AMSC/CMSC 660: Scientific Computing I

Homework Collection (HWs 1 - 12)

Codes for HW collection: bisection.m, hybrid.m, LJ7findmin.m, LJ7_NonlinCG_random_initial_conf.m, cat.txt, symplectic_demo.m

Final Exam


Introduction: Computer Arithmetic and Errors

  • Computer numbers
  • Floating point arithmetic
  • Sources of errors
  • Stability and Conditioning

Refs: (1)  Bindel and Goodman, Principles of scientific computing  (Chapter 2)

(2) G.W. Stewart, Afternotes on numerical analysis, SIAM 1996  (Lecture 7) 

Matrix Factorization

  • Matrix Norms
  • Eigenvalues and eigenvectors
  • Singular Value Decomposition
  • Condition numbers
  • LU decomposition
  • Cholesky factorization
  • Least Squares and QR factorization

Refs: (1)  Bindel and Goodman, Principles of scientific computing  (Chapter 4 and 5)

(2)  J. Demmel "Applied Numerical Linear Algebra" 

Nonlinear Systems

  • Newton's method and variants
  • Continuation
  • Globally Convergent Methods 

Refs: (1)  Bindel and Goodman, Principles of scientific computing (Chapter 6)

(2) J. Nocedal and S. Wright, "Numerical Optimization"  (Chapter 11) 

(3) G.W. Stewart, Afternotes on numerical analysis, SIAM 1996 (Lecture 5, Hybrid Method) 


Ordinary Differential Equations

  • Consistency, Stability Convergence
  • Linear Stability Theory
  • Runge-Kutta Methods
  • Multistep Methods (Adams, BDF)
  • Symplectic Methods for Integrating Hamiltonian systems

Refs: (1) John Strain, Lectures on Numerical solutions of ODE  (Consistency, Stability, Convergence, Runge-Kutta methods and multistep methods, linear stability theory,stiff problems)

(2) E. Hairer, S.~P. Norsett, G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Revised Edition, Springer, 2000 

(3) Symplectic methods: Erns Hairer, Geometric Numerical Integration. Lecture 1Lecture 2Lecture 3Lecture 4 Lecture 5.

(4) M. P. Allen's talk in the workshop "Computational methods for statistical mechanics - at the interface between mathematical statistics and molecular simulation", June 2 - 6, 2014, Edinburgh, Scotland 

(5) Lecture notes on symplectic methods: SymplecticMethods.pdf

Monte-Carlo Methods

  • Basic statistics: random numbers, pseudo-random numbers
  • Mean, variance, central limit theorem
  • Monte-Carlo Integration, convergence        Codes:  MCint.mMCnsphere.m
  • Variance reduction, importance sampling
  • Metropolis and Metropolis-Hastings algorithms
  • Simulated annealing                                    

       Code:  traveling_salesman.m  (based on Ref. (4))   

       Solution found by this code: TravelingSalesman.pdf

Refs: (1) Lecture notes: MonteCarlo.pdf (updated 12/8/2015 at 10:58 AM)

(2)  Bindel and Goodman, Principles of scientific computing (Chapter 9)

(3) A. Chorin, O. Hald, Stochastic Tools in Mathematics and Science, Third Edition, Springer, 2013 (2nd edition is also fine, it is available via UMD library:

(4) S. Kirkpatrick; C. D. Gelatt; M. P. Vecchi, Optimization by Simulated Annealing, Science, New Series, Vol. 220, No. 4598. (May 13, 1983), pp. 671-680

(5) David J. Wales, Jonathan P. K. Doye, Global Optimization by Basin-Hopping and the Lowest Energy Structures of Lennard-Jones Clusters Containing up to 110 Atoms,  J. Phys. Chem. A 1997, 101, 5111-5116 

 Copyright 2010, 2015 , 2017 by Maria Cameron