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.