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**1 - 6**of**6**### NLEIGS: A CLASS OF ROBUST FULLY RATIONAL KRYLOV METHODS FOR NONLINEAR EIGENVALUE PROBLEMS

, 2013

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### residual inverse iteration

, 2010

"... A complete characterization of the convergence factor can be very useful when analyzing the asymptotic convergence of an iterative method. We will here establish a formula for the convergence factor of the method called residual inverse iteration, which is a method for nonlinear eigenvalue problems ..."

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A complete characterization of the convergence factor can be very useful when analyzing the asymptotic convergence of an iterative method. We will here establish a formula for the convergence factor of the method called residual inverse iteration, which is a method for nonlinear eigenvalue problems and a generalization of the well known inverse iteration. The formula for the convergence factor is explicit and only involves quantities associated with the eigenvalue the iteration converges to, in particular the eigenvalue and eigenvector. Besides deriving the explicit formula we also use the formula to characterize the convergence of the method. In particular, we derive a formula for the first order expansion when the shift is close to the eigenvalue. The residual inverse iteration allows some freedom in the choice of a vector rk. In the analysis we characterize the convergence for different choices of rk. We use the explicit formula for the first order expansion to show that the convergence factor approaches zero when the shift approaches the eigenvalue for an arbitrary choice

### Modeling of Fluids and Waves with Analytics and Numerics

, 2013

"... Capillary instability (Plateau-Rayleigh instability) has been playing an important role in experimental work such as multimaterial fiber drawing and multilayer particle fabrication. Motivated by complex multi-fluid geometries currently being explored in these applications, we theoretically and compu ..."

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Capillary instability (Plateau-Rayleigh instability) has been playing an important role in experimental work such as multimaterial fiber drawing and multilayer particle fabrication. Motivated by complex multi-fluid geometries currently being explored in these applications, we theoretically and computationally studied capillary instabilities in concentric cylindrical flows of N fluids with arbitrary viscosities, thicknesses, densities, and surface tensions in both the Stokes regime and for the full Navier-Stokes problem. The resulting mathematical model, based on linear-stability analysis, can quickly predict the breakup lengthscale and timescale of concentric cylindrical fluids, and provides useful guidance for material selections and design parameters in fiberdrawing experiments. A three-fluid system with competing breakup processes at very different length scales is demonstrated with a full Stokes flow simulation. In the second half of this thesis, we study large-scale PDE-constrained microcavity topology optimization. Applications such as lasers and nonlinear devices require optical microcavities with long lifetimes Q and small modal volumes V. While most

### J. Fluid Mech. Preprint: DOI: 10.1017/jfm.2011.260 1

"... Linear stability analysis of capillary ..."