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Numerical Mathematics Mini-symposia / Special Sessions in 47^th BAMC – 4-7 April 2005

Tuesday - Numer Math (I)	14:00-14:45	A1	Prof Chris J Budd (Invited) Scale Free Adaptive Methods for Partial Differential Equations	University of Bath	Numerical PDE
	14:45-15:15	A2	Dr Keith Davey Successive Preconditioning For The Iterative Solution Of Non-Linear Algebraic Equations	University of Manchester	Nonlinear Iterations
	15:15-15:20		(Break)
	15:20-15:50	A3	Mr J Savage / Dr K Chen / Prof W G Li Fast multilevel methods for image processing	Liverpool University	Multigrid Methods/Image
	15:50-16:20	A4	Dr Peter J. Young Network Analysis through Numerical Solution of Computer-generated Mathematical Equations	DSTL	Linear/Nonlinear Iterations

Wednesday - Numer Math (II)	09:15-09:45	B1	Dr Sapna Somani A Wavelet Preconditioner For Two Dimensional Elliptic Problem	University of Baroda, INDIA	Wavelet Poissons
	09:45-10:15	B2	Dr A. Shidfar/R. Pourgholi An approximate stable solution for an inverse heat conduction problem	University of Sci and Tech IRAN	Numerical Inverse Probs
	10:15-10:30		(Break)
	10:30-11:00	B3	Dr Dugald B Duncan / Penny J Davies Numerical approximations of time domain electromagnetic scattering	Heriot-Watt University	Numerical Scattering
	11:00-11:30	B4	Dr Stephen Langdon / Prof S.N. Chandler-wilde A boundary element method for high frequency scattering by convex polygons	Reading University	Numerical Scattering

Thursday - Numer Math (III)	09:15-09:40	C1	Prof Ronald Smith / M K Bowen Compact Schemes for Evolution Equations	Loughborough University	High Order Time Stepping
	09:40-10:05	C2	Prof Nev J Ford / Dr Patricia M. Lumb Detecting small solutions to a class of multi-delay differential equations: Two approaches	University College Chester	Delay ODEs
	10:05-10:30	C3	Dr N B Petrovskaya The Impact of Grid Cell Geometry on the Least-Squares Gradient reconstruction	Birmingham University	Numerical Least-squares
	10:30-10:45		(Break)
	10:45-11:10	C4	Dr Teijo Arponen 2-tensor invariants in numerical integration	Warwick University	Numerical Symplectic ODE
	11:10-11:35	C5	Dr. A.Lukyanov A combined BIE-FE method for dynamic wetting	Birmingham University	FEM-BEM coupling

Prof Chris J Budd (Invited)

University of Bath

Title = Scale Free Adaptive Methods for Partial Differential

equations

Abstract =

One of the key qualitative features of the solution of a nonlinear partial differential equation is the evolution of behaviour on many different length scales. Examples are the continuum of length scales in turbulence modelling, the formation of shocks in fluid mechanics and the appearance of singularities in nonlinear optics. Traditional numerical methods perform poorly in such situations as the discretisation imposes an artificial length scale on the problem. Ideally, to give the correct qualitative behaviour a numerical method should perform equally well at all length scales. I will show in this talk that by using a mixture of ideas from Lie group theory to meteorology it is possible to construct

adaptive numerical methods which can effectively capture many different scales of behaviour. I will illustrate this with examples from optics and combustion.

Dr K. DAVEY

University of Manchester

\title{Successive Preconditioning For The Iterative Solution Of

Non-Linear Algebraic Equations}

\abstract{ABSTRACT

Ill-conditioned algebraic equations involving positive definite matrices arise in many physical representations. A particular problem of interest is ring rolling simulation using a constrained-flow formulation and the finite element method, as this gives rise to poorly-conditioned non-linear equations that require repeated solution. The imposition of a volume-constancy constraint has the effect of producing an eigen-spectrum with a cluster of relatively small eigenvalues. This makes traditional iterative solution methods such as the conjugate gradient method unreliable, typically failing to converge. Some success has been reported with the quasi-Newton method but again the eigenvalue distribution makes for an unreliable performance. Direct methods are reliable but the associated computational costs are extreme making the analysis impracticable. This work is concerned with a method that unifies the conjugate gradient and the quasi-Newton method and is called the Successive Preconditioned Conjugate Gradient Method (SPCGM). Both the conjugate gradient and quasi-Newton methods possess quadratic termination, i.e. for exact arithmetic the minimum for a quadratic function can be found within a maximum of n iterations for an n-degree problem. This property is shown to be preserved with the SPCGM, which in its extreme form collapses to either the conjugate gradient or the quasi-Newton method. A feature of the method is the automatic development of a suitable preconditioner that adapts as the equations change. In the context of non-linear problems the SPCGM can be classified as an inexact Newton method [1] but possesses the added feature that the preconditioner developments over linear and non-linear iterations. The method has similarities with the method of van der Vorst and Vuik [2], which is founded on rank-one matrix updates [3], and that of Axelsson and Vassilevski [4]. The new features of the proposed method are: the linkage with quasi-Newton methods, the use of rank-two updates, and preconditioners developing over linear and non-linear iterations. The performance of the solver is compared against well-known alternatives for varying problem sizes. The approach is shown to be generic but in particular makes ring rolling simulation a practicable proposition.

[1] C. T. Kelly, 'Iterative Methods for Linear and Non-linear Equations', SIAM Frontiers in Applied Mathematics, Vol. 16, (1995).

[2] H. A. van der Vorst and C. Vuik, 'GMRESR: A Family of Nested GMRES Methods', Numerical Linear Algebra and Applications, Vol. 1, pp. 369-386, (1994).

[3] T. Eirola and O. Nevanlanna, 'Accelerating with Rank-One Updates', Linear Algebra and its Applications, Vol. 121, pp. 511-520, (1989).

[4] O. Axelsson and P. S. Vassilevski, 'A Black Box Generalised Conjugate Gradient Solver with Inner Iterations and Variable-step Preconditioning', SIAM, J. Matrix Anal. Appl., Vol. 12, pp 625-644, (1991).

Mr J Savage / Dr K Chen / Prof W G Li,

University of Liverpool,

\title{Fast multilevel methods for image processing}

ABSTRACT:

The variational partial differential equation (PDE) approach for image denoising restoration leads to PDEs with nonlinear and highly non-smooth coefficients. Such PDEs present severe difficulties for the convergence of standard multigrid methods. In this talk we shall present some of our recent work on developing fast algorithms in several ways: (i) the linear multigrid methods in nonlinear iterations; (ii) fully nonlinear multigrid methods; (Iii) the algebraic multigrid methods (AMGs); (iv) the multigrid for the primal-dual system; (v) the multigrid methods for the optimisation formulation. Experiments are shown to demonstrate the improvements obtained.

Dr Peter J. Young

DSTL

\title{Network Analysis through Numerical Solution of Computer-generated Mathematical Equations}

\abstract{This presentation describes research that has employed automatic generation of mathematical equations in symbolic form on a computer for problem analysis and solution. The research was initiated through the development of a computer model to permit analysis of call blocking for a generic communications network in which inter-nodal bandwidths and Poisson distributed, traffic calling patterns were specified. A novel approach was developed in which the equations describing probability of call blocking were automatically constructed in symbolic form for a user-specified communications network. The resulting set of coupled, nonlinear equations was then solved using an iterative numerical technique. The symbolic form of a single equation consisted of nested collections of pointers to network variables and mathematical operators. Given their symbolic forms, the equations were parsed permitting recalculation of network variables from earlier estimates until convergence was achieved. Verification of the computer-generated equations was performed through comparison to equations derived by hand for simple network configurations. The approach quickly demonstrated its utility when larger, more complex networks were considered in which the algebraic form of the equations becomes increasingly complicated. Validation of the network communications model with Monte Carlo simulation results identified limitations of the theoretical model (based on application of the Erlang B loss formula) for situations involving conditional probabilities. Methods were then devised to automatically construct a Markov process to represent the various states the network could reside in. Again, equations representing state probabilities for the system at statistical equilibrium were constructed in symbolic form and solved numerically through an iterated approach. The research illustrates the potential in using a computer to automatically generate the mathematical equations associated with a network-based problem, which can then be solved numerically.}

Dr Sapna Somani University of Baroda, INDIA

\title{A Wavelet Preconditioner For Two Dimensional Elliptic Problem}

\abstract{In this paper we have made an attempt to solve Poisson's

equation in two dimension using preconditioning concepts. We consider the representation

of elliptic differential operators in wavelet bases and preconditioner based on wavelet

methods. The method for solving Poisson equation for two dimensions and three dimensions

has been implemented in Matlab based on work of G. Beylkin [1]. In Galerkin approach we

get a ill-conditioned system. Here, we have made an attempt to show that the condition

number of the reduced matrix is of size O(1) using preconditioning.

References

[1]. G. Beylkin: On the representation of operators in bases of compactly

supported wavelets, SIAM J. on Numerical Analysis, vol. 6, No. 6, pp. 1716-1740

[2] I. Daubechies, Orthonormal bases of compactly supported wavelets,Comm.Pure.Appl.

Math. 41(1988), 909-996

Dr A. Shidfar/R. Pourgholi, University of Sci and Tech IRAN

An approximate stable solution for an inverse heat conduction problem

\abstract{

This paper deals the determination of a stable solution for an Ill-posed Inverse heat conduction problem (IHCP). By using the Chebyshev polynomials based function an approximate stable solution to the IHCP will be determined in finite or infinite domain.

(Keywords: Inverse heat conduction problem, Existence, Stability;

AMS Subject Classification: 35K05, 58G11, 93D05)}

Dr Dugald B Duncan / Dr Penny J Davies Heriot-Watt University

\title{Numerical approximations of time domain electromagnetic scattering}

\abstract{We derive and analyse collocation approximations of retarded potential integral equations (RPIEs) arising as models of

scattering of waves from thin wire and strip antennas. We Fourier

analyse the temporal stability of spatially exact piecewise

constant and linear in time approximations of these three RPIEs.

Numerical results are presented for practical schemes that are

piecewise constant or linear in time and space, and these are in

close agreement with the predictions of the Fourier analysis.}

Dr Stephen Langdon / Prof S.N. Chandler-Wilde

Reading University

\title{A boundary element method for high frequency scattering by convex polygons}

\abstract{Standard boundary or finite element methods for problems of high frequency acoustic scattering by convex polygons have a computational cost that grows in direct proportion to the frequency of the incident wave. Here we present a novel Galerkin boundary element method, using plane wave basis functions on a graded mesh, for which the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to frequency.}

Prof Ronald Smith / M K Bowen

Loughborough University

\title{Compact Schemes for Evolution Equations}

\abstract{A method is given for the construction of high-accuracy compact schemes for the numerical solution of linear evolution equations. It brings together exact time stepping and expansions for the error in difference approximations to spatial derivatives. The accuracy permits the use of a smaller computational module than is usual. Three points in $x$ suffice for the linear Korteweg de Vries equation with damping

\partial_t c+\lambda \, c+\, u\, \partial_xc -\kappa\partial_x^2 c +\frac{1}{6}h^2u\partial_x^3c =0

\quad {\rm with} \quad \kappa,\, h \ge 0.

Direct numerical modelling of $\partial_x^3 c$ would have required at least four points in $x$. For the severe test-case of a delta-function initial condition, the computations fail to replicate the sub-grid oscillatory tail far to the rear. However, the resolvable part oscillatory tail and the forward skewness caused by the $\partial_x^3 c$ term are accurately replicated.

}

Prof Neville J Ford / Dr Patricia M. Lumb

University College Chester

\title{Detecting small solutions to a class of multi-delay differential equations: Two approaches}

\abstract{Certain non-autonomous delay differential equations are known to admit small solutions (solutions that are not identically zero but which tend to zero more quickly than any exponential such that $\lim_{t\rightarrow \infty}e^{kt}x(t)=0$, for all $k \in \bf{R}$). The detection of non-trivial small solutions is a key tool for the mathematical analyst.\newline

\noindent We consider periodic delay differential equations of the form

\dot x(t)=\sum_{j=0}^{m}b_j(t)x(t-jw), x_s=\phi, \mbox{ } t\ge s,

where $b_j$, $j=0,1,...,m$ are continuous periodic functions with period $\omega$.\newline

We are interested in detecting the presence, or otherwise, of so-called \emph{small} solutions using a numerical method.

In our earlier work we have successfully detected the presence of small solutions to linear periodic delay differential equations with a single delay. We show how our approach can be adapted for the multi-delay equation. We then develop a more sophisticated approach using Floquet solutions. We compare the two approaches and discuss the ease and clarity with which the presence, or otherwise, of small solutions is detected. We present illustrative examples.

We have developed an algorithm that automates the detection of small solutions to linear periodic delay differential equations with a single delay. We indicate how our algorithm is adapted for equations with multiple delays.

}

Dr N B Petrovskaya

Birmingham University

\title{The Impact of Grid Cell Geometry on the Least-Squares Gradient reconstruction}

\abstract{Numerical solution of many modern problems in physics and

engineering often requires to approximate the solution gradient on

grid cells which are almost degenerate. We consider the problem of

the least-squares gradient approximation on two-dimensional

unstructured grids with "bad" cells. It will be discussed how the

accuracy of the least-squares approximation depends on the cell

geometry. We analyze a simple geometry and demonstrate that

introducing weight coefficients into the problem may essentially

help to improve the accuracy of the least-squares approximation.

Based on the results of our analysis, a heuristic choice of the

weights in a general least-squares procedure is suggested. Our

approach is illustrated by numerical tests.

}

Dr Teijo Arponen

Warwick University

\title{2-tensor invariants in numerical integration}

\abstract{In numerical geometric integration of ODEs one is concerned about preserving geometrical properties of the flow under discretization. The best known examples are Hamiltonian ODEs which are ubiquitous in applications. Today it is well known that conserving the symplectic structure is the correct way to solve these Hamiltonian ODEs numerically. Similar results are known for Lie-group equations, to mention another class of applications.

But the key question in preserving "the" structure is, how do we

define structure? In Hamiltonian case the answer is known: it's the

symplectic structure. In this talk, I introduce 2-tensor-invariants.

These are defined for any ODE and they are and algebraic generalization of symplectic structure.

This is work in progress and I present some intermediate results which show this to be a promising approach to develop geometric integrators to new classes of equations.

Dr. A.Lukyanov

Birmingham University

\title{A combined BIE-FE method for dynamic wetting}

The method is applied to curtain coating where it allows one to resolve the fine dynamics near the contact line and incorporate the ``extra'' physics of interface formation, whereas the BIE component of the method makes it capable of handling a wide range of shapes of the flow domain with significant economy in computational time and memory. A combined Boundary Integral Equation (BIE) and Finite Element (FE) technique is developed and applied to curtain coating by high-viscosity liquids in the regime of creeping flow. The idea of the new method is to combine advantages and compensate limitations inherent in the BIEM and FEM. The BIEM uses the exact solution of the Stokes equations in the bulk thus, on the one hand, reducing the problem to integral equations along the boundary of the flow domain but, on the other hand, it applies only when this boundary is smooth. The FEM can handle domains with boundaries having such features as contact angles or cusps and it is not limited to the Stokes equations (hence can incorporate extra physics that can be important near such features). However, since the FEM makes it necessary to tessellate the whole domain and solve the problem there, this becomes a nontrivial and time-consuming task for domains with complex shapes. The combined BIE-FE method employs finite elements to deal with the regions comprising singularities of curvature of the boundary and utilizes the boundary-integral representation for the rest of the flow. The two methods are coupled along an internal boundary using mutually compatible representation of the unknowns there.

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