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Evolutionary partial differential equations are usually solved by decretization in time and space, and by applying a marching in time procedure to data and algorithms potentially parallelized in the spatial domain.
An algorithm for spectral decomposition is presented which does not require knowledge of eigenvalues and eigenvectors. A set of eigenprojectors are defined which covers the entire spectrum of a matrix, and special attention is given to the projection on the zero eigenvalue. Some useful applications are discussed in the paper.
An efficient algorithm is presented for computing particle paths, streak lines and time lines in time-dependent flows with moving curvilinear grids. The integration, velocity interpolation and step-size control are all performed in physical space which avoids the need to transform the velocity field into computational space. This leads to higher accuracy because there are no Jacobian matrix approximations or expensive matrix inversions. Integration accuracy is maintained using an adaptive step-size control scheme which is regulated by the path line curvature. The problem of cell-searching, point location and interpolation in physical space is simplified by decomposing hexahedral cells into tetrahedral cells. This enables the point location to be done analytically and substantially faster than with a Newton-Raphson iterative method. Results presented show this algorithm is up to six times faster than particle tracers which operate on hexahedral cells yet produces almost identical particle trajectories.
Computational algorithms for structural analysis on parallel-vector supercomputers are reviewed. These parallel algorithms, developed by the authors, are for the assembly of structural equations, 'out-of-core' strategies for linear equation solution, massively distributed-memory equation solution, unsymmetric equation solution, general eigen-solution, geometrically nonlinear finite element analysis, design sensitivity analysis for structural dynamics, optimization algorithm and domain decomposition. The source code for many of these algorithms is available from NASA Langley.
Striping has been observed in the MetOp-A Microwave Humidity Sounder (MHS) data since its degradation in November 2018. However, accurate striping quantification and characterization remain challenging due to the large scene dynamics observed at W/G bands of MHS. Here we have developed a set of novel algorithms for striping quantification, decomposition, characterization, and simulation. Our algorithm extracts striping from the warm-load and cold-space scenes that are relatively stable. We break down the striping into two parts of thermal and 1/f noise and quantify their absolute magnitude and relative ratio. We found a significant increase in striping at 157 GHz that has more than quadrupled by October 2019 relative to its normal level. Regardless of the degradation, the ratio of thermal and 1/f noise remains the same. Our simulation reproduces all the characteristics of striping against observation. It is shown that 1/f noise generates sharp, non-periodic stripes, while thermal noise also generates stripes but with smoother band features. The latter is due to the periodic calibration that has a chopping effect. The striping percentage, defined as the ratio of 1/f to total noise, shows no dependence on the scene temperature. Striping is pronounced not only in 157 GHz but also in 89 and190 GHz with the striping percentage over 50%, while lower in183 GHz of 20%. The results provide insights for quantifying and understanding striping. Our algorithm can be applied to other radiometers and to simulate striping for evaluating its impact in data assimilation and science products.
A novel domain decomposition method based on a hybrid variational principle is presented. Prior to any computation, a given finite element mesh is torn into a set of totally disconnected submeshes. First, an incomplete solution is computed in each subdomain. Next, the compatibility of the displacement field at the interface nodes is enforced via discrete, polynomial and/or piecewise polynomial Lagrange multipliers. In the static case, each floating subdomain induces a local singularity that is resolved very efficiently. The interface problem associated with this domain decomposition method is, in general, indefinite and of variable size. A dedicated conjugate projected gradient algorithm is developed for solving the latter problem when it is not feasible to explicitly assemble the interface operator. When implemented on local memory multiprocessors, the proposed methodology requires less interprocessor communication than the classical method of substructuring. It is also suitable for parallel/vector computers with shared memory and compares favorably with factorization based parallel direct methods.
Stochastic approximation procedure minimizing mean square error criterion for system identification, estimation and decomposition of mixtures
We describe a new interactive parallel Adaptive Mesh Refinement (AMR) framework written in the Python programming language. This new framework, PyAMR, hides the details of parallel AMR data structures and algorithms (e.g., domain decomposition, grid partition, and inter-process communication), allowing the user to focus on the development of algorithms for advancing the solution of a systems of partial differential equations on a single uniform mesh. We demonstrate the use of PyAMR by simulating the pairwise coalescence of magnetic islands using the resistive Hall MHD equations. Techniques for coupling different physics models on different levels of the AMR grid hierarchy are discussed.
Least-squares (LS) estimation is a basic operation in many signal processing problems. Given y = Ax + v, where A is a m x n coefficient matrix, y is a m x 1 observation vector, and v is a m x 1 zero mean white noise vector, a simple least-squares solution is finding the estimated vector x which minimizes the norm of /Ax-y/. It is well known that for an ill-conditioned matrix A, solving least-squares problems by orthogonal triangular (QR) decomposition and back substitution has robust numerical properties under finite word length effect since 2-norm is preserved. Many fast algorithms have been proposed and applied to systolic arrays. Gentleman-Kung (1981) first presented the trianglular systolic array for a basic Givens reduction. McWhirter (1983) used this array structure to find the least-squares estimation errors. Then by geometric approach, several different systolic array realizations of the recursive least-squares estimation algorithms of Lee et al (1981) were derived by Kalson-Yao (1985). Basic QR decomposition algorithms are considered in this paper and it is found that under a one-row time updating situation, the Householder transformation degenerates to a simple Givens reduction. Next, an improved least-squares estimation algorithm is derived by considering a modified version of fast Givens reduction. From this approach, the basic relationship between Givens reduction and Modified-Gram-Schmidt transformation can easily be understood. This improved algorithm also has simpler computational and inter-cell connection complexities while compared with other known least-squares algorithms and is more realistic for systolic array implementation.
One of the main heritage tools used in scientific and engineering data spectrum analysis is the Fourier Integral Transform and its high performance digital equivalent - the Fast Fourier Transform (FFT). Both carry strong a-priori assumptions about the source data, such as linearity, of being stationary, and of satisfying the Dirichlet conditions. A recent development at the National Aeronautics and Space Administration (NASA) Goddard Space Flight Center (GSFC), known as the Hilbert-Huang Transform (HHT), proposes a novel approach to the solution for the nonlinear class of spectrum analysis problems. Using a-posteriori data processing based on the Empirical Mode Decomposition (EMD) sifting process (algorithm), followed by the normalized Hilbert Transform of the decomposition data, the HHT allows spectrum analysis of nonlinear and nonstationary data. The EMD sifting process results in a non-constrained decomposition of a source real value data vector into a finite set of Intrinsic Mode Functions (IMF). These functions form a near orthogonal adaptive basis, a basis that is derived from the data. The IMFs can be further analyzed for spectrum interpretation by the classical Hilbert Transform. A new engineering spectrum analysis tool using HHT has been developed at NASA GSFC, the HHT Data Processing System (HHT-DPS). As the HHT-DPS has been successfully used and commercialized, new applications post additional questions about the theoretical basis behind the HHT and EMD algorithms. Why is the fastest changing component of a composite signal being sifted out first in the EMD sifting process? Why does the EMD sifting process seemingly converge and why does it converge rapidly? Does an IMF have a distinctive structure? Why are the IMFs near orthogonal? We address these questions and develop the initial theoretical background for the HHT. This will contribute to the developments of new HHT processing options, such as real-time and 2-D processing using Field Programmable Array (FPGA) computational resources, enhanced HHT synthesis, and broaden the scope of HHT applications for signal processing.
A Parallel Hierarchical algorithm for Global Routing (PHIGURE) is presented. The router is based on the work of Burstein and Pelavin, but has many extensions for general global routing and parallel execution. Main features of the algorithm include structured hierarchical decomposition into separate independent tasks which are suitable for parallel execution and adaptive simplex solution for adding feedthroughs and adjusting channel heights for row-based layout. Alternative decomposition methods and the various levels of parallelism available in the algorithm are examined closely. The algorithm is described and results are presented for a shared-memory multiprocessor implementation.
One of the major applications of the domain decomposition time marching algorithm is the coupling of the Navier-Stokes systems with Boltzmann equations in order to compute transitional flows. Another important application is the coupling of a global Navier-Stokes problem with a local one in order to use different modelizations and/or discretizations. Both of these applications involve a global Navier-Stokes system with nonstandard boundary conditions. The purpose of this work is to prove, using the classical Leray-Schauder theory, that these boundary conditions are admissible and lead to a well posed problem.
The aerodynamic computer code, OVERFLOW, with a multi-zone overset grid feature, has been parallelized to enhance its performance on distributed and shared memory paradigms. Practical application benchmarks have been set to assess the efficiency of code's parallelism on high-performance architectures. The code's performance has also been experimented with in the context of the distributed computing paradigm on distant computer resources using the Information Power Grid (IPG) toolkit, Globus. Two parallel versions of the code, namely OVERFLOW-MPI and -MLP, have developed around the natural coarse grained parallelism inherent in a multi-zonal domain decomposition paradigm. The algorithm invokes a strategy that forms a number of groups, each consisting of a zone, a cluster of zones and/or a partition of a large zone. Each group can be thought of as a process with one or multithreads assigned to it and that all groups run in parallel. The -MPI version of the code uses explicit message-passing based on the standard MPI library for sending and receiving interzonal boundary data across processors. The -MLP version employs no message-passing paradigm; the boundary data is transferred through the shared memory. The -MPI code is suited for both distributed and shared memory architectures, while the -MLP code can only be used on shared memory platforms. The IPG applications are implemented by the -MPI code using the Globus toolkit. While a computational task is distributed across multiple computer resources, the parallelism can be explored on each resource alone. Performance studies are achieved with some practical aerodynamic problems with complex geometries, consisting of 2.5 up to 33 million grid points and a large number of zonal blocks. The computations were executed primarily on SGI Origin 2000 multiprocessors and on the Cray T3E. OVERFLOW's IPG applications are carried out on NASA homogeneous metacomputing machines located at three sites, Ames, Langley and Glenn. Plans for the future will exploit the distributed parallel computing capability on various homogeneous and heterogeneous resources and large scale benchmarks. Alternative IPG toolkits will be used along with sophisticated zonal grouping strategies to minimize the communication time across the computer resources.
We generalize Harten's multiresolution representation to biorthogonal multiwavelets. Several variants are considered. For example, a given array of discrete point values is transformed to point values and derivatives or point 'values and cell averages'. Compact Hermite interpolation is used in the decomposition and reconstruction algorithm. The resulting basis functions that are symmetric or skewsymmetric, compact, and smooth with optimal order accuracy. Harten's approach has several advantages: the multiresolution scheme is inherently discrete, non-periodic boundary conditions are easy to implement, and the representation can be extended to unstructured grids in bounded domains. We demonstrate the compression features of the new mutliwavelets by application to variable scale piecewise smooth functions with jump discontinuities typical of numerical solutions of nonlinear hyperbolic conservation laws.
The emerging concepts of Urban Air Mobility (UAM) and Advanced Air Mobility (AAM) open a new paradigm for urban air transportation. A big challenge is that these new aerial vehicles will quickly saturate the already crowded aviation spectrum, which is an essential resource to ensure reliable communications for safe operations. In this paper, we consider an air transportation system where multiple aerial vehicles are operated to transport passengers or cargo from different sources to destinations along their pre-defined paths. During the flight, the minimum communication Quality of Service (QoS) requirement must be achieved to ensure flight safety. Our objective is to minimize the average mission completion time by jointly optimizing the velocity selection and spectrum allocation for all aerial vehicles. We formulate the optimization problem as a multi-stage Markov Decision Process (MDP) where the optimization variables are coupled together. A multi-agent Deep Reinforcement Learning (DRL) based solution is proposed where Value Decomposition Networks (VDN) algorithm is utilized to take discrete actions. Additionally, we propose a heuristic greedy algorithm as a baseline solution. Simulation results show that our learning based solution outperforms the heuristic greedy algorithm and another Orthogonal Multiple Access (OMA) solution in minimizing the mission completion time.
The emerging concepts of Urban Air Mobility (UAM) and Advanced Air Mobility (AAM) open a new paradigm for urban air transportation. A big challenge is that these new aerial vehicles will quickly saturate the already crowded aviation spectrum, which is an essential resource to ensure reliable communications for safe operations. In this paper, we consider an air transportation system where multiple aerial vehicles are operated to transport passengers or cargo from different sources to destinations along their pre-defined paths. During the flight, the minimum communication Quality of Service (QoS) requirement must be achieved to ensure flight safety. Our objective is to minimize the average mission completion time by jointly optimizing the velocity selection and spectrum allocation for all aerial vehicles. We formulate the optimization problem as a multi-stage Markov Decision Process (MDP) where the optimization variables are coupled together. A multi-agent Deep Reinforcement Learning (DRL) based solution is proposed where Value Decomposition Networks (VDN) algorithm is utilized to take discrete actions. Additionally, we propose a heuristic greedy algorithm as a baseline solution. Simulation results show that our learning based solution outperforms the heuristic greedy algorithm and another Orthogonal Multiple Access (OMA) solution in minimizing the mission completion time.
A time efficient algorithm for image registration between two images that differ in translation is discussed. The algorithm is based on coarse-fine strategy using wavelet decomposition of both the images. The wavelet decomposition serves two different purposes: (1) its high frequency components are used to detect feature points (corner points here) and (2) it provides coarse-to-fine structure for making the algorithm time efficient. The algorithm is based on detecting the corner points from one of the images called reference image and computing corresponding points from the other image called test image by using local correlations using 7x7 windows centered around the corner points. The corresponding points are detected at the lowest decomposition level in a search area of about 11x11 (depending on the translation) and potential points of correspondence are projected onto higher levels. In the subsequent levels the local correlations are computed in a search area of no more than 3x3 for refinement of the correspondence.
Systems health management (SHM) is an important set of technologies aimed at increasing system safety and reliability by detecting, isolating, and identifying faults; and predicting when the system reaches end of life (EOL), so that appropriate fault mitigation and recovery actions can be taken. Model-based SHM approaches typically make use of global, monolithic system models for online analysis, which results in a loss of scalability and efficiency for large-scale systems. Improvement in scalability and efficiency can be achieved by decomposing the system model into smaller local submodels and operating on these submodels instead. In this paper, the global system model is analyzed offline and structurally decomposed into local submodels. We define a common model decomposition framework for extracting submodels from the global model. This framework is then used to develop algorithms for solving model decomposition problems for the design of three separate SHM technologies, namely, estimation (which is useful for fault detection and identification), fault isolation, and EOL prediction. We solve these model decomposition problems using a three-tank system as a case study.