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All sourcesNASA NTRSDOE OSTI
NASA NTRS2009

Highly Scalable Matching Pursuit Signal Decomposition Algorithm

Matching Pursuit Decomposition (MPD) is a powerful iterative algorithm for signal decomposition and feature extraction. MPD decomposes any signal into linear combinations of its dictionary elements or atoms . A best fit atom from an arbitrarily defined dictionary is determined through cross-correlation. The selected atom is subtracted from the signal and this procedure is repeated on the residual in the subsequent iterations until a stopping criterion is met. The reconstructed signal reveals the waveform structure of the original signal. However, a sufficiently large dictionary is required for an accurate reconstruction; this in return increases the computational burden of the algorithm, thus limiting its applicability and level of adoption. The purpose of this research is to improve the scalability and performance of the classical MPD algorithm. Correlation thresholds were defined to prune insignificant atoms from the dictionary. The Coarse-Fine Grids and Multiple Atom Extraction techniques were proposed to decrease the computational burden of the algorithm. The Coarse-Fine Grids method enabled the approximation and refinement of the parameters for the best fit atom. The ability to extract multiple atoms within a single iteration enhanced the effectiveness and efficiency of each iteration. These improvements were implemented to produce an improved Matching Pursuit Decomposition algorithm entitled MPD++. Disparate signal decomposition applications may require a particular emphasis of accuracy or computational efficiency. The prominence of the key signal features required for the proper signal classification dictates the level of accuracy necessary in the decomposition. The MPD++ algorithm may be easily adapted to accommodate the imposed requirements. Certain feature extraction applications may require rapid signal decomposition. The full potential of MPD++ may be utilized to produce incredible performance gains while extracting only slightly less energy than the standard algorithm. When the utmost accuracy must be achieved, the modified algorithm extracts atoms more conservatively but still exhibits computational gains over classical MPD. The MPD++ algorithm was demonstrated using an over-complete dictionary on real life data. Computational times were reduced by factors of 1.9 and 44 for the emphases of accuracy and performance, respectively. The modified algorithm extracted similar amounts of energy compared to classical MPD. The degree of the improvement in computational time depends on the complexity of the data, the initialization parameters, and the breadth of the dictionary. The results of the research confirm that the three modifications successfully improved the scalability and computational efficiency of the MPD algorithm. Correlation Thresholding decreased the time complexity by reducing the dictionary size. Multiple Atom Extraction also reduced the time complexity by decreasing the number of iterations required for a stopping criterion to be reached. The Course-Fine Grids technique enabled complicated atoms with numerous variable parameters to be effectively represented in the dictionary. Due to the nature of the three proposed modifications, they are capable of being stacked and have cumulative effects on the reduction of the time complexity.

Christensen, Daniel↗
DOE OSTI2024

Parallel Randomized Tucker Decomposition Algorithms

The Tucker tensor decomposition is a natural extension of the singular value decomposition (SVD) to multiway data. Here, we propose to accelerate Tucker tensor decomposition algorithms by using randomization and parallelization. We present two algorithms that scale to large data and many processors, significantly reduce both computation and communication cost compared to previous deterministic and randomized approaches, and obtain nearly the same approximation errors. The key idea in our algorithms is to perform randomized sketches with Kronecker-structured random matrices, which reduces computation compared to unstructured matrices and can be implemented using a fundamental tensor computational kernel. We provide probabilistic error analysis of our algorithms and implement a new parallel algorithm for the structured randomized sketch. Our experimental results demonstrate that our combination of randomization and parallelization achieves accurate Tucker decompositions much faster than alternative approaches. We observe up to a 16X speedup over the fastest deterministic parallel implementation on 3D simulation data.

Tucker decompositions↗
NASA NTRS2010

Decomposition Algorithm for Global Reachability on a Time-Varying Graph

A decomposition algorithm has been developed for global reachability analysis on a space-time grid. By exploiting the upper block-triangular structure, the planning problem is decomposed into smaller subproblems, which is much more scalable than the original approach. Recent studies have proposed the use of a hot-air (Montgolfier) balloon for possible exploration of Titan and Venus because these bodies have thick haze or cloud layers that limit the science return from an orbiter, and the atmospheres would provide enough buoyancy for balloons. One of the important questions that needs to be addressed is what surface locations the balloon can reach from an initial location, and how long it would take. This is referred to as the global reachability problem, where the paths from starting locations to all possible target locations must be computed. The balloon could be driven with its own actuation, but its actuation capability is fairly limited. It would be more efficient to take advantage of the wind field and ride the wind that is much stronger than what the actuator could produce. It is possible to pose the path planning problem as a graph search problem on a directed graph by discretizing the spacetime world and the vehicle actuation. The decomposition algorithm provides reachability analysis of a time-varying graph. Because the balloon only moves in the positive direction in time, the adjacency matrix of the graph can be represented with an upper block-triangular matrix, and this upper block-triangular structure can be exploited to decompose a large graph search problem. The new approach consumes a much smaller amount of memory, which also helps speed up the overall computation when the computing resource has a limited physical memory compared to the problem size.

Kuwata, Yoshiaki↗
NASA NTRS1975

A two-level trajectory decomposition algorithm featuring optimal intermediate target selection

A decomposition algorithm is presented that optimizes complex missions by partitioning the trajectory into natural segments such as ascent or entry. Each segment defines a full-rank targeting subproblem. These are solved sequentially using the Newton-Raphson algorithm. The master problem, representing the complete mission, is to determine subproblem targets and master-problem controls that optimize the mission objective subject to intersegment constraints. The gradient projection algorithm solves this problem using derivatives obtained analytically from finite-difference subproblem sensitivities. Thus, the mission is optimized by coordinating the solution of tractible subproblems. Computational results for a synchronous equatorial mission are included.

Petersen, F. M.↗
NASA NTRS1977

A two-level trajectory decomposition algorithm featuring optimal intermediate target selection

A decomposition algorithm is presented which optimizes complex missions by partitioning the trajectory into natural segments such as ascent or entry. Each segment defines a full-rank targeting subproblem. These are solved sequentially using the Newton-Raphson algorithm. The master problem, representing the complete mission, is to determine subproblem targets and master-problem controls that optimize the mission objective subject to intersegment constraints. The gradient projection algorithm solves this problem using derivatives obtained analytically from finite-difference subproblem sensitivities. Thus, the mission is optimized by coordinating the solution of tractible subproblems. Computational results for a synchronous equatorial mission are included.

Petersen, F. M.↗
NASA NTRS1988

Domain decomposition algorithms and computational fluid dynamics

Some of the new domain decomposition algorithms are applied to two model problems in computational fluid dynamics: the two-dimensional convection-diffusion problem and the incompressible driven cavity flow problem. First, a brief introduction to the various approaches of domain decomposition is given, and a survey of domain decomposition preconditioners for the operator on the interface separating the subdomains is then presented. For the convection-diffusion problem, the effect of the convection term and its discretization on the performance of some of the preconditioners is discussed. For the driven cavity problem, the effectiveness of a class of boundary probe preconditioners is examined.

Chan, Tony F.↗
DOE OSTI2020

Decomposition Algorithms for Scalable Quantum Annealing

The presented python code provides wrapper functions for two decomposition algorithms: One algorithm to decompose Maximum Clique problems and one to decompose Minimum Vertex Cover problems. The functions take as input a networkx graph object, and decompose either problem on the input graph recursively into subproblems such that the optimal solution can be constructed from the optimal solutions of both subproblems. The recursion ends as soon as the subproblems reach a pre-specified size limit by the user, and they can be solved using any method provided in advance by the user as external function. This includes exact solvers or an adiabatic quantum annealer.

Pelofske, Elijah↗
DOE OSTI2020

Decomposition Algorithms for Solving NP-hard Problems on a Quantum Annealer

NP-hard problems such as the maximum clique or minimum vertex cover problems, two of Karp’s 21 NP-hard problems, have several applications in computational chemistry, biochemistry and computer network security. Adiabatic quantum annealers can search for the optimum value of such NP-hard optimization problems, given the problem can be embedded on their hardware. However, this is often not possible due to certain limitations of the hardware connectivity structure of the annealer. This paper studies a general framework for a decomposition algorithm for NP-hard graph problems aiming to identify an optimal set of vertices. Our generic algorithm allows us to recursively divide an instance until the generated subproblems can be embedded on the quantum annealer hardware and subsequently solved. Furthermore, the framework is applied to the maximum clique and minimum vertex cover problems, and we propose several pruning and reduction techniques to speed up the recursive decomposition. The performance of both algorithms is assessed in a detailed simulation study.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2004

An Integrated Centroid Finding and Particle Overlap Decomposition Algorithm for Stereo Imaging Velocimetry

An integrated algorithm for decomposing overlapping particle images (multi-particle objects) along with determining each object s constituent particle centroid(s) has been developed using image analysis techniques. The centroid finding algorithm uses a modified eight-direction search method for finding the perimeter of any enclosed object. The centroid is calculated using the intensity-weighted center of mass of the object. The overlap decomposition algorithm further analyzes the object data and breaks it down into its constituent particle centroid(s). This is accomplished with an artificial neural network, feature based technique and provides an efficient way of decomposing overlapping particles. Combining the centroid finding and overlap decomposition routines into a single algorithm allows us to accurately predict the error associated with finding the centroid(s) of particles in our experiments. This algorithm has been tested using real, simulated, and synthetic data and the results are presented and discussed.

McDowell, Mark↗
NASA NTRS1996

Domain Decomposition Algorithms for First-Order System Least Squares Methods

Least squares methods based on first-order systems have been recently proposed and analyzed for second-order elliptic equations and systems. They produce symmetric and positive definite discrete systems by using standard finite element spaces, which are not required to satisfy the inf-sup condition. In this paper, several domain decomposition algorithms for these first-order least squares methods are studied. Some representative overlapping and substructuring algorithms are considered in their additive and multiplicative variants. The theoretical and numerical results obtained show that the classical convergence bounds (on the iteration operator) for standard Galerkin discretizations are also valid for least squares methods.

Pavarino, Luca F.↗
DOE OSTI2023

Optimization under uncertainty of a hybrid waste tire and natural gas feedstock flexible polygeneration system using a decomposition algorithm

Market uncertainties motivate the development of flexible polygeneration systems that are able to adjust operating conditions to favor production of the most profitable product portfolio. However, this operational flexibility comes at the cost of higher capital expenditure. A scenario-based two-stage stochastic nonconvex Mixed-Integer Nonlinear Programming (MINLP) approach lends itself naturally to optimizing these trade-offs. This work studies the optimal design and operation under uncertainty of a hybrid feedstock flexible polygeneration system producing electricity, methanol, dimethyl ether, olefins or liquefied (synthetic) natural gas. A recently developed C++ based software framework (named GOSSIP) is used for modeling the optimization problem as well as its efficient solution using the Nonconvex Generalized Benders Decomposition (NGBD) algorithm. Two different cases are studied: The first uses estimates of the means and variances of the uncertain parameters from historical data, whereas the second assesses the impact of increased uncertain parameter volatility. The value of implementing flexible designs characterized by the value of the stochastic solution (VSS) is in the range of 260–405 M$ for a scale of approximately 893 MW of thermal input. Increased price volatility around the same mean results in higher expected net present value and VSS as operational flexibility allows for asymmetric exploitation of price peaks.

42 ENGINEERING↗
DOE OSTI2024

An Incremental Tensor Train Decomposition Algorithm

We present a new algorithm for incrementally updating the tensor train decomposition of a stream of tensor data. This new algorithm, called the tensor train incremental core expansion (TT-ICE) improves upon the current state-of-the-art algorithms for compressing in tensor train format by developing a new adaptive approach that incurs significantly slower rank growth and guarantees compression accuracy. This capability is achieved by limiting the number of new vectors appended to the TT-cores of an existing accumulation tensor after each data increment. These vectors represent directions orthogonal to the span of existing cores and are limited to those needed to represent a newly arrived tensor to a target accuracy. We provide two versions of the algorithm: TT-ICE and TT-ICE accelerated with heuristics (TT-ICE*). Here, we provide a proof of correctness for TT-ICE and empirically demonstrate the performance of the algorithms in compressing large-scale video and scientific simulation datasets. Compared to existing approaches that also use rank adaptation, TT-ICE* achieves 57× higher compression and up to 95% reduction in computational time.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1988

Domain decomposition algorithms and computation fluid dynamics

In the past several years, domain decomposition was a very popular topic, partly motivated by the potential of parallelization. While a large body of theory and algorithms were developed for model elliptic problems, they are only recently starting to be tested on realistic applications. The application of some of these methods to two model problems in computational fluid dynamics are investigated. Some examples are two dimensional convection-diffusion problems and the incompressible driven cavity flow problem. The construction and analysis of efficient preconditioners for the interface operator to be used in the iterative solution of the interface solution is described. For the convection-diffusion problems, the effect of the convection term and its discretization on the performance of some of the preconditioners is discussed. For the driven cavity problem, the effectiveness of a class of boundary probe preconditioners is discussed.

Chan, Tony F.↗
NASA NTRS1998

A Parallel Non-Overlapping Domain-Decomposition Algorithm for Compressible Fluid Flow Problems on Triangulated Domains

This paper considers an algebraic preconditioning algorithm for hyperbolic-elliptic fluid flow problems. The algorithm is based on a parallel non-overlapping Schur complement domain-decomposition technique for triangulated domains. In the Schur complement technique, the triangulation is first partitioned into a number of non-overlapping subdomains and interfaces. This suggests a reordering of triangulation vertices which separates subdomain and interface solution unknowns. The reordering induces a natural 2 x 2 block partitioning of the discretization matrix. Exact LU factorization of this block system yields a Schur complement matrix which couples subdomains and the interface together. The remaining sections of this paper present a family of approximate techniques for both constructing and applying the Schur complement as a domain-decomposition preconditioner. The approximate Schur complement serves as an algebraic coarse space operator, thus avoiding the known difficulties associated with the direct formation of a coarse space discretization. In developing Schur complement approximations, particular attention has been given to improving sequential and parallel efficiency of implementations without significantly degrading the quality of the preconditioner. A computer code based on these developments has been tested on the IBM SP2 using MPI message passing protocol. A number of 2-D calculations are presented for both scalar advection-diffusion equations as well as the Euler equations governing compressible fluid flow to demonstrate performance of the preconditioning algorithm.

Barth, Timothy J.↗