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Introduction to Multigrid Method

Introduction to Multigrid Method. Presented by: Bogojeska Jasmina. The ultimate upshot of MLAT. The amount of computational work should be proportional to the amount of real physical changes in the computed system!

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Introduction to Multigrid Method

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

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

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

  5. Approximation with the finite difference method JASS, 2005, St. Petersburg

  6. Matrix Form JASS, 2005, St. Petersburg

  7. Testing Ground II • Two-dimensional boundary value problem JASS, 2005, St. Petersburg

  8. Approximation with the finite difference method JASS, 2005, St. Petersburg

  9. Matrix Form JASS, 2005, St. Petersburg

  10. Matrix Form II JASS, 2005, St. Petersburg

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

  12. Some Notations and Definitions JASS, 2005, St. Petersburg

  13. Stationary Linear Iterations JASS, 2005, St. Petersburg

  14. Assymptotic Convergence Factor JASS, 2005, St. Petersburg

  15. Jacobi Relaxation JASS, 2005, St. Petersburg

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

  17. Fourier Modes JASS, 2005, St. Petersburg

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

  19. Numerical Example JASS, 2005, St. Petersburg

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

  22. Why? JASS, 2005, St. Petersburg

  23. Why? 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? 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: JASS, 2005, St. Petersburg

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

  28. Coarse Grids JASS, 2005, St. Petersburg

  29. Coarse Grids JASS, 2005, St. Petersburg

  30. Coarse Grids JASS, 2005, St. Petersburg

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

  32. Correction Scheme JASS, 2005, St. Petersburg

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

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

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

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

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

  38. Full Weighting JASS, 2005, St. Petersburg

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

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

  41. V-Cycle JASS, 2005, St. Petersburg

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

  43. Storage Costs JASS, 2005, St. Petersburg

  44. Computational Costs JASS, 2005, St. Petersburg

  45. Convergence Analysis JASS, 2005, St. Petersburg

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

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

  48. Full Multigrid JASS, 2005, St. Petersburg

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

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

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