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At least 37 records · Page 2

A Block-Based Triangle Counting Algorithm on Heterogeneous Environments

Triangle counting is a fundamental building block in graph algorithms. In this paper, we propose a block-based triangle counting algorithm to reduce data movement during both sequential and parallel execution. Our block-based formulation makes the algorithm naturally suitable for heterogeneous architectures. The problem of partitioning the adjacency matrix of a graph is well-studied. Our task decomposition goes one step further: it partitions the set of triangles in the graph. By streaming these small tasks to compute resources, we can solve problems that do not fit on a device. We demonstrate the effectiveness of our approach by providing an implementation on a compute node with multiple sockets, cores and GPUs. The current state-of-the-art in triangle enumeration processes the Friendster graph in 2.1 seconds, not including data copy time between CPU and GPU. Using that metric, our approach is 20 percent faster. When copy times are included, our algorithm takes 3.2 seconds. This is 5.6 times faster than the fastest published CPU-only time.

97 MATHEMATICS AND COMPUTING↗

HARP: A Dynamic Inertial Spectral Partitioner

Partitioning unstructured graphs is central to the parallel solution of computational science and engineering problems. Spectral partitioners, such recursive spectral bisection (RSB), have proven effecfive in generating high-quality partitions of realistically-sized meshes. The major problem which hindered their wide-spread use was their long execution times. This paper presents a new inertial spectral partitioner, called HARP. The main objective of the proposed approach is to quickly partition the meshes at runtime in a manner that works efficiently for real applications in the context of distributed-memory machines. The underlying principle of HARP is to find the eigenvectors of the unpartitioned vertices and then project them onto the eigerivectors of the original mesh. Results for various meshes ranging in size from 1000 to 100,000 vertices indicate that HARP can indeed partition meshes rapidly at runtime. Experimental results show that our largest mesh can be partitioned sequentially in only a few seconds on an SP2 which is several times faster than other spectral partitioners while maintaining the solution quality of the proven RSB method. A parallel WI version of HARP has also been implemented on IBM SP2 and Cray T3E. Parallel HARP, running on 64 processors SP2 and T3E, can partition a mesh containing more than 100,000 vertices into 64 subgrids in about half a second. These results indicate that graph partitioning can now be truly embedded in dynamically-changing real-world applications.

Simon, Horst D.↗

Minimizing Optimal Transport for Functions with Fixed-Size Nodal Sets

Consider the class of zero-mean functions with fixed L ∞ and L 1 norms and exactly N ϵ N nodal points. Which functions f minimize W p (f + ,f – ), the Wasserstein distance between the measures whose densities are the positive and negative parts? We provide a complete solution to this minimization problem on the line and the circle, which provides sharp constants for previously proven “uncertainty principle”-type inequalities, i.e., lower bounds on N • W p (f + ,f – ). We further show that, while such inequalities hold in many metric measure spaces, they are no longer sharp when the non-branching assumption is violated; indeed, for metric star-graphs, the optimal lower bound on W p (f + ,f – ) is not inversely proportional to the size of the nodal set, N. Here, based on similar reductions, we make connections between the analogous problem of minimizing W p (f + ,f – ) for f defined on Ω C R d with an equivalent optimal domain partition problem.

97 MATHEMATICS AND COMPUTING↗

Dynamic Load Balancing for Adaptive Computations on Distributed-Memory Machines

Dynamic load balancing is central to adaptive mesh-based computations on large-scale parallel computers. The principal investigator has investigated various issues on the dynamic load balancing problem under NASA JOVE and JAG rants. The major accomplishments of the project are two graph partitioning algorithms and a load balancing framework. The S-HARP dynamic graph partitioner is known to be the fastest among the known dynamic graph partitioners to date. It can partition a graph of over 100,000 vertices in 0.25 seconds on a 64- processor Cray T3E distributed-memory multiprocessor while maintaining the scalability of over 16-fold speedup. Other known and widely used dynamic graph partitioners take over a second or two while giving low scalability of a few fold speedup on 64 processors. These results have been published in journals and peer-reviewed flagship conferences.

Source record↗

A remark about pointed bubbles

The polymer expansion is a formal algebraic identity between a partition function and logarithm in statistical physics problems. The expansion gives a systematic method to control the free energy or to establish exponential tree-graph decay of connected correlations. Here, the convergence properties of the polymer expansion are analyzed in connection with three practical examples, including: intersecting bonds in chemical polymer chains; a connected closed hypersurface built from the (d-1)-faces of the d-dimensional unit cubes; and the set of Feynamn diagrams in the perturbation series of the Euclidean field theory partition function Z. The example of connected polymer chains is generalized to apply to other lattice models, including n-state Ising models at high temperature; short range lattice gases at high temperature; and weak coupling lattice field and gauge theories.

Garabedian, P. R.↗

Scalable Approaches to Selecting Key Entities in Large Networked Infrastructure Systems

This work aims at bringing advances in discrete optimization algorithms to solving practical engineering problems at scale. Often times, in many engineering design problems, there is a need to select a small set of influential or representative elements from a large ground set of entities in an optimal fashion. Submodular optimization provides for a formal way to solve such problems. Common examples with infrastructure systems involve sensor placement and identification of key entities with certain objectives. However, scaling these approaches to large infrastructure systems can be challenging because of the high computational complexity of the overall framework that include the optimization algorithms as well as high-complexity compute-oracles that provide the necessary objective function values. In this work, we explore a well-studied and widely-applicable paradigm, namely leader-selection in a multi-agent networked setting in the context of scalable methodologies. We demonstrate novel frameworks that utilize variations of accelerated submodular optimization algorithms along with linear-algebraic methods that can help accelerate the oracle computations. We further explore this combination in conjunction with graph partitioning paradigms to take advantage of the accelerated algorithms in a distributed setting. Finally we demonstrate the key findings on a practical problem in an operational setting. For this, we leverage an example road network with approximately 18k nodes and 27k edges in a traffic control application, where we seek a limited number of k=200 key intersections. This problem can be solved in a serial setting in just under 5 hours providing more than 2 orders of magnitude speed-up over methods that do not consider acceleration techniques.

Visweswara Sathanur, Arun↗

Applications of the Dulmage–Mendelsohn decomposition for debugging nonlinear optimization problems

Nonlinear modeling and optimization is a valuable tool for aiding decisions by engineering practitioners, but programming an optimization problem based on a complex electrical, mechanical, or chemical process is a time-consuming and error-prone activity. Therefore, there is a need for model analysis and debugging tools that can detect and diagnose modeling errors. One such tool is the Dulmage–Mendelsohn decomposition, which identifies structurally under- and over-determined subsets in systems of equations and variables by partitioning the bipartite graph of the system. This work provides the necessary background to understand the Dulmage–Mendelsohn decomposition and its application to the analysis of nonlinear optimization problems, demonstrates its use in diagnosing a variety of modeling errors, and introduces software implementations for analyzing nonlinear optimization problems in the Pyomo and JuMP algebraic modeling languages.

42 ENGINEERING↗

Parametric binary dissection

Binary dissection is widely used to partition non-uniform domains over parallel computers. This algorithm does not consider the perimeter, surface area, or aspect ratio of the regions being generated and can yield decompositions that have poor communication to computation ratio. Parametric Binary Dissection (PBD) is a new algorithm in which each cut is chosen to minimize load + lambda x(shape). In a 2 (or 3) dimensional problem, load is the amount of computation to be performed in a subregion and shape could refer to the perimeter (respectively surface) of that subregion. Shape is a measure of communication overhead and the parameter permits us to trade off load imbalance against communication overhead. When A is zero, the algorithm reduces to plain binary dissection. This algorithm can be used to partition graphs embedded in 2 or 3-d. Load is the number of nodes in a subregion, shape the number of edges that leave that subregion, and lambda the ratio of time to communicate over an edge to the time to compute at a node. An algorithm is presented that finds the depth d parametric dissection of an embedded graph with n vertices and e edges in O(max(n log n, de)) time, which is an improvement over the O(dn log n) time of plain binary dissection. Parallel versions of this algorithm are also presented; the best of these requires O((n/p) log(sup 3)p) time on a p processor hypercube, assuming graphs of bounded degree. How PBD is applied to 3-d unstructured meshes and yields partitions that are better than those obtained by plain dissection is described. Its application to the color image quantization problem is also discussed, in which samples in a high-resolution color space are mapped onto a lower resolution space in a way that minimizes the color error.

Bokhari, Shahid H.↗

Gauges, loops, and polynomials for partition functions of graphical models

Graphical models represent multivariate and generally not normalized probability distributions. Computing the normalization factor, called the partition function, is the main inference challenge relevant to multiple statistical and optimization applications. The problem is #P-hard that is of an exponential complexity with respect to the number of variables. Here, aimed at approximating the partition function, we consider multi-graph models where binary variables and multivariable factors are associated with edges and nodes, respectively, of an undirected multi-graph. We suggest a new methodology for analysis and computations that combines the Gauge function technique from Chertkov and Chernyak with the technique developed in Anari and Oveis Gharan 2017 arXiv:1702.02937; Gurvits 2011 arXiv:1106.2844; Straszak and Vishnoi 2017 55th Annual Allerton Conf. on Communication, Control, and Computing, based on the recent progress in the field of real stable polynomials. We show that the Gauge function, representing a single-out term in a finite sum expression for the partition function which achieves extremum at the so-called belief-propagation gauge, has a natural polynomial representation in terms of gauges/variables associated with edges of the multi-graph. Moreover, Gauge function can be used to recover the partition function through a sequence of transformations allowing appealing algebraic and graphical interpretations. Algebraically, one step in the sequence consists of the application of a differential operator over gauges associated with an edge. Graphically, the sequence is interpreted as a repetitive elimination/contraction of edges resulting in multi-graph models on decreasing in size (number of edges) graphs with the same partition function as in the original multi-graph model. Even though the complexity of computing factors in the sequence of the derived multi-graph models and respective Gauge functions grow exponentially with the number of eliminated edges, polynomials associated with the new factors remain bi-stable if the original factors have this property. Moreover, we show that BP estimations in the sequence do not decrease, each low-bounding the partition function.

97 MATHEMATICS AND COMPUTING↗

A Machine Learning Approach to Improve Air Traffic Management Initiatives

Collaborating closely with commercial air carriers and related organizations, the Federal Aviation Administration(FAA) regulates air traffic and ensures the safety and efficiency of air operations. Air traffic controllers make strategic decisions, such as delaying, rerouting, or canceling flights, partly based on guidance provided by the FAA’s Air TrafficControl System Command Center (ATCSCC). The guidance includes, among other things, control measures known asTraffic Management Initiatives (TMIs) designed to enhance safety and improve operational efficiency. TMIs play a crucial role in managing the demand and capacity within the U.S. National Airspace System (NAS). Two major TMIs that are routinely used (primarily to mitigate the adverse effects of bad weather) are Ground Delay Programs (GDPs) andGround Stops (GSs). In a GDP, flights destined for airports facing thunderstorm activity experience delays at their origin airports. This proactive approach minimizes the risk of routing aircraft through hazardous weather conditions and also replaces (fuel burning) airborne delays with ground delays. In a GS, a temporary restriction is imposed on the departure or arrival of aircraft at a specific airport or within a designated airspace. Although other TMIs (e.g., miles-in-trail) are also implemented as part of (air) traffic flow management in the NAS, the focus of this work is on GDPs and GSs. Since TMIs, by design, lead to flight delays or cancellations, it is crucial to put in place the right set of parameters(e.g., scope and duration of the GDP). For example, when the end time of a GDP extends beyond what is necessary, it imposes unnecessary delays on departing flights. This situation could occur as a result of inaccurate prediction of the(required) duration of the GDP based on the weather forecast. On the other hand, if a GDP ends prematurely before the underlying capacity constraints are resolved at the destination airport, it may result in airborne holding. The delicate balance lies in matching the termination of the GDP precisely with the resolution of capacity constraints, avoiding both the imposition of unnecessary ground delays and the need for airborne holding due to premature program termination.Failing to specify the right parameters for TMIs also leads to flight delays, creating a significant obstacle in managing the increasing traffic volumes causing increased work load for the controllers. To address this issue, we propose the integration of Machine Learning (ML) models in the traffic flow management(TFM) pipeline. In current operations, decisions are made by human experts based on extensive training, historical patterns, available traffic and weather data. Since we have an abundance of data from past events that tell us the likely impact of various TMIs, by ingesting historical data, properly trained ML models can offer valuable insights and aid human decision-making. With the FAA increasingly exploring advanced analytics, ML emerges as a focal point for enhancing TFM within the National Airspace System (NAS). As a first step, this study aims to provide traffic controllers with decision-making support for the issuance and adjustment of TMIs. Data analytics and machine learning have been previously employed to address some of the challenges associated with TMIs. Numerous studies have concentrated on various facets of TMI issuance, exploring factors influencing TMI parameters, including arrival rate, airport capacity, and delay prediction. For example, using weather forecasts, several statistical methods were used to produce probabilistic capacity profiles which in conjunction with deterministic models provided insights into the GDP planning process [1–4]. The downside of using deterministic models is that they rely on fixed inputs and predetermined rules, which lack the ability to account for the inherent uncertainty and variability present in real-world scenarios. In a separate series of studies, researchers aimed to predict the occurrences of GDPs and GSs. The majority of these studies utilized various supervised learning methods, including Decision Trees, Naive Bayes, Support VectorMachines, and Random Forests to analyze the influence of weather conditions and arrival demand on TMI incidents[5–8]. However, these studies primarily focused on predicting the incidence of TMIs without explicitly addressing the scope of TMIs, including their duration and their geographical coverage. Furthermore, the emphasis of these studies was largely on GDPs, given their higher frequency and longer duration when compared to GSs. A limited number of studies focused on predicting the parameters of TMIs, specifically addressing their duration and extent. In one such study focusing on optimizing the TMI parameters at San Francisco International Airport (SFO),the authors utilized a probabilistic forecast of fog [9]. They simulated various capacity scenarios based on the (fog)burn-off forecasts, selecting GDP parameters that minimized airborne and overall ground delays. However, this approach exclusively emphasizes stratus (fog) burn-off as the primary determinant of GDP and GS, neglecting other influential factors like severe weather events, runway closures, lower capacity than traffic demand, and other important variables. Given the complexity of predicting the TMI and determining its scope, we seek a more holistic approach. We aim to consider all significant factors that could impact TMIs and their parameters. What sets this research apart is the fusion of all data sources relevant to the issuance and adjustment of TMIs and it represents the first comprehensive attempt to optimize TMIs in this manner. Since this comprehensive solution involves various aspects, we break down the problem into smaller components and input all parameters into a unified model called the “TMI Adjuster”. Figure 1 shows the overall framework and the list of datasets used in each model. The objective of the TMI Adjuster module is to deliver reliable, consistent and expedited recommendations for the progression, adjustment, and termination of TMIs. The ML solution entails developing a pipeline capable of predicting the necessity of a TMI (e.g., GS or GDP) along with its various parameters. For example, in the case of a GS, this includes the scope of the GS either in terms of distance from the destination airport or based on pre-defined airspace sectors. Here, scope refers to those regions and departing airports that are subject to the GS. In this paper, we concentrate on the issuance of GSs in the three major airports in the New York area — LaGuardia(LGA), John F. Kennedy International (JFK), and Newark Liberty International (EWR). We fuse traffic, weather and other relevant aviation data from years 2017 to 2019 to train and validate the ML models. In particular, we use the following datasets: •Terminal Aerodrome Forecast (TAF): meteorological forecasts specific to each airport, issued four times a day, covering predefined time periods. •TMI data: includes all GSs and GDPs along with their respective parameters. •Aviation System Performance Metrics (ASPM): includes traffic related data such as aircraft delays, arrival, and departure rates. •Notices to Airmen (NOTAMs): utilized to extract runway closure data and manage interdependencies between terminals in close proximity. •Flight cancellation data •Airspace Flow Programs (AFP): includes information on flight airborne holdings caused by TMIs. The data preprocessing entails transforming ASPM, TMI, AFP, NOTAMs, and weather data into an hourly format and consolidating all datasets by merging them based on date and time as the primary key. The TMI Adjuster framework comprises two parallel models: one dedicated to GS and a second model focused on GDP. As previously mentioned, our specific focus is on the GS model as a multi-classification problem. In this framework, each data point of the GS model input summarizes ten hours of data. Specifically, the data loader for the GS model generates the input and output of the model as follows: at a given time step, the input includes the actual traffic, weather, and TMI data from the two-hour window before the time step, alongside the weather forecast and scheduled traffic for the next 8 hours starting from the time step. Based on this information, the output of the GS model for each time interval consists of three dimensions. The first dimension represents a binary decision on whether there should be a GS in place for the next hour or not. The second dimension is related to the scope of the GS in the United States, and the third dimension is related to the scope of the GS in Canada (i.e., to determine if the GS impacts airports in Canada).One of the challenges with TMI modeling is the sparsity of TMI events, particularly regarding its scope. To address this challenge in the scope of the GS model output, we implement grouping. The GS scope for the US region is defined based on a list of centers that should be included when the GS is in place. With 20 centers in the US, we utilized historical data to group them into 4 categories. In particular, we summarized our historical data in a graph format where nodes represent centers, and link weights are defined based on the co-occurrence of centers in the scope parameter ofTMIs. By identified strongly connected components in this graph, we were able to partition the centers into four groups. We consider two model structures for the GS Model. Firstly, a hierarchical classification model [10], where the human decision-making for a GS is of hierarchical nature. The decision-maker first decides whether there is a need fora GS, and if the answer is yes, determines the scope. A hierarchical classification model organizes the problem into a class hierarchy, typically a tree or a Directed Acyclic Graph (DAG) structure, and considers the dependency of the decision in the previous step to the next component [10]. Here, we employ the local classifier per level approach, which involves training one multi-class classifier for each level of the class hierarchy. The second structure is the independent structure. In this setting, as the name suggests, we do not consider the dependency of the decisions in the different dimensions of the output of the model. Instead, for each dimension, we train a multi-class classifier independently. Table 1 summarizes GS model statistics for training, validation and testing. The table documents the effect of limiting data to the time steps when there was actually a TMI in place or when a TMI had just terminated. This resulted in a more balanced distribution of the GS class(GS positive class)versus “No GS”(GS negative class), which might help the training process. While JFK and LGA follow very similar distributions, with 40% and 42% GS positive class respectively, EWR has proportionally fewer GS incidents at 28%. Our subsequent phase involves evaluating the performance of both hierarchical structure and independent structure using different state-of-the-art multi-class classifier models such as Random Forest, Decision Trees, K-nearest Neighbors, and Logistic Regression and forecast the duration and scope of the GSs.

Farzan Masrour Shalmani↗

A differentiable approach to the maximum independent set problem using dataless neural networks

The success of machine learning solutions for reasoning about discrete structures has brought attention to its adoption within combinatorial optimization algorithms. Such approaches generally rely on supervised learning by leveraging datasets of the combinatorial structures of interest drawn from some distribution of problem instances. Reinforcement learning has also been employed to find such structures. Here, in this paper, we propose a different approach in that no data is required for training the neural networks that produce the solution. In this sense, what we present is not a machine learning solution, but rather one that is dependent on neural networks and where backpropagation is applied to a loss function defined by the structure of the neural network architecture as opposed to a training dataset. In particular, we reduce the popular combinatorial optimization problem of finding a maximum independent set to a neural network and employ a dataless training scheme to refine the parameters of the network such that those parameters yield the structure of interest. Additionally, we propose a universal graph reduction procedure to handle large-scale graphs. The reduction exploits community detection for graph partitioning and is applicable to any graph type and/or density. Experimental results on both real and synthetic graphs demonstrate that our proposed method performs on par or outperforms state-of-the-art learning-based methods in terms of the size of the found set without requiring any training data.

97 MATHEMATICS AND COMPUTING↗

Tools for analysis of optimization models

The software is code for analyzing, debugging, and solving optimization models. The code implements several algorithms and provides convenient APIs to apply these algorithms to optimization models. The algorithms are primarily based on graph theory. They compute well-known partitions of graphs, and use these partitions to provide the user information about their optimization model, including diagnosing certain types of structural modeling errors. Some algorithms may be used as a subroutine to solve an optimization problem, and may call an optimization solver to facilitate this. Well-known linear algebra routines are called to provide further diagnostics. The software contains no data and no models other than toy models used for testing.

Parker, Robert↗

Distributed state-space generation of discrete-state stochastic models

High-level formalisms such as stochastic Petri nets can be used to model complex systems. Analysis of logical and numerical properties of these models of ten requires the generation and storage of the entire underlying state space. This imposes practical limitations on the types of systems which can be modeled. Because of the vast amount of memory consumed, we investigate distributed algorithms for the generation of state space graphs. The distributed construction allows us to take advantage of the combined memory readily available on a network of workstations. The key technical problem is to find effective methods for on-the-fly partitioning, so that the state space is evenly distributed among processors. In this paper we report on the implementation of a distributed state-space generator that may be linked to a number of existing system modeling tools. We discuss partitioning strategies in the context of Petri net models, and report on performance observed on a network of workstations, as well as on a distributed memory multi-computer.

Ciardo, Gianfranco↗

Automated problem scheduling and reduction of synchronization delay effects

It is anticipated that in order to make effective use of many future high performance architectures, programs will have to exhibit at least a medium grained parallelism. A framework is presented for partitioning very sparse triangular systems of linear equations that is designed to produce favorable preformance results in a wide variety of parallel architectures. Efficient methods for solving these systems are of interest because: (1) they provide a useful model problem for use in exploring heuristics for the aggregation, mapping and scheduling of relatively fine grained computations whose data dependencies are specified by directed acrylic graphs, and (2) because such efficient methods can find direct application in the development of parallel algorithms for scientific computation. Simple expressions are derived that describe how to schedule computational work with varying degrees of granularity. The Encore Multimax was used as a hardware simulator to investigate the performance effects of using the partitioning techniques presented in shared memory architectures with varying relative synchronization costs.

Saltz, Joel H.↗

Scalable statistical inference of photometric redshift via data subsampling

Handling big data has largely been a major bottleneck in traditional statistical models. Consequently, when accurate point prediction is the primary target, machine learning models are often preferred over their statistical counterparts for bigger problems. But full probabilistic statistical models often outperform other models in quantifying uncertainties associated with model predictions. We develop a data-driven statistical modeling framework that combines the uncertainties from an ensemble of statistical models learned on smaller subsets of data carefully chosen to account for imbalances in the input space. We demonstrate this method on a photometric redshift estimation problem in cosmology, which seeks to infer a distribution of the redshift—the stretching effect in observing the light of far-away galaxies—given multivariate color information observed for an object in the sky. Our proposed method performs balanced partitioning, graph-based data subsampling across the partitions, and training of an ensemble of Gaussian process models.

data subsampling↗

QSPIN: A High Level Java API for Quantum Computing Experimentation

QSPIN is a high level Java language API for experimentation in QC models used in the calculation of Ising spin glass ground states and related quadratic unconstrained binary optimization (QUBO) problems. The Java API is intended to facilitate research in advanced QC algorithms such as hybrid quantum-classical solvers, automatic selection of constraint and optimization parameters, and techniques for the correction and mitigation of model and solution errors. QSPIN includes high level solver objects tailored to the D-Wave quantum annealing architecture that implement hybrid quantum-classical algorithms [Booth et al.] for solving large problems on small quantum devices, elimination of variables via roof duality, and classical computing optimization methods such as GPU accelerated simulated annealing and tabu search for comparison. A test suite of documented NP-complete applications ranging from graph coloring, covering, and partitioning to integer programming and scheduling are provided to demonstrate current capabilities.

Quantu↗

Data-driven Whitney forms for structure-preserving control volume analysis

Control volume analysis models physics via the exchange of generalized fluxes between subdomains. Here, we introduce a scientific machine learning framework adopting a partition of unity architecture to identify physically-relevant control volumes, with generalized fluxes between subdomains encoded via Whitney forms. The approach provides a differentiable parameterization of geometry which may be trained in an end-to-end fashion to extract reduced models from full field data while exactly preserving physics. The architecture admits a data-driven finite element exterior calculus allowing discovery of mixed finite element spaces with closed form quadrature rules. An equivalence between Whitney forms and graph networks reveals that the geometric problem of control volume learning is equivalent to an unsupervised graph discovery problem. The framework is developed for manifolds in arbitrary dimension, with examples provided for H(div) problems in $\mathbb{R}$ establishing convergence and structure preservation properties. Finally, we consider a lithium-ion battery problem where we discover a reduced finite element space encoding transport pathways from high-fidelity microstructure resolved simulations. The approach reduces the 5.89M finite element simulation to 136 elements while reproducing pressure to under 0.1% error and preserving conservation.

97 MATHEMATICS AND COMPUTING↗

Partitioning of unstructured problems for parallel processing

Many large-scale computational problems are based on unstructured computational domains. Primary examples are unstructured grid calculations based on finite volume methods in computational fluid dynamics, or structural analysis problems based on finite element approximations. The question of how to distribute such unstructured computational domains over a large number of processors in a MIMD machine with distributed memory is addressed. A graph theoretical framework for these problems is established. Based on this framework three decomposition algorithms are introduced. In particular a new decomposition algorithm is discussed, which is based on the computation of an eigenvector of the Laplacian matrix associated with the graph. Numerical comparisons on large-scale two- and three-dimensional problems demonstrate the superiority of the new spectral bisection algorithm.

Simon, H. D.↗