The unsigned p-Willmore functional introduced in the work of Mondino [2011] generalizes important geometric functionals, which measure the area and Willmore energy of immersed surfaces. Presently, techniques from the work of Dziuk [2008] are adapted to compute the first variation of this functional as a weak-form system of equations, which are subsequently used to develop a model for the p-Willmore flow of closed surfaces in R 3 . This model is amenable to constraints on surface area and enclosed volume and is shown to decrease the p-Willmore energy monotonically. In addition, a penalty-based regularization procedure is formulated to prevent artificial mesh degeneration along the flow; inspired by a conformality condition derived in the work of Kamberov et al. [1996], this procedure encourages angle-preservation in a closed and oriented surface immersion as it evolves. Following this, a finite-element discretization of both procedures is discussed, an algorithm for running the flow is given, and an application to mesh editing is presented.

We present an approach to analyze $C^1(\mathbb{R}^m)$ functions that addresses limitations present in the Active Subspaces (AS) method of Constantine et al. (2014; 2015). Under appropriate hypotheses, our Active Manifolds (AM) method identifies a 1-D curve in the domain (the active manifold) on which nearly all values of the unknown function are attained, which can be exploited for approximation or analysis, especially when $m$ is large (high-dimensional input space). We provide theorems justifying our AM technique and an algorithm permitting functional approximation and sensitivity analysis. Using accessible, low-dimensional functions as initial examples, we show AM reduces approximation error by an order of magnitude compared to AS, at the expense of more computation. Following this, we revisit the sensitivity analysis by Glaws et al. (2017), who apply AS to analyze a magnetohydrodynamic power generator model, and compare the performance of AM on the same data. Our analysis provides detailed information not captured by AS, exhibiting the influence of each parameter individually along an active manifold. Overall, AM represents a novel technique for analyzing functional models with benefits including: reducing $m$-dimensional analysis to a 1-D analogue, permitting more accurate regression than AS (at more computational expense), enabling more informative sensitivity analysis, and granting accessible visualizations (2-D plots) of parameter sensitivity along the AM.