CSE Courses

Engineering at Illinois Engineering at Illinois

CS 450 | CSE 401 | MATH 450 | ECE 491 – Numerical Analysis

Research Projects

Here is a list of possible projects. You are welcome to suggest a project yourself, possibly related to an application arising in your own research. The projects listed can usually be done in various ways, so multiple people can work on different variants of the same project. Most of the projects listed are an indication of what might be done; the exact project will be decided after some discussion with the instructor. More projects may be added, but don't wait too long to decide, as you will need to start in time to complete your project by the end of the term.

Linear Equations

  1. Adaptive GMRES method
  2. Hybrid BiCGSTAB/GMRES method
  3. Experiment with loss of orthogonality in CG
  4. Incomplete LU-type (ILU) preconditioners to improve convergence of iterative methods
  5. Incomplete Choleski (IC) preconditioners to improve convergence of iterative methods
  6. Block-ILU or -IC preconditioners
  7. Multi-grid method (various possibilities)
  8. Test MATLAB's sparse direct solvers

Eigenvalue Problems

  1. Test MATLAB's sparse eigenvalue solver
  2. Implement and test variant(s) of Jacobi-Davidson method

Nonlinear Equations

  1. Robust Newton methods (various possibilities)
  2. Newton-Krylov methods (hybrid Newton-GMRES type method)
  3. Broyden's method or other secant updating methods

Optimization

  1. Simplex method for linear programming
  2. Interior point method for linear programming
  3. Broyden's method or other secant updating methods
  4. Robust Newton methods (various possibilities)

Discretization of ODEs or PDEs

  1. One-dimensional, adaptive discretization of wave equation
  2. Discretization of one-dimensional channel flow
  3. Two-dimensional Galerkin finite element discretization of PDE on quadrilaterals (various possibilities)
  4. Two-dimensional Galerkin finite element discretization of PDE on triangles (various possibilities)
  5. Test various MATLAB ODE methods
  6. Test MATLAB's PDE toolbox

Applications
  1. Medical image reconstruction (various possibilities)
  2. Experiments with pattern formation in biology (various possibilities)
  3. Image processing using SVD, FFT, or wavelets