APPLICATION OF REDUCED ORDER MODELING TECHNIQUES TO PROBLEMS IN HEATCONDUCTION, ISOELECTRIC FOCUSING AND DIFFERENTIAL ALGEBRAIC EQUATIONS

Abstract

This thesis focuses on applying and augmenting `Reduced Order Modeling'

(ROM) techniques to large scale problems. ROM refers to the set of mathematical

techniques that are used to reduce the computational expense of conventional modeling techniques, like finite element and finite difference methods, while minimizing the loss of accuracy that typically accompanies such a reduction.

The first problem that we address pertains to the prediction of the level of

heat dissipation in electronic and MEMS devices. With the ever decreasing feature

sizes in electronic devices, and the accompanied rise in Joule heating, the electronics industry has, since the 1990s, identified a clear need for computationally cheap heat transfer modeling techniques that can be incorporated along with the electronic design process. We demonstrate how one can create reduced order models for simulating heat conduction in individual components that constitute an idealized electronic device. The reduced order models are created using Krylov Subspace Techniques (KST). We introduce a novel `plug and play' approach, based on the small gain theorem in control theory, to interconnect these component reduced order models (according to the device architecture) to reliably and cheaply replicate

whole device behavior. The final aim is to have this technique available commercially

as a computationally cheap and reliable option that enables a designer to

optimize for heat dissipation among competing VLSI architectures.

Another place where model reduction is crucial to better design is Isoelectric

Focusing (IEF) - the second problem in this thesis - which is a popular technique

that is used to separate minute amounts of proteins from the other constituents

that are present in a typical biological tissue sample. Fundamental questions about

how to design IEF experiments still remain because of the high dimensional and

highly nonlinear nature of the differential equations that describe the IEF process

as well as the uncertainty in the parameters of the differential equations. There

is a clear need to design better experiments for IEF without the current overhead

of expensive chemicals and labor. We show how with a simpler modeling of the

underlying chemistry, we can still achieve the accuracy that has been achieved in

existing literature for modeling small ranges of pH (hydrogen ion concentration)

in IEF, but with far less computational time. We investigate a further reduction

of time by modeling the IEF problem using the Proper Orthogonal Decomposition

(POD) technique and show why POD may not be sufficient due to the underlying

constraints.

The final problem that we address in this thesis addresses a certain class of

dynamics with high stiffness - in particular, differential algebraic equations. With

the help of simple examples, we show how the traditional POD procedure will fail to

model certain high stiffness problems due to a particular behavior of the vector field

which we will denote as twist. We further show how a novel augmentation to the traditional POD algorithm can model-reduce problems with twist in a computationally

cheap manner without any additional data requirements.