Scottish Numerical Methods Network 2019
Fourth workshop: University of Strathclyde, 27 September 2019
Iterative Methods for Partial Differential Equations
09:30-10:00 Registration and coffee
10:05-11:05 Zdenek Strakoš (Charles University in Prague)
Krylov subspace methods, minimal polynomials and clustering of eigenvalues: intriguing relationships and motivating challenges [abstract]
11:05-11:45 John Pearson (Edinburgh)
Fast Solvers and Multilevel Circulant Preconditioners for Fractional Differential Equation Constrained Optimization [abstract]
11:45-12:25 Iain Smears (UCL)
Time-parallel iterative solvers for parabolic evolution equations [abstract]
12:25-14:20 Lunch break
14:20-15:00 Niall Bootland (Strathclyde)
On coarse spaces for solving the heterogeneous Helmholtz equation with domain decomposition methods [abstract]
15:00-16:00 Paola Antonietti (Politecnico di Milano)
Fast solution techniques for high-order Discontinuous Galerkin method on polygonal/polyhedral grids [abstract]
Zdenek Strakoš, Krylov subspace methods, minimal polynomials and clustering of eigenvalues: intriguing relationships and motivating challenges
Since Krylov subspace methods can be formulated using polynomials satisfying certain orthogonality/optimality properties, it is natural to use in the description of their behaviour arguments involving minimal polynomials of the system matrix and clustering of the matrix eigenvalues. Nonlinearity of the underlying phenomena however makes the relationships very subtle and the argumentation intriguing. This contribution will attempt to review the topic with presenting the circumstances where the minimal polynomial and eigenvalue clustering arguments provide the rigorously justified insight. On the other hand, we present limitations and theoretical challenges related their general use. They can motivate further research and lead to a step towards better understanding of the Krylov subspace methods behaviour.
John Pearson, Fast Solvers and Multilevel Circulant Preconditioners for Fractional Differential Equation Constrained Optimization
It is well recognised that multilevel Toeplitz and multilevel circulant matrices arise in a range of mathematical applications. In particular, when seeking numerical solutions of partial differential equation and fractional differential equations, it is often valuable to apply multilevel circulant preconditioners to accelerate iterative methods. In this talk we discuss the advantages of exploiting multilevel circulant properties of matrices arising in certain structured optimization problems, specifically where fractional differential equations themselves act as constraints.
Iain Smears, Time-parallel iterative solvers for parabolic evolution equations
Time-parallel solvers are iterative methods for the discrete systems originating from evolution problems, and they can provide effective alternatives to the sequential time-stepping approach on parallel computers. Although time-parallel methods can applied to classical time-discretisation schemes, they are based on a global space-time formulation of the problem that is similar to the approach adopted in recent space-time discretisation methods. We present how the inf-sup stability of space-time formulations of continuous and discretized parabolic problems provides an effective approach to the construction and rigorous analysis of parallel-in-time solvers. Considering the implicit Euler discretization of a general linear parabolic evolution equation with time-dependent self-adjoint spatial operators, we show that the standard nonsymmetric time-global system can be equivalently reformulated as an original symmetric saddle-point system that remains inf-sup stable in the same norms. We then propose and analyse an inexact Uzawa method for the saddle-point reformulation based on an efficient parallel-in-time preconditioner. The preconditioner is non-intrusive and easy to implement in practice, and we prove robust spectral bounds, leading to convergence rates that are independent of the number of time-steps, final time, or spatial mesh sizes. Large scale parallel numerical experiments demonstrate the efficiency of the method.
Niall Bootland, On coarse spaces for solving the heterogeneous Helmholtz equation with domain decomposition methods
The development of effective solvers for high frequency wave propagation problems, such as those described by the Helmholtz equation, presents significant challenges. One promising class of solvers for such problems are parallel domain decomposition methods, however, an appropriate coarse space is typically required in order to obtain robust behaviour (scalable with respect to the number of domains, weakly dependant on the wave number but also on the heterogeneity of the physical parameters). In this talk we introduce a coarse space based on generalised eigenproblems in the overlap (GenEO) for the Helmholtz equation. Numerical results within FreeFEM demonstrate convergence that is effectively independent of the wave number and contrast in the heterogeneous coefficient as well as good performance for minimal overlap.
Paola Antonietti, Fast solution techniques for high-order Discontinuous Galerkin method on polygonal/polyhedral grids
In this talk, we present a survey of fast solution algorithms for high-order discontinuous Galerkin finite element methods which employ general polygonal/polyhedral elements. In particular, we present and analyse a family of multigrid schemes on nested and non-nested agglomerated meshes as well as Schwarz-type domain decomposition preconditioners. The performance of the propose solvers will be tested on both two- and three-dimensional test cases. The design of efficient quadrature rules for the numerical approximation of integrals of polynomial functions over general polytopic elements that do not require the explicit construction of a sub-tessellation into triangular/tetrahedral elements will also be discussed.