Numerical analysis course syllabi (June 2002)

Syllabi for the numerical analysis courses in the Department of Mathematics at Penn State. The actual syllabus of any course may depend on the instructor; the date next to each course is when that syllabus was written.

Undergraduate courses

MATH/CSE 451. Numerical Computations (June 2002)
MATH/CSE 455. Introduction to Numerical Analysis I (June 2002)
MATH/CSE 456. Introduction to Numerical Analysis II (June 2002)

Graduate courses

MATH/CSE 523 and MATH/CSE 524. Numerical Analysis (June 2002)
MATH/CSE 552. Numerical Solution of Partial Differential Equations (June 2002)
MATH/CSE 556. Finite Element Methods (June 2002)

Undergraduate courses

MATH/CSE 451. Numerical Computations (June 2002)

Brief description

451. (CSE) NUMERICAL COMPUTATIONS (3) Algorithms for interpolation, numerical integration, numerical solution of nonlinear equations, linear systems and ordinary differential equations emphasizing computational properties and implementation. Students may take only one course for credit from MATH 451 and 455. Prerequisites: CMPSC 201C, 201F, or CSE 103; MATH 230 or 231.

Instructions for students

This is an undergraduate course introducing most of the basic and classical numerical algorithms. The course is more focused on the study and implementation of these basic algorithms. The students will have some to complete several computing projects in addition to other homeworks.

More detailed syllabus

Actual syllabus may depend on the instructor.

MATH/CSE 451--Numerical Computations

Computer arithmetic
1. Representation of numbers in different bases
2. Floating point representation
3. Loss of significiance
Numerical solution of nonlinear equations.
1. Bisection method
2. Newton's method
3. Secant method
Polynomial interpolation
1. Lagrange interpolation
2. Errors in Polynomial Interpolation
3. Estimating derivatives and Richardson Extrapolation
Numerical integration
1. Trapezoid rule
2. Romberg Algorithm
3. Adaptive Simpson's quadrature scheme
4. Gaussian quadrature formulae
Direct methods for linear systems.
1. Gaussian elimination.
2. Gaussian Elimination with scaled partial pivoting
3. Tri-diagonal and banded systems
4. LU-Factorization
Piece-wise polynomial interpolation. Splines
1. First degree and second degree splines
2. Natural cubic splines
Numerical solution of ordinary differential equations (ODE)
1. Taylor series methods for ODE
2. Runge Kutta methods
3. Methods for first order systems of ODE

Textbook:

Numerical Mathematics and Computing, by Ward Cheney and David Kincaid, published by Brooks/Cole publishing Company, 2000. ISBN 0-534-35184-0

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MATH/CSE 455. Introduction to Numerical Analysis I (June 2002)

Brief description

455. (CSE) INTRODUCTION TO NUMERICAL ANALYSIS I (3) Floating point computation, numerical rootfinding, interpolation, numerical quadrature, direct methods for linear systems. Students may take only one course for credit from MATH 451 and 455. Prerequisites: CMPSC 201C, 201F, or CSE 103; MATH 220; MATH 230 or 231.

Instruction for students

The course will provide an introduction to the basics of the modern numerical analysis and its techniques when applied to various problems of analysis and algebra. Various numerical techniques and algorithms for some classical problems will be considered. The focus will be on their efficient computer implementation, robustness and reliability. Some essential theoretical properties of these numerical techniques will also be studied in more detail.

More detailed syllabus

Actual syllabus may depend on the instructor.

MATH/CSE 455--Introduction to Numerical Analysis I

Computer arithmetic
1. Floating point numbers and roundoff errors.
2. Absolute and relative errors: Loss of significance.
3. Conditioning: Stable and unstable computations.
Numerical methods for nonlinear equations:
1. Bisection, Newton's and Secant methods.
2. Fixed point iterations and convergence rate.
Direct methods for systems of linear equations
1. Matrices and vectors. Norms, condition numbers and convergence matrices.
2. The Cholesky Factorization.
3. Gaussian elimination with scaled partial pivoting.
4. Direct methods for banded and sparse matrices.
Approximation of functions
1. Polynomial interpolation.
2. Errors in polynomial interpolation.
3. Splines.
Numerical differentiation and integration
1. Numerical differentiation
2. Trapezoid rule. Romberg Algorithm.
3. Simpson's rule.
4. Gaussian quadrature formulae.

Textbook:

Numerical Analysis: Mathematics of Scientific Computing, Second Edition, by David Kincaid and Ward Cheney, Brooks/Cole Publishing Co. 1996, ISBN 0-534-33892-5.

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MATH/CSE 456. Introduction to Numerical Analysis II (June 2002)

Brief description

456. (CSE) INTRODUCTION TO NUMERICAL ANALYSIS II (3) Polynomial and piecewise polynomial approximation, matrix least squares problems, numerical solution of eigenvalue problems, numerical solution of ordinary differential equations. Prerequisite: MATH 455.

Instruction for students

This course is a follow-up to Introduction to Numerical Analysis I. It will provide further introduction to the basics of the modern numerical techniques and the supporting mathematical theory.

More detailed syllabus

Actual syllabus may depend on the instructor.

MATH/CSE 456--Introduction to Numerical Analysis II

Polynomial approximation
1. Polynomial approximation. Weierstrass' approximation theorem.
2. Best approximation and least squares method
3. Trigonometric interpolation and Fast Fourier Transform.
Iterative methods for systems of linear equations.
1. Basic iterative methods for linear systems.
2. Convergence of the basic iterative methods.
Numerical methods for eigenvalue problems
1. Power method.
2. Schur theorem.
3. The symmetric and non-symmetric QR algorithm.
Numerical methods for ordinary differential equations
1. Taylor series methods for ODEs.
2. Runge--Kutta methods for ODEs.
3. Local and global errors: stability.
4. Boundary value problems. Finite differences.
5. Variational principle and introduction to finite element method.

Textbook:

Numerical Analysis: Mathematics of Scientific Computing, Second Edition, by David Kincaid and Ward Cheney, Brooks/Cole Publishing Co. 1996, ISBN 0-534-33892-5.

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Graduate courses

MATH/CSE 523 and MATH/CSE 524. Numerical Analysis (June 2002)

Brief description

523. NUMERICAL ANALYSIS I. Approximation and Interpolation; Numerical Quadrature; Direct Methods of Numerical Linear Algebra; Numerical solution of nonlinear systems and optimization

524. NUMERICAL ANALYSIS II. Numerical Solution of Ordinary Differential Equations; Numerical Solution of Partial Differential Equations; Some Iterative Methods of Numerical Linear Algebra

Instruction for students

These courses form a two-semester graduate level introduction to numerical analysis. They will be mainly focused on the design and the analysis of classical as well as recently developed numerical algorithms and techniques, for the solution of variety of problems in mathematial analysis and algebra. In short, this course will provide an introduction to the basics of the mathematical theory behind scientific and engineering computing. The students who take this course should have a very good and stable knowledge of single and multivariable calculus, linear algebra and be familiar with basic facts from functional, real and complex analysis and the theory of partial differential equations.

More detailed syllabus

Actual syllabus may depend on the instructor.

\subsectionMATH 523--First semester of two semester graduate course. Numerical analysis I

Approximation and Interpolation; Numerical Quadrature; Direct Methods of Numerical Linear Algebra; Numerical solution of nonlinear systems and optimization
1. Polynomial Approximation
2. Lagrange Interpolation
3. Least Squares Polynomial Approximation
4. Piecewise polynomial approximation and interpolation
5. The Fast Fourier Transform
6. Multipole method for special dense matrix vector product
Numerical Quadrature
1. Basic quadrature
2. The Peano Kernel Theorem
3. Richardson Extrapolation
4. Asymptotic error expansions
5. Romberg Integration
6. Gaussian Quadrature
7. Adaptive quadrature
8. Monte Carlo methods for higher dimensional integrals.
Direct Methods of Numerical Linear Algebra
1. Triangular systems
2. Gaussian elimination and LU decomposition
3. Pivoting
4. Backward error analysis
5. Conditioning and roundoff errors.
Numerical solution of nonlinear systems and optimization
1. One-point iteration
2. Newton's method
3. Unconstrained minimization
4. Newton's method
5. Line search methods
6. Conjugate gradients

\subsectionMATH 524--Second semester of two semester graduate course. Numerical analysis II

Numerical Solution of Ordinary Differential Equations
1. Euler's Method
2. Linear multistep methods
3. One step methods
4. Stiffness
Numerical Solution of Partial Differential Equations
1. BVPs for 2nd order elliptic PDEs
2. The five-point discretization of the Laplacian
3. Finite element methods
4. Difference methods for the heat equation
5. Difference methods for hyperbolic equations
6. Hyperbolic conservation laws
Some Iterative Methods of Numerical Linear Algebra
1. Classical iterations
2. Multigrid methods

Textbook

Analysis of Numerical Methods, by Eugene Isaacson and Herbert Bishop Keller; Dover Publications 1994.
Introduction to Numerical Analysis, by J. Stoer and R. Bulirsch; Springer-Verlag 1980. ISBN 0-387-90420-4.

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MATH/CSE 552. Numerical Solution of Partial Differential Equations (June 2002)

Brief description

552. (CSE 552) Numerical Solution of Partial Differential Equations (3) Finite difference methods for elliptic, parabolic, and hyperbolic differential equations. Solutions techniques for discretized systems. Finite element methods for elliptic problems. Prerequisite: MATH 402 or 404; MATH (CSE) 451 or 456.

Instructions to students

This is an introductory graduate course on numerical methods for partial differential equations. It is designed mainly for graduate students in department of mathematics. The course should also be appropriate for non-math graduate students who are good at advanced calculus and linear algebra. The students are required to do computing projects in addition to theoretical homework problmes.

All the graduate students in computational and applied math program are required to take this course.

MATH/CSE 552--More detailed syllabus

A review of basic methods for the Poisson Equations on regular domain (1 week)
- The finite difference method
- The finite element method
- The finite volume method
Second order elliptic boundary value equations (5 weeks)
1. A review of qualitative properties (2 hour)
  - Maximal principle
  - Existence and uniqueness (existence of classical and weak solutions)
  - Regularity (H² regularity of the solutions for smooth or convex domain)
2. The finite difference method (4 hours)
  - Basic finite difference schemes
  - Discrete maximal principle and M-matrices
  - Error estimates
  - Broundary treatments
3. The finite element method (5 hours)
  - Linear finite element methods
  - Error estimates: estimates in H¹ norm and L² norm
  - Construction of more general finite element methods
4. The finite volume method (3 hours)
  - Basic finite volume schemes
  - Conservation properties
  - Relation with finite element method
  - Error estimates
  - High order finite volume method
Direct and iterative methods for solving the discrete systems (2 weeks)
1. Direct methods (1)
2. Sparse matrix data structure (1)
3. Basic iterative methods (2)
  - Basic iterative methods
  - Jacobi and Gauss-Seidel methods
4. The method of subspace corrections and its convergence properties (2)
5. Conjugate gradient methods and preconditioning (2)
The multigrid method (2 weeks)
1. Introduction of the algorthm using one dimensional problem (2 hours)
2. Algorithmic details for /-cycle, \-cycle, (2 hours) V-cycle and W-cycle algorithms
3. Convergence analysis using the method of subspace corrections (2 hours)
Parabolic and hyperbolic problems (4 weeks)
1. Model problems and stability estimates (2 hours)
2. Examples of the methods of lines (2 hours)
3. The Lax-Richtmyer equivalence theorem (1 hour)
4. Stability analysis (2 hours)
  - Discrete Fourier series (.5)
  - von Neumann stability analysis (.5)
  - The Kreiss matrix theory (1)
5. Consistency, convergence and error estimates (1)
Convection dominated problems (1 week)
1. The failure of standard discretization
2. Monotone schemes and Godunov theorem
3. Higher order methods
4. Nonlinear problems

Textbooks and references

There are many text books available, but there is no single one that would fit the aforementioned syllabus.

Major references

Hackbusch, W., Elliptic differential equations : theory and numerical treatment Berlin ; New York : Springer-Verlag, c1992.
good reference for elliptic problems
Strikwerda, John C., Finite difference schemes and partial differential equations / John C. Strikwerda. Pacific Grove, Calif. : Wadsworth & Brooks/Cole Advanced Books & Software, c1989.
good reference for linear parabolic and hyperbolic problems
Johnson, Claes, Numerical solution of partial differential equations by the finite element method / Claes Johnson. Cambridge [Cambridgeshire] ; New York: Cambridge University Press, c1987.
Overall good reference for finite element method for this course; but not enough materials for theoretical stuffs.
Susanne Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, New York : Springer-Verlag, c1994.
Chapter 3 is a good reference on the construction of a finite element space; other parts of the book are too theoretical/technical for this course
Jinchao Xu, Lecture notes for MATH/CSE 552
covers materials that can not be found in the above three books and other major text books, especially good materials for iterative and multigrid methods, finite volume methods

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MATH/CSE 556. Finite Element Methods (June 2002)

Brief description

556. (CSE 556) Finite Element Methods (3) Sobolev spaces, variational formulations of boundary value problems; piecewise polynomial approximation theory, convergence and stability, special methods and applications. Prerequisite: Math 412, MATH 501 and 502, or consent from instructor.

Instructions to students

This is a graduate course on finite element method and theory for partial differential equations. It is designed for graduate students in department of mathematics. The students who take the course should have a good knowlege of multivariable caclulus, real, comples and functional analysis, linear algebra and basic knowlege of partial differential equations.

More detailed syllabus

Actual syllabus may depend on the instructor.

Preliminaries
- Linear algebra
- Sobolev spaces
- Bramble-Hilbert Lemma
- Babuska-Brezzi theory
Definition of finite element triple
H(id)=L² elements
H(grad)=H¹ elements and second order elliptic boundary value problems
H(curl) elements and Maxwell equations
H(div) elements and and mixed finite element methods
Exact sequence relating H(grad), H(curl), H(div) and H(id) elements; Helmholtz decomposition
H² elements and biharmonic equations
Non-conforming elements
Generalized finite element method; Mortar elements
Stokes equations and Navier-Stokes equations
Multigrid methods and convergence analysis

References

P. Ciarlet, The finite element method for elliptic problems North-Holland, Amsterdam 1978 (it was recently published by SIAM)
Preliminaries, finite element triple, H(grad) and H² elements, non-conforming elements
Vivette Girault, Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations : theory and algorithms, Berlin ; New York : Springer-Verlag, c1986.
H(curl) and H(div) elements
Jinchao Xu, Lecture notes for MATH/CSE 552
multigrid methods

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