Introduction to Scientific Computing II From Gaussian Elimination to Relaxation Methods Dr. Miriam Mehl

Typical SLE sparse band structure

Example

Gaussian Elimination

Gaussian Elimination

Gaussian Elimination

Gaussian Elimination

Gaussian Elimination

Gaussian Elimination

Gaussian Elimination

Gaussian Elimination

Gaussian Elimination (LU)

Gaussian Elimination (LU)

Gaussian Elimination (LU)

Gaussian Elimination (LU)

Gaussian Elimination (LU)

Gaussian Elimination (LU)

Gaussian Elimination (LU)

Gaussian Elimination (LU)

Gaussian Elimination (LU)

Gaussian Elimination – Costs

Gaussian Elimination – Costs O(1/h)2

Gaussian Elimination – Costs O(1/h)2*(1/h)2

Gaussian Elimination – Costs O(1/h)2*(1/h)2 + O(1/h)

Gaussian Elimination – Costs O(1/h)2*(1/h)2 + O(1/h)*(1/h)2

Gaussian Elimination – Costs O(1/h)2*(1/h)2 + O(1/h)*(1/h)2 + O(1/h)

Gaussian Elimination – Costs O(1/h)2*(1/h)2 + O(1/h)*(1/h)2 + O(1/h)*(1/h)2

Gaussian Elimination – Costs 2D: O(1/h)4 3D: O(1/h)7

Gaussian Elimination – Costs 2D h runtime (HLRB2, 62 TFlop/s) 2-7 0.02 sec 2-8 0.27 sec 2-9 4.4 sec 2-10 1 min 16 sec 2-11 18 min 55 sec 2-12 5 h 02 min 40 sec 2-13 3 d 8 h 37 min 15 sec hallo

Gaussian Elimination – Costs 3D hallo h runtime (HLRB2, 62 TFlop/s) 2-6 4 min 44 sec 2-7 10 h 05 min 24 sec 2-8 53 d 19 h 21 min 17 sec 2-9 18 a 313 d 21 h 54 min 22 sec

Iterative Solvers – Principle series of approximations costs per iteration? convergence? stopping criterion?

Relaxation Methods – Gauss-Seidel

Relaxation Methods – Jacobi