Comlab

It is well known that discrete solutions to the convection-diffusion

equation contain nonphysical oscillations when boundary layers are present

but not resolved by the discretisation. For the Galerkin finite element

method with linear elements on a uniform 1D grid, a precise statement as

to exactly when such oscillations occur can be made, namely, that for a

problem with mesh size h, constant advective velocity and different values

at the left and right boundaries, oscillations will occur if the mesh

P\'{e}clet number $P_e$ is greater than one. In 2D, however, the situation

is not so well understood. In this talk, we present an analysis of a 2D

model problem on a square domain with grid-aligned flow which enables us

to clarify precisely when oscillations occur, and what can be done to

prevent them. We prove the somewhat surprising result that there are

oscillations in the 2D problem even when $P_e$ is less than 1. Also, we show that there

are distinct effects arising from differences in the top and bottom

boundary conditions (equivalent to those seen in 1D), and the non-zero

boundaries parallel to the flow direction.

A systematic approach to error control and mesh adaptation for

optimal control of systems governed by PDEs is presented.

Starting from a coarse mesh, the finite element spaces are successively

enriched in order to construct suitable discrete models.

This process is guided by an a posteriori error estimator which employs

sensitivity factors from the adjoint equation.

We consider different examples with the stationary Navier-Stokes

equations as state equation.

We discuss two filters that are frequently used to smooth data.

One is the (nonlinear) median filter, that chooses the median

of the sample values in the sliding window. This deals effectively

with "outliers" that are beyond the correct sample range, and will

never be chosen as the median. A straightforward implementation of

the filter is expensive for large windows, particularly in two dimensions

(for images).

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The second filter is linear, and known as "Savitzky-Golay". It is

frequently used in spectroscopy, to locate positions and peaks and

widths of spectral lines. This filter is based on a least-squares fit

of the samples in the sliding window to a polynomial of relatively

low degree. The filter coefficients are unlike the equiripple filter

that is optimal in the maximum norm, and the "maxflat" filters that

are central in wavelet constructions. Should they be better known....?

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We will discuss the analysis and the implementation of both filters.

The asymptotic rate of convergence of the trapezium rule is

defined, for smooth functions, by the Euler-Maclaurin expansion.

The extension to other methods, such as Gauss rules, is straightforward;

this talk begins with some special cases, such as Periodic functions, and

functions with various singularities.

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Convergence rates for ODEs (Initial and Boundary value problems)

and for PDEs are available, but not so well known. Extension to singular

problems seems to require methods specific to each situation. Some of

the results are unexpected - to me, anyway.

In the past few years we have developed some expertise in solving optimization

problems that involve large scale simulations in various areas of Computational

Geophysics and Engineering. We will discuss some of those applications here,

namely: inversion of seismic data, characterization of piezoelectrical crystals

material properties, optimal design of piezoelectrical transducers and

opto-electronic devices, and the optimal design of steel structures.

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A common theme among these different applications is that the goal functional

is very expensive to evaluate, often, no derivatives are readily available, and

some times the dimensionality can be large.

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Thus parallelism is a need, and when no derivatives are present, search type

methods have to be used for the optimization part. Additional difficulties can

be ill-conditioning and non-convexity, that leads to issues of global

optimization. Another area that has not been extensively explored in numerical

optimization and that is important in real applications is that of

multiobjective optimization.

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As a result of these varied experiences we are currently designing a toolbox

to facilitate the rapid deployment of these techniques to other areas of

application with a minimum of retooling.