Interprocessor communication with limited memory

A long-standing open question associated with the use of collocation methods for boundary value ordinary differential equations is concerned with the development of a high order continuous solution approximation to augment the high order discrete solution approximation, obtained at the mesh points which subdivide the problem interval. It is well known that the use of collocation at Gauss points leads to solution approximations at the mesh points for which the global error is O(h2k), where k is the number of collocation points used per subinterval and h is the subinterval size. This discrete solution is said to be superconvergent. The collocation solution also yields a C0 continuous solution approximation that has a global error of O(hk+1). In this paper, we show how to efficiently augment the superconvergent discrete collocation solution to obtain C1 continuous "superconvergent" interpolants whose global errors are O(h2k ). The key ideas are to use the theoretical framework of continuous Runge--Kutta schemes and to augment the collocation solution with inexpensive monoimplicit Runge--Kutta stages. Specific schemes are derived for k = 1, 2, 3, and 4. Numerical results are provided to support the theoretical analysis.

Task-parallel problems are difficult to implement efficiently in parallel because they are asynchronous and unpredictable. The difficulties are compounded on distributedmemory computers where interprocessor communication can impose a substantial overhead. A few languages and libraries have been proposed that are specifically designed to support this kind of computation. However, one big challenge still remains: to make those tools understood and used by scientists. engineers, and others who want to exploit the power of parallel computers without spending much effort in mastering those tools. The PMESC programming paradigm and library presented here are designed to make programming on distributed-memory computers easy to understand and to make efficient parallel code easy to produce. The paradigm provides a methodology for structuring task-parallel problems that allows the separation of different phases in the computation. The library provides support for those phases that are application-independent allowing the users to concentrate on the applicationspecific oues.

A local search approximation algorithm for k-means clustering

In k-means clustering we are given a set of n data points in d-dimensional space ℜd and an integer k, and the problem is to determine a set of k points in ℜd, called centers, to minimize the mean squared distance from each data point to its nearest center. No exact polynomial-time algorithms are known for this problem. Although asymptotically efficient approximation algorithms exist, these algorithms are not practical due to the extremely high constant factors involved. There are many heuristics that are used in practice, but we know of no bounds on their performance.We consider the question of whether there exists a simple and practical approximation algorithm for k-means clustering. We present a local improvement heuristic based on swapping centers in and out. We prove that this yields a (9+e)-approximation algorithm. We show that the approximation factor is almost tight, by giving an example for which the algorithm achieves an approximation factor of (9-e). To establish the practical value of the heuristic, we present an empirical study that shows that, when combined with Lloyd's algorithm, this heuristic performs quite well in practice.

We present a quite simple, fast and practical algorithm to find a short cyclic tour that visits a set of points distributed on the plane. The algorithm runs in O(n log n) time with O(n) space, and is simple enough to easily implement on resource restricted machines. It constructs a tour essentially by axis-sorts of the points and takes a kind of the 'fixed dissection strategy,' though it neither tries to find best tours in subregions nor optimizes the order among subregions. As well as the worst-case approximation ratio of produced tours, we show that the algorithm is a 'probabilistic' constant-ratio approximation algorithm for uniform random distributions. We made computational comparisons of our algorithm, Karp's partitioning algorithm, Lin-Kernighan local search, Arora's randomized PTAS, etc. The results indicate that in running time our algorithm overwhelms existing ones, and the average approximation ratio is better or competitive.

Scheduling directed acyclic graphs with optimal duplication strategy on homogeneous multiprocessor systems

It is tacitly accepted that, for practical basis sets consisting of N functions, solution of the two-electron Coulomb problem in quantum mechanics requires storage of O(N(4)) integrals in the small N limit. For localized functions, in the large N limit, or for planewaves, due to closure, the storage can be reduced to O(N(2)) integrals. Here, it is shown that the storage can be further reduced to O(N(2/3)) for separable basis functions. A practical algorithm, that uses standard one-dimensional Gaussian-quadrature sums, is demonstrated. The resulting algorithm allows for the simultaneous storage, or fast reconstruction, of any two-electron Coulomb integral required for a many-electron calculation on processors with limited memory and disk space. For example, for calculations involving a basis of 9171 planewaves, the memory required to effectively store all Coulomb integrals decreases from 2.8 Gbytes to less than 2.4 Mbytes.

This paper presents a systematic method for generating optimal helicopter flight trajectories in real time for cockpit display. Helicopter flights in the event of a single engine failure are critical, and display of optimal flight paths on-board is desirable. Due to the large number of states and constraints, however, numerical solutions on a mainframe computer can take a long tune. In this paper, optimal solutions are first obtained off-line using a ground computer. For a given set of initial helicopter states and flight parameters, optimal solutions are fitted by a Fourier series. The coefficients of the Fourier approximations are then interpolated as functions of initial flight states and parameters. The approximate optimal trajectories can be quickly generated on-board using the proposed method. Introduction Engine failure is a major safety issue for all helicopters, especially during takeoff and landing operations. Due to the complexity of the situation and the dependence of safety on many parameters, optimal flight trajectories can provide useful guidance for pilot operation. In particular, on-board display of optimal trajectories is desirable. This paper presents a method for generating optimal flight trajectories in real-tune that can be used for on-board display. Depending on its capability and performance during an engine failure, a transport helicopter can be certified as Category A or Category B. The Federal Aviation Administration (FAA) grants Category B certification to a single or multi-engine helicopter with gross weight less than 20,000 Ib and capable of landing safely in case of one or all engine failures. Category A certification, on the other hand, is granted to multi-engine helicopters with independent engine systems. Category A helicopters must be able to either land safely or continue flight in case of One Engine Inoperation (OEI). This makes Category A helicopters uniquely suited for remote places where emergency landing sites are not available. I Ph.D. Candidate, Center for Control Sciences, Member AIAA * Assistant Professor, Dept. of Aerospace Eng. and Mechanics, Member AIAA + Civil Rotorcraft Group Leader, Rotorcraft and Powered Lift Branch The decision to continue the flight or to land represents an energy management problem. It has been shown''' that this decision depends on initial conditions at the time of failure as well as flight parameters. Due to the numerous initial variables involved in this decision, a method to communicate all this information to the pilot instantaneously, at engine failure, is needed. This method must keep the emergency procedure simple and manageable. In particular, on-board cockpit display of optimal flight trajectories is useful. This requires a reliable real-tune solution of optimal trajectories, on board, with limited time, limited computing power, and limited memory available. If possible, an analytical solution of optimal trajectories is desirable. Unfortunately, this can only be achieved for systems of low order and where the equations of motion and the cost function are simple. Therefore, for a complex system, an open-loop numerical procedure becomes necessary. Research efforts were made to solve optimal control problems in real-time. Schultz et al. used calculus of variation and energy state method to generate optimal trajectories on-board for the National Aerospace Plane (NASP). In his study, the equations of motion of the NASP are less complex than those of the helicopter, and an interval of time of few minutes is allowed. Bryson used neighboring control theory to generate near-optimal solutions that could be used in real-time. This method is elegant but requires a large amount of memory to store the needed perturbation matrices. Also, it is valid on a relatively small neighborhood of the nominal trajectory. Perturbation methods and table lookups are often used if only few parameters are involved. In the current problem, the equations of motion of the helicopter are complex and too many path equality and inequality constraints are present. Also, limitations on tune, computing power, and memory available make all the above methods unsuitable for this problem. This paper proposes a technique of real-time optimal trajectory generation based on approximation and

In this paper, we describe a very simple (1+/spl epsi/)-approximation algorithm for the multicommodity flow problem. The algorithm runs in time that is polynomial in N (the number of nodes in the network) and /spl epsiv//sup -1/ (the closeness of the approximation to optimal). The algorithm is remarkable in that it is much simpler than all known polynomial time flow algorithms (including algorithms for the special case of one-commodity flow). In particular, the algorithm does not rely on augmenting paths, shortest paths, min-cost paths, or similar techniques to push flow through a network. In fact, no explicit attempt is ever made to push flow towards a sink during the algorithm. Because the algorithm is so simple, it can be applied to a variety of problems for which centralized decision making and flow planning is not possible. For example, the algorithm can be easily implemented with local control in a distributed network and it can be made tolerant to link failures. In addition, the algorithm appears to perform well in practice. Initial experiments using the DIMACS generator of test problems indicate that the algorithm performs as well as or better than previously known algorithms, at least for certain test problems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The onset of wireless networks globally has thrust researchers in academia and industry to solve problems related to this ever-growing field. In this paper, we study the multi-constrained opportunistic wireless scheduling problem in cognitive radio networks. Given a collection of secondary user communication links, the channel state of each link is unknown due to the unpredictable primary users’ activities, but can be estimated by exploring the channel state transitions and channel state feedback. A scheduling algorithm is used to decide a subset of links to transmit each time with both interference-free constraints and power budget constraints. The objective of this paper is to design a scheduling algorithm to optimize the average reward over a long time horizon. Current existing approaches cannot satisfyingly provide solutions for the wireless opportunistic scheduling problem when considering multiple constraints. In this work, we adopt the paradigm of the restless multi-armed bandit and propose a fast and simple approximation algorithm. The performance of the proposed algorithm is verified with a small approximation bound for the multi-constrained wireless opportunistic wireless scheduling problem.

Matchings for a random bipartite graph are considered. Each of the alpha M nodes on one side of the graph is directly connected to Q nodes chosen randomly and uniformly from among the M nodes on the other side of the graph. The size matchings found by two simple approximation algorithms, as well as the size of the maximum matching when Q=2, are asymptotically determined in the limit as Q tends to infinity with alpha fixed. The work is motivated by a distributed communications protocol for accessing a silent receiver. The theory of approximating slow Markov random walks by ordinary differential equations is used for the analysis. >

Randomized approximation algorithms for set multicover problems with applications to reverse engineering of protein and gene networks

An upper dominating set is a minimal dominating set in a graph. In the Upper Dominating Set problem, the goal is to find an upper dominating set of maximum size. We study the complexity of parameterized algorithms for Upper Dominating Set, as well as its sub-exponential approximation. First, we prove that, under ETH, k -Upper Dominating Set cannot be solved in time \(O(n^{o(k)})\) (improving on \(O(n^{o(\sqrt{k})})\)), and in the same time we show under the same complexity assumption that for any constant ratio r and any \(\varepsilon > 0\), there is no r-approximation algorithm running in time \(O(n^{k^{1-\varepsilon }})\). Then, we settle the problem’s complexity parameterized by pathwidth by giving an algorithm running in time \(O^*(6^{pw})\) (improving the current best \(O^*(7^{pw})\)), and a lower bound showing that our algorithm is the best we can get under the SETH. Furthermore, we obtain a simple sub-exponential approximation algorithm for this problem: an algorithm that produces an r-approximation in time \(n^{O(n/r)}\), for any desired approximation ratio \(r 1\) and \(\varepsilon > 0\), no algorithm can output an r-approximation in time \(n^{(n/r)^{1-\varepsilon }}\). Hence, we completely characterize the approximability of the problem in sub-exponential time.

This paper considers the following problem: given an edgeweighted graphG = (V, E, w) and disjointk-subsetsU p ofV, find a minimum weighted set of edgesE′ ⊆E such that its removal disconnects the graph intok parts and each part contains exactly one vertex from eachU p for 1 ≤p ≤r. This generalizes some well-known NP-hard problems. In this paper, we first apply greedy local search algorithm to obtain better approximation solutions. Then we give a randomized local search algorithm which produces a solution within a factor (1 + e) with the probability at least (1 - 1/e) for any small e. Simple near-optimum approximation algorithms are also proposed. Analogously, there is a maximumk-multiway cut problem with the same restrictions.

Aerodynamic optimization design of wing shape is a complicated problem, so the computational efficiency of the method is very important. Genetic Algorithms (GAs) are strong tools for solving the large-scale optimization problems. They have very strong global search capability, but poor local search capability. This makes GAs converge very slowly especially when they are used for such complex question as the inverse design of 3d wing shape. The genetic operators are specially designed and Simultaneous Perturbation Stochastic Approximation (SPSA) is introduced into Stochastic Gradient Genetic Algorithm (SGGA). In fact, SPSA is a simple, easily implemented and efficient stochastic approximation algorithm. Also SGGA is used for the inverse design of wing shape. And flow field is calculated by means of wing supercritical analysis program-the software of simplification version of FLO22VSE. The paper explores a high efficient method for the inverse design of wing shape. The experiments show satisfactory results.