Iterative Solvers for Linear Systems


 Hilary Hardy
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1 9th SimLab Course on Parallel Numerical Simulation, Iterative Solvers for Linear Systems Bernhard Gatzhammer Chair of Scientific Computing in Computer Science Technische Universität München Germany
2 Outline PoissonEquation: From the Model to an Equation System Direct Solvers vs. Iterative Solvers Jacobi and GaussSeidel Method Residual and Smoothness Multigrid Solver
3 Outline PoissonEquation: From the Model to an Equation System Direct Solvers vs. Iterative Solvers Jacobi and GaussSeidel Method Residual and Smoothness Multigrid Solver
4 The Poisson Equation An Elliptic PDE General: 2 R d, ± 2 R d 1, u :! R u = f in ( = r 2 = r : : : 2 d = f in u = g on ± 1D: = f in 2D: 2 R = f in
5 Why the Poisson Equation is of Importance The Laplace operator is part of many important PDEs: Heat equation: diffusion of heat
6 Why the Poisson Equation is of Importance The Laplace operator is part of many important PDEs: Flow dynamics: friction between molecules
7 Why the Poisson Equation is of Importance The Laplace operator is part of many important PDEs: Structural mechanics: displacement of membrane structures
8 Why the Poisson Equation is of Importance The Laplace operator is part of many important PDEs: Electrics: electric, magnetic potentials
9 Why the Poisson Equation is of Importance The Laplace operator is part of many important PDEs: Financial mathematics: Diffusive process for option pricing
10 Why the Poisson Equation is of Importance The Laplace operator is part of many important PDEs: Heat equation: diffusion of heat Flow dynamics: friction between molecules Structural mechanics: displacement of membrane structures Electrics: electric, magnetic potentials Financial mathematics: Diffusive process for option pricing...
11 Discretization with Finite Differences I Find solution to Poisson equation numerically on a computer: Computer has only a finite amount of memory/power discretization Finite Differences as simple discretization Idea: Replace derivative by difference u( + h) u() + h u() u( h) h
12 Discretization with Finite Differences II Use forward and backward differences for 2 nd derivatives: forward diff.: backward 2 ¼ u( + h) u() h u( + h) u() h u( + h) 2u() + u( h) h 2 u( + h h) u( h) h h
13 Grid, Stencil, and Matri 1D Regular Cartesian grid: Nodal equation: Stencil: h i = ih u i 1 2u i + u i+1 = h 2 f i i 1 i i+1 System matri: A = 1 h A 2
14 Grid, Stencil, and Matri 2D Regular Cartesian grid: Nodal equation: h h u i 1;j + u i+1;j + u i;j 1 + u i;j+1 4u i;j = h 2 f i;j i;j = (ih; jh) Stencil: 1 i;j i 1;j i;j 1 i;j 1 i+1;j
15 Grid, Stencil, and Matri 2D System matri: h h A = 1 h B A 1 1 4
16 Discretized Poisson Equation We obtain a system of equations to be solved: Au = f, A 2 R N N, u; f 2 R N N is the number of grid unknowns, i.e. equations f contains sink/source terms and boundary influences We want to obtain u = A 1 f
17 Outline PoissonEquation: From the Model to an Equation System Direct Solvers vs. Iterative Solvers Jacobi and GaussSeidel Method Residual and Smoothness Multigrid Solver
18 Direct Solvers of Systems of Linear Equations Popular direct solvers for Au = f : U = Gaussian elimination LU decomposition Cholesky factorization T N ::: T 2 Au = T N ::: T 2 f, Uu = f 0 A = LU, LUu = f, Uu = L 1 f A = U T U compute u = A 1 f directly up to numerical accuracy computational compleity of GEM in 2D: O(h 6 ), in 3D: O(h 9 ) GEM optimized for bandstructure: in 2D: O(h 4 ), in 3D: O(h 7 )
19 Runtime of GEM for Solving Poisson 2D h N runtime (33 TFlop/s) sec sec min 23 sec min 32 sec h 28 min 38 sec d 7 h 28 min 10 sec
20 Runtime of GEM for Solving Poisson 3D h N runtime (33 TFlop/s) min 53 sec h 57 min 26 sec d 26 h 10 min 53 sec a 156 d 57 h 53 min 04 sec
21 Iterative Solvers of Systems of Linear Equations In contrast to direct solvers, iterative solvers Compute approimate solutions u k ¼ A 1 f Improve the approimation in an iterative process One iteration is cheap to compute, in 2D O(h 2 ), in 3D O(h 3 ) Fipoint iteration: Types of methods: u k+1 = F(u k ) k!1! u = F(u) Splitting methods, Krylovsubspace methods, Multigrid methods
22 Outline PoissonEquation: From the Model to an Equation System Direct Solvers vs. Iterative Solvers Jacobi and GaussSeidel Method Residual and Smoothness Multigrid Solver
23 Splitting Methods Split up matri A: Update rule: Au = f (S + T )u = f Su k+1 = f T u k u k+1 = S 1 (f T u k ) u k+1 = F (u k ) Matri S has to be of simple form to to get u k+1
24 The Jacobi Method Choose: Splitting: Update rule: S = diag(a) := D A A = S + T = D A + (A D A ) Au = f (D A + (A D A ))u = f D A u k+1 = (f (A D A )u k ) u k+1 = D 1 A (f (A D A)u k ) u k+1 i = 1 a ii (f i NX a ij u k j ) j=1 j6=i
25 Jacobi Method Eample 1D Heat Equation Stationary heat equation with u as temperature: Boundaries have zero temperature, no 2 2 = f in f = 0 Solution is known: u = 0 u k+1 i = 1 a ii (f i NX j=1 j6=i a ij u j ) =) u k+1 i = 1 2 (uk i 1 + u k i+1)
26 Jacobi Method Eample 1D Heat Equation Unit domain: := [0; 1]; h = 1 8 ; N = A = B A 1 2
27 Jacobi Method Eample 1D Heat Equation Random initial value u 0 = rand(n)
28 Jacobi Method Eample 1D Heat Equation k = 0 0 u k+1 = (uk i 1 + u k i+1) i
29 Jacobi Method Eample 1D Heat Equation k = 1
30 Jacobi Method Eample 1D Heat Equation k = 2
31 Jacobi Method Eample 1D Heat Equation k = 3
32 Jacobi Method Eample 1D Heat Equation k = 4
33 Jacobi Method Eample 1D Heat Equation k = 5
34 Jacobi Method Eample 1D Heat Equation k = 6
35 Jacobi Method Eample 1D Heat Equation k = 7
36 Jacobi Method Eample 1D Heat Equation k = 8
37 Jacobi Method Eample 1D Heat Equation k = 8 Initial values
38 The GaussSeidel Method Choose: Splitting: Update rule: S = L(A) := L A A = S + T = L A + (A L A ) Au = f (L A + (A L A ))u = f L A u k+1 = (f (A L A )u k ) u k+1 = L 1 A (f (A L A)u k ) u k+1 i = 1 a ii (f i Xi 1 j=1 a ij u k+1 j NX j=i+1 a ij u k j )
39 GaussSeidel Method Eample 1D Heat Equation Stationary heat equation with u as temperature: Boundaries have zero temperature, no 2 2 = f in f = 0 Solution is known: u = 0 u k+1 i = 1 a ii (f i Xi 1 j=1 a ij u k+1 j NX j=i+1 a ij u k j ) =) u k+1 i = 1 2 (uk+1 i 1 + uk i+1)
40 GaussSeidel Method Eample 1D Heat Equation Random initial value u 0 = rand(n)
41 GaussSeidel Method Eample 1D Heat Equation k = 0
42 GaussSeidel Method Eample 1D Heat Equation k = 0
43 GaussSeidel Method Eample 1D Heat Equation k = 0
44 GaussSeidel Method Eample 1D Heat Equation k = 0
45 GaussSeidel Method Eample 1D Heat Equation k = 0
46 GaussSeidel Method Eample 1D Heat Equation k = 0
47 GaussSeidel Method Eample 1D Heat Equation k = 0
48 GaussSeidel Method Eample 1D Heat Equation k = 1
49 GaussSeidel Method Eample 1D Heat Equation k = 2
50 GaussSeidel Method Eample 1D Heat Equation k = 3
51 GaussSeidel Method Eample 1D Heat Equation k = 4
52 GaussSeidel Method Eample 1D Heat Equation k = 5
53 GaussSeidel Method Eample 1D Heat Equation k = 6
54 GaussSeidel Method Eample 1D Heat Equation k = 7
55 GaussSeidel Method Eample 1D Heat Equation k = 8
56 GaussSeidel Method Eample 1D Heat Equation k = 8 Initial values
57 GaussSeidel Method Eample 1D Heat Equation k = 8 Jacobi after 8 iterations
58 Outline PoissonEquation: From the Model to an Equation System Direct Solvers vs. Iterative Solvers Jacobi and GaussSeidel Method Residual and Smoothness Multigrid Solver
59 Residual Residual: Au = f Au k 6= f f Au k = res k Splitting: Au = (S + T )u = f u k+1 = S 1 (f T u k ) u k+1 = S 1 (f (A S)u k ) u k+1 = S 1 (f Au k ) + u k u k+1 = u k + S 1 res k
60 Residual for 1D Heat Equation Eample Local influences form residual: u i 1 2u i + u i+1 = 0 u k i 1 2u k i + u k i+1 = res k i GS k = 0: high residual oscillatory error/fct values GS k = 8: low residual smooth error/fct values
61 Smoothness Observation Smoothnes depends on grid resolution h : Error reduction h = 1 : 8 h = 1 : 4 k! k + 1 h = 1 : 2
62 Outline PoissonEquation: From the Model to an Equation System Direct Solvers vs. Iterative Solvers Jacobi and GaussSeidel Method Residual and Smoothness Multigrid Solver
63 Geometric Multigrid Idea Only high error frequencies are removed effectively Frequencies depend on grid resolution Idea: Use several grid levels Use splitting method to smooth error on each grid level
64 Multigrid Eample 1D Heat Equation Random initial value u 0 = rand(n)
65 Multigrid Eample 1D Heat Equation k = 0, N = 1
66 Multigrid Eample 1D Heat Equation k = 0, N = 1
67 Multigrid Eample 1D Heat Equation k = 0, N = 3
68 Multigrid Eample 1D Heat Equation k = 0, N = 3
69 Multigrid Eample 1D Heat Equation k = 0, N = 7
70 Multigrid Eample 1D Heat Equation k = 1
71 Thank you for your attention!
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