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Current Research

Proper Orthogonal Decomposition (POD)
Proper Orthogonal Decomposition (POD) is a technique which has been used successfully to reduce the computation time in various fields of engineering. In my Masters Thesis, POD is applied to electromagnetic field computations, specifically to the simulation of electromagnetic diffusion by the time-domain finite element method. The standard method requires a large matrix equation to be solved at every time step. POD is applied to greatly reduce the size of the matrices. Both 1-D and 2-D test cases are considered. Applying POD reduces the matrix dimension for a 2-D problem from 2,535 to just 5, with negligible loss of accuracy. http://mcgill.worldcat.org/oclc/741292337

Finite Element Method (FEM)
The Finite Element Method (FEM) is a good choice for solving partial differential equations over complicated domains. The FEM can be derived in two different ways: by variational principles, or by the method of weighted residuals. The former involves a functional whose minimum is the solution of a PDE and associated boundary conditions. The latter uses integrals of the weighted error (residual) over the problem domain. Both usually end up with identical equations. The field is discretized using meshes of simple geometric shapes (e.g., triangles, tetrahedra) called "elements". The FEM can handle both eigenanalysis (source free) and deterministic (driven) problems.