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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Multilevel Combinatorial Optimization across Quantum Architectures

Emerging quantum processors provide an opportunity to explore new approaches for solving traditional problems in the post Moore’s law supercomputing era. However, the limited number of qubits makes it infeasible to tackle massive real-world datasets directly in the near future, leading to new challenges in utilizing these quantum processors for practical purposes. Furthermore, hybrid quantum-classical algorithms that leverage both quantum and classical types of devices are considered as one of the main strategies to apply quantum computing to large-scale problems. In this article, we advocate the use of multilevel frameworks for combinatorial optimization as a promising general paradigm for designing hybrid quantum-classical algorithms. To demonstrate this approach, we apply this method to two well-known combinatorial optimization problems, namely, the Graph Partitioning Problem, and the Community Detection Problem. We develop hybrid multilevel solvers with quantum local search on D-Wave’s quantum annealer and IBM’s gate-model based quantum processor. We carry out experiments on graphs that are orders of magnitude larger than the current quantum hardware size, and we observe results comparable to state-of-the-art solvers in terms of quality of the solution.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Randomized Cholesky Preconditioning for Graph Partitioning Applications

A graph is a mathematical representation of a network; we say it consists of a set of vertices, which are connected by edges. Graphs have numerous applications in various fields, as they can model all sorts of connections, processes, or relations. For example, graphs can model intricate transit systems or the human nervous system. However, graphs that are large or complicated become difficult to analyze. This is why there is an increased interest in the area of graph partitioning, reducing the size of the graph into multiple partitions. For example, partitions of a graph representing a social network might help identify clusters of friends or colleagues. Graph partitioning is also a widely used approach to load balancing in parallel computing. The partitioning of a graph is extremely useful to decompose the graph into smaller parts and allow for easier analysis. There are different ways to solve graph partitioning problems. For this work, we focus on a spectral partitioning method which forms a partition based upon the eigenvectors of the graph Laplacian (details presented in Acer, et. al.). This method uses the LOBPCG algorithm to compute these eigenvectors. LOBPCG can be accelerated by an operator called a preconditioner. For this internship, we evaluate a randomized Cholesky (rchol) preconditioner for its effectiveness on graph partitioning problems with LOBPCG. We compare it with two standard preconditioners: Jacobi and Incomplete Cholesky (ichol). This research was conducted from August to December 2021 in conjunction with Sandia National Laboratories.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

A Fast and Scalable Genetic Algorithm-Based Approach for Planning of Microgrids in Distribution Networks

As a result of climate change, extreme weather events are occurring more frequently and with increasing impact. This trend poses a significant challenge for distribution utilities and system operators to ensure that there is uninterrupted power supply to critical loads in their networks; thus, the level of proactive preparation of the distribution system to be able to handle severe impacts of extreme weather events represents the system's resilience. One method that distribution system planners can use to prepare for future extreme events is to plan multiple microgrids which can use local generation as much as possible to supply critical loads. But partitioning an existing distribution system such that multiple feasible islands are planned and which are capable of supporting critical loads is still challenging for distribution systems - first, because of the size of the network graph partitioning problem and, second, because of the difficulty in properly formulating the desired attributes of such islands or microgrids. Therefore, this paper presents a genetic algorithm based approach that facilitates incorporating multiple objectives for grid partitioning by formulating two types of problems - node allocation and edge elimination - and it considers multiple topological and resilience-enhancing objectives. The performance of the proposed genetic algorithm-based approach is numerically evaluated on multiple test systems as well as on a real distribution feeder in Colorado, United States.

genetic algorithm↗
DOE OSTI2020

Detecting multiple communities using quantum annealing on the D-Wave system

A very important problem in combinatorial optimization is the partitioning of a network into communities of densely connected nodes; where the connectivity between nodes inside a particular community is large compared to the connectivity between nodes belonging to different ones. This problem is known as community detection, and has become very important in various fields of science including chemistry, biology and social sciences. The problem of community detection is a twofold problem that consists of determining the number of communities and, at the same time, finding those communities. This drastically increases the solution space for heuristics to work on, compared to traditional graph partitioning problems. In many of the scientific domains in which graphs are used, there is the need to have the ability to partition a graph into communities with the “highest quality” possible since the presence of even small isolated communities can become crucial to explain a particular phenomenon. We have explored community detection using the power of quantum annealers, and in particular the D-Wave 2X and 2000Q machines. It turns out that the problem of detecting at most two communities naturally fits into the architecture of a quantum annealer with almost no need of reformulation. This paper addresses a systematic study of detecting two or more communities in a network using a quantum annealer.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2017

Applying Graph Theory to Problems in Air Traffic Management

Graph theory is used to investigate three different problems arising in air traffic management. First, using a polynomial reduction from a graph partitioning problem, it isshown that both the airspace sectorization problem and its incremental counterpart, the sector combination problem are NP-hard, in general, under several simple workload models. Second, using a polynomial time reduction from maximum independent set in graphs, it is shown that for any fixed e, the problem of finding a solution to the minimum delay scheduling problem in traffic flow management that is guaranteed to be within n1-e of the optimal, where n is the number of aircraft in the problem instance, is NP-hard. Finally, a problem arising in precision arrival scheduling is formulated and solved using graph reachability. These results demonstrate that graph theory provides a powerful framework for modeling, reasoning about, and devising algorithmic solutions to diverse problems arising in air traffic management.

computational complexity↗
NASA NTRS2017

Applying Graph Theory to Problems in Air Traffic Management

Graph theory is used to investigate three different problems arising in air traffic management. First, using a polynomial reduction from a graph partitioning problem, it is shown that both the airspace sectorization problem and its incremental counterpart, the sector combination problem are NP-hard, in general, under several simple workload models. Second, using a polynomial time reduction from maximum independent set in graphs, it is shown that for any fixed e, the problem of finding a solution to the minimum delay scheduling problem in traffic flow management that is guaranteed to be within n1-e of the optimal, where n is the number of aircraft in the problem instance, is NP-hard. Finally, a problem arising in precision arrival scheduling is formulated and solved using graph reachability. These results demonstrate that graph theory provides a powerful framework for modeling, reasoning about, and devising algorithmic solutions to diverse problems arising in air traffic management.

precision arrival scheduling↗
DOE OSTI2022

A Fast and Scalable Genetic Algorithm-Based Approach for Planning of Microgrids in Distribution Networks: Preprint

As a result of climate change, extreme weather events are occurring more frequently and with increasing impact. This trend poses a significant challenge for distribution utilities and system operators to ensure that there is uninterrupted power supply to critical loads in their networks; thus, the level of proactive preparation of the distribution system to be able to handle severe impacts of extreme weather events represents the system's resilience. One method that distribution systems use to prepare for extreme events is to form multiple microgrids and thereby isolate themselves from the grid supply by using local generation as much as possible to supply critical loads. But partitioning an existing system into multiple feasible islands capable of supporting critical loads is still challenging for distribution systems - first, because of the size of the graph partitioning problem and, second, because of the difficulty in properly formulating the desired attributes of such islands or microgrids. Therefore, this paper presents a genetic algorithmbased approach that facilitates incorporating multiple objectives for grid partitioning by formulating two types of problems - node allocation and edge elimination - and it considers multiple topological and resilience-enhancing objectives. The performance of the proposed genetic algorithm-based approach is numerically evaluated on multiple test systems as well as on a real distribution feeder in Colorado, USA.

edge elimination↗
DOE OSTI2020

Optimized Lie–Trotter–Suzuki decompositions for two and three non-commuting terms

Lie–Trotter–Suzuki decompositions are an efficient way to approximate operator exponentials exp ( t H ) when H is a sum of n (non-commuting) terms which, individually, can be exponentiated easily. They are employed in time-evolution algorithms for tensor network states, digital quantum simulation protocols, path integral methods like quantum Monte Carlo, and splitting methods for symplectic integrators in classical Hamiltonian systems. Here, we provide optimized decompositions up to order t 6 . The leading error term is expanded in nested commutators (Hall bases) and we minimize the 1-norm of the coefficients. For n = 2 terms, several of the optima we find are close to those in McLachlan (1995). Generally, our results substantially improve over unoptimized decompositions by Forest, Ruth, Yoshida, and Suzuki. We explain why these decompositions are sufficient to efficiently simulate any one- or two-dimensional lattice model with finite-range interactions. This follows by solving a partitioning problem for the interaction graph.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Toward computing bounds for Ramsey numbers using quantum annealing

Quantum annealing is a powerful tool for solving and approximating combinatorial optimization problems, such as graph partitioning, community detection, centrality, routing problems, and more. In this paper we explore the use of quantum annealing as a tool for use in exploring combinatorial mathematics research problems. We consider the monochromatic triangle problem and the Ramsey number problem, both examples of graph coloring. Conversion to quadratic unconstrained binary optimization (QUBO) form is required to run on quantum hardware. While the monochromatic triangle problem is quadratic by nature, the Ramsey number problem requires the use of order reduction methods for a quadratic formulation. The goal is to provide a method for producing special colorings of graphs which if successful would provide lower bounds for certain Ramsey numbers. We discuss implementations, limitations, and results when running on the D-Wave Advantage quantum annealer.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2002

Graph Partitioning for Parallel Applications in Heterogeneous Grid Environments

The problem of partitioning irregular graphs and meshes for parallel computations on homogeneous systems has been extensively studied. However, these partitioning schemes fail when the target system architecture exhibits heterogeneity in resource characteristics. With the emergence of technologies such as the Grid, it is imperative to study the partitioning problem taking into consideration the differing capabilities of such distributed heterogeneous systems. In our model, the heterogeneous system consists of processors with varying processing power and an underlying non-uniform communication network. We present in this paper a novel multilevel partitioning scheme for irregular graphs and meshes, that takes into account issues pertinent to Grid computing environments. Our partitioning algorithm, called MiniMax, generates and maps partitions onto a heterogeneous system with the objective of minimizing the maximum execution time of the parallel distributed application. For experimental performance study, we have considered both a realistic mesh problem from NASA as well as synthetic workloads. Simulation results demonstrate that MiniMax generates high quality partitions for various classes of applications targeted for parallel execution in a distributed heterogeneous environment.

Bisws, Rupak↗
DOE OSTI2022

Enhancing scalability of a matrix-free eigensolver for studying many-body localization

We propose several techniques to enhance the parallel scalability of a matrix-free eigensolver designed for studying many-body localization (MBL) of quantum spin chain models with nearest-neighbor interactions and on-site disorder. This type of problem is computationally challenging because the dimension of the associated Hamiltonian matrix grows exponentially with respect to the number of spins L, and we need to average over different realizations of the random disorder to obtain relevant statistical behavior. For each disorder realization, we need to compute eigenvalues from different regions of the spectrum and their corresponding eigenvectors. In previous work, the interior eigenstates for a single eigenvalue problem are computed via the shift-and-invert Lanczos algorithm. Due to the extremely high memory footprint of the LU factorizations, this technique is not well suited for large L’s. For example, we need thousands of compute nodes on modern high performance computing infrastructures to go beyond L = 24. The matrix-free approach does not suffer from this memory bottleneck, however, its scalability is limited by a computation and communication load imbalance. To reduce this imbalance and to significantly enhance the scalability of the matrix-free eigensolver, we reorder the matrix and leverage the consistent space runtime, CSPACER. We also show its efficiency in managing irregular communication patterns at scale compared to optimized MPI non-blocking two-sided and one-sided RMA implementation variants. This effort enables us to study MBL for spin chains with a larger number of spins. The efficiency and effectiveness of the proposed algorithm is demonstrated by computing eigenstates on a massively parallel many-core high performance computer.

METIS↗
NASA NTRS1994

On bottleneck partitioning k-ary n-cubes

Graph partitioning is a topic of extensive interest, with applications to parallel processing. In this context graph nodes typically represent computation, and edges represent communication. One seeks to distribute the workload by partitioning the graph so that every processor has approximately the same workload, and the communication cost (measured as a function of edges exposed by the partition) is minimized. Measures of partition quality vary; in this paper we consider a processor's cost to be the sum of its computation and communication costs, and consider the cost of a partition to be the bottleneck, or maximal processor cost induced by the partition. For a general graph the problem of finding an optimal partitioning is intractable. In this paper we restrict our attention to the class of k-art n-cube graphs with uniformly weighted nodes. Given mild restrictions on the node weight and number of processors, we identify partitions yielding the smallest bottleneck. We also demonstrate by example that some restrictions are necessary for the partitions we identify to be optimal. In particular, there exist cases where partitions that evenly partition nodes need not be optimal.

Nicol, David M.↗
NASA NTRS1989

Efficient mapping algorithms for scheduling robot inverse dynamics computation on a multiprocessor system

Two efficient mapping algorithms for scheduling the robot inverse dynamics computation consisting of m computational modules with precedence relationship to be executed on a multiprocessor system consisting of p identical homogeneous processors with processor and communication costs to achieve minimum computation time are presented. An objective function is defined in terms of the sum of the processor finishing time and the interprocessor communication time. The minimax optimization is performed on the objective function to obtain the best mapping. This mapping problem can be formulated as a combination of the graph partitioning and the scheduling problems; both have been known to be NP-complete. Thus, to speed up the searching for a solution, two heuristic algorithms were proposed to obtain fast but suboptimal mapping solutions. The first algorithm utilizes the level and the communication intensity of the task modules to construct an ordered priority list of ready modules and the module assignment is performed by a weighted bipartite matching algorithm. For a near-optimal mapping solution, the problem can be solved by the heuristic algorithm with simulated annealing. These proposed optimization algorithms can solve various large-scale problems within a reasonable time. Computer simulations were performed to evaluate and verify the performance and the validity of the proposed mapping algorithms. Finally, experiments for computing the inverse dynamics of a six-jointed PUMA-like manipulator based on the Newton-Euler dynamic equations were implemented on an NCUBE/ten hypercube computer to verify the proposed mapping algorithms. Computer simulation and experimental results are compared and discussed.

Lee, C. S. G.↗
NASA NTRS1990

Partitioning sparse matrices with eigenvectors of graphs

The problem of computing a small vertex separator in a graph arises in the context of computing a good ordering for the parallel factorization of sparse, symmetric matrices. An algebraic approach for computing vertex separators is considered in this paper. It is shown that lower bounds on separator sizes can be obtained in terms of the eigenvalues of the Laplacian matrix associated with a graph. The Laplacian eigenvectors of grid graphs can be computed from Kronecker products involving the eigenvectors of path graphs, and these eigenvectors can be used to compute good separators in grid graphs. A heuristic algorithm is designed to compute a vertex separator in a general graph by first computing an edge separator in the graph from an eigenvector of the Laplacian matrix, and then using a maximum matching in a subgraph to compute the vertex separator. Results on the quality of the separators computed by the spectral algorithm are presented, and these are compared with separators obtained from other algorithms for computing separators. Finally, the time required to compute the Laplacian eigenvector is reported, and the accuracy with which the eigenvector must be computed to obtain good separators is considered. The spectral algorithm has the advantage that it can be implemented on a medium-size multiprocessor in a straightforward manner.

Pothen, Alex↗
DOE OSTI2020

Virtual-Voltage Partition-Based Approach to Optimal Transmission Switching

This article deals with optimal transmission switching (OTS) problems involving binary decisions about network topology and nonconvex power flow constraints. We adopt a semidefinite programming formulation for the optimal power flow (OPF) problem that, however, remains nonconvex due to the presence of discrete variables and bilinear products between the decision variables. To tackle the latter, we introduce a physically inspired, virtual-voltage approximation that leads to provable lower and upper bounds on the solution of the original problem. To deal with the exponential complexity caused by the discrete variables, we introduce a graph partition-based algorithm that breaks the problem into several parallel mixed-integer subproblems of smaller size. Simulations on the IEEE bus test cases demonstrate the high degree of accuracy and affordable computational requirements of our approach.

42 ENGINEERING↗
DOE OSTI2022

A Block-Based Triangle Counting Algorithm on Heterogeneous Environments

Triangle counting is a fundamental building block in graph algorithms. In this article, 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↗