Numerical Analysis I

AMSC/CMSC 666, Fall 2020


Course Information

Lecture TuTh 2-3:15pm Room 4122 CSIC Bldg. #406
InstructorProfessor Eitan Tadmor
Contacttel.: x5-0648   e-mail:
Office HoursBy appointment (e-mail: )
CSCAMM 4122 CSIC Bldg. #406
Teaching Assistant
Final
Grading40% Homework; 60% Final


Course Description

  1. Approximation Theory.

    1. General overview. Least-squares vs. the uniform norm
    2. Least squares I. Fourier expansions. Chebyshev polynomials
    3. Least squares II. The finite-dimensional case: QR, SVD, PCA (LS rank approximations)
    4. Least squares III. Gauss quadrature
    5. Min-Max approximations. Chebyshev interpolant. Runge effect. error estimates.

  2. Numerical Solution of Initial-Value Problems

    1. General overview. Systems of ODEs. Existence, uniqueness & stability.

    2. Euler's method. Multistep methods -- stability and instability.
      • Predictor-Corrector methods:. Adams-Bashforth-Moulton schemes

    3. Consistency, stability and convergence. Dahlquist theory.

    4. Runge-Kutta methods
      • Local time stepping and error estimates. RK4 and RKF5.
      • Stability and Convergence of Runge-Kutta methods

    5. Strong-Stability Preserving (SSP) schemes. Methods for stiff equations
  3. Iterative Methods for Linear Systems of Equations

    1. General overview. Boundary-value problems. Stiff systems.

    2. Classical methods: Jacobi, Gauss-Siedel, SOR, ...

    3. Energy functionals and gradinet methods
      • Steepest descent. Conjugate gradient. ADI and dimensional splitting.

    4. Non-stationary methods. Local vs. global methods

    5. Preconditioners

    6. Multigrid mthods*; GMRES*

  4. Numerical Optimization

    1. General overview. Fixed point iterations, low-order and Newton's methods

    2. Steepest descent

    3. Line search methods. Trust regions and Wolfe conditions

    4. Newton's and quasi-Newton methods: Rank one/two modifications

    5. Rates of converegence

    6. Nonlinear Least-sqaures. Conjugate gradient, Gauss-Newton.



    References

    GENERAL TEXTBOOKS

    K. Atkinson, An INTRODUCTION to NUMERICAL ANALYSIS, Wiley, 1987

    S. Conte & C. deBoor, ELEMENTARY NUMERICAL ANALYSIS, McGraw-Hill
    User friendly; Shows how 'it' works; Proofs, exercises and notes

    G. Dahlquist & A. Bjorck, NUMERICAL METHODS, Prentice-Hall,
    User friendly; Shows how 'it' works; Exercises

    A. Ralston & P. Rabinowitz, FIRST COURSE in NUMERICAL ANALYSIS, 2nd ed., McGraw-hill,
    Detailed; Scholarly written; Comprehensive; Proofs exercises and notes

    J. Stoer & R. Bulrisch, INTRODUCTION TO NUMERICAL ANALYSIS, 2nd ed., Springer
    detailed account on approximation, linear solvers & eigen-solvers, ODE solvers,..

    APPROXIMATION THEORY

    E. W. Cheney, INTRODUCTION TO APPROXIMATION THEORY
    Classical

    P. Davis, INTERPOLATION & APPROXIMATION, Dover
    Very readable

    T. Rivlin, AN INTRODUCTION to the APPROXIMATION of FUNCTIONS
    Classical

    R. DeVore & G. Lorentz, CONSTRUCTIVE APPROXIMATION, Springer
    A detailed account from classical theory to the modern theory; everything; Proofs exercises and notes

    NUMERICAL INTEGRATION

    F. Davis & P. Rabinowitz, NUMERICAL INTEGRATION,
    Everything...

    NUMERICAL SOLUTION Of INITIAL-VALUE PROBLEMS

    E. Hairer, S.P. Norsett and G. Wanner, SOLVING ODEs I: NONSTIFF PROBLEMS, Springer-Verlag, Berlin. 1991, (2nd ed)
    Everything - the modern version

    A. Iserles, A FIRST COURSE in the NUMERICAL ANALYSIS of DEs, Cambridge Texts

    W. Gear, NUMERICAAL INITIAL VALUE PROBLEMS in ODEs, 1971
    The classical reference on theory and applications

    Lambert, COMPUTATIONAL METHODS for ODEs, 1991
    Detailed discussion of ideas and practical implementation

    Shampine and Gordon, COMPUTER SOLUTION of ODES, 1975
    Adams methods and practial implementation of ODE "black box" solvers

    Butcher, NUMERICAL ANALYSIS of ODEs, 1987
    Comprehensive discussion on Runge-Kutta methods

    (mainly) ITERATIVE SOLUTION OF LINEAR SYSTEMS

    A. Householder, THE THEORY OF MATRICES IN NUMERICAL ANALYSIS
    The theoretical part by one of the grand masters; Outdated in some aspects

    G. H. Golub & Van Loan, MATRIX COMPUTATIONS,
    The basic modern reference

    Y. Saad, ITERATIVE METHODS for SPARSE LINEAR SYSTEMS,
    PWS Publishing, 1996. (Available on line at http://www-users.cs.umn.edu/~saad/books.html)

    R. Varga, MATRIX ITERATIVE ANALYSIS,
    Classical reference for the theory of iterations

    James Demmel, APPLIED NUMERICAL LINEAR ALGEBRA,
    SIAM, 1997

    Kelley, C. T., ITERATIVE METHODS for LINEAR and NONLINEAR EQUATIONS,
    SIAM 1995

    NUMERICAL OPTMIZATION

    J. Nocedal, S. Wright, NUMERICAL OPTIMIZATION
    Springer, 1999

    T. Kelly, ITERATIVE METHODS for OPTIMIZATION
    SIAM
    Eitan Tadmor