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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

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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
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, ……
computational science10
Computational Science
  • Computational Mathematics – Scientific computing/numerical analysis
  • Computational Physics
  • Computational Chemistry
  • Computational Biology
  • Computational Fluid Dynamics
  • Computational Enginnering
  • Computational Materials Sciences
  • ……...
steps for solving a practical problems
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
computational mathematics
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,….
history
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, ……
milestone algorithms
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
milestone algorithms15
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
top 10 algorithms
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

contents
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
contents18
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, ……
how to do it well
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
numerical error
Numerical error
  • Example 1:
  • Example 2:
  • Example 3:
  • Example 4:
numerical error21
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.
absolute error
Absolute error
  • Absolute error:
  • Absolute error bound (not unique!!):
relative error
Relative error
  • An example:
  • Relative error:
  • Relative error bound:
significant digits
Significant digits
  • An example
  • Definition: n significant digits
  • Method:
    • Write in the standard form
    • Count the number of digits after decimal
error spreading an example
Error spreading: An example
  • Algorithm 1:
    • Use 4 significant digits for practical computation
    • Results
error spreading an example27
Error spreading: An example
  • Algorithm 2
    • Result
    • Truncation error analysis
convergence and its rate
Convergence and its rate
  • Numerical integration
  • Exact solution
numerical methods
Numerical methods
  • Composite midpoint rule
  • Composite Simpson’s rule
  • Composite trapezoidal rule
  • Error estimate
observations
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!!!
numerical differentiation
Numerical Differentiation
  • Numerical differentiation
  • The total error
numerical differentiation35
Numerical Differentiation
  • Total error depends
    • Truncation error:
    • Round-off error:
    • Minimizer of E(h):
    • Double precision:
    • Clear region to observe truncation error:
how to determine order of accuracy
How to determine order of accuracy
  • Numerical approximation or method
  • How to determine p and C??
    • By plot log E(h) vs log h