Maria K. Cameron

AMSC/CMSC 660: Scientific Computing I

HW collection (HWs 1 - 12)

Codes for HW collections: 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

• [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] Randall J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM 2007, ( freely available online via the UMD library), Chapter 9, Chapter 10, Appendix E


• [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

• Line search methods (steepest descend, Newton's, BFGS, SR1)


• Trust region methods (Newton's, BFGS, SR1)


• Nonlinear conjugate gradient methods (Fletcher - Reeves, Polak - Ribiere)

Refs:

• [1] J. Nocedal and S. Wright, "Numerical Optimization" (Chapters 2, 3, 4, 5)

Ordinary Differential Equations

• Consistency, Stability Convergence


• Linear Stability Theory


• Runge-Kutta Methods


• Multistep Methods (Adams, BDF)


• Symplectic Methods for Integrating Hamiltonian systems

• [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 1, Lecture 2, Lecture 3, Lecture 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 symplecticmethods: SymplecticMethods.pdf

Monte-Carlo Methods

• Basic statistics: random numbers, pseudo-random numbers


• Mean, variance, central limit theorem


• Monte-Carlo Integration, convergience (Codes: MCint.m, MCnsphere.m)


• Variance reduction, importance smapling


• Metropolis and Metropolis-Hastings algorithms


• Simulated annealing

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

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