1. Introduction to Multigrid Method
Presented by: Bogojeska Jasmina
15/11/2018
JASS, 2005, St. Petersburg

2. The ultimate upshot of MLAT
The amount of computational work should be proportional to the amount of real physical changes in the computed system!
In fully developped Multigrid processes the amount of computations should be determined only by the amount of real physical information
15/11/2018
JASS, 2005, St. Petersburg

3. Content Model Problems Basic Iterative Schemes The Multigrid Method
Is everything really that simple???
15/11/2018
JASS, 2005, St. Petersburg

4. Testing Ground
One-dimensional boundary value problem describing the steady-state temperature distribution in a long uniform rod
Grid:
15/11/2018
JASS, 2005, St. Petersburg

5. Approximation with the finite difference method
15/11/2018
JASS, 2005, St. Petersburg

6. Matrix Form
15/11/2018
JASS, 2005, St. Petersburg

7. Testing Ground II Two-dimensional boundary value problem 15/11/2018
JASS, 2005, St. Petersburg

8. Approximation with the finite difference method
15/11/2018
JASS, 2005, St. Petersburg

9. Matrix Form
15/11/2018
JASS, 2005, St. Petersburg

10. Matrix Form II
15/11/2018
JASS, 2005, St. Petersburg

11. Content Model Problems Basic Iterative Schemes The Multigrid Method
Is everything really that simple???
15/11/2018
JASS, 2005, St. Petersburg

12. Some Notations and Definitions
15/11/2018
JASS, 2005, St. Petersburg

13. Stationary Linear Iterations
15/11/2018
JASS, 2005, St. Petersburg

14. Assymptotic Convergence Factor
15/11/2018
JASS, 2005, St. Petersburg

15. Jacobi Relaxation
15/11/2018
JASS, 2005, St. Petersburg

16. Gauss-Seidel Relaxation
Components of the new approximation are used as soon as they are calculated – reduced storage requirements
15/11/2018
JASS, 2005, St. Petersburg

17. Fourier Modes
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JASS, 2005, St. Petersburg

18. Fourier Modes I
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JASS, 2005, St. Petersburg

19. Numerical Example
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JASS, 2005, St. Petersburg

20. Numerical Example I
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JASS, 2005, St. Petersburg

21. Observation
Standard iterations converge quickly as long as the error has high-frequency components
BUT the slow elimination of the low frequency components of the error degrades the performance
15/11/2018
JASS, 2005, St. Petersburg

22. Why?
15/11/2018
JASS, 2005, St. Petersburg

23. Why?
15/11/2018
JASS, 2005, St. Petersburg

24. Conclusion
The eigenvalue associated with the smoothest mode will always be close to 1 (esspecially for smaller grid spacing)
No value of can reduce the smooth components of the error effectively
What value of damps best the oscillatory components of the error?
15/11/2018
JASS, 2005, St. Petersburg

25. Smoothing Factor
Smoothing factor - the largest absolute value among the eigenvalues in the upper half of the spectrum (the oscillatory modes) of the iteration matrix:
Smoothing property for weighted Jacobi after 35 iteration sweeps:
15/11/2018
JASS, 2005, St. Petersburg

26. Content Model Problems Basic Iterative Schemes The Multigrid Method
Is everything really that simple???
15/11/2018
JASS, 2005, St. Petersburg

27. Elements of Multigrid Coarse Grids Nested Iteration Correction Scheme
Interpolation Operator
Restriction Operator
Two-Grid Correction Scheme
V-Cycle Scheme
Full Multigrid V-Cycle - FMG
15/11/2018
JASS, 2005, St. Petersburg

28. Coarse Grids
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JASS, 2005, St. Petersburg

29. Coarse Grids
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JASS, 2005, St. Petersburg

30. Coarse Grids
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JASS, 2005, St. Petersburg

31. Nested Iteration
Compute an improved initial guess for the fine-grid relaxation
15/11/2018
JASS, 2005, St. Petersburg

32. Correction Scheme
15/11/2018
JASS, 2005, St. Petersburg

33. Interpolation Operator (1D)
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JASS, 2005, St. Petersburg

34. Interpolation Operator (1D)
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JASS, 2005, St. Petersburg

35. Interpolation Operator (1D)
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JASS, 2005, St. Petersburg

36. Interpolation Operator (1D)
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JASS, 2005, St. Petersburg

37. Restriction Operator (1D)
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JASS, 2005, St. Petersburg

38. Full Weighting
15/11/2018
JASS, 2005, St. Petersburg

39. Two-Grid Correction Scheme
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JASS, 2005, St. Petersburg

40. Two-Grid Correction Scheme
15/11/2018
JASS, 2005, St. Petersburg

41. V-Cycle
15/11/2018
JASS, 2005, St. Petersburg

42. V-Cycle - Recursive
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JASS, 2005, St. Petersburg

43. Storage Costs
15/11/2018
JASS, 2005, St. Petersburg

44. Computational Costs
15/11/2018
JASS, 2005, St. Petersburg

45. Convergence Analysis
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JASS, 2005, St. Petersburg

46. Converging to Level of Truncation
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JASS, 2005, St. Petersburg

47. Full Multigrid V-Cycle
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JASS, 2005, St. Petersburg

48. Full Multigrid
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JASS, 2005, St. Petersburg

49. Full Multigrid - Recursive
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JASS, 2005, St. Petersburg

50. Costs of Full Multigrid
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JASS, 2005, St. Petersburg

51. Building
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JASS, 2005, St. Petersburg

52. Variational Properties
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JASS, 2005, St. Petersburg

53. Spectral Properties of the Restriction Operator
15/11/2018
JASS, 2005, St. Petersburg

54. Spectral Properties of the Interpolation Operator
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JASS, 2005, St. Petersburg

55. Two-Grid Correction Scheme
15/11/2018
JASS, 2005, St. Petersburg

56. Algebraic Analysis
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JASS, 2005, St. Petersburg

57. Spectral and Algebraic Decompozition
Spectral decompozition:
Algebraic decompozition:
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JASS, 2005, St. Petersburg

58. How it works?
15/11/2018
JASS, 2005, St. Petersburg

59. How it works?
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JASS, 2005, St. Petersburg

60. How it works?
15/11/2018
JASS, 2005, St. Petersburg

61. Is everything really so simple???
Anisotropic operators and grids
Discontinuous or anisotropic coefficients
Nonlinear problems
Non-scalar PDE systems
High order discretization
Algebraic Turbulence models
Chemicaly reacting flows
Shocks
Small-scale singularities
15/11/2018
JASS, 2005, St. Petersburg