Browsing by Author "Elmor L. Peterson, Committee Member"

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Bivariate Cubic L1 Splines and Applications
(2007-11-06) Zhang, Wei; Shu-Cherng Fang, Committee Chair; Elmor L. Peterson, Committee Member; Henry L.W. Nuttle, Committee Member; Xiuli Chao, Committee Member; John E. Lavery, Committee Member
Bivariate cubic L1 splines can provide shape-preserving surfaces for various applications. Using the reduced Hsieh-Clough-Tocher (rHCT) elements on the triangulated irregular networks (TINs), we model a bivariate cubic L1 spline as the solution to a nonsmooth convex programming problem. This problem is a generalized geometric programming (GGP) problem, whose dual problem is to optimize a linear objective function over convex cubic constraints. Using a linear programming transformation, a dual optimal solution can be converted to a desired primal solution. For computational efficiency, we further develop a compressed primal-dual interior-point method to directly calculate an approximated primal optimal solution. This compressed primal-dual algorithm can handle terrain data over hundreds-by-hundreds grids using a personal computer. However, for real-life applications, terrain data are given in thousands-by-thousands grids. To meet the computational challenge, we establish a "non-iterative" domain decomposition principle to reduce the computational requirements. We have also conducted computational experiments to show that the proposed domain decomposition principle can handle large size data for real terrain applications.
Metaheuristics for solving the Dial-a-Ride problem
(2004-08-10) Chan, Sook-Yee Edna; Elmor L. Peterson, Committee Member; John R. Stone, Committee Member; John W. Baugh Jr., Committee Chair; Yahya Fathi, Committee Member
Many transit agencies face the problem of generating routes and schedules to meet customer requests consisting of either pickup or dropoff requests using an available fleet of vehicles. The Dial-a-Ride Problem (DARP) is a mathematical model that closely approximates the problem faced by these agencies. The problem is a generalization of the well-known Pickup and Delivery Vehicle Routing Problem or Vehicle Routing Problem with Time Window. However, due to the high level of service required by this type of transportation service, additional operational constraints must be considered. While the DARP can be solved exactly by various techniques, exact approaches for the solution to real-world problems (typically consisting of hundreds of requests) are not practical. The time required is often excessive as the problem is NP-hard. In this thesis, we develop heuristics that find high quality solutions in a reasonable amount of computer time for the many-to-many, advanced reservation, multi-vehicle, single-depot, static DARP. The objectives considered include the minimization of total travel time and excess ride time, and the problem is subjected to maximum ride time, route duration, vehicle capacity, and wait time constraints. The cluster-first route-second approach is adopted. Clustering is performed using either Tabu Search (TS) or Scatter Search (SS) while routing is performed via insertion. The class of insertion heuristics has been extensively applied to the DARP. Earlier algorithms focused on feasible insertions but recently, heuristics that allow infeasible insertions to be considered during searches have been introduced. In this research, two insertion heuristics are considered: IRAU, which assigns requests only when they are feasible, and IRDU, which assigns all requests even if they result in infeasibilities. Comparison studies show that the benefit of using a particular algorithm depends on the statistical properties of the data sets used. Overall, the algorithms generated better solutions than a previously published real-world (322-request) problem and found the optimal solutions for constructed (32-request and 80-request) problems with known optimal solutions.
Theory and algorithms for cubic L1 splines
(2003-02-09) Cheng, Hao; Shu-Cherng Fang, Committee Chair; Henry L.W. Nuttle, Committee Co-Chair; Yahya Fathi, Committee Member; John E. Lavery, Committee Member; Elmor L. Peterson, Committee Member; Hien T. Tran, Committee Member
In modern geometric modeling, one of the requirements for interpolants is that they 'preserve shape well.' Shape preservation has often been associated with preservation of monotonicity and convexity/concavity. While shape preservation cannot yet be defined quantitatively, it is generally agreed that shape preservation involves eliminating extraneous non-physical oscillation. Classical splines, which exhibit extraneous oscillation, do not 'preserve shape well.' Recently, Lavery introduced a new class of cubic L1 splines. Empirical experiment has shown that cubic L1 splines are cable of providing C¹-smooth, shape-preserving, multi-scale interpolation of arbitrary data, including data with abrupt changes in spacing and magnitude, with no need for monotonicity or convexity constraints, node adjustment or other user input. However, the shape-preserving capability of cubic L1 splines has not been proved theoretically. The currently available algorithm only provides an approximation to the coefficients of cubic L1 splines. To lay the groundwork for theoretical analysis and the development of an exact algorithm, this dissertation proposes to treat cubic L1 spline problems in a geometric programming framework. Such a framework leads to a geometric dual problem with a linear objective function and convex quadratic constraints. It also provides a linear system for dual-to-primal conversion. We prove that cubic L1 splines preserve shape well, in particular, in eliminating non-physical oscillations, without review of raw data or any human intervention. We also show that cubic L1 splines perform well for multi-scale data, as well as preserve linearity and convexity/concavity under mild conditions. An exact algorithm based on the geometric programming model is proposed for solving cubic L1 splines. It decomposes the geometric programming problem into several independent small-sized sub-problems and applies a specialized active set algorithm to solve the sub-problems. The algorithm is numerically stable and highly parallelizable. It requires only simple algebraic operations.