Advisor: Prof. Zhiqin Lu

Classical ergodic averages give good norm approximations, but these averages are not necessarily giving the best norm approximation among all possible averages. We consider
1) what the optimal Cesaro norm approximation can be in terms of the transformation and the function,
2) when these optimal Cesaro norm approximations are comparable to the norm of the usual ergodic average, and
3) oscillatory behavior of these norm approximations.

 In this work, we develop a cascadic multigrid method for the elliptic eigenvalue problems and show its optimality under certain assumptions.  We also develop an algebraic variant of the cascadic multigrid method for the fast computation of the Fiedler vector of a graph Laplacian, namely, the eigenvector corresponding to the second smallest eigenvalue, and explore the applicability of such an eigensolver to the graph partition and drawing.  Numerical tests for practical graphs are presented to show the efficiency of the proposed cascadic multigrid method. This is a joint work with J. Urschel, J. Xu, and L. Zikatanov at Penn State University.