1 / 37

MA5233: Computational Mathematics

MA5233: Computational Mathematics Weizhu Bao Department of Mathematics & Center for Computational Science and Engineering National University of Singapore Email: bao@math.nus.edu.sg URL: http://www.math.nus.edu.sg/~bao Computational Science Third paradigm for Discovery in Science

bernad
Télécharger la présentation

MA5233: Computational Mathematics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. MA5233: Computational Mathematics Weizhu Bao Department of Mathematics & Center for Computational Science and Engineering National University of Singapore Email: bao@math.nus.edu.sg URL: http://www.math.nus.edu.sg/~bao

  2. Computational Science • Third paradigm for • Discovery in Science • Solving scientific & engineering problems • Interdisciplinary • Problem-driven • Mathematical models • Numerical methods • Algorithmic aspects— computer science • Programming • Software • Applications, ……

  3. Dynamics of soliton in quantum physics

  4. Wave interaction in plasma physics

  5. Wave interaction in particle physics

  6. Vortex-pair dynamics in superfluidity

  7. Vortex-dipole dynamics in superfluidity

  8. Vortex lattice dynamics in superfluidity

  9. Vortex lattice dynamics in BEC

  10. Computational Science • Computational Mathematics – Scientific computing/numerical analysis • Computational Physics • Computational Chemistry • Computational Biology • Computational Fluid Dynamics • Computational Enginnering • Computational Materials Sciences • ……...

  11. Steps for solving a practical problems • Physical or engineering problems • Mathematical model – physical laws • Analytical methods – existence, regularity, solution, … • Numerical methods – discretization • Programming -- coding • Results -- computing • Compare with experimental results

  12. Computational Mathematics • Numerical analysis/Scientific computing • A branch of mathematics interested in constructive methods • Obtain numerically the solution of mathematical problems • Theory or foundation of computational science • Develop new numerical methods • Computational error analysis: • Stability • Convergence • Convergence rate or order of accuracy,….

  13. History • Numerical analysis can be traced back to a symposium with the title ``Problems for the Numerical Analysis of the Future, UCLA, July 29-31, 1948. • Volume 15 in Applied Mathematics Series, National Bureau of Standards • Boom of research and applications: Fluid flow, weather prediction, semiconductor, physics, ……

  14. Milestone Algorithms • 1901: Runge-Kutta methods for ODEs • 1903: Cholesky decomposition • 1926: Aitken acceleration process • 1946: Monte Carlo method • 1947: The simplex algorithm • 1955: Romberg method • 1956: The finite element method

  15. Milestone algorithms • 1957: The Fortran optimizing compiler • 1959: QR algorithm • 1960: Multigrid method • 1965: Fast Fourier transform (FFT) • 1969: Fast matrix manipulations • 1976: High Performance computing & packages: LAPACK, LINPACK – Matlab • 1982: Wavelets • 1982: Fast Multipole method

  16. Top 10 Algorithms • 1946: Monte Carlo method • 1947: Simplex method for linear programming • 1950: Krylov subspace iterative methods • 1951: Decompositional approach for matrix computation • 1957: Fortran optimizing compiler • 1959-61: QR algorithms • 1962: Quicksort • 1965: Fast Fourier Transform (FFT) • 1977: Integer relation detection algorithm • 1982: Fast multipole algorithm http://amath.colorado.edu/resources/archive/topten.pdf

  17. Contents • Basic numerical methods • Round-off error • Function approximation and interpolation • Numerical integration and differentiation • Numerical linear algebra • Linear system solvers • Eigenvalue probems • Numerical ODE • Nonlinear equations solvers & optimization

  18. Contents • Numerical PDE • Finite difference method (FDM) • Finite element method (FEM) • Finite volume method (FVM) • Spectral method • Problem driven research: • Computational Fluid dynamics (CFD) • Computational physics • Computational biology, ……

  19. How to do it well • Three key factors • Master all kinds of different numerical methods • Know and aware the progress in the applied science • Know and aware the progress in PDE or ODE • Ability for a graduate student • Solve problem correctly • Write your results neatly • Speak your results well and clear – presentation • Find good problems to solve

  20. Numerical error • Example 1: • Example 2: • Example 3: • Example 4:

  21. Numerical error • Truncation error or error of the method • Round-off error: due to finite digits of numbers in computer • Numerical errors for practical problems • Truncation error • Round-off error • Model error & observation error & empirical error etc.

  22. Absolute error • Absolute error: • Absolute error bound (not unique!!):

  23. Relative error • An example: • Relative error: • Relative error bound:

  24. Absolute error bounds for basic operations • Suppose • Error bounds

  25. Significant digits • An example • Definition: n significant digits • Method: • Write in the standard form • Count the number of digits after decimal

  26. Error spreading: An example • Algorithm 1: • Use 4 significant digits for practical computation • Results

  27. Error spreading: An example • Algorithm 2 • Result • Truncation error analysis

  28. Convergence and its rate • Numerical integration • Exact solution

  29. Numerical methods • Composite midpoint rule • Composite Simpson’s rule • Composite trapezoidal rule • Error estimate

  30. Numerical results

  31. Numerical errors

  32. Observations • Before h0 • Truncation error is too large !! • After h1 • Round-off error is dominated!! • Between h0 and h1 • Clear order of accuracy is observed for the method • We can observe clear convergence rate for proper region of the mesh size!!!

  33. Numerical Differentiation • Numerical differentiation • The total error

  34. Numerical Differentiation

  35. Numerical Differentiation • Total error depends • Truncation error: • Round-off error: • Minimizer of E(h): • Double precision: • Clear region to observe truncation error:

  36. How to determine order of accuracy • Numerical approximation or method • How to determine p and C?? • By plot log E(h) vs log h

  37. How to determine order of accuracy • By quotation

More Related