## AMSC/CMSC 660: Scientific Computing I

Homework Collection (HWs 1 - 12)

Codes for HW collection:

Final Exam

Syllabus

Introduction: Computer Arithmetic and Errors

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

Matrix Factorization

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

Nonlinear Systems

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

(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
• 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)

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

(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)

(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)

(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