University of Manchester

In this presentation we examine the convergence characteristics of a

Krylov subspace solver preconditioned by a new indefinite

constraint-type preconditioner, when applied to discrete systems

arising from low-order mixed finite element approximation of the

classical biharmonic problem. The preconditioning operator leads to

preconditioned systems having an eigenvalue distribution consisting of

a tightly clustered set together with a small number of outliers. We

compare the convergence characteristics of a new approach with the

convergence characteristics of a standard block-diagonal Schur

complement preconditioner that has proved to be extremely effective in

the context of mixed approximation methods.

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In the second part of the presentation we are concerned with the

efficient parallel implementation of proposed algorithm on modern

shared memory architectures. We consider use of the efficient parallel

"black-box'' solvers for the Dirichlet Laplacian problems based on

sparse Cholesky factorisation and multigrid, and for this purpose we

use publicly available codes from the HSL library and MGNet collection.

We compare the performance of our algorithm with sparse direct solvers

from the HSL library and discuss some implementation related issues.

The study of the finite precision behaviour of numerical algorithms dates back at least as far as Turing and Wilkinson in the 1940s. At the start of the 21st century, this area of research is still very active.

We focus on some topics of current interest, describing recent developments and trends and pointing out future research directions. The talk will be accessible to those who are not specialists in numerical analysis.

Specific topics intended to be addressed include

• Floating point arithmetic: correctly rounded elementary functions, and the fused multiply-add operation.
• The use of extra precision for key parts of a computation: iterative refinement in fixed and mixed precision.
• Gaussian elimination with rook pivoting and new error bounds for Gaussian elimination.
• Automatic error analysis.
• Application and analysis of hyperbolic transformations.

The aim of this talk is to give some understanding of the theory of matrix $p$'th roots (solutions to the nonlinear matrix equation $X^{p} = A$), to explain how and how not to compute roots, and to describe some applications. In particular, an application in finance will be described concerning roots of transition matrices from Markov models.

In this talk we discuss the design of efficient numerical methods for solving symmetric indefinite linear systems arising from mixed approximation of elliptic PDE problems with associated constraints. Examples include linear elasticity (Navier-Lame equations), steady fluid flow (Stokes' equations) and electromagnetism (Maxwell's equations).

The novel feature of our iterative solution approach is the incorporation of error control in the natural "energy" norm in combination with an a posteriori estimator for the PDE approximation error. This leads to a robust and optimally efficient stopping criterion: the iteration is terminated as soon as the algebraic error is insignificant compared to the approximation error. We describe a "proof of concept" MATLAB implementation of this algorithm, which we call EST_MINRES, and we illustrate its effectiveness when integrated into our Incompressible Flow Iterative Solution Software (IFISS) package (http://www.manchester.ac.uk/ifiss/).

In the last few years, there has been renewed interest in stochastic

finite element methods (SFEMs), which facilitate the approximation

of statistics of solutions to PDEs with random data. SFEMs based on

sampling, such as stochastic collocation schemes, lead to decoupled

problems requiring only deterministic solvers. SFEMs based on

Galerkin approximation satisfy an optimality condition but require

the solution of a single linear system of equations that couples

deterministic and stochastic degrees of freedom. This is regarded as

a serious bottleneck in computations and the difficulty is even more

pronounced when we attempt to solve systems of PDEs with

random data via stochastic mixed FEMs.

In this talk, we give an overview of solution strategies for the

saddle-point systems that arise when the mixed form of the Darcy

flow problem, with correlated random coefficients, is discretised

via stochastic Galerkin and stochastic collocation techniques. For

the stochastic Galerkin approach, the systems are orders of

magnitude larger than those arising for deterministic problems. We

report on fast solvers and preconditioners based on multigrid, which

have proved successful for deterministic problems. In particular, we

examine their robustness with respect to the random diffusion

coefficients, which can be either a linear or non-linear function of

a finite set of random parameters with a prescribed probability

distribution.