Course Information

Lecturer:

Associate Professor Timothy Moroney, Queensland University of Technology

Synopsis:

Sparse matrices arise in a great many applications across science, engineering, statistics, business, and beyond.  Depending on the application, these matrices may be very large, with many millions of rows and columns.  Exploiting the sparsity of these matrices is essential for overcoming the scaling on storage and floating point calculations that would otherwise render even problems with dimensions in the thousands utterly impractical to solve.

Typical operations that are required on large, sparse matrices in practice are:

  • Solving linear systems
  • Computing eigenvalues and eigenvectors
  • Computing functions of a matrix (e.g. the matrix exponential)

A remarkably versatile family of numerical methods called Krylov subspace methods can be applied to all three of these operations, and in doing so require only the action of the matrix in the form of matrix-vector products.  By placing such minimal requirements on the means by which a matrix is utilised, these methods facilitate wide-ranging possibilities for how matrices can be stored (or not!) and accessed across modern computational architectures, paving the way for many of today’s high performance codes.

This course will cover Krylov subspace methods for all three problems: linear systems, eigenvalue problems and matrix functions, from their derivation through to efficient numerical implementations.

Course overview:

  • Krylov subspace methods for eigenvalue problems
  • Krylov subspace methods for linear systems
  • Preconditioning
  • Restarting and deflation
  • Jacobian-free Newton-Krylov methods
  • Krylov subspace methods for matrix functions

Teaching approach:

This will be a very hands-on course.  Students will have plenty of time in the computer laboratory to explore these algorithms and their implementations.

Prerequisites:

Students should be familiar with undergraduate linear algebra, and have experience with a high-level programming language (MATLAB will be used throughout the course, but familiarity with any other similar environment will be a suitable prerequisite).

Lecturer Biographies

Iterative methods for sparse matrices

Associate Professor Timothy Moroney, Queensland University of Technology

Tim is Associate Professor in Applied and Computational Mathematics at QUT and Senior Fellow of the Higher Education Academy, with more than a decade of experience in teaching and research in computational linear algebra.  His research interests include designing preconditioners for large linear systems, efficient computation of matrix functions on parallel architectures, and applications of large, sparse linear and nonlinear problems in solving PDEs.  Tim is the 2017 David Gardiner QUT Teacher of the Year, and has been previously recognised with multiple Vice-Chancellor’s awards for teaching and an Australian Awards for University Teaching citation nomination.

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