Numerical Linear Algebra (NUI Galway 2013/2014)

Here is the sparsity pattern for a matrix arising from the numerical solution to a partial differential equation using a sparse grid finite element method (joint work with my Ph.D. student, Stephen Russell).

The image on the left is based on using a "natural" lexicographic ordering for the nodes in the mesh. If this matrix is factored using a standard Cholskey method, then many new non-zero entries in creased in the factors.

One can reduce the this number by first permuting the rows and columns of the matrix. The second image from the left is permuted using Matlab's implementation of the approximate minimum degree permutation. The third uses the column approximate minimum degree algorithm of Larimore and Davis. The fourth uses METIS. The sparsity patterns of the corresponding factors are shown below.

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