Approximation algorithms for Capacitated Facility Location Problem with Penalties

In this paper, we address the problem of capacitated facility location problem with penalties (CapFLPP) paid per unit of unserved demand. In case of uncapacitated FLP with penalties demands of a client are either entirely met or are entirely rejected and penalty is paid. In the uncapacitated case, there is no reason to serve a client partially. Whereas, in case of CapFLPP, it may be beneficial to serve a client partially instead of not serving at all and, pay the penalty for the unmet demand. Charikar et. al. \cite{charikar2001algorithms}, Jain et. al. \cite{jain2003greedy} and Xu- Xu \cite{xu2009improved} gave $3$, $2$ and $1.8526$ approximation, respectively, for the uncapacitated case . We present $(5.83 + \epsilon)$ factor for the case of uniform capacities and $(8.532 + \epsilon)$ factor for non-uniform capacities

Discrete functional inequalities on lattice graphs

In this thesis, we study problems at the interface of analysis and discrete mathematics. We discuss analogues of well known Hardy-type inequalities and Rearrangement inequalities on the lattice graphs Z^d, with a particular focus on behaviour of sharp constants and optimizers. In the first half of the thesis, we analyse Hardy inequalities on Z^d, first for d=1 and then for d >= 3. We prove a sharp weighted Hardy inequality on integers with power weights of the form n^\alpha. This is done via two different methods, namely 'super-solution' and 'Fourier method'. We also use Fourier method to prove a weighted Hardy type inequality for higher order operators. After discussing the one dimensional case, we study the Hardy inequality in higher dimensions (d >= 3). In particular, we compute the asymptotic behaviour of the sharp constant in the discrete Hardy inequality, as d \rightarrow \infty. This is done by converting the inequality into a continuous Hardy-type inequality on a torus for functions having zero average. These continuous inequalities are new and interesting in themselves. In the second half, we focus our attention on analogues of Rearrangement inequalities on lattice graphs. We begin by analysing the situation in dimension one. We define various notions of rearrangements and prove the corresponding Polya-Szego inequality. These inequalities are also applied to prove some weighted Hardy inequalities on integers. Finally, we study Rearrangement inequalities (Polya-Szego) on general graphs, with a particular focus on lattice graphs Z^d, for d >=2. We develop a framework to study these inequalities, using which we derive concrete results in dimension two. In particular, these results develop connections between Polya-Szego inequality and various isoperimetric inequalities on graphs.Open Acces

Comprehensive STATCOM Control For Distribution And Transmission System Applications

This thesis presents the development of a comprehensive STATCOM controller for load compensation, voltage regulation and voltage balancing in electric power distribution and transmission networks. The behavior of this controller is first validated with published results. Subsequently, the performance of this STATCOM controller is examined in a realistic Hydro One distribution feeder for accomplishing the compensation of both mildly and grossly unbalanced loads, and balancing of network voltages using PSCAD/EMTDC software. The STATCOM voltage control function is utilized for increasing the connectivity of wind plants in the same distribution feeder. The thesis further presents a frequency scanning technique for simple and rapid identification of the potential of subsynchronous resonance in induction generator based wind farms connected to series compensated lines, utilizing MATLAB software. This technique is validated by published eigenvalue analysis results. The voltage control performance of the developed comprehensive STATCOM controller is then demonstrated for different scenarios in the modified IEEE First SSR Benchmark transmission system for mitigating subsynchronous resonance in series compensated wind farms using industry grade PSCAD/EMTDC software

Generalizing Deep Learning Methods for Particle Tracing Using Transfer Learning

Particle tracing is a very important method for scientific visualization of vector fields, but it is computationally expensive. Deep learning can be used to speed up particle tracing, but existing deep learning models are domain-specific. In this work, we present a methodology to generalize the use of deep learning for particle tracing using transfer learning. We demonstrate the performance of our approach through a series of experimental studies that address the most common simulation design scenarios: varying time span, Reynolds number, and problem geometry. The results show that our methodology can be effectively used to generalize and accelerate the training and practical use of deep learning models for visualization of unsteady flows

Hardy inequalities for antisymmetric functions

We study Hardy inequalities for antisymmetric functions in three different settings: euclidean space, torus and the integer lattice. In particular, we show that under the antisymmetric condition the sharp constant in Hardy inequality increases substantially and grows as d^4 as d \rightarrow \infty in all cases. As a side product, we prove Hardy inequality on a domain whose boundary forms a corner at the point of singularity x=0.Comment: 20 page