 95
16 Multiplicative Schwarz DD 96
17 Multiplicative Schwarz DD 97
18 Multiplicative Schwarz DD 98
19 Multiplicative Schwarz DD with Coarse Grid 99
20 Multiplicative Schwarz DD with Coarse Grid 100
21 Multiplicative Schwarz DD with Coarse Grid 101
22 Multiplicative Schwarz DD with Coarse Grid 102
23 Multiplicative Schwarz DD with Coarse Grid 103
24 Multiplicative Schwarz DD with Coarse Grid 104
25 Overlapping Schwarz, the nonmatching case 105
26 Overlapping Schwarz, the multigrid case 106
27 Overlapping Schwarz, the multigrid case 107
28 Overlapping Schwarz, the multigrid case 108
29 Overlapping Schwarz, the multigrid case 109
30 Overlapping Schwarz, the multigrid case 110
31 Overlapping Schwarz, the multigrid case 111
32 Overlapping Schwarz, the multigrid case 112
33 Geometric Multigrid 113
34 Algebraic Multigrid 114
35 Algebraic Multigrid 115
36 Algebraic Multigrid: Requirements 116
37 Algebraic Multigrid: Requirements 117
38 Algebraic Multigrid: Requirements 118
39 Algebraic Multigrid: Formulation 119
40 Algebraic Multigrid: Formulation 120
41 Algebraic Multigrid: Formulation 121
42 Algebraic Multigrid: Formulation 122
43 Algebraic Multigrid: Formulation 123
44 Gram-Schmidt Orthonormalization 124
45 Algebraic Multigrid: Formulation 125
46 Algebraic Multigrid: Formulation 126
47 Algebraic Multigrid: Formulation 127
48 Algebraic Multigrid: Formulation 128
49 Algebraic Multigrid: 1D example 129
50 Algebraic Multigrid: 1D example 130
51 Algebraic Multigrid: 1D example 131
52 Smoothed Aggregation AMG (SA-AMG) 132
53 CG with SA-AMG preconditioner: 3D elasticity example 133
54 CG with SA-AMG preconditioner: 3D elasticity example 134
55 Fin 135
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