On The Applications Of Numerical Methods For Elliptic Partial Differential Equations PDF Download

"The goal of this dissertation is to explore and demonstrate the applications of numerical methods for elliptic partial differential equations (PDEs). The numerical methods presented, as we will see, are applicable in a variety of contexts, ranging from computational geometry to machine learning. The general analytic framework of this dissertation is viscosity solutions for elliptic PDEs. The corresponding numerical framework belongs to Barles and Souganidis, with emphasis on its reinterpretation using elliptic finite difference schemes in lieu of monotone schemes. The first problem considered was building a multi-criteria anomaly detection algorithm that can be applied in a real-time setting. The algorithm was centered around a recently discovered PDE continuum limit for nondominated sorting. By exploiting the relatively low computational cost of numerically approximating the PDE we developed an efficient method to detect anomalies in two-dimensional data in real-time. We also derived a transport equation which characterizes sorting points within nondominated layers. This allowed us to add to our algorithm the ability of classifying anomalies. Our algorithm has an inherent ability to adapt to changes in the trend of data. In addition to demonstrating the effectiveness of our algorithm on synthetic and real data, we presented probabilistic arguments proving convergence rates for the PDE-based ranking.The second problem addressed the issue of computing the quasiconvex envelope of a given function. In a series of papers written by Barron, Goebel, and Jensen, first- and second-order differential operators characterizing quasiconvexity were rigourously developed. These characterizations, arising in the form of PDEs, unfortunately prove intractable in light of existing numerical methods. Hence, attempting to generate the quasiconvex envelope using these operators with an obstacle term, in a manner similar to Oberman, is not prudent. Our solution to this, and consequently our contribution, came two-fold (each of which is its own article, respectively): (i) a first-order nonlocal line solver which can compute the quasiconvex envelope in one dimension, and for which the extension to arbitrary dimensions follows naturally; (ii) a second-order operator which offers a more relaxed notion of quasiconvexity, and is more obliging to numerical approximation. Convergence of the algorithms presented in both solutions is proven, and numerical examples validating the arguments presented therein are demonstrated." --