Generalized Method of Moments (gmm)

In the electromagnetic simulation of realistic structures, the spatial representation of the domain being analyzed depends not only on the  frequency of interest but also on the need to capture possible fine geometric features. Such mixed scales cause havoc in standard integral equation based solvers on three fronts; (i) discretized integral equations become poorly conditioned as the size of the element becomes smaller, (ii) the function spaces used do not optimally represent the underlying physics, and (iii) the overall computational burden is exceedingly large. This largely limits the applicability of the existing methods. The proposed project seeks to develop a demonstrably unified, robust and accurate solution methodology that is well conditioned over a wide range of frequencies and, at the same time, has the flexibility to handle complicated (and possibly near singular) geometries. This is achieved by (i) developing a well conditioned integral equation scheme (that are Fredholm equations of the second kind) with provable bounds on convergence rates and accuracy to solve for electromagnetic quantities over a large range of spatial frequencies; (ii) enlarging the approximation space used for representing the unknown quantity so as to include the local physics; (iii) designing a scheme that permits seamless interplay between a variety of basis functions to model the unknown quantities to be used with the above integral equation scheme; (iv) deriving error bounds and convergence estimates on these schemes to demonstrate clear and easy user control over the error, and (iv) developing a domain decomposition framework so that these schemes can be integrated seamlessly with classical integral equation and finite element methods to solve electrically large problems. The educational objective is to develop a publicly available set of tutorials/teaching modules based on this research. (NSF : Award Number 0811197)

rewind

The possibility of engineered nano-systems has captured the imagination of both scientists and the lay public. So much so, that Feynman’s words “there is plenty of room at the bottom” has pretty much become a clich´e. Over the past few years, research into nano-related topics has increased exponentially, with studies ranging from fundamental understanding of the underlying physics to development of experimental techniques, and to building actual devices. One should stress that a bulk of theoretical research has largely focused on developing a fundamental understanding; deservedly so, as the problem is exceedingly complex. Now, with the current advances, interesting questions that could be asked are as follows: (i) what conditions are necessary to affect nano growth? (ii) can we modulate the temporal variation of the growth condition to achieve more control? (iii) can portions of the “device” be grown in place? More simply, given a performance criterion, it is possible to arrive at the necessary shapes and distributions of nano-structures. Rhetorically speaking, we would like develop the necessary mathematics and software to enable us to “rewind” from a final state to statistically possible pathways that could lead to this state.

More Research ...

The two problems above are what I am working on right now. Please feel free to check out my older research here. Alternatively, you can visit the webpages of either the NDEL lab or the Electromagnetics lab both of which I am closely associated with

Some (more) Qoutes I Like

Steal from one and it is plagarism, Steal from many and it is research.
-Albert Einstein (1879 - 1955)

2+2 = 4, except when 2 is very large
-Anon

There was once a man called Maxwell,
Four equations, he once did tell,
Why ? No one knows...
But at 28 my graying hair shows,
He has made my life a living hell.
-Ich (1980-)