Rutherford Appleton Laboratory, nr Didcot

We present algorithms for dense LU and QR factorizations that minimize the cost of communication. One of today's challenging technology trends is the increased communication cost. This trend predicts that arithmetic will continue to improve exponentially faster than bandwidth, and bandwidth exponentially faster than latency. The new algorithms for dense QR and LU factorizations greatly reduce the amount of time spent communicating, relative to conventional algorithms.

This is joint work with James Demmel, Mark Hoemmen, Julien Langou, and Hua Xiang.

A scale-invariant moving finite element method is proposed for the

adaptive solution of nonlinear partial differential equations. The mesh

movement is based on a finite element discretisation of a scale-invariant

conservation principle incorporating a monitor function, while the time

discretisation of the resulting system of ordinary differential equations

may be carried out using a scale-invariant time-stepping. The accuracy and

reliability of the algorithm is tested against exact self-similar

solutions, where available, and a state-of-the-art $h$-refinement scheme

for a range of second and fourth order problems with moving boundaries.

The monitor functions used are the dependent variable and a monitor

related to the surface area of the solution manifold.

The convergence of iterative methods for solving the linear system Ax = b with a Hermitian positive definite matrix A depends on the condition number of A: the smaller the latter the faster the former. Hence the idea to multiply the equation by a matrix T such that the condition number of TA is much smaller than that of A. The above is a common interpretation of the technique known as preconditioning, the matrix T being referred to as the preconditioner for A.
The eigenvalue computation does not seem to benefit from the direct application of such a technique. Indeed, what is the point in replacing the standard eigenvalue problem Ax = λx with the generalized one TAx = λTx that does not appear to be any easier to solve? It is hardly surprising then that modern eigensolvers, such as ARPACK, do not use preconditioning directly. Instead, an option is provided to accelerate the convergence to the sought eigenpairs by applying spectral transformation, which generally requires the user to supply a subroutine that solves the system (A−σI)y = z, and it is entirely up to the user to employ preconditioning if they opt to solve this system iteratively.
In this talk we discuss some alternative views on the preconditioning technique that are more general and more useful in the convergence analysis of iterative methods and that show, in particular, that the direct preconditioning approach does make sense in eigenvalue computation. We review some iterative algorithms that can benefit from the direct preconditioning, present available convergence results and demonstrate both theoretically and numerically that the direct preconditioning approach has advantages over the two-level approach. Finally, we discuss the role that preconditioning can play in the a posteriori error analysis, present some a posteriori error estimates that use preconditioning and compare them with commonly used estimates in terms of the Euclidean norm of residual.

Many industrial flow problems, expecially in the minerals and process

industries, are very complex, with strong interactions between phases

and components, and with very different length and time scales. This

presentation outlines the algorithms used in the CFX-5 software, and

describes the extension of its coupled solver approach to some

multi-scale industrial problems. including Population Balance modelling

to predict size distributions of a disperse phase. These results will be

illustrated on some practical industrial problems.