MA5233: Computational Mathematics

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MA5233 Computational Mathematics

• Weizhu Bao
• Department of Mathematics
• Center for Computational Science and
  Engineering
• National University of Singapore
• Email bao_at_math.nus.edu.sg
• URL http//www.math.nus.edu.sg/bao

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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,

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Dynamics of soliton in quantum physics
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Wave interaction in plasma physics
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Wave interaction in particle physics
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Vortex-pair dynamics in superfluidity
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Vortex-dipole dynamics in superfluidity
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Vortex lattice dynamics in superfluidity
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Vortex lattice dynamics in BEC
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Computational Science

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

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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

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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,.

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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,

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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

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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

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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/top
  ten.pdf

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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

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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,

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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

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Numerical error

• Example 1
• Example 2
• Example 3
• Example 4

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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.

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Absolute error

• Absolute error
• Absolute error bound (not unique!!)

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Relative error

• An example
  
• Relative error
• Relative error bound

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Absolute error bounds for basic operations

• Suppose
  
• Error bounds

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Significant digits

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

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Error spreading An example

• Algorithm 1
• Use 4 significant digits for practical
  computation
• Results

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Error spreading An example

• Algorithm 2
• Result
• Truncation error analysis

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Convergence and its rate

• Numerical integration
  
• Exact solution

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Numerical methods

• Composite midpoint rule
• Composite Simpsons rule
• Composite trapezoidal rule
• Error estimate

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Numerical results
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Numerical errors
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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!!!

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Numerical Differentiation

• Numerical differentiation
• The total error

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Numerical Differentiation
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Numerical Differentiation

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

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How to determine order of accuracy

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

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How to determine order of accuracy

• By quotation