To help people communicate with each other, there is a
Course Forum.
Please check this regularly since I will also post announcements
there.
If you have questions or problems with the homework or notes, please
post them rather than emailing them to me. This way everyone can
see them.
Upload Process:
The file upload page is located at
http://www.math.umd.edu/upload
.
The student will select the instructor and class for uploads, fill
out their name, email address, choose the file to upload, and add any
comments.
The size limit on uploaded files is currently 5 MB.
The uploaded filenames have the following format:
FirstnameLastname-YYYYMMDDHHMM.ext. The file extension will match the
extension of the original uploaded file.
Prerequisite:
AMSC/CMSC 666 or permission of instructor.
Course Description:
Nonlinear systems of equations, ordinary differential equations,
boundary value problems.
I. Nonilinear Systems (Newton's Method and Variations)
Nonlinear systems of equations; Existences and uniqueness;
Newton's method in 1D; Order of convergence; Modification of Newton's
method that are globally convergent;
Multivariable calculus; Frechet and Gateaux derivativies;
Integration of vector-valued functions;
Neumann series; Contraction mapping theorem; Inverse function theorem;
Fixed point iterations;
Newton's method in high-D and convergence theorem; Chord iteration;
Secant method;
Broyden's method; Inverse Broden mehtod;
II. Numerical Optimization
unconstrained minimization, some preliminaries (gradient, Hessian, etc.)
Newton's method
line search methods: Golden section search, steepest descent methods and conjugate gradient methods
convergence of conjugate gradient methods
quasi-Newton's methods, Davidson-Fletcher-Powell(DFP) method,
Broyden, Fletcher, Gordearb and Shannd(BFGS) method, Broyden family,
nonlinear conjugate gradient method (Fletcher-Beeves, Polak-Ribiere),
Memoryless quasi-Newton methods,
constrained optimization; quasi-Newton methods
III. Initial Value Problems for Ordinary Differential Equations
review of ODE theory: existence; uniqueness; stability,
model problems and their stability,
finite-time blow-up for nonlinear equations, revised existence results,
solution to first-order linear equation, general solutions to a
linear homogeneous high-order ODE, Gronwall inequality, a lemma
Euler's method: derivation, truncation error and consistency, convergence
and error estimates, numerical stability and rounding errors, asymptotic expansion
linear multistep methods: some examples, general definition: local discretization
error; consistency; convergence; and stability
linear multistep methods: necessary and sufficient for consistency and for
order-m discretization error, example of high order but divergent method,
a convergence theoreom
One step methods
Runge-Kutta methods
Stiffness
IV. Numerical Solution of Partial Differential Equations: Finite Difference Methods
boundary value problems of second elliptic problems, five-point
discretization of Laplacian, maximum principle, convergence and
error estimate
initial and boundary value problem for the heat equation, explicit
and implicit time discretization, consistency, stability, and
convergence
second order hyperbolic equations, finite difference methods
hyperbolic equations of conservation laws, upwinding
V. Numerical Solution of Partial Differential Equations: Finite Element Methods
weak formulation, finite elements, interpolation errors
error in energy norm and L2 norm, dual argument