Office: MATH 4105. Office Hours: Tue. 11-12, Wed. 3:30 - 4:30
Teaching
AMSC 667 / CMSC 667 : Numerical Analysis II, Tue.-Th. 9:30-10:45 AM, MATH 1308
AMSC 667/ CMSC 667 Syllabus
Numerical Solution of Initial Value Problems (3 weeks)
-- Consistency, stability and convergence analysis
-- Runge-Kutta methods
-- Error estimates and stepsize control
-- Multistep methods
-- Methods for stiff systems
References: J. Strain, [SB] Sections 7.1-7.2, Atkinson Chapter 6.
HW 1, Due Th. Feb. 9: Download J. Strain's Lectures 1,2, and 3. Solve exercises: Lec. 1, #1 ,2; Lec. 2, # 2, 4 (to be done in MATLAB); Lec. 3, #2,3
Handout: Regions of Absolute Stability of Explicit Runge-Kutta Methods
HW 2, Due Th. Feb. 16. Supplementary codes: AMSC667VDPol.m, FindStableCycles.m
HW3, Due Th. Feb. 23. Supplementary codes: RASplot.m, RKmRAS.m
Numerical Solution of Boundary Value Problems (4 weeks)
-- Two-point boundary value problems
-- Finite difference methods
-- Variational formulation and the finite element method
-- Introduction to error analysis
-- Discretization methods for multidimensional problems
References: [SB] Sections 7.4-7.7, [LT] Chapters 4-5.
Iterative Methods (4 weeks)
-- The conjugate gradient method for symmetric positive-systems
-- Preconditioning techniques and relation to stationary iterative methods
(such as Jacobi, Gauss-Seidel, SOR)
-- Application to boundary value problems
-- Convergence analysis
-- Multigrid
-- Krylov subspace methods for indefinite and nonsymmetric systems
References: [SB] Chapter 8, [Saad] Chapters 4, 6, 9, 10, 13,
[Kelley-Iter] Chapters 1-3.
J. Nocedal and S. Wrigth, "Numerical Optimization", Chapter 5: Conjugate Gradient Method
K. W. Morton and D. F. Mayers, "Numerical Solution of PDE's", Chapter 7: Jacobi, Gausss-Seidel, SOR, Multigrid, Convergence analysis
HW6, Due Th. March 15. Supplementary code: AMSC667Mcommittor.m
-- Linear algebra concepts (similarity transformations,
singular value decomposition, Rayleigh quotients)
-- Power and inverse power methods
-- QR algorithm
-- Lanczos and Arnoldi methods
References: [SB] Chapter 6, [Stewart] Chapters 1-2, 5.
James W. Demmel, "Applied Numerical Linear Algebra", Chapters 3, 4, 5, 7 (freely available online via the UMD library)
HW8, Due Th. April 5. Supplementary code: multigrid.m
References:
[Atkinson] K. Atkinson, An Introduction to Numerical Analysis, 2nd Edition,
John Wiley & Sons, 1989.
[Kelley-Iter] C. T. Kelley, Iterative Methods for Linear and Nonlinear
Equations, SIAM, 1995.
[Kelley-Opt] C. T. Kelley, Iterative Methods for Optimization, SIAM, 1999.
[LT] S. Larsson and V. Thomee, Partial Differential Equations with Numerical
Methods, Springer-Verlag, 2005.
[Saad] Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM 2003.
[Stewart] G. W. Stewart, Matrix Algorithms, Volume II: Eigensystems, SIAM
2001.
[SB] J. Stoer and R. Bulirsch, Introduction to Numerical Anaysis, 3rd
Edition, Springer-Verlag, 2002.
AMSC666: Numerical Analysis I Fall 2011, MWF 12:00-12:50, CHE2145
Textbooks:
1. J. Stoer, R. Bulirsch, "Introduction to Numerical Analysis"
2. A. Ralston, P. Rabinowitz, "A First Course in Numerical Analysis"
AMSC666 Syllabus
(1) Approximation Theory
(a) General overview: norms, Weierstrass’ density theorem, Bernstein polynomials.
Notes on Bernstein polynomials and Weierstrass' density theorem
(b) Least Squares Approximations
• A general overview: Gramm mass matrix, ill-conditioning of monomials in L2.
• Generalized Fourier expansions: Bessel’s inequality, Parseval’s equality; orthog-
onal polynomials: Legendre, Chebyshev.
Homework Assignment #1. Due Friday, Sept. 16
Supplementary materials:
- MATLAB code for computing Legendre coefficients
• Discrete expansions
(c) Interpolation
• Lagrange and Newton interpolants: divided differences, equi-distant points, syn-
thetic calculus, forward, backward, and centered formulas.
Homework Assignment #2, Due Friday, Sept. 23
Homework Assignment #3, Due Friday, Sept. 30
• Error estimates: Runge effect, region of analyticity.
• Trigonometric interpolation: FFT, truncation+aliasing, error estimates.
(d) Min-Max approximations: Chebyshev interpolant and relationship
to the min-max polynomial, economization.
(e) Approximation with derivatives and rational approximations:
Hermite interpolation, splines, Pade.
Homework Assignment #4, Due Friday, Oct. 7
Homework Assignment #5, Due Friday, Oct. 14
(2) Numerical Differentiation and Integration
(a) Numerical Differentiation
• Polynomial and spline interpolants - error estimates.
• Richardson extrapolation.
(b) Gaussian quadratures
(c) Numerical integration with equidistant points
• Newton-Cotes formulas, Peano kernel theorem
• Composite Simpson’s Rule.
• Romberg and adaptive integration.
Homework Assignment # 6, Due Friday, Oct. 21
Homework Assignment #7, Due Friday, Oct. 28
Homework Assignment #8, Due Friday, Nov. 4
C-code for the adaptive Simpson rule
(3) Nonlinear systems
(a) Fixed point methods.
(b) Newton's iterations.
(c) Modified Newton's methods, Broyden's method.
(d) Rate of convergence
Literature: J. Nocedal, S. Wright, "Numerical Optimization". Electronic copy is available for free. Search it in the UMD Library.
Homework Assignment #9, Due Monday Nov. 14
Homework Assignment #10, Due Monday Nov. 21
Matlab code for Broyden's method
(4) Optimization
(a) Nonlinear least squares methods.
(b) Steepest descent and conjugate gradient methods.
(c) Newton's methods and quasi-Newton methods (DFP, BFGS).
(d) Rate of convergence.
(e) Gauss-Newton and Levenberg-Marquardt methods.
Homework Assignment #11, Due Monday Nov. 28
Homework Assignment #12, Due Monday Dec. 5
Matlab code and data files: ico2fcc.m, ico38.txt, fcc38.txt
Homework Assignment #13, Due Monday Dec. 12
MATH462. Partial Differential Equations. SPRING 2011
Announcements
Final: Thursday, May 12 (based on Chapters 1, 2, 3, 4, 5 ,6), CHM 2201, 8AM - 10AM
Midterm: March 17 (based on Chapters 1, 2, and 3)
Class Room change: Chemistry building: CHM2201
Office hours: MATH Bldg, room 4105, Tue. 3 - 4, Wed. 11 - 12
HW1, Due Feb. 10th : 1.2 # 3, 11, 13; 1.3 # 1, 5; 1.4 # 3, 6
HW2, Due Feb. 17th : 1.5 # 2, 5; 1.6 # 1, 2, 5, 6
HW3, Due Feb. 24th : 2.1 # 2, 3, 5 (take c=a=1), 6, 10; 2.2 # 5
HW4, Due Mar. 3rd : 2.3 # 4, 6; 2.4 # 9, 10, 15, 18
HW5, Due Mar. 10th : 2.5 # 2; 3.1 # 1, 3; 3.2 # 2 (a=c=1), 3
HW6, Due Mar. 17th : 3.3 # 2, 3; 3.4 # 2, 3, 12 (use any method), 14
HW7, Due Mar. 31st : 2.1 # 8; 2.2 # 6; 2.4 # 19
HW8, Due Apr. 7th : 4.1 # 3, 5, 6; 4.2 # 2, 3, 4
HW9, Due Apr. 14th: 5.1 # 8, 9, 10, 11; 5.2 # 11, 17
HW10, Due Apr. 21st: 5.3 # 3, 5, 6, 10, 15
HW11, Due Apr. 28st: 5.4 # 3, 8, 9, 11, 13, 16
HW12, Due May 5th: 6.1 # 6, 8, 9, 11; 6.3 # 3; 6.4 # 5
Syllabus
Math462. Partial Differential Equations for Scientists and Engineers
Maria Cameron
Spring 2011
APM 0101
Textbook: Walter A. Strauss, Partial differential Equations. An Introduction. Second Edition. John Wiley and Sons, Ltd, 2007
1. Chapter 1. Where PDE’s come from
• What is a Partial Differential Equation?
• First-Order Linear Equations
• Flows, Vibrations, and Diffusions
• Initial and Boundary Conditions
• Well-Posed Problems
• Types of Second order Equations
2. Chapter 2. Waves and Diffusions
• The Wave Equation
• Causality and Energy
• The Diffusion Equation
• Diffusion on the whole line
• Comparison of Waves and Diffusions
3. Chapter 3. Reflections and Sources
• Diffusion on the half-line
• Reflections of waves
• Diffusion with a Source
• Waves with a Source
Midterm Exam
4. Chapter 4. Boundary Problems
• Separation of Variables, the Dirichlet Condition
• The Neumann Condition
5. . Chapter 5. Fourier Series
• The Coefficients
• Even, Odd, Periodic, and Complex Functions
• Orthogonality and General Fourier Series
• Completeness
• The Gibbs Phenomenon
6. Chapter 6. Harmonic functions
• Laplace’s Equation
• Rectangles and Cubes
• Poisson’s Formula
• Circles, Wedges, and Annuli
7. Chapter 14. Nonlinear PDE’s
• The Burgers Equation and Shock waves
• The Korteweg-deVries Equation and Solitons
Final Exam
MATH462. Partial Differential Equations, FALL 2010
Announcements
Final Exam: Monday, Dec. 13, ANS 412, 8AM - 10 AM, cumulative.
Midterm Exam: Thursday, Oct. 14. Based on chapters 1, 2, and 3.
Office hours: MATH Bldg, room 4105, Wed. : 11AM - 1 PM
HW 1: 1.1 # 3; 1.2 # 5,11,13
HW 2: 1.3 # 5; 1.4 # 3, 6; 1.5 # 4; 1.6 # 1
HW 3: 1.6 # 5, 6; 2.1 # 2, 5 (take a=c=1), 6; 2.2 # 3;
HW 4: 2.1 # 10; 2.2 # 5; 2.3 # 6; 2.4 # 9, 10, 16;
HW 5: 2.3 # 4; 3.1 # 1; 3.2 # 2 (a=c=1), 3; 3.3 # 2, 3;
HW 6: 2.1 # 7, 8; 2.2 # 6; 2.3 # 3; 2.5 # 2;
HW 7: 4.1 # 3, 5, 6;
HW 8: 4.2 # 2, 3, 4;
HW 9, 10, DUE TH. NOV. 11: 5.1 # 8, 10, 11; 5.2 # 11; 5.3 # 2, 5, 6, 15
HW 11: 5.4 # 3, 8, 9, 11, 12, 18
HW 12, DUE TH. DEC. 2: 6.1 # 6, 8, 9, 11; 6.3 # 1, 3
HW 13: 14.1 # 3, 5, 8, 10; 14.2 # 1