University of Cambridge

interpolate values of a smooth function $f(x)$, $x \!\in\! {\cal R}^d$.

It is known that, if the data are the values $f(jh)$, $j \in {\cal Z}^d$,

where $h$ is the spacing between data points and ${\cal Z}^d$ is the

set of points in $d$ dimensions with integer coordinates, then the

accuracy of the interpolant is of magnitude $h^{d+2}$. This beautiful

result, due to Buhmann, will be explained briefly. We will also survey

some recent findings of Bejancu on Lagrange functions in two dimensions

when interpolating at the integer points of the half-plane ${\cal Z}^2

\cap \{ x : x_2 \!\geq\! 0 \}$. Most of our attention, however, will

be given to the current research of the author on interpolation in one

dimension at the points $h {\cal Z} \cap [0,1]$, the purpose of the work

being to establish theoretically the apparent deterioration in accuracy

at the ends of the range from ${\cal O} ( h^3 )$ to ${\cal O} ( h^{3/2}

)$ that has been observed in practice. The analysis includes a study of

the Lagrange functions of the semi-infinite grid ${\cal Z} \cap \{ x :

x \!\geq\! 0 \}$ in one dimension.

framework of practical interior point methods for

solving linear programming problems. We consider

such an algorithm and a second order variant of it.

We address the problem of the convergence of

the sequences of iterates generated by the two algorithms

to the analytic centre of the optimal primal-dual set.

Lagrangian function do not apply to objective or constraint functions

which are defined by means of a combinatorial selection structure. Such

selection structures can be explicit, for example in the case of "min",

"max" or "if" statements in function evaluations, or implicit as in the

case of inverse optimisation problems where the combinatorial structure is

induced by the possible selections of active constraints. The resulting

optimisation problems are typically neither convex nor smooth and do not

fit into the standard framework of nonlinear optimisation. Users typically

treat these problems either through a mixed-integer reformulation, which

drastically reduces the size of tractable problems, or by employing

nonsmooth optimisation methods, such as bundle methods, which are

typically based on convex models and therefore only allow for weak

convergence results. In this talk we argue that the classical Lagrangian

theory and SQP methodology can be extended to a fairly general class of

nonlinear programs with combinatorial constraints. The paper is available

at http://www.eng.cam.ac.uk/~ss248/publications.

Generators submit bids to an Independent System Operator (ISO) to indicate

how much power they can produce depending on price. The ISO takes these bids

with demand forecasts and minimizes the total cost of power production

subject to feasibility of distribution in the electrical network.

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Each generator can optimise its bid using a bilevel program or

mathematical program with equilibrium (or complementarity) constraints, by

taking the ISOs problem, which contains all generators bid information, at

the lower level. This leads immediately to a game between generators, where

a Nash equilibrium - at which each generator's bid maximises its profit

provided that none of the other generators changes its bid - is sought.

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In particular, we examine the idealised model of Berry et al (Utility

Policy 8, 1999), which gives a bilevel game that can be modelled as an

"equilibrium problem with complementarity constraints" or EPCC.

Unfortunately, like bilevel games, EPCCs on networks may not have Nash

equilibria in the (common) case when one or more of links of the network is

saturated (at maximum capacity). Nevertheless we explore some theory and

algorithms for this problem, and discuss the economic implications of

numerical examples where equilibria are found for small electricity

networks.

the container and experiments finding no such dependence. A possible resolution involves stratification originating from the spreading of the front between the clear liquid above and the suspension below. One theory describes the spreading front by a nonlinear diffusion equation

$\frac{\partial \phi}{\partial t} = D \frac{\partial }{\partial z}(\phi^{4/5}(\frac{\partial \phi}{\partial z})^{2/5})$.

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Experiments and computer simulations find differently.

the integral of a multivariate highly oscillatory function. We begin with a review

of the asymptotic and Filon-type methods developed by Iserles and Nørsett. Using a

method developed by Levin as a point of departure we will construct a new method that

uses the same information as the Filon-type method, and obtains the same asymptotic

order, while not requiring moments. This allows us to integrate over nonsimplicial

domains, and with complicated oscillators.

Strong horizontal gradients of density are responsible for the occurence of a large number of (often catastrophic) flows, such as katabatic winds, dust storms, pyroclastic flows and powder-snow avalanches. For a large number of applications, the overall density contrast in the flow remains small and simulations are carried in the Boussinesq limit, where density variations only appear in the body-force term. However, pyroclastic flows and powder-snow avalanches involve much larger density contrasts, which implies that the inhomogeneous Navier-Stokes equations need to be solved, along with a closure equation describing the mass diffusion. We propose a Lagrange-Galerkin numerical scheme to solve this system, and prove optimal error bounds subject to constraints on the order of the discretization and the time-stepping. Simulations of physical relevance are then shown.

It is well known that the computation of a few extreme eigenvalues, and the corresponding eigenvectors, of a symmetric matrix A can be rewritten as computing extrema of the Rayleigh quotient of A. However, since the Rayleigh quotient is a homogeneous function of degree zero, its extremizers are not isolated. This difficulty can be remedied by restricting the search space to a well-chosen manifold, which brings the extreme eigenvalue problem into the realm of optimization on manifolds. In this presentation, I will show how a recently-proposed generalization of trust-region methods to Riemannian manifolds applies to this problem, and how the resulting algorithms compare with existing ones.

I will also show how the Joint Diagonalization problem (that is, approximately diagonalizing a collection of symmetric matrices via a congruence transformation) can be tackled by a differential geometric approach. This problem has an important application in Independent Component Analysis.