Judith M. Ford

Computer simulation of grain-boundary diffusion creep

Under externally applied stress, polycrystalline materials may deform as a result of matter diffusing from grain boundaries in compression to those in relative tension. We model the behaviour of a general structure under several different boundary conditions and show how the stress function along each grain boundary may be obtained by solving a system of linear equations. We further show how knowledge of the stresses enables us to predict the position of the grain boundaries after a small time-step δt and use this to simulate changes in the grain structure over a finite time interval. Our simulations indicate that grain-size distribution is of importance with regard to the onset of buckling. We are also able to track the movement of individual grains during deformation and our results show that, in an irregular grain structure, diffusion creep may cause significant grain rotation.

Combining Kronecker Product Approximation with Discrete Wavelet Transforms to Solve Dense, Function-Related Linear Systems

A new solution technique is proposed for linear systems with large dense matrices of a certain class including those that come from typical integral equations of potential theory. This technique combines Kronecker product approximation and wavelet sparsification for the Kronecker product factors. The user is only required to supply a procedure for computation of each entry of the given matrix. nThe main sources of efficiency are the incomplete cross approximation procedure adapted from the mosaic-skeleton method of the second author and data-sparse preconditioners (the incomplete LU decomposition with dynamic choice of the fill-in structure with a prescribed threshold and the inverse Kronecker product preconditioner) constructed for the sum of Kronecker products of sparsified finger-like matrices computed by the discrete wavelet transform. In some model, but quite representative, examples the new technique allowed us to solve dense systems with more than 1 million unknowns in a few minutes on a personal computer with 1 Gbyte operative memory.

Wavelet-based Preconditioners for Dense Matrices with Non-Smooth Local Features

We present algorithms for the detection of local non-smooth features within a dense matrix and show how, by isolating such features, we are able to use wavelet compression to design preconditioners for the corresponding dense linear system. We illustrate our approach with examples from the solution of elastohydrodynamic lubrication problems and boundary integral equations.

A new wavelet transform preconditioner for iterative solution of elastohydrodynamic lubrication problems

We develop a novel preconditioning strategy, based on a non-standard, Discrete Wavelet Transform (DWT), for the dense, non-symmetric, linear systems that must be solved when Newtons method is used in the solution of Elastohydrodynamic Lubrication (EHL) problems. Simple band preconditioners and sparse preconditioners based on standard DWT have been found to be of limited value for EHL problems, since they may be singular, give poor convergence or be expensive to apply. We present algorithms for preconditioner design based on detecting non-smooth diagonal bands within an otherwise smooth matrix and applying a non-standard DWT to compress the part of the matrix away from the band. We illustrate, by numerical examples, the improvements that can be made when our methods are used.

Matrix approximations and solvers using tensor products and non-standard wavelet transforms related to irregular grids

Dense large-scale matrices coming from integral equations and tensor-product grids can be approximated by a sum of Kronecker products with further sparsification of the factors via discrete wavelet transforms, which results in reduced storage and computational costs and also in good preconditioners in the case of uniform one-dimensional grids. However, irregular grids lead to a loss of approximation quality and, more significantly, to a severe deterioration in efficiency of the preconditioners that have been considered previously (using a sparsification of the inverse to one Kronecker product or an incomplete factorization approach). In this paper we propose to use non-standard wavelet transforms related to the irregular grids involved and, using numerical examples, we show that the new transforms provide better compression than the Daubechies wavelets. A further innovation is a scaled two-level circulant preconditioner that performs well on irregular grids. The proposed approximation and preconditioning techniques have been applied to a hypersingular integral equation modelling flow around a thin aerofoil and made it possible to solve linear systems with more than 1 million unknowns in 15–20 minutes even on a personal computer.

Bifurcations in approximate solutions of stochastic delay differential equations

We consider stochastic delay differential equations of the form interpreted in the Ito sense, with Y(t)=Φ(t) for t∈[t0-τ,t0] (here, W(t) is a standard Wiener process and τ>0 is the constant lag, or time-lag). We are interested in bifurcations (that is, changes in the qualitative behavior of solutions of these equations) and we draw on insights from the related deterministic delay differential equation, for which there is a substantial body of known theory, and numerical results that enable us to discuss where changes occur in the behavior of the (exact and approximate) solutions of the equation. Rather diverse components of mathematical background are necessary to understand the questions of interest. In this paper we first review some deterministic results and some basic elements of the stochastic analysis that (i) suggests lines of investigation for the stochastic case and (ii) are expected to facilitate the theoretical investigation of the stochastic problem. We then present the results of numerical experiments that illustrate some of the complexities that arise when considering bifurcations in stochastic delay differential equations. They give prima facie evidence for certain convergence properties of the bifurcation points estimated using the Euler–Maruyama method for the equations considered. We conclude by drawing attention to a number of open questions in the field.

Flexible parallelization of fast wavelet transforms

In this paper, we present a new parallel algorithm for fast wavelet transforms (FWT) of a matrix of arbitrary size using any given number of parallel processors. The main idea in achieving the optimal load balancing is through a complexity analysis and flops minimization. This makes parallel implementation of FWT feasible and efficient on distributed memory machines with only a small number of processors and on local area networks. The new algorithm is tested by numerical experiments.

Simulation of grain-boundary diffusion creep: analysis of some new numerical techniques

We consider the simulation of deformation of polycrystalline materials by grain–boundary diffusion creep. For a given network of grain boundaries intersecting at nodes, with appropriate boundary conditions, we can calculate the rate at which material will be dissolved or deposited along each grain boundary and hence predict the rate at which each grain will move to accommodate this dissolution/deposition. We discuss two numerical methods for simulating the network changes over a finite time–interval, based on using the movement of adjacent grain boundaries over a small time–interval to estimate the velocities of the nodes. (The second of these methods has enabled us to speed up solution by 100 times in typical experiments compared with a naive forward–Euler approach.) We show that the accuracy with which the node velocities can be estimated is dependent only on the precision of the machine with which they are computed and deduce that, for all practical purposes, the lack of precise node velocity values does not detract from the quality of our solution. Finally, we consider the underlying stability of the problem under various different boundary conditions and conclude that our methods have the potential for providing useful insight into the effect of grain size and shape on deformation in polycrystalline materials.

On A Recursive Schur Preconditioner For Iterative Solution Of A Class Of Dense Matrix Problems

There are currently several distinct preconditioning methods for dense matrices based on applying a wavelet transform to obtain a matrix with a large number of small entries. A sparse preconditioner for this transformed matrix can be formed by setting to zero entries that are assumed to be unimportant. The effectiveness of the preconditioner depends on retaining the most important entries and on ensuring that they are positioned conveniently within the transformed matrix. In this paper we present a new, recursive preconditioning strategy that takes into account more of the significant entries without greatly increasing cost and outperforms existing methods in certain cases.

Solving linear systems using wavelet compression combined with Kronecker product approximation

AbstractnDiscrete wavelet transform approximation is an established means of approximating dense linear systems arising from discretization of differential and integral equations defined on a one-dimensional domain. For higher dimensional problems, approximation with a sum of Kronecker products has been shown to be effective in reducing storage and computational costs. We have combined these two approaches to enable solution of very large dense linear systems by an iterative technique using a Kronecker product approximation represented in a wavelet basis. Further approximation of the system using only a single Kronecker product provides an effective preconditioner for the system. Here we present our methods and illustrate them with some numerical examples. This technique has the potential for application in a range of areas including computational fluid dynamics, elasticity, lubrication theory and electrostatics.n