Applied Math PhD comprehensive exam | Manda Winlaw

Preconditioning is often used in iterative methods for linear systems to speed up convergence.  Preconditioning can also be used to increase the speed of convergence for nonlinear problems. Hager and Zhang (2006) show how a linear preconditioner can be applied to the nonlinear conjugate gradient algorithm.  We introduce a nonlinearly preconditioned nonlinear conjugate gradient (PNCG) algorithm. We then  apply this algorithm to the canonical rank-R tensor decomposition problem using the alternating least squares (ALS) algorithm as the nonlinear preconditioner. As background, we discuss the canonical tensor decomposition and the ALS algorithm. We also present several existing algorithms that provide a basis for our PNCG algorithm before presenting our algorithm and the numerical results.

With the continued interest in so-called "interaction'" networks such as social networks, collaboration networks, citation networks, and computer traffic networks, there has been continued interest in developing models that can accurately capture the underlying properties. Our goal is to build a graph generator that can produce random graphs that match the network community structure as measured by the clustering coefficient. Our graph should also match most of the other graph properties considered of interest and as such we present some of the properties that are often looked at when building graph generators. We also examine some of the existing graph generators used to model real-world networks. Then we present our own graph generating algorithm that attempts to match the number of triangles in real-world networks. and show some preliminary results.

William W. Hager and Hongchao Zhang. A survey of nonlinear conjugate gradient methods. Pacific Journal of Optimization, 2:35–58, 2006.