Strengthening the integrality gap for the capacitated facility location problem with LP-based rounding algorithms

This thesis studies the capacitated facility location problem, in which all clients have unit demand and all facilities have integral capacity. A linear relaxation is researched, with corresponding integrality gap bounded by a constant. Recently, such a linear relaxation has been found and proven using an LP-bounding algorithm. The formulation of the relaxation and the proof were very complex and intuitively hard to understand, however. Therefore, this thesis provides a simpler, more formulation and proof. This thesis has two main contributions. First, a structured overview of all the theory prior to the construction of the relaxation is provided. To do so, the minimum knapsack problem is treated, which is a simplied version of the capacitated facility location problem. An LP-based rounding algorithm is presented to illustrate general ow-network techniques for facility location problems. Second, the rounding algorithm for the capacitated facility location problem is illustrated and explained more accessible to readers less familiar with LP-based rounding algorithms. The existing rounding algorithm for the capacitated facility location problem is treated, illustrated and extended with Matlab code. The rounding algorithm proves an integral solution for the capacitated facility location can be constructed from the linear optimal solution, with cost no more than 288 times the cost of the fractional optimal solution. This proves that the integrality gap of the proposed relaxation is bounded by 288.Electrical Engineering, Mathematics and Computer ScienceDelft Institute of Applied Mathematic

Dispersion and Nonlinearity Identification for Single-Mode Fibers Using the Nonlinear Fourier Transform

Efficient fiber-optic communication requires precise knowledge of the fiber coefficients, but these often change over time due to factors such as aging or bending. We propose a novel algorithm that identifies the average second-order dispersion and Kerr nonlinearity coefficient of a fiber, without employing any special training signals. Instead, ordinary input and output data recorded during normal operation is used. To the best of our knowledge, this is the first such algorithm. The algorithm is based on the nonlinear Fourier spectrum of the signal, which is known to evolve trivially as the signal propagates through an idealized model of the fiber. The algorithm varies the values of the fiber coefficients until the corresponding nonlinear Fourier spectrum at transmitter and receiver match optimally. We test the algorithm on simulated transmission data over a 1600 km link, and accurately identify the fiber coefficients. The identification algorithm is in particular well suited for providing a fiber model for nonlinear Fourier transform-based communication.Accepted Author ManuscriptTeam Sander Wahl

Fibre model identification for nonlinear Fourier transform-based transmission

Fibre-optic transceivers based on nonlinear Fourier transforms approximate the link by a lossless path-average model. We propose a new method that identifies a suitable lossless model from time-domain input-output data when fibre parameters are unknown. No special training signals are needed.Team Sander Wahl

Extension of B-spline Material Point Method for unstructured triangular grids using Powell–Sabin splines

The Material Point Method (MPM) is a numerical technique that combines a fixed Eulerian background grid and Lagrangian point masses to simulate materials which undergo large deformations. Within the original MPM, discontinuous gradients of the piecewise-linear basis functions lead to the so-called grid-crossing errors when particles cross element boundaries. Previous research has shown that B-spline MPM (BSMPM) is a viable alternative not only to MPM, but also to more advanced versions of the method that are designed to reduce the grid-crossing errors. In contrast to many other MPM-related methods, BSMPM has been used exclusively on structured rectangular domains, considerably limiting its range of applicability. In this paper, we present an extension of BSMPM to unstructured triangulations. The proposed approach combines MPM with C1-continuous high-order Powell–Sabin spline basis functions. Numerical results demonstrate the potential of these basis functions within MPM in terms of grid-crossing-error elimination and higher-order convergence.Team Raf Van de PlasNumerical Analysi

Fast Single-Mode Fiber Nonlinearity Monitoring: An Experimental Comparison Between Split-Step and Nonlinear Fourier Transform-Based Methods

We experimentally investigate the problem of monitoring the Kerr-nonlinearity coefficient $\gamma$ from transmitted and received data for a single-mode fiber link of 1600 km length. We compare the accuracy and speed of three different approaches. First, a standard split-step Fourier method is used to predict the output at various $\gamma$ values, which are then compared to the measured output. Second, a recently proposed nonlinear Fourier transform (NFT)-based method, which matches solitonic eigenvalues in the transmitted and received signals for various $\gamma$ values. Third, a novel fast version of the NFT-based method, which only matches the highest few eigenvalues. Although the NFT-based methods do not scale with link length, we demonstrate that the SSFM-based method is significantly faster than the basic NFT-based method for the considered link of 1600 km, and outperforms even the faster version. However, for a simulated link of 8000 km, the fast NFT-based method is shown to be faster than the SSMF-based method, although at the cost of a small loss in accuracy.Team Sander WahlsTeam Michel Verhaege

Experimental validation of nonlinear Fourier transform-based Kerr-nonlinearity identification over a 1600km SSMF link

Recently, a nonlinear Fourier transform-based Kerr-nonlinearity identification algorithm was demonstrated for a 1000 km NZDSF link with accuracy of 75%. Here, we demonstrate an accuracy of 99% over 1600 km SSMF. Reasons for improved accuracy are discussed.Accepted Author ManuscriptTeam Sander Wahl