Faculty & Staff Researchhttp://hdl.handle.net/1911/751722018-06-23T07:35:11Z2018-06-23T07:35:11ZBilevel Clique Interdiction and Related ProblemsBecker, Timothy Josephhttp://hdl.handle.net/1911/1022512018-06-19T17:51:22Z2017-05-01T00:00:00ZBilevel Clique Interdiction and Related Problems
Becker, Timothy Joseph
I introduce a formulation of the bilevel clique interdiction problem. Interdiction, a military term, describes the removal of enemy resources. The single level clique interdiction problem describes the attempt of an attacker to interdict a maximum number of cliques. The bilevel form of the problem introduces a defender who attempts to minimize the number of cliques interdicted by the attacker. An algorithm and formulation for the bilevel clique interdiction problem has not previously been investigated. I start by introducing a formulation and a column-generation algorithm to solve the problem of bilevel interdiction of a minimum clique transversal and move forward to the creation of a delayed row-and-column generation algorithm for bilevel clique interdiction. Next, I introduce a formulation and algorithm to solve the bilevel interdiction of a maximum stable set problem. Bilevel interdiction of a maximum stable set is choosing a maximum stable set, but with a defender who is attempting to minimize the maximum stable set that can be chosen by the interdictor. I introduce a deterministic formulation and a delayed column generation algorithm. Additionally, I introduce a stochastic formulation of the problem. I solve this problem using a cross-decomposition method that involves L-shaped cuts into a master problem as well as new "clique" cuts for the inner problem. Lastly, I define new classes of valid inequalities and facets for the clique transversal polytope. The valid inequalities come from two graph structures who have a closed form for their vertex cover number, which we use as a specic case for nding a minimum clique transversal. The first class of facets are just the maximal clique constraints of the clique transversal polytope. The next class contains an odd hole with distinct cliques on each edge of the hole. Another similar class contains an odd clique with distinct maximal cliques on the edges of one of its spanning cycles. The fourth class contains a clique with distinct maximal cliques on every edge of the initial clique, while the last class is a prism graph with distinct maximal cliques on every edge of the prism.
This work was also published as a Rice University thesis/dissertation: http://hdl.handle.net/1911/96109
2017-05-01T00:00:00ZNovel Techniques for the Zero-Forcing and p-Median Graph Location ProblemsFast, Caleb C.http://hdl.handle.net/1911/1022542018-06-19T17:51:22Z2017-05-01T00:00:00ZNovel Techniques for the Zero-Forcing and p-Median Graph Location Problems
Fast, Caleb C.
This thesis presents new methods for solving two graph location problems, the p-Median problem and the zero-forcing problem. For the p-median problem, I present a branch decomposition based method that finds the best p-median solution that is limited to some input support graph. The algorithm can be used to either find an integral solution from a fractional linear programming solution, or it can be used to improve on the solutions given by a pool of heuristics. In either use, the algorithm compares favorably in running time or solution quality to state-of-the-art heuristics. For the zero-forcing problem, this thesis gives both theoretical and computational results. In the theoretical section, I show that the branchwidth of a graph is a lower bound on its zero-forcing number, and I introduce new bounds on the zero-forcing iteration index for cubic graphs. This thesis also introduces a special type of graph structure, a zero-forcing fort, that provides a powerful tool for the analysis and modeling of zero-forcing problems. In the computational section, I introduce multiple integer programming models for finding minimum zero-forcing sets and integer programming and combinatorial branch and bound methods for finding minimum connected zero-forcing sets. While the integer programming methods do not perform better than the best combinatorial method for the basic zero-forcing problem, they are easily adapted to the connected zero-forcing problem, and they are the best methods for the connected zero-forcing problem.
This work was also published as a Rice University thesis/dissertation: http://hdl.handle.net/1911/96074
2017-05-01T00:00:00ZHermite Methods for the Simulation of Wave PropagationVargas, Arturohttp://hdl.handle.net/1911/1022522018-06-19T17:51:22Z2017-05-01T00:00:00ZHermite Methods for the Simulation of Wave Propagation
Vargas, Arturo
Simulations of wave propagation play a crucial role in science and engineering. In applications of geophysics, they are the engine of many seismic imaging algorithms. For electrical engineers, they can be a useful tool for the design of radars and antennas. In these applications achieving high fidelity, simulations are challenging due to the inherent issues in modeling highly oscillatory waves and the associated high computational cost of high-resolution simulations. Thus the ideal numerical method should be able to capture high-frequency waves and be suitable for parallel computing. In both seismic applications and computational electromagnetics the Yee scheme, a finite difference time domain (FDTD) method, is the method of choice for structured grids. The scheme has the benefit of being easy to implement but performs poorly in the presence of high-frequency waves. High order accurate FDTD methods may be derived but ultimately rely on neighboring grid points when approximating derivative. In contrast to FDTD methods, the Hermite methods of Goodrich and co-authors (2006) use Hermite interpolation and a staggered (dual) grid to construct high order accurate numerical methods for first order hyperbolic equations. These methods achieve high order approximations in both time and space by reconstructing local polynomials within cells of the computational domain and employing Hermite-Taylor time stepping. The resulting schemes are able to evolve the solution locally within a cell making them ideal for parallel computing. Building on the original Hermite methods this thesis focuses on two goals: (1) the development of new Hermite methods and (2) their implementation on modern computing architectures. To accomplish the first objective, this thesis presents two variations of Hermite methods which are designed to simplify the scheme while preserving the favorable features. The first variation is a family of Hermite methods which do not require a dual grid. These methods eliminate the need for storing dual coefficients while maintaining optimal convergence rates. The second type of variation are Hermite methods which use leapfrog time-stepping. These schemes propagate the solution with less computation than the original scheme and may be used for either first or second order equations. To address the second objective, this thesis presents algorithms which take advantage of the many-core architecture of graphics processing units (GPU). As threedimensional simulations can easily exceed the memory of a single GPU, techniques for partitioning the data across multiple GPUs are presented. Finally, this thesis presents numerical results and performance studies which confim the accuracy and efficiency of the proposed Hermite methods for linear and nonlinear wave equations.
This work was also published as a Rice University thesis/dissertation: http://hdl.handle.net/1911/96153
2017-05-01T00:00:00ZGPU-Accelerated Discontinuous Galerkin Methods on Hybrid Meshes: Applications in Seismic ImagingWang, Zhenghttp://hdl.handle.net/1911/1022502018-06-19T17:51:22Z2017-05-01T00:00:00ZGPU-Accelerated Discontinuous Galerkin Methods on Hybrid Meshes: Applications in Seismic Imaging
Wang, Zheng
Seismic imaging is a geophysical technique assisting in the understanding of subsurface structure on a regional and global scale. With the development of computer technology, computationally intensive seismic algorithms have begun to gain attention in both academia and industry. These algorithms typically produce high-quality subsurface images or models, but require intensive computations for solving wave equations. Achieving high-fidelity wave simulations is challenging: first, numerical wave solutions may suffer from dispersion and dissipation errors in long-distance propagations; second, the efficiency of wave simulators is crucial for many seismic applications. High-order methods have advantages of decreasing numerical errors efficiently and hence are ideal for wave modelings in seismic problems. Various high order wave solvers have been studied for seismic imaging. One of the most popular solvers is the finite difference time domain (FDTD) methods. The strengths of finite difference methods are the computational efficiency and ease of implementation, but the drawback of FDTD is the lack of geometric flexibility. It has been shown that standard finite difference methods suffer from first order numerical errors at sharp media interfaces. In contrast to finite difference methods, discontinuous Galerkin (DG) methods, a class of high-order numerical methods built on unstructured meshes, enjoy geometric flexibility and smaller interface errors. Additionally, DG methods are highly parallelizable and have explicit semi-discrete form, which makes DG suitable for large-scale wave simulations. In this dissertation, the discontinuous Galerkin methods on hybrid meshes are developed and applied to two seismic algorithms---reverse time migration (RTM) and full waveform inversion (FWI). This thesis describes in depth the steps taken to develop a forward DG solver for the framework that efficiently exploits the element specific structure of hexahedral, tetrahedral, prismatic and pyramidal elements. In particular, we describe how to exploit the tensor-product property of hexahedral elements, and propose the use of hex-dominant meshes to speed up the computation. The computational efficiency is further realized through a combination of graphics processing unit (GPU) acceleration and multi-rate time stepping. As DG methods are highly parallelizable, we build the DG solver on multiple GPUs with element-specific kernels. Implementation details of memory loading, workload assignment and latency hiding are discussed in the thesis. In addition, we employ a multi-rate time stepping scheme which allows different elements to take different time steps. This thesis applies DG schemes to RTM and FWI to highlight the strengths of the DG methods. For DG-RTM, we adopt the boundary value saving strategy to avoid data movement on GPUs and utilize the memory load in the temporal updating procedure to produce images of higher qualities without a significant extra cost. For DG-FWI, a derivation of the DG-specific adjoint-state method is presented for the fully discretized DG system. Finally, sharp media interfaces are inverted by specifying perturbations of element faces, edges and vertices.
This work was also published as a Rice University thesis/dissertation: http://hdl.handle.net/1911/96151
2017-05-01T00:00:00Z