Math 270B (Numerical Approximation and Nonlinear Equations)

Course Topics: Numerical Approximation and Nonlinear Equations
Instructor: Prof. Michael Holst (5739 AP&M, mholst@math.ucsd.edu; Office Hours: MW 2-3pm)
Term: Winter 2013
Lecture: 11:00a-11:50a MWF, APM 2402
TA: Shi (Fox) Cheng (5768 AP&M, scheng@ucsd.edu; Office Hours: Tue-Thu 3-4pm)
Main Class Webpage: http://ccom.ucsd.edu/~mholst/teaching/ucsd/270b_w13/index.html
Textbook(s): Theoretical Numerical Analysis: A Functional Analysis Framework, Second Edition, by K. Atkinson and W. Han. Springer-Verlag, New York.


CATALOG DESCRIPTION: 270B. Numerical Approximation and Nonlinear Equations (4)
Iterative methods for nonlinear systems of equations, Newton’s method. Unconstrained and constrained optimization. The Weierstrass theorem, best uniform approximation, least-squares approximation, orthogonal polynomials. Polynomial interpolation, piecewise polynomial interpolation, piecewise uniform approximation. Numerical differentiation: divided differences, degree of precision. Numerical quadrature: interpolature quadrature, Richardson extrapolation, Romberg Integration, Gaussian quadrature, singular integrals, adaptive quadrature. Prerequisites: Math. 270A or consent of instructor.


COURSE INFORMATION: Many of the advances of modern science have been made possible only through the sophisticated use of computer modeling. The mathematical foundation of the computer modeling techniques now used in all areas of mathematics, engineering, and science is known as numerical analysis (sometimes referred to as computational mathematics, numerical mathematics, or scientific computing). The Math 270ABC series at UCSD provides a graduate level overview of some of the foundation topics in numerical analysis. Math 270A deals with various aspects of numerical linear algebra, including direct methods for solving systems of equations involving dense and sparse matrices, iterative methods for solving linear systems, and methods for solving eigenvalue problems. Math 270B focuses on numerical methods for solving nonlinear equations and optimization problems, and on classical and modern approximation theory as needed for analyzing numerical methods. Math 270C concentrates on bringing the tools from 270A and 270B together to develop numerical methods for solving initial value problems (IVPs) and boundary value problems (BVPs) in ordinary differential equations (ODEs). The topics in 270A and the first half of 270B may be viewed as preparation for Math 271ABC (numerical optimization), whereas the topics in the entire series, with emphasis on the topics in the second half of 270B and the topics in 270C, may be viewed as preparation for 272ABC (numerical PDE) and 273ABC (advanced techniques in computational and applied mathematics).


GRADES, HOMEWORKS, EXAMS, AND IMPORTANT DATES: Course information, such as any homework assignments given out, exam dates, and so forth, will be maintained on this course webpage. Note that I sometimes make changes to the lecture schedule and homework assignments as the quarter progresses, depending on how far I get each week in the lectures. Therefore, check this webpage regularly. I will periodically (about every 2-3 weeks) give out homeworks to help you prepare for the final exam. The last week I will finish any remaining material, and then focus on reviewing the material from the class for the final exam. The final exam will be based on the material I cover in class (a subset of the textbook), supplemented with some background material I will lecture on at the beginning of the course (most of which is also in the textbook).

Important dates:
  • Review for Final (Optional): Sat 3/16/2013, 2:00p-4:00p, TA Office (5768 AP&M; may move to one of the lecture halls nearby)
  • Final Exam Time/Place: Mon 3/18/2013, 11:30a-2:29p, 2402 AP&M
  • NOTE: You ARE allowed to bring a 1-page crib sheet (both sides) to the 270B final. HOWEVER, you will not be allowed to use a crib sheet on the qual exam in the Spring that covers this same material (270B is approximately 1/3 of the qual).
Homeworks: I will give out some homework problems that will be very similar to problems that will appear on the final exam (and later the qual exam). The TA will be available to help you work the problems as needed. We will collect and mark some of the homework problems to give you feedback, but your grade for the course will be based only on your final examination at the end of the quarter.

Turning in the homeworks: Please just do not give all of your homeworks to the TA toward the end of the quarter; he will not have time to give you any feedback before the final. To make best use of his help, try to do the homeworks within a couple of weeks of their posting, and then give the homeworks to the TA; he will then be able to give you some marked versions back within a week.

Homework Problems: (Assigned problems are exercises appearing in the textbook)
  • HW1 (Chapters 1 and 2, Covering Weeks 1-2):
    - Topic: Linear Spaces and Linear Operators on Normed (and Inner-Product) Spaces
    - Section 1.1: 1.1.1, 1.1.2
    - Section 1.2: 1.2.3, 1.2.4, 1.2.5, 1.2.6, 1.2.7
    - Section 1.3: 1.3.2, 1.3.4, 1.3.8, 1.3.9
    - Section 2.2: 2.2.3, 2.2.7, 2.2.8
  • HW2 (Chapter 5 and Holst Notes, Covering Weeks 3-6):
    - Topic: Nonlinear Equations and Optimization
    - Homework 2 as a PDF file can be found [ here ]
    - NOTE: I handed out type-written notes on optimization to supplement the nonlinear equations material in Chapter 5.
  • HW3 (Chapters 3, Covering Weeks 7-10):
    - Topic: Approximation Theory
    - Homework 3 as a PDF file can be found [ here ]
NOTE: The textbook has quite a bit of material in it that is relevant for advanced topics in numerical analysis, such as the approximation of solutions to partial differential equations. We devote an entire year to this topic in Math 272ABC, and you can view the text as a great background reference for taking that course later. However, in Math 270ABC we are focusing mainly on the solution to linear (270A) and nonlinear (270B) equations, on approximation theory (270B), and on the numerical solution to ordinary differential quations (270C), and therefore will only use the portions of the text that are relevant to what we need on 270ABC. In particular, there is some advanced material on functions spaces (Lp and Sobolev spaces) in Chapter 1 (and later on in Chapter 7), and on operators (compact operators in particular) in Chapter 2 that you are welcome to read, but I will not assign homeworks on this material, and will therefore not put this material on the final or the qual. Our focus in 270B will be on the sections we need from Chapters 1 and 2 that I assign homework problems from as listed above, and then the sections of Chapters 5, 3, and (time permitting) 4 that I will assign homework problems from as listed above.


SCHEDULE OF LECTURES: Our schedule in Math 270B will be roughly as follows. In weeks 1-2 we will cover just part of the material form Chapters 1 and 2, mainly focusing on obtaining background on normed and inner-product spaces, and the basics of linear operators on those spaces. In weeks 3-6 we will cover iterative methods for solving nonlinear systems of equations (primarily Newton's method), methods for unconstrained optimization, and methods for equality-constrained optimization. Our approach will be to develop the theory abstractly in Banach and Hilbert spaces, and then look at specific examples from application areas such as algebraic systems and partial differential equations. In weeks 7-10 we will cover classical approximation theory: abstract approximation theory in Banach and Hilbert spaces, interpolation of real-valued functions of a single real variable by polynomials, numerical differentiation, and numerical quadrature. Time permitting, we will discuss modern topics in approximation theory at the very end of the course, including wavelets and applied harmonic analysis.