MATH 62263/72263: Numerical Solution of Large Sparse Linear Systems

TIME & PLACE: Tu + Th 2:15 - 3:30 pm, MSB 276.

TEXT:

as well as journal papers distributed in class. The book is available at

A hard copy can be ordered from SIAM at https://www.siam.org/books/ if you so desire.

INSTRUCTOR:

COURSE OBJECTIVES:
• Many computational problems that arise in science and engineering give rise to linear systems of equations with a very large matrix. These problems cannot be solved by using Gaussian elimination or Cholesky factorization, because these methods require too many arithmetic floating point operations to be practical. Also, the storage required by the matrix factors determined by Gaussian elimination or Cholesky factorization can be prohibitive. It is the purpose of this course to discuss numerical methods suitable for the solution of large linear systems of equations. These methods are iterative and do not call for the factorization of the matrix. Instead, they "learn" about the matrix by evaluating matrix-vector products. In particular, only some of the entries of the matrix have to be stored simultaneously. Therefore the storage requirement is modest also when the matrix is very large. The iterative methods are designed to determine an accurate approximation of the solution in a linear space of fairly low dimension. Generally these linear spaces are Krylov subspaces. We will describe properties in terms of oblique or orthogonal projections, and discuss techniques for choosing suitable subspaces. Most of the course will present and analyze properties of state-of-the art iterative methods, including the conjugate gradient method, GMRES, BiCGStab, IDR, and LSQR. The role of preconditioning will be discussed and so will storage schemes for large sparse matrices. Towards the end of the course, we will also discuss some techniques for the computation of a few eigenvalues and associated eigenvectors of a very large matrix, because several of the techniques for the solution of large linear systems of equations also, after minor modifications, can be applied to eigenvalue computation.
CLASS OPERATION:
• Homework will be assigned regularly and collected at the end of each major section. The homework will include programming and applyzing some of the methods discussed. Journal papers that complement the textbook will be distributed. Course assignments will include reading and presenting state-of-the-art papers on iterative methods.
• The programming languages MATLAB and GNU Octave will be used for illustrations, however, students may use their programming language of choice for their homework. MATLAB is available on the the Math/CS Network. It is beneficial for students to have an account on this network. GNU Octave is a public domain language very similar to MATLAB, and can be used for homework assignments. Instructions on how to install GNU Octave on your PC are available here.
REGISTRATION INFORMATION:
• Information can be found at this web site. It is important that everyone registers in time, otherwise late registration fees may have to be paid; click here for further details. Plagiarism, i.e., presenting someone else's work as your own is discussed here. This includes finding the answer of a homework problem in a book or in someone else's assignment, and copying it. Plagiarism, of course, is unacceptable.
STUDENTS WITH DISABILITIES:
• will be provided reasonable accommodations to ensure their equal access to course content. Futher information can be found at this web site.