Description of the Workshop

The application of state-of-the-art iterative methods from numerical linear algebra has enabled the fast and efficient solution of a range of problems from crucial real-world applications. For the past few decades, Krylov subspace methods such as the method of Conjugate Gradients and GMRES (along with multigrid methods) have been workhorse algorithms for solving the linear systems of equations arising in computational sciences and engineering. Many such problems can be treated using approximation techniques wherein the core computational cost comes from the solution of a linear matrix equation whose coefficient matrix is often hyper-scale (with the number of unknowns being in the tens of millions or more) and sparse, thus beyond the reach of all but the most sophisticated computational algorithms. This linear system generally arises from a process of discretization, whereby the underlying continuum, infinite-dimensional problem is mapped to a discrete, finite dimensional problem whose solution, when mapped back to continuum space, provides a good approximation to the solution of the original problem.

A large body of work focuses on analyzing all aspects of the convergence behavior of Krylov subspace methods, developing effective preconditioners for them, and designing new methods. Much of this work has been done from the perspective of the finite dimensional problem, where one implicitly considers the discretization to have no role in the discussion of the iterative methods or how one judges the method to have converged (i.e., in what norm we measure convergence). The majority of the work in the literature takes this approach, but it is by no means universally so. For example, the development of a preconditioner is sometimes guided by properties of the underlying continuous problem or carried out specifically to obtain preconditioned linear systems whose condition number is independent of the discretization mesh. Furthermore, there has been an effort by some authors more recently to approach Krylov subspace methods directly from this continuum perspective. One can then reconsider what the role and interpretation of the preconditioner is and its relationship to the scalar product, with respect to the underlying continuum problem and physics.

The view that the underlying continuous problem should be borne in mind here is an exciting alternative advocated by some authors, which opens up new avenues through which to analyze existing methods and preconditioners and provide insight into developing new ones. The purpose of this workshop is to advance this way of thinking about iterative methods to younger researchers while bringing together current leaders in the field to have fruitful discussions and share their latest research. This workshop will combine introductory lectures on interrelated subtopics from

  • those in the field who have been leading the development of iterative methods from the continuum-level point-of-view
  • those who have developed new analytic insights into the behavior of these algorithms
  • developers of preconditioners based on the structure of the continuum-level problem
  • those who have proposed new algorithms based on this continuum level point-of-view.

Featured Speakers

  • Victorita Dolean (University of Strathclyde/Nice) read more
    Reader in applied mathematics at the University of Strathclyde working, applied mathematician/computational scientist working in the fields of: numerical analysis and scientific computing with relevant experise in domain decomposition methods for systems of PDEs and design of multi-domain parallel algorithms as well as having made significant contributions in many other fields.
  • Maya Neytcheva (Uppsala University) read more
    Professor of high-performance computing at Uppsala University, Sweden with expertise in Iterative methods for solving discretized partial differential equations – elliptic and parabolic (time-dependent) PDEs, convection-diffusion problems, Stokes and Navier-Stokes problems and indefinite systems, linear elasticity and visco-elasticity, Helmholtz equations, robust preconditioners.
  • Catherine Powell (University of Manchester) read more
    Reader in applied mathematics at the University of Manchester with expertise in numerical analysis (linear algebra, fast solvers, error estimation) related to the numerical solution of partial differential equations (PDEs) with particular interest in in the numerical solution of PDE models with uncertain/random inputs, and efficient algorithms for uncertainty quantification (UQ).
  • Zdenek Strakoš (Charles University Prague) read more
    Professor of mathematics at Charles University in Prague with with many contributions in numerical analysis and iterative matrix-free methods, particularly Krylov subspace methods.
  • Walter Zulehner (Johannes Kepler University, Austria) read more
    Associate professor of mathematics at Johannes Kepler University in Linz, Austria with research interests in numerical analysis and numerical partial differential equations and, in particular, in the development of preconditioners using properties of the underlying continuum PDE model.

Updates

Registration and financial support

Download important dates into your calendar here!

We invite all mathematicians interested in attending the event to fill in an online registration form (see below). That form will be available until May 20, 2019. We encourage all participants to fill it in as soon as possible; this would be very helpful for optimal planning of the event.

For early-career participants, there will be the possibility to present a poster or (for a limited number) give a talk.  The deadline for registration to have your abstract considered is April 19, 2019. In the registration form, there is the possiblity to give a title and abstract, for those who are interested.  Decisions about who will give talks versus present posters will be made by the organizing committee.

We have funding available to support early-career participants; preference is given to participants who present a poster or give a contributed talk. If you wish to apply for financial support, we require you to fill in the form and submit your CV and list of publications together with estimate of required funding before April 19, 2019 to beyonddiscrete2019@maths.tcd.ie. We shall get in touch with all participants who requested funding soon after that.