Optimal location with equitable loads

The problem considered in this paper is to find p locations for p facilities such that the weights attracted to each facility will be as close as possible to one another. We model this problem as minimizing the maximum among all the total weights attracted to the various facilities. We propose solution procedures for the problem on a network, and for the special cases of the problem on a tree or on a path. The complexity of the problem is analyzed, O(n) algorithms and an O(pn ³) dynamic programming algorithm are proposed for the problem on a path respectively for p=2 and p>2 facilities. Heuristic algorithms (two types of a steepest descent approach and tabu search) are proposed for its solution. Extensive computational results are presented.     

On Conditional Covering Problem

The conditional covering problem (CCP) aims to locate facilities on a graph, where the vertex set represents both the demand points and the potential facility locations. The problem has a constraint that each vertex can cover only those vertices that lie within its covering radius and no vertex can cover itself. The objective of the problem is to find a set that minimizes the sum of the facility costs required to cover all the demand points. An algorithm for CCP on paths was presented by Horne and Smith (Networks 46(4):177–185, 2005). We show that their algorithm is wrong and further present a correct O(n ³) algorithm for the same. We also propose an O(n ²) algorithm for the CCP on paths when all vertices are assigned unit costs and further extend this algorithm to interval graphs without an increase in time complexity.     

Parallel algorithms for permutation graphs

In this paper, parallel algorithms are presented for solving some problems on permutation graphs. The coloring problem is solved inO(log² n) time usingO(n ³/logn) processors on the CREW PRAM, orO(logn) time usingO(n ³) processors on the CRCW PRAM. The weighted clique problem, the weighted independent set problem, the cliques cover problem, and the maximal layers problem are all solved with the same complexities. We can also show that the longest common subsequence problem belongs to the class NC.     

Extrapolation and superconvergence of the Steklov eigenvalue problem

On the basis of a transform lemma, an asymptotic expansion of the bilinear finite element is derived over graded meshes for the Steklov eigenvalue problem, such that the Richardson extrapolation can be applied to increase the accuracy of the approximation, from which the approximation of O(h ^3.5) is obtained. In addition, by means of the Rayleigh quotient acceleration technique and an interpolation postprocessing method, the superconvergence of the bilinear finite element is presented over graded meshes for the Steklov eigenvalue problem, and the approximation of O(h ³) is gained. Finally, numerical experiments are provided to demonstrate the theoretical results.     

On a decision rule using dichotomies for identifying the nonnegligible parameter in certain linear models

Consider the class of linear models (with uncorrelated observation, each having variance σ²), in which it is known that at most k (location) parameters are negligible, but it is not known which are negligible. The problem is to identify the nonnegligible parameters. In this paper, for k = 1, and under certain restrictions on the model, a technique is developed for solving this problem, which has the feature of requiring (in an information theoretic sense) the minimum amount of computation. (It can “search through” 2^m objects, using m “steps.”) The technique consists of dichotomizing the set of parameters (one known subset possibly containing the nonnegligible element, and the other not), using chi-square variables. A method for computing the probability that the correct parameter is identified, is presented, and an important application to factorial search designs is established.     

A new implementation of an algorithm for the optimal assignment problem: An improved version of Munkres' algorithm

Existing implementations of Munkres' algorithm for the optimal assignment problem are shown to requireO(n ⁴) time in the worstn×n case. A new implementation is presented which runs in worst-case timeO(n ³) and compares favorably in performance with the algorithm of Edmonds and Karp for this problem.The results of this paper were obtained by the author while at the Department of Computer Science, Cornell University. This work was supported in part by a Vanderbilt University Research Council Grant.     

Efficient Sequential Quadratic Programming Implementations for Equality-Constrained Discrete-Time Optimal Control

Efficient sequential quadratic programming (SQP) implementations are presented for equality-constrained, discrete-time, optimal control problems. The algorithm developed calculates the search direction for the equality-based variant of SQP and is applicable to problems with either fixed or free final time. Problem solutions are obtained by solving iteratively a series of constrained quadratic programs. The number of mathematical operations required for each iteration is proportional to the number of discrete times N. This is contrasted by conventional methods in which this number is proportional to N ³. The algorithm results in quadratic convergence of the iterates under the same conditions as those for SQP and simplifies to an existing dynamic programming approach when there are no constraints and the final time is fixed. A simple test problem and two application problems are presented. The application examples include a satellite dynamics problem and a set of brachistochrone problems involving viscous friction.     

Segment Intersection Searching Problems in General Settings

Abstract. We consider segment intersection searching amidst (possibly intersecting) algebraic arcs in the plane. We show how to preprocess n arcs in time O(n ^2+ɛ ) into a data structure of size O(n ^2+ɛ ) , for any ɛ >0 , such that the k arcs intersecting a query segment can be counted in time O( log n) or reported in time O( log n+k) . This problem was extensively studied in restricted settings (e.g., amidst segments, circles, or circular arcs), but no solution with comparable performance was previously presented for the general case of possibly intersecting algebraic arcs. Our data structure for the general case matches or improves (sometimes by an order of magnitude) the size of the best previously presented solutions for the special cases. As an immediate application of this result, we obtain an efficient data structure for the triangular windowing problem, which is a generalization of triangular range searching. As another application, the first substantially subquadratic algorithm for a red—blue intersection counting problem is derived. We also describe simple data structures for segment intersection searching among disjoint arcs, and for ray shooting among possibly intersecting arcs.     

Analysis of the immersed boundary method for a finite element Stokes problem

Convergence results are presented for the immersed boundary (IB) method applied to a model Stokes problem. As a discretization method, we use the finite element method. First, the immersed force field is approximated using a regularized delta function. Its error in the W^{?1, p} norm is examined for 1 ≤ p < n/(n ? 1), with n representing the space dimension. Subsequently, we consider IB discretization of the Stokes problem and examine the regularization and discretization errors separately. Consequently, error estimate of order h^{1 ? α} in the W^{1, 1} × L¹ norm for the velocity and pressure is derived, where α is an arbitrary small positive number. The validity of those theoretical results is confirmed from numerical examples.     

The almost global and global existence for quasi-linear wave equations with multiple-propagation speeds in high dimensions

In this paper, we consider the Cauchy problem for systems of quasi-linear wave equations with multiple propagation speeds in spatial dimensions n ≥ 4. The problem when the nonlinearities depend on both the unknown function and their derivatives is studied. Based on some Klainerman-Sideris type weighted estimates and space-time L ² estimates, the results that the almost global existence for space dimensions n = 4 and global existence for n ≥ 5 of small amplitude solutions are presented.     

Increasing the Weight of Minimum Spanning Trees

The problems of computing the maximum increase in the weight of the minimum spanning trees of a graph caused by the removal of a given number of edges, or by finite increases in the weights of the edges, are investigated. For the case of edge removals, the problem is shown to be NP-hard and an Ω(1/log k)-approximation algorithm is presented for it, where (input parameter) k > 1 is the number of edges to be removed. The second problem is studied, assuming that the increase in the weight of an edge has an associated cost proportional to the magnitude of the change. An O(n³m² log(n²/m)) time algorithm is presented to solve it.     

Convergence analysis of a hp-finite element approximation of the time-harmonic Maxwell equations with impedance boundary conditions in domains with an analytic boundary

We consider a nonconforming hp -finite element approximation of a variational formulation of the time-harmonic Maxwell equations with impedance boundary conditions proposed by Costabel et al. The advantages of this formulation is that the variational space is embedded in H¹ as soon as the boundary is smooth enough (in particular it holds for domains with an analytic boundary) and standard shift theorem can be applied since the associated boundary value problem is elliptic. Finally in order to perform a wavenumber explicit error analysis of our problem, a splitting lemma and an estimation of the adjoint approximation quantity are proved by adapting to our system the results from Melenk and Sauter obtained for the Helmholtz equation. Some numerical tests that illustrate our theoretical results are also presented. Analytic regularity results with bounds explicit in the wavenumber of the solution of a general elliptic system with lower order terms depending on the wavenumber are needed and hence proved.     

Finite difference schemes for the fourth-order parabolic equations with different boundary value conditions

In this paper, the fourth-order parabolic equations with different boundary value conditions are studied. Six kinds of boundary value conditions are proposed. Several numerical differential formulae for the fourth-order derivative are established by the quartic interpolation polynomials and their truncation errors are given with the aid of the Taylor expansion with the integral remainders. Effective difference schemes are presented for the third Dirichlet boundary value problem, the first Neumann boundary value problem and the third Neumann boundary value problem, respectively. Some new embedding inequalities on the discrete function spaces are presented and proved. With the method of energy analysis, the unique solvability, unconditional stability and unconditional convergence of the difference schemes are proved. The convergence orders of derived difference schemes are all O(τ² + h²) in appropriate norms. Finally, some numerical examples are provided to confirm the theoretical results.     

Application of Rothe’s Method to a Parabolic Inverse Problem with Nonlocal Boundary Condition

We consider an inverse problem for the determination of a purely time-dependent source in a semilinear parabolic equation with a nonlocal boundary condition. An approximation scheme for the solution together with the well-posedness of the problem with the initial value u₀ ∈ H¹(Ω) is presented by means of the Rothe time-discretization method. Further approximation scheme via Rothe’s method is constructed for the problem when u₀ ∈ L²(Ω) and the integral kernel in the nonlocal boundary condition is symmetric.

     

Single Supplier Scheduling for Multiple Deliveries

The problem of scheduling the production and delivery of a supplier to feed the production of F manufacturers is studied. The orders fulfilled by the supplier are delivered to the manufacturers in batches of the same size. The supplier's production line has to be set up whenever it switches from processing an order of one manufacturer to an order of another manufacturer. The objective is to minimize the total setup cost, subject to maintaining continuous production for all manufacturers. The problem is proved to be NP-hard. It is reduced to a single machine scheduling problem with deadlines and jobs belonging to F part types. An O(NlogF) algorithm, where N is the number of delivery batches, is presented to find a feasible schedule. A dynamic programming algorithm with O(N ^F /F ^F–2) running time is presented to find an optimal schedule. If F=2 and setup costs are unit, an O(N) time algorithm is derived.     

Embedding Free Burnside Groups in Finitely Presented Groups

We construct an embedding of a free Burnside group B(m,n) of odd exponent n > 2⁴⁸ and rank m >1 in a finitely presented group with some special properties. The main application of this embedding is an easy construction of finitely presented nonamenable groups without noncyclic free subgroups (which provides a new finitely presented counterexample to the von Neumann problem on amenable groups). As another application, we construct weakly finitely presented groups of odd exponent n ≫ 1 which are not locally finite.     

Matrix decomposition algorithms for the finite element Galerkin method with piecewise Hermite cubics

Matrix decomposition algorithms (MDAs) employing fast Fourier transforms are developed for the solution of the systems of linear algebraic equations arising when the finite element Galerkin method with piecewise Hermite bicubics is used to solve Poisson’s equation on the unit square. Like their orthogonal spline collocation counterparts, these MDAs, which require O(N ²logN) operations on an N×N uniform partition, are based on knowledge of the solution of a generalized eigenvalue problem associated with the corresponding discretization of a two-point boundary value problem. The eigenvalues and eigenfunctions are determined for various choices of boundary conditions, and numerical results are presented to demonstrate the efficacy of the MDAs. Weiwei Sun was supported in part by a grant from City University of Hong Kong (Project No. CityU 7002110).     

A Priori Error Estimates of Crank–Nicolson Finite Volume Element Method for a Hyperbolic Optimal Control Problem

In this article, a Crank–Nicolson linear finite volume element scheme is developed to solve a hyperbolic optimal control problem. We use the variational discretization technique for the approximation of the control variable. The optimal convergent order O(h² + k²) is proved for the numerical solution of the control, state and adjoint‐state in a discrete L²‐norm. To derive this result, we also get the error estimate (convergent order O(h² + k²)) of Crank–Nicolson finite volume element approximation for the second‐order hyperbolic initial boundary value problem. Numerical experiments are presented to verify the theoretical results.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1331–1356, 2016