Instructor: Francisco BernalAn integrated circuit can be modelled as a twodimensional domain where the voltages are piecewise constant. On an insulating platform (the waffer) several electronic, conducting components are placed which stand at various voltages. Estimation of the electric fields in the integrated circuit require that a Laplace equation be solved in the whole domain. It is critical, from the point of view of design, that the electric field does not 'spike' anywhere, since this would give rise to spurious electric currents which would yield the integrated circuit useless. The industry is therefore interested in cheap estimates of the electric field at specific locations in the integrated circuit in the design phase, when many different possible layouts are considered prior to manufacture. Since the number of components is in the order of tenths of millions (or more), standard tools such as finite elements are ruled out for this task.
This kind of problem, however, is ideally suited for stochastic solution methods of PDEs, because: a) the solution is only required at specific points, and b) a modest accuracy suffices. Moreover, the typically rectangular shape of electronic components allow for improvements to the standard stochastic method, such as the socalled 'walk on cubes' algorithm.
COMMENTS: This problem is not only of industrial relevance but very topical (Intel announced recently the first 3D chip ever).
Preferred mathematical background:  a basic background in electrostatics (basically the connection between the electric field, the electric potential, and Laplace equation)
 basic notions of probability and statistics
 basic notions of programming and basic numerical schemes such as Euler's method for IVPs.
 there is plenty of room for modelling and imagination here, because since just a rough solution is required, it has to be decided which features are the most relevant (and which may be left out) in order to speed up the algorithm.
