Research

Having a Think

© Angela Askham.

I work primarily in the field of scientific computing, which broadly seeks to explore and understand scientific phenomena with the aid of a computer. The primary goal of such research is to improve the software and algorithms available for the computer-aided design of modern devices, control systems, and therapeutics.

Integral equation methods for PDEs

An integral equation method utilizes either an exact or approximate inverse of a partial differential equation to analytically precondition the problem before it is discretized on a computer. Such an approach results in high-order-accurate, robust, and fast solvers for a range of PDEs. Research in this area is focused on (1) expanding the number of PDEs for which integral equation methods apply (e.g. by developing new integral representations), (2) developing quadrature rules to numerically evaluate the necessary singular integrals in the method, and (3) the fast solution of the (typically) dense linear systems that result.

My Ph.D. thesis focused on integral-equation methods for the solution of inhomogeneous elliptic partial differential equations in complex geometry. It can be downloaded here, with known errata here.

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Reduced order modeling

Direct numerical simulation of complex physical systems, e.g. the fluid flow around an intricate geometry, can require an incredibly large number of degrees of freedom to represent the solution. Often, however, we are actually concerned with how the behavior of such systems changes as a function of a small number of system parameters. The reduced order modeling paradigm is based on the idea that because the space of system parameters is lower dimensional, it is sometimes possible to represent all possible solutions of interest in terms of a relatively small set of basis functions. This reduced basis can then be used to design an approximately optimal control system or to perform short term future-state prediction.

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Optimization methods

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