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NASA NTRS1996

Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems

Multigrid methods are very efficient iterative solvers for system of algebraic equations arising from finite element and finite difference discretization of elliptic boundary value problems. The main principle of multigrid methods is to complement the local exchange of information in point-wise iterative methods by a global one utilizing several related systems, called coarse levels, with a smaller number of variables. The coarse levels are often obtained as a hierarchy of discretizations with different characteristic meshsizes, but this requires that the discretization is controlled by the iterative method. To solve linear systems produced by existing finite element software, one needs to create an artificial hierarchy of coarse problems. The principal issue is then to obtain computational complexity and approximation properties similar to those for nested meshes, using only information in the matrix of the system and as little extra information as possible. Such algebraic multigrid method that uses the system matrix only was developed by Ruge. The prolongations were based on the matrix of the system by partial solution from given values at selected coarse points. The coarse grid points were selected so that each point would be interpolated to via so-called strong connections. Our approach is based on smoothed aggregation introduced recently by Vanek. First the set of nodes is decomposed into small mutually disjoint subsets. A tentative piecewise constant interpolation (in the discrete sense) is then defined on those subsets as piecewise constant for second order problems, and piecewise linear for fourth order problems. The prolongation operator is then obtained by smoothing the output of the tentative prolongation and coarse level operators are defined variationally.

Vanek, Petr↗
NASA NTRS1996

Multilevel Algorithm for Atmospheric Data Assimilation

A multiscale algorithm for the problem of optimal statistical interpolation of observed data has been developed. This problem includes the calculation of the vector of the 'analyzed' (best estimated) atmosphere flow field w(sup a) by the formula: w(sup a) = w(sup f) + P(sup f) H(sup T) y, where the quantity y is defined by the equation (H P(sup f) H(sup T) + R)y = w(sup o) - H w(sup f), using the given model forecast first guess w(sup f) and the vector of observations w(sup o); H is an interpolation operator from the regular grid to the observation network, P(sup f) is the forecast error covariance matrix, and R is the observation error covariance matrix. At this initial stage the case of univariate analysis of single level radiosonde height data is considered. The matrix R is assumed to be diagonal, and the matrix P(sup f) is assumed to be given by the formula P(sub ij)(sup f) = sigma(sub i)(sup f) mu(sub ij) sigma(sub j)(sub f), where mu(sub ij) is a smooth, decreasing function of the distance between the i-th and the j-th points. In this paper we describe a multiscale iterative process based on a multiresolution, simultaneous displacement technique and a localized variational calculation of iteration parameters.

Brandt, Achi↗
NASA NTRS1994

Progress in Grid Generation: From Chimera to DRAGON Grids

Hybrid grids, composed of structured and unstructured grids, combines the best features of both. The chimera method is a major stepstone toward a hybrid grid from which the present approach is evolved. The chimera grid composes a set of overlapped structured grids which are independently generated and body-fitted, yielding a high quality grid readily accessible for efficient solution schemes. The chimera method has been shown to be efficient to generate a grid about complex geometries and has been demonstrated to deliver accurate aerodynamic prediction of complex flows. While its geometrical flexibility is attractive, interpolation of data in the overlapped regions - which in today's practice in 3D is done in a nonconservative fashion, is not. In the present paper we propose a hybrid grid scheme that maximizes the advantages of the chimera scheme and adapts the strengths of the unstructured grid while at the same time keeps its weaknesses minimal. Like the chimera method, we first divide up the physical domain by a set of structured body-fitted grids which are separately generated and overlaid throughout a complex configuration. To eliminate any pure data manipulation which does not necessarily follow governing equations, we use non-structured grids only to directly replace the region of the arbitrarily overlapped grids. This new adaptation to the chimera thinking is coined the DRAGON grid. The nonstructured grid region sandwiched between the structured grids is limited in size, resulting in only a small increase in memory and computational effort. The DRAGON method has three important advantages: (1) preserving strengths of the chimera grid; (2) eliminating difficulties sometimes encountered in the chimera scheme, such as the orphan points and bad quality of interpolation stencils; and (3) making grid communication in a fully conservative and consistent manner insofar as the governing equations are concerned. To demonstrate its use, the governing equations are discretized using the newly proposed flux scheme, AUSM+, which will be briefly described herein. Numerical tests on representative 2D inviscid flows are given for demonstration. Finally, extension to 3D is underway, only paced by the availability of the 3D unstructured grid generator.

Liou, Meng-Sing↗
NASA NTRS1997

An optical systems analysis approach to image resampling

All types of image registration require some type of resampling, either during the registration or as a final step in the registration process. Thus the image(s) must be regridded into a spatially uniform, or angularly uniform, coordinate system with some pre-defined resolution. Frequently the ending resolution is not the resolution at which the data was observed with. The registration algorithm designer and end product user are presented with a multitude of possible resampling methods each of which modify the spatial frequency content of the data in some way. The purpose of this paper is threefold: (1) to show how an imaging system modifies the scene from an end to end optical systems analysis approach, (2) to develop a generalized resampling model, and (3) empirically apply the model to simulated radiometric scene data and tabulate the results. A Hanning windowed sinc interpolator method will be developed based upon the optical characterization of the system. It will be discussed in terms of the effects and limitations of sampling, aliasing, spectral leakage, and computational complexity. Simulated radiometric scene data will be used to demonstrate each of the algorithms. A high resolution scene will be "grown" using a fractal growth algorithm based on mid-point recursion techniques. The result scene data will be convolved with a point spread function representing the optical response. The resultant scene will be convolved with the detection systems response and subsampled to the desired resolution. The resultant data product will be subsequently resampled to the correct grid using the Hanning windowed sinc interpolator and the results and errors tabulated and discussed.

Lyon, Richard G.↗
NASA NTRS1998

Conversion from Engineering Units to Telemetry Counts on Dryden Flight Simulators

Dryden real-time flight simulators encompass the simulation of pulse code modulation (PCM) telemetry signals. This paper presents a new method whereby the calibration polynomial (from first to sixth order), representing the conversion from counts to engineering units (EU), is numerically inverted in real time. The result is less than one-count error for valid EU inputs. The Newton-Raphson method is used to numerically invert the polynomial. A reverse linear interpolation between the EU limits is used to obtain an initial value for the desired telemetry count. The method presented here is not new. What is new is how classical numerical techniques are optimized to take advantage of modem computer power to perform the desired calculations in real time. This technique makes the method simple to understand and implement. There are no interpolation tables to store in memory as in traditional methods. The NASA F-15 simulation converts and transmits over 1000 parameters at 80 times/sec. This paper presents algorithm development, FORTRAN code, and performance results.

Fantini, Jay A.↗
NASA NTRS1999

Parallel Grid Manipulations in Earth Science Calculations

The National Aeronautics and Space Administration (NASA) Data Assimilation Office (DAO) at the Goddard Space Flight Center is moving its data assimilation system to massively parallel computing platforms. This parallel implementation of GEOS DAS will be used in the DAO's normal activities, which include reanalysis of data, and operational support for flight missions. Key components of GEOS DAS, including the gridpoint-based general circulation model and a data analysis system, are currently being parallelized. The parallelization of GEOS DAS is also one of the HPCC Grand Challenge Projects. The GEOS-DAS software employs several distinct grids. Some examples are: an observation grid- an unstructured grid of points at which observed or measured physical quantities from instruments or satellites are associated- a highly-structured latitude-longitude grid of points spanning the earth at given latitude-longitude coordinates at which prognostic quantities are determined, and a computational lat-lon grid in which the pole has been moved to a different location to avoid computational instabilities. Each of these grids has a different structure and number of constituent points. In spite of that, there are numerous interactions between the grids, e.g., values on one grid must be interpolated to another, or, in other cases, grids need to be redistributed on the underlying parallel platform. The DAO has designed a parallel integrated library for grid manipulations (PILGRIM) to support the needed grid interactions with maximum efficiency. It offers a flexible interface to generate new grids, define transformations between grids and apply them. Basic communication is currently MPI, however the interfaces defined here could conceivably be implemented with other message-passing libraries, e.g., Cray SHMEM, or with shared-memory constructs. The library is written in Fortran 90. First performance results indicate that even difficult problems, such as above-mentioned pole rotation- a sparse interpolation with little data locality between the physical lat-lon grid and a pole rotated computational grid- can be solved efficiently and at the GFlop/s rates needed to solve tomorrow's high resolution earth science models. In the subsequent presentation we will discuss the design and implementation of PILGRIM as well as a number of the problems it is required to solve. Some conclusions will be drawn about the potential performance of the overall earth science models on the supercomputer platforms foreseen for these problems.

Sawyer, W.↗
NASA NTRS1999

Student Support for Research in Hierarchical Control and Trajectory Planning

Generally, classical polynomial splines tend to exhibit unwanted undulations. In this work, we discuss a technique, based on control principles, for eliminating these undulations and increasing the smoothness properties of the spline interpolants. We give a generalization of the classical polynomial splines and show that this generalization is, in fact, a family of splines that covers the broad spectrum of polynomial, trigonometric and exponential splines. A particular element in this family is determined by the appropriate control data. It is shown that this technique is easy to implement. Several numerical and curve-fitting examples are given to illustrate the advantages of this technique over the classical approach. Finally, we discuss the convergence properties of the interpolant.

Martin, Clyde F.↗
NASA NTRS1999

Artifacts Introduced in the Point Evaluation of Functions Expanded into a Degree 360 Spherical Harmonic Series

An expansion of a function initially given in 1deg cells was carried out to degree 360 by using 30'cells whose value was initially assigned to be the value of the 1deg cell in which it fell. The evaluation of point values of the function from the degree 360 expansion revealed spurious patterns attributed to the coefficients from degree 181 to 360. Expansion of the original function in 1deg cells to degree 180 showed no problems in the point evaluation. Mean 1deg values computed from both degree 180 to 360 expansions showed close agreement with the original function. The artifacts could be removed if the 30' values were interpolated by spline procedures from adjacent I' cells. These results led to an examination of the gravity anomalies and geoid undulations from EGM96 in areas where I' values were "split up" to form 30'cells. The area considered was 75degS to 85degS, 100degE to 120degE where the split up cells were basically south of 81 degS. A small, latitude related, and possibly spurious effect might be detectable in anomaly variations in the region. These results suggest that point values of a function computed from a high degree expansion may have spurious signals unless the cell size is compatible with the maximum degree of expansion. The spurious signals could be eliminated by using a spline interpolation procedure to obtain the 30'values from the 1deg values.

Rapp, R.↗
NASA NTRS1999

Development of Three-Dimensional DRAGON Grid Technology

For a typical three dimensional flow in a practical engineering device, the time spent in grid generation can take 70 percent of the total analysis effort, resulting in a serious bottleneck in the design/analysis cycle. The present research attempts to develop a procedure that can considerably reduce the grid generation effort. The DRAGON grid, as a hybrid grid, is created by means of a Direct Replacement of Arbitrary Grid Overlapping by Nonstructured grid. The DRAGON grid scheme is an adaptation to the Chimera thinking. The Chimera grid is a composite structured grid, composing a set of overlapped structured grids, which are independently generated and body-fitted. The grid is of high quality and amenable for efficient solution schemes. However, the interpolation used in the overlapped region between grids introduces error, especially when a sharp-gradient region is encountered. The DRAGON grid scheme is capable of completely eliminating the interpolation and preserving the conservation property. It maximizes the advantages of the Chimera scheme and adapts the strengths of the unstructured and while at the same time keeping its weaknesses minimal. In the present paper, we describe the progress towards extending the DRAGON grid technology into three dimensions. Essential and programming aspects of the extension, and new challenges for the three-dimensional cases, are addressed.

Zheng, Yao↗
NASA NTRS1997

Annual Sea Level Variability Induced by Changes in Sea Ice Extent and Accumulation on Ice Sheets: An Assessment Based on Remotely Sensed Data

Changes of mean annual net accumulation at the surface on the grounded ice sheets of East Antarctica, West Antarctica, and Greenland in response to variations in sea ice extent are estimated using grid-point values 100 km apart. The data bases are assembled principally by bilinear interpolation of remotely sensed brightness temperature (Nimbus-5 ESMR, Nimbus-7 SMMR), surface temperature (Nimbus-7 THIR), and surface elevation (ERS-1 radar altimeter). These data, complemented by field data where remotely sensed data are not available, are used in multivariate analyses in which mean annual accumulation (derived from firn emissivity) is the dependent variable; the independent variables are latitude, surface elevation, mean annual surface temperature, and mean annual distance to open ocean (as a source of energy and moisture). The last is the shortest distance measured between a grid point and the mean annual position of the 10% sea ice concentration boundary, and is used as an index of changes in sea ice extent as well as of mean concentration. Stepwise correlation analyses indicate that variations in sea ice extent of +/-50 km would lead to changes in accumulation inversely of +/-4% on East Antarctica, +/- 10% on West Antarctica, and +4% on Greenland. These results are compared with those obtained in a previous study using visually interpolated values from contoured compilations of field data; they substantiate the findings for the Antarctic ice sheets (+/-4% on East Antarctica, +/-9% in West Antarctica), and suggest a reduction by one half of the probable change of accumulation on Greenland (from +/-8%). The results also suggest a reduction of the combined contribution to sea level variability to +/- 0.19 mm/a (from +/- 0.22 mm/a).

Zwally, H. J.↗
NASA NTRS2000

Crack Turning in Integrally Stiffened Aircraft Structures

Current emphasis in the aircraft industry toward reducing manufacturing cost has created a renewed interest in integrally stiffened structures. Crack turning has been identified as an approach to improve the damage tolerance and fail-safety of this class of structures. A desired behavior is for skin cracks to turn before reaching a stiffener, instead of growing straight through. A crack in a pressurized fuselage encounters high T-stress as it nears the stiffener--a condition favorable to crack turning. Also, the tear resistance of aluminum alloys typically varies with crack orientation, a form of anisotropy that can influence the crack path. The present work addresses these issues with a study of crack turning in two-dimensions, including the effects of both T-stress and fracture anisotropy. Both effects are shown to have relation to the process zone size, an interaction that is central to this study. Following an introduction to the problem, the T-stress effect is studied for a slightly curved semi-infinite crack with a cohesive process zone, yielding a closed form expression for the future crack path in an infinite medium. For a given initial crack tip curvature and tensile T-stress, the crack path instability is found to increase with process zone size. Fracture orthotropy is treated using a simple function to interpolate between the two principal fracture resistance values in two-dimensions. An extension to three-dimensions interpolates between the six principal values of fracture resistance. Also discussed is the transition between mode I and mode II fracture in metals. For isotropic materials, there is evidence that the crack seeks out a direction of either local symmetry (pure mode I) or local asymmetry (pure mode II) growth. For orthotropic materials the favored states are not pure modal, and have mode mixity that is a function of crack orientation.

Pettit, Richard Glen↗
NASA NTRS2000

On Fast Post-Processing of Global Positioning System Simulator Truth Data and Receiver Measurements and Solutions Data

Post-Processing of data related to a Global Positioning System (GPS) simulation is an important activity in qualification of a GPS receiver for space flight. Because a GPS simulator is a critical resource it is desirable to move off the pertinent simulation data from the simulator as soon as a test is completed. The simulator data files are usually moved to a Personal Computer (PC), where the post-processing of the receiver logged measurements and solutions data and simulated data is performed. Typically post-processing is accomplished using PC-based commercial software languages and tools. Because of commercial software systems generality their general-purpose functions are notoriously slow and more than often are the bottleneck problem even for short duration experiments. For example, it may take 8 hours to post-process data from a 6-hour simulation. There is a need to do post-processing faster, especially in order to use the previous test results as feedback for a next simulation setup. This paper demonstrates that a fast software linear interpolation algorithm is applicable to a large class of engineering problems, like GPS simulation data post-processing, where computational time is a critical resource and is one of the most important considerations. An approach is developed that allows to speed-up post-processing by an order of magnitude. It is based on improving the post-processing bottleneck interpolation algorithm using apriori information that is specific to the GPS simulation application. The presented post-processing scheme was used in support of a few successful space flight missions carrying GPS receivers. A future approach to solving the post-processing performance problem using Field Programmable Gate Array (FPGA) technology is described.

Kizhner, Semion↗
NASA NTRS2000

An Automated Approach to Very High Order Aeroacoustic Computations in Complex Geometries

Computational aeroacoustics requires efficient, high-resolution simulation tools. And for smooth problems, this is best accomplished with very high order in space and time methods on small stencils. But the complexity of highly accurate numerical methods can inhibit their practical application, especially in irregular geometries. This complexity is reduced by using a special form of Hermite divided-difference spatial interpolation on Cartesian grids, and a Cauchy-Kowalewslci recursion procedure for time advancement. In addition, a stencil constraint tree reduces the complexity of interpolating grid points that are located near wall boundaries. These procedures are used to automatically develop and implement very high order methods (>15) for solving the linearized Euler equations that can achieve less than one grid point per wavelength resolution away from boundaries by including spatial derivatives of the primitive variables at each grid point. The accuracy of stable surface treatments is currently limited to 11th order for grid aligned boundaries and to 2nd order for irregular boundaries.

Dyson, Rodger W.↗
NASA NTRS2000

The Surface Pressure Response of a NACA 0015 Airfoil Immersed in Grid Turbulence: Characteristics of the Turbulence - Volume 1

Two grids have been developed for the Virginia Tech 6 ft x 6 ft Stability wind tunnel for the purpose of generating homogeneous isotropic turbulent flows for the study of unsteady airfoil response. The first, a square bi-planar grid with a 12" mesh size and an open area ratio of 69.4%, was mounted in the wind tunnel contraction. The second grid, a metal weave with a 1.2 in. mesh size and an open area ratio of 68.2% was mounted in the tunnel test section. Detailed statistical and spectral measurements of the turbulence generated by the two grids are presented for wind tunnel free stream speeds of 10, 20, 30 and 40 m/s. These measurements show the flows to be closely homogeneous and isotropic. Both grids produce flows with a turbulence intensity of about 4% at the location planned for the airfoil leading edge. Turbulence produced by the large grid has an integral scale of some 3.2 inches here. Turbulence produced by the small grid is an order of magnitude smaller. For wavenumbers below the upper limit of the inertial subrange, the spectra and correlations measured with both grids at all speeds can be represented using the von Karman interpolation formula with a single velocity and length scale. The spectra maybe accurately represented over the entire wavenumber range by a modification of the von Karman interpolation formula that includes the effects of dissipation. These models are most accurate at the higher speeds (30 and 40 m/s).

Bereketab, Semere↗
NASA NTRS2001

Impact of Lidar Wind Sounding on Mesoscale Forecast

An Observing System Simulation Experiment (OSSE) was conducted to study the impact of airborne lidar wind sounding on mesoscale weather forecast. A wind retrieval scheme, which interpolates wind data from a grid data system, simulates the retrieval of wind profile from a satellite lidar system. A mesoscale forecast system based on the PSU/NCAR MM5 model is developed and incorporated the assimilation of the retrieved line-of-sight wind. To avoid the "identical twin" problem, the NCEP reanalysis data is used as our reference "nature" atmosphere. The simulated space-based lidar wind observations were retrieved by interpolating the NCEP values to the observation locations. A modified dataset obtained by smoothing the NCEP dataset was used as the initial state whose forecast was sought to be improved by assimilating the retrieved lidar observations. Forecasts using wind profiles with various lidar instrument parameters has been conducted. The results show that to significantly improve the mesoscale forecast the satellite should fly near the storm center with large scanning radius. Increasing lidar firing rate also improves the forecast. Cloud cover and lack of aerosol degrade the quality of the lidar wind data and, subsequently, the forecast.

Miller, Timothy L.↗
NASA NTRS2000

On Utilization of NEXRAD Scan Strategy Information to Infer Discrepancies Associated With Radar and Rain Gauge Surface Volumetric Rainfall Accumulations

To evaluate the Tropical Rainfall Measuring Mission (TRMM) monthly Ground Validation (GV) rain map, 42 quality controlled tipping bucket rain gauge data (1 minute interpolated rain rates) were utilized. We have compared the gauge data to the surface volumetric rainfall accumulation of NEXRAD reflectivity field, (converting to rain rates using a 0.5 dB resolution smooth Z-R table). The comparison was carried out from data collected at Melbourne, Florida during the month of July 98. GV operational level 3 (L3 monthly) accumulation algorithm was used to obtain surface volumetric accumulations for the radar. The gauge records were accumulated using the 1 minute interpolated rain rates while the radar Volume Scan (VOS) intervals remain less than or equal to 75 minutes. The correlation coefficient for the radar and gauge totals for the monthly time-scale remain at 0.93, however, a large difference was noted between the gauge and radar derived rain accumulation when the radar data interval is either 9 minute, or 10 minute. This difference in radar and gauge accumulation is being explained in terms of the radar scan strategy information. The discrepancy in terms of the Volume Coverage Pattern (VCP) of the NEXRAD is being reported where VCP mode is ascertained using the radar tilt angle information. Hourly radar and gauge accumulations have been computed using the present operational L3 method supplemented with a threshold period of +/- 5 minutes (based on a sensitivity analysis). These radar and gauge accumulations are subsequently improved using a radar hourly scan weighting factor (taking ratio of the radar scan frequency within a time bin to the 7436 total radar scans for the month). This GV procedure is further being improved by introducing a spatial smoothing method to yield reasonable bulk radar to gauge ratio for the hourly and daily scales.

Roy, Biswadev↗
NASA NTRS2001

Automated Approach to Very High-Order Aeroacoustic Computations

Computational aeroacoustics requires efficient, high-resolution simulation tools. For smooth problems, this is best accomplished with very high-order in space and time methods on small stencils. However, the complexity of highly accurate numerical methods can inhibit their practical application, especially in irregular geometries. This complexity is reduced by using a special form of Hermite divided-difference spatial interpolation on Cartesian grids, and a Cauchy-Kowalewski recursion procedure for time advancement. In addition, a stencil constraint tree reduces the complexity of interpolating grid points that am located near wall boundaries. These procedures are used to develop automatically and to implement very high-order methods (> 15) for solving the linearized Euler equations that can achieve less than one grid point per wavelength resolution away from boundaries by including spatial derivatives of the primitive variables at each grid point. The accuracy of stable surface treatments is currently limited to 11th order for grid aligned boundaries and to 2nd order for irregular boundaries.

Dyson, Rodger W.↗
NASA NTRS1994

A Navier-Strokes Chimera Code on the Connection Machine CM-5: Design and Performance

We have implemented a three-dimensional compressible Navier-Stokes code on the Connection Machine CM-5. The code is set up for implicit time-stepping on single or multiple structured grids. For multiple grids and geometrically complex problems, we follow the 'chimera' approach, where flow data on one zone is interpolated onto another in the region of overlap. We will describe our design philosophy and give some timing results for the current code. A parallel machine like the CM-5 is well-suited for finite-difference methods on structured grids. The regular pattern of connections of a structured mesh maps well onto the architecture of the machine. So the first design choice, finite differences on a structured mesh, is natural. We use centered differences in space, with added artificial dissipation terms. When numerically solving the Navier-Stokes equations, there are liable to be some mesh cells near a solid body that are small in at least one direction. This mesh cell geometry can impose a very severe CFL (Courant-Friedrichs-Lewy) condition on the time step for explicit time-stepping methods. Thus, though explicit time-stepping is well-suited to the architecture of the machine, we have adopted implicit time-stepping. We have further taken the approximate factorization approach. This creates the need to solve large banded linear systems and creates the first possible barrier to an efficient algorithm. To overcome this first possible barrier we have considered two options. The first is just to solve the banded linear systems with data spread over the whole machine, using whatever fast method is available. This option is adequate for solving scalar tridiagonal systems, but for scalar pentadiagonal or block tridiagonal systems it is somewhat slower than desired. The second option is to 'transpose' the flow and geometry variables as part of the time-stepping process: Start with x-lines of data in-processor. Form explicit terms in x, then transpose so y-lines of data are in-processor. Form explicit terms in y, then transpose so z-lines are in processor. Form explicit terms in z, then solve linear systems in the z-direction. Transpose to the y-direction, then solve linear systems in the y-direction. Finally transpose to the x direction and solve linear systems in the x-direction. This strategy avoids inter-processor communication when differencing and solving linear systems, but requires a large amount of communication when doing the transposes. The transpose method is more efficient than the non-transpose strategy when dealing with scalar pentadiagonal or block tridiagonal systems. For handling geometrically complex problems the chimera strategy was adopted. For multiple zone cases we compute on each zone sequentially (using the whole parallel machine), then send the chimera interpolation data to a distributed data structure (array) laid out over the whole machine. This information transfer implies an irregular communication pattern, and is the second possible barrier to an efficient algorithm. We have implemented these ideas on the CM-5 using CMF (Connection Machine Fortran), a data parallel language which combines elements of Fortran 90 and certain extensions, and which bears a strong similarity to High Performance Fortran. We make use of the Connection Machine Scientific Software Library (CMSSL) for the linear solver and array transpose operations.

Jespersen, Dennis C.↗