Showing papers presented at "Conference on Scientific Computing in 2016"

Coordinated Platoon Routing in a Metropolitan Network.

Abstract: Platooning vehicles—connected and automated vehicles traveling with small intervehicle distances—use less fuel because of reduced aerodynamic drag. Given a network defined by vertex and edge sets and a set of vehicles with origin/destination nodes/times, we model and solve the combinatorial optimization problem of coordinated routing of vehicles in a manner that routes them to their destination on time while using the least amount of fuel. Common approaches decompose the platoon coordination and vehicle routing into separate problems. Our model addresses both problems simultaneously to obtain the best solution. We use modern modeling techniques and constraints implied from analyzing the platoon routing problem to address larger numbers of vehicles and larger networks than previously considered. While the numerical method used is unable to certify optimality for candidate solutions to all networks and parameters considered, we obtain excellent solutions in approximately one minute for much larger networks and vehicle sets than previously considered in the literature.

Graph Partitioning Methods for Fast Parallel Quantum Molecular Dynamics

Abstract: We study a graph partitioning problem motivated by the simulation of the physical movement of multi-body systems on an atomistic level, where the forces are calculated from a quantum mechanical description of the electrons. Several advanced algorithms have been published in the literature for such simulations that are based on evaluations of matrix polynomials. We aim at efficiently parallelizing these computations by using a special type of graph partitioning. For this, we represent the zero-nonzero structure of a thresholded matrix as a graph and partition that graph into several components. The matrix polynomial is then evaluated for each separate submatrix corresponding to the subgraphs and the evaluated submatrix polynomials are used to assemble the final result for the full matrix polynomial. The paper provides a rigorous definition as well as a mathematical justification of this partitioning problem. We use several algorithms to compute graph partitions and experimentally evaluate their performance with respect to the quality of the partition obtained with each method and the time needed to produce it.

On Stable Marriages and Greedy Matchings

Abstract: Research on stable marriage problems has a long and mathematically rigorous history, while that of exploiting greedy matchings in combinatorial scientific computing is a younger and less developed research field. We consider the relationships between these two areas. In particular we show that several problems related to computing greedy matchings can be formulated as stable marriage problems and as a consequence several recently proposed algorithms for computing greedy matchings are in fact special cases of well known algorithms for the stable marriage problem. However, in terms of implementations and practical scalable solutions on modern hardware, work on computing greedy matchings has made considerable progress. We show that due to this strong relationship many of these results are also applicable for solving stable marriage problems. This is further demonstrated by designing and testing efficient multicore as well as GPU algorithms for the stable marriage problem.

A New 3/2-Approximation Algorithm for the b -Edge Cover Problem.

Abstract: We describe a 3/2-approximation algorithm, LSE, for computing a b-Edge Cover of minimum weight in a graph with weights on the edges. The b-Edge Cover problem is a generalization of the better-known Edge Cover problem in graphs, where the objective is to choose a subset C of edges in the graph such that at least a specified number b(v) of edges in C are incident on each vertex v. In the weighted b-Edge Cover problem, we minimize the sum of the weights of the edges in C. We prove that the LSE algorithm computes the same b-Edge Cover as the one obtained by the Greedy algorithm for the problem. However, the Greedy algorithm requires edges to be sorted by their effective weights, and these weights need to be updated after each iteration. These requirements make the Greedy algorithm sequential and impractical for massive graphs. The LSE algorithm avoids the sorting step, and is amenable for parallelization. We implement the algorithm on a serial machine and compare its performance against a collection of approximation algorithms for the b-Edge Cover problem. Our results show that the LSE algorithm is 3× to 5× faster than the Greedy algorithm on a serial processor. The approximate edge covers obtained by the LSE algorithm have weights greater by at most 17% of the optimal weight for problems where we could compute the latter. We also investigate the relationship between the b-Edge Cover and the b-Matching problems, show that the latter has a faster implementation since edge weights are static in this algorithm, and obtain a heuristic solution for the former from the latter.

Estimating Current-Flow Closeness Centrality with a Multigrid Laplacian Solver.

Abstract: Matrices associated with graphs, such as the Laplacian, lead to numerous interesting graph problems expressed as linear systems. One field where Laplacian linear systems play a role is network analysis, e. g. for certain centrality measures that indicate if a node (or an edge) is important in the network. One such centrality measure is current-flow closeness. To allow network analysis workflows to profit from a fast Laplacian solver, we provide an implementation of the LAMG multigrid solver in the NetworKit package, facilitating the computation of current-flow closeness values or related quantities. Our main contribution consists of two algorithms that accelerate the current-flow computation for one node or a reasonably small node subset significantly. One sampling-based algorithm provides an unbiased estimation of the related electrical farness, the other one is based on the Johnson-Lindenstrauss transform. Our inexact algorithms lead to very accurate results in practice. Thanks to them one is now able to compute an estimation of current-flow closeness of one node on networks with tens of millions of nodes and edges within seconds or a few minutes. From a network analytical point of view, our experiments indicate that current-flow closeness can discriminate among different nodes significantly better than traditional shortest-path closeness and is also considerably more resistant to noise -- we thus show that two known drawbacks of shortest-path closeness are alleviated by the current-flow variant.

Highly efficient surface normal integration

Abstract: The integration of surface normals for the computation of a surface in 3D space is a classic problem in computer vision. However, even nowadays it is still a challenging task to device a method that combines the flexibility to deal with non-trivial computational domains with high accuracy, robustness and computational efficiency. In this paper we propose to use for the first time in the literature Krylov subspace solvers as a main step in tackling the task. While these methods can be very efficient, they may only show their full potential when combined with a numerical preconditioning and even more importantly, a suitable initialization. To address the latter issue we propose to compute this initial state via a recently developed fast marching integrator. Numerical experiments prove the benefits of this novel combination.

On matrix symmetrization and sparse direct solvers

Abstract: We investigate algorithms for finding column permutations of sparse matrices in order to have large diagonal entries and to have many entries symmetrically positioned around the diagonal. The aim is to improve the memory and running time requirements of a certain class of sparse direct solvers. We propose efficient algorithms for this purpose by combining two existing approaches and demonstrate the effect of our findings in practice using a direct solver. In particular, we show improvements in a number of components of the running time of a sparse direct solver with respect to the state of the art on a diverse set of matrices.

An Empirical Study of Cycle Toggling Based Laplacian Solvers

Abstract: Author(s): Deweese, Kevin; Gilbert, John R; Miller, Gary; Peng, Richard; Xu, Hao Ran; Xu, Shen Chen | Abstract: We study the performance of linear solvers for graph Laplacians based on the combinatorial cycle adjustment methodology proposed by [Kelner-Orecchia-Sidford-Zhu STOC-13]. The approach finds a dual flow solution to this linear system through a sequence of flow adjustments along cycles. We study both data structure oriented and recursive methods for handling these adjustments. The primary difficulty faced by this approach, updating and querying long cycles, motivated us to study an important special case: instances where all cycles are formed by fundamental cycles on a length $n$ path. Our methods demonstrate significant speedups over previous implementations, and are competitive with standard numerical routines.

Turning points of nonlinear circuits

Abstract: Bifurcation theory plays a key role in the qualitative analysis of dynamical systems. In nonlinear circuit theory, bifurcations of equilibria describe qualitative changes in the local phase portrait near an operating point, and are important from both an analytical and a numerical point of view. This work is focused on quadratic turning points, which, in certain circumstances, yield saddle-node bifurcations. Algebraic conditions guaranteeing the existence of this kind of points are well-known in the context of explicit ordinary differential equations (ODEs). We transfer these conditions to semiexplicit differential-algebraic equations (DAEs), in order to impose them to branch-oriented models of nonlinear circuits. This way, we obtain a description of the conditions characterizing these turning points in terms of the underlying circuit digraph and the devices’ characteristics.

RS-IMEX splitting for the isentropic Euler equations

Abstract: Approximating solutions to singularly perturbed differential equations necessitates the use of stable integrators. One famous approach is to split the equation into stiff and non-stiff parts. Treating stiff parts implicitly, non-stiff ones explicitly leads to so-called IMEX schemes. These schemes are supposed to exhibit very good accuracy and uniform stability, however, not every (seemingly reasonable) splitting induces a stable algorithm. In this paper, we present a new IMEX-splitting based on a reference solution (RS) applied to the isentropic Euler equations.