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

Syllabus

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 http://math.nyu.edu/faculty/shelley/Classes/SciComp/BindelGoodman.pdf  (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 http://math.nyu.edu/faculty/shelley/Classes/SciComp/BindelGoodman.pdf  (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 http://math.nyu.edu/faculty/shelley/Classes/SciComp/BindelGoodman.pdf (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) 

Optimization

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 http://math.nyu.edu/faculty/shelley/Classes/SciComp/BindelGoodman.pdf (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: http://link.springer.com.proxy-um.researchport.umd.edu/book/10.1007%2F978-1-4419-1002-8)

(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, 2018  by Maria Cameron