SEARCH · Search NASA

Results for “sampling algorithm”

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.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 361 records · Page 20

All sources NASA NTRS DOE OSTI

DOE OSTI2026

Quantum-Inspired Bayesian Sampling for Uncertainty Quantification and Machine Learning (Final Technical Report)

With increasing simulation and measurement data, machine learning and artificial intelligence have been widely used in computational decision-making of complex engineering systems. The resulting tools, such as uncertainty quantification solvers, reinforcement learning, and physics-informed machine learning, have achieved great success in critical DOE tasks such as material discovery and design, energy system modeling and control, and numerical weather and climate prediction. A core topic in scientific machine learning and artificial intelligence is Bayesian inference: given an observed data set, people want to estimate the posterior distribution of a (possibly large) number of hidden parameters. Due to the flexibility and weak assumptions, Bayesian sampling has been the mainstream Bayesian inference solvers despite the rapid progress of approximate Bayesian inference. Classical Bayesian sampling methods such as Markov-chain Monte Carlo suffer from a low-acceptance rate due to the random walk nature, therefore state-of-the-art techniques use Hamiltonian Monte Carlo and its variants to efficiently draw posterior samples in a high dimension. The key idea of Hamiltonian Monte Carlo and its variants is to simulate the Hamiltonian dynamics of a classical particle with a fixed mass, and their performance significantly degrades when the posterior distribution is highly spiky or has multiple modes. Leveraging the idea of quantum physics, this project has investigated new theory, algorithms and applications of Bayesian inference (especially Bayesian sampling). The main results include: (1) novel quantum-inspired Bayesian sampling methods that can lead to better accuracy for challenging multi-modal or spiky distributions, (2) more scalable machine learning framework leveraging tensor-compressed Bayesian inference, and (3) Bayesian and sampling approaches for verifying the robustness of continuous and binary neural networks.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Intrepid MCMC: Metropolis-Hastings with exploration

In engineering examples, one often encounters the need to sample from unnormalized distributions with complex shapes that may also be implicitly defined through a physical or numerical simulation model, making it computationally expensive to evaluate the associated density function. For such cases, MCMC has proven to be an invaluable tool. Random-walk Metropolis Methods (also known as Metropolis-Hastings (MH)), in particular, are highly popular for their simplicity, flexibility, and ease of implementation. However, most MH algorithms suffer from significant limitations when attempting to sample from distributions with multiple modes (particularly disconnected ones). Here, in this paper, we present Intrepid MCMC - a novel MH scheme that utilizes a simple coordinate transformation to significantly improve the mode-finding ability and convergence rate to the target distribution of random-walk Markov chains while retaining most of the simplicity of the vanilla MH paradigm. Through multiple examples, we showcase the improvement in the performance of Intrepid MCMC over vanilla MH for a wide variety of target distribution shapes. We also provide an analysis of the mixing behavior of the Intrepid Markov chain, as well as the efficiency of our algorithm for increasing dimensions. A thorough discussion is presented on the practical implementation of the Intrepid MCMC algorithm. Finally, its utility is highlighted through a Bayesian parameter inference problem for a two-degree-of-freedom oscillator under free vibration.

97 - MATHEMATICS AND COMPUTING↗

NASA NTRS1980

The SASS scattering coefficient algorithm

This paper describes the algorithms used to convert engineering unit data obtained from the Seasat-A satellite scatterometer (SASS) to radar scattering coefficients and associated supporting parameters. A description is given of the instrument receiver and related processing used by the scatterometer to measure signal power backscattered from the earth's surface. The applicable radar equation used for determining scattering coefficient is derived. Sample results of SASS data processed through current algorithm development facility (ADF) scattering coefficient algorithms are presented which include scattering coefficient values for both water and land surfaces. Scattering coefficient signatures for these two surface types are seen to have distinctly different characteristics. Scattering coefficient measurements of the Amazon rain forest indicate the usefulness of this type of data as a stable calibration reference target.

Bracalente, E. M.↗

NASA NTRS2011

A Ground Validation Network for the Global Precipitation Measurement Mission

A prototype Validation Network (VN) is currently operating as part of the Ground Validation System for NASA's Global Precipitation Measurement (GPM) mission. The VN supports precipitation retrieval algorithm development in the GPM prelaunch era. Postlaunch, the VN will be used to validate GPM spacecraft instrument measurements and retrieved precipitation data products. The period of record for the VN prototype starts on 8 August 2006 and runs to the present day. The VN database includes spacecraft data from the Tropical Rainfall Measuring Mission (TRMM) precipitation radar (PR) and coincident ground radar (GR) data from operational meteorological networks in the United States, Australia, Korea, and the Kwajalein Atoll in the Marshall Islands. Satellite and ground radar data products are collected whenever the PR satellite track crosses within 200 km of a VN ground radar, and these data are stored permanently in the VN database. VN products are generated from coincident PR and GR observations when a significant rain event occurs. The VN algorithm matches PR and GR radar data (including retrieved precipitation data in the case of the PR) by calculating averages of PR reflectivity (both raw and attenuation corrected) and rain rate, and GR reflectivity at the geometric intersection of the PR rays with the individual GR elevation sweeps. The algorithm thus averages the minimum PR and GR sample volumes needed to "matchup" the spatially coincident PR and GR data types. The result of this technique is a set of vertical profiles for a given rainfall event, with coincident PR and GR samples matched at specified heights throughout the profile. VN data can be used to validate satellite measurements and to track ground radar calibration over time. A comparison of matched TRMM PR and GR radar reflectivity factor data found a remarkably small difference between the PR and GR radar reflectivity factor averaged over this period of record in stratiform and convective rain cases when samples were taken from high in the atmosphere. A significant difference in PR and GR reflectivity was found in convective cases, particularly in convective samples from the lower part of the atmosphere. In this case, the mean difference between PR and corrected GR reflectivity was −1.88 dBZ. The PR-GR bias was found to increase with the amount of PR attenuation correction applied, with the PR-GR bias reaching −3.07 dBZ in cases where the attenuation correction applied is greater than 6 dBZ. Additional analysis indicated that the version 6 TRMM PR retrieval algorithm underestimates rainfall in case of convective rain in the lower part of the atmosphere by 30%-40%.

Schwaller, Mathew R.↗

NASA NTRS2011

A Ground Validation Network for the Global Precipitation Measurement Mission

A prototype Validation Network (VN) is currently operating as part of the Ground Validation System for NASA's Global Precipitation Measurement (GPM) mission. The VN supports precipitation retrieval algorithm development in the GPM prelaunch era. Postlaunch, the VN will be used to validate GPM spacecraft instrument measurements and retrieved precipitation data products. The period of record for the VN prototype starts on 8 August 2006 and runs to the present day. The VN database includes spacecraft data from the Tropical Rainfall Measuring Mission (TRMM) precipitation radar (PR) and coincident ground radar (GR) data from operational meteorological networks in the United States, Australia, Korea, and the Kwajalein Atoll in the Marshall Islands. Satellite and ground radar data products are collected whenever the PR satellite track crosses within 200 km of a VN ground radar, and these data are stored permanently in the VN database. VN products are generated from coincident PR and GR observations when a significant rain event occurs. The VN algorithm matches PR and GR radar data (including retrieved precipitation data in the case of the PR) by calculating averages of PR reflectivity (both raw and attenuation corrected) and rain rate, and GR reflectivity at the geometric intersection of the PR rays with the individual GR elevation sweeps. The algorithm thus averages the minimum PR and GR sample volumes needed to "matchup" the spatially coincident PR and GR data types. The result of this technique is a set of vertical profiles for a given rainfall event, with coincident PR and GR samples matched at specified heights throughout the profile. VN data can be used to validate satellite measurements and to track ground radar calibration over time. A comparison of matched TRMM PR and GR radar reflectivity factor data found a remarkably small difference between the PR and GR radar reflectivity factor averaged over this period of record in stratiform and convective rain cases when samples were taken from high in the atmosphere. A significant difference in PR and GR reflectivity was found in convective cases, particularly in convective samples from the lower part of the atmosphere. In this case, the mean difference between PR and corrected GR reflectivity was -1.88 dBZ. The PR-GR bias was found to increase with the amount of PR attenuation correction applied, with the PR-GR bias reaching -3.07 dBZ in cases where the attenuation correction applied is greater than 6 dBZ. Additional analysis indicated that the version 6 TRMM PR retrieval algorithm underestimates rainfall in case of convective rain in the lower part of the atmosphere by 30%-40%.

Schwaller, Mathew R.↗

NASA NTRS1997

A Comparison of Soil Moisture Retrieval Models Using SIR-C Measurements over the Little Washita River Watershed

Six SIR-C L-band measurements over the Little Washita River watershed in Chickasha, Oklahoma during 11-17 April 1994 have been analyzed for studying the change of soil moisture in the region. Two algorithms developed recently for estimation of moisture content in bare soil were applied to these measurements and the results were compared with those sampled on the ground. There is a good agreement between the values of soil moisture estimated by either one of the algorithms and those measured from ground sampling for bare or sparsely vegetated fields. The standard error from this comparison is on the order of 0.05-0.06 cu cm/cu cm, which is comparable to that expected from a regression between backscattering coefficients and measured soil moisture. Both algorithms provide a poor estimation of soil moisture or fail to give solutions to areas covered with moderate or dense vegetation. Even for bare soils the number of pixels that bear no numerical solution from the application of either one of the two algorithms to the data is not negligible. Results from using one of these algorithms indicate that the fraction of these pixels becomes larger as the bare soils become drier. The other algorithm generally gives a larger fraction of these pixels when the fields are vegetation-covered. The implication and impact of these features are discussed in this article.

Wang, J. R.↗

NASA NTRS2010

Photon Counting Using Edge-Detection Algorithm

New applications such as high-datarate, photon-starved, free-space optical communications require photon counting at flux rates into gigaphoton-per-second regimes coupled with subnanosecond timing accuracy. Current single-photon detectors that are capable of handling such operating conditions are designed in an array format and produce output pulses that span multiple sample times. In order to discern one pulse from another and not to overcount the number of incoming photons, a detection algorithm must be applied to the sampled detector output pulses. As flux rates increase, the ability to implement such a detection algorithm becomes difficult within a digital processor that may reside within a field-programmable gate array (FPGA). Systems have been developed and implemented to both characterize gigahertz bandwidth single-photon detectors, as well as process photon count signals at rates into gigaphotons per second in order to implement communications links at SCPPM (serial concatenated pulse position modulation) encoded data rates exceeding 100 megabits per second with efficiencies greater than two bits per detected photon. A hardware edge-detection algorithm and corresponding signal combining and deserialization hardware were developed to meet these requirements at sample rates up to 10 GHz. The photon discriminator deserializer hardware board accepts four inputs, which allows for the ability to take inputs from a quadphoton counting detector, to support requirements for optical tracking with a reduced number of hardware components. The four inputs are hardware leading-edge detected independently. After leading-edge detection, the resultant samples are ORed together prior to deserialization. The deserialization is performed to reduce the rate at which data is passed to a digital signal processor, perhaps residing within an FPGA. The hardware implements four separate analog inputs that are connected through RF connectors. Each analog input is fed to a high-speed 1-bit comparator, which digitizes the input referenced to an adjustable threshold value. This results in four independent serial sample streams of binary 1s and 0s, which are ORed together at rates up to 10 GHz. This single serial stream is then deserialized by a factor of 16 to create 16 signal lines at a rate of 622.5 MHz or lower for input to a high-speed digital processor assembly. The new design and corresponding hardware can be employed with a quad-photon counting detector capable of handling photon rates on the order of multi-gigaphotons per second, whereas prior art was only capable of handling a single input at 1/4 the flux rate. Additionally, the hardware edge-detection algorithm has provided the ability to process 3-10 higher photon flux rates than previously possible by removing the limitation that photoncounting detector output pulses on multiple channels being ORed not overlap. Now, only the leading edges of the pulses are required to not overlap. This new photon counting digitizer hardware architecture supports a universal front end for an optical communications receiver operating at data rates from kilobits to over one gigabit per second to meet increased mission data volume requirements.

Gin, Jonathan W.↗

NASA NTRS2020

Image Moment-Based Extended Object Tracking for Complex Motions

A novel image moment-based model for shape estimation and tracking of an extended target moving with a complex trajectory is presented. The proposed extended object tracking algorithm is based on multiple noisy measurement points sampled from the target at each time step. The shape of the object, approximated by an ellipse, is estimated using a combination of image moments. Dynamic models of image moments for constant velocity and coordinated turn motions are mathematically derived. An unscented Kalman filter - interacting multiple model (UKF-IMM) method is used to track the object and estimate its shape. A likelihood function based on average log-likelihood is derived for the IMM filter. Simulation results of the proposed UKF-IMM algorithm with the image momentbased models are presented that show the estimation of the shape of the object moving in a complex trajectory. The intersection over union (IoU), and the root mean square errors (RMSEs) of the position and velocity of the centroid of the ellipse are used as metrics. The comparison results of the proposed algorithm with a benchmark algorithm from literature based on the IoU and RMSE metrics are presented.

Extended object tracking↗

NASA NTRS2011

Receiver Gain Modulation Circuit

A receiver gain modulation circuit (RGMC) was developed that modulates the power gain of the output of a radiometer receiver with a test signal. As the radiometer receiver switches between calibration noise references, the test signal is mixed with the calibrated noise and thus produces an ensemble set of measurements from which ensemble statistical analysis can be used to extract statistical information about the test signal. The RGMC is an enabling technology of the ensemble detector. As a key component for achieving ensemble detection and analysis, the RGMC has broad aeronautical and space applications. The RGMC can be used to test and develop new calibration algorithms, for example, to detect gain anomalies, and/or correct for slow drifts that affect climate-quality measurements over an accelerated time scale. A generalized approach to analyzing radiometer system designs yields a mathematical treatment of noise reference measurements in calibration algorithms. By treating the measurements from the different noise references as ensemble samples of the receiver state, i.e. receiver gain, a quantitative description of the non-stationary properties of the underlying receiver fluctuations can be derived. Excellent agreement has been obtained between model calculations and radiometric measurements. The mathematical formulation is equivalent to modulating the gain of a stable receiver with an externally generated signal and is the basis for ensemble detection and analysis (EDA). The concept of generating ensemble data sets using an ensemble detector is similar to the ensemble data sets generated as part of ensemble empirical mode decomposition (EEMD) with exception of a key distinguishing factor. EEMD adds noise to the signal under study whereas EDA mixes the signal with calibrated noise. It is mixing with calibrated noise that permits the measurement of temporal-functional variability of uncertainty in the underlying process. The RGMC permits the evaluation of EDA by modulating the receiver gain using an external signal. Without the RGMC, samples of calibrated references from radiometers form an ensemble data set of the natural occurring fluctuations within a receiver. By driving the gain of an otherwise stable receiver with an external signal, the conceptual framework and generalization of the mathematics of EDA can be tested. A series of measurements was conducted to evaluate and characterize the performance of the RGMC. Test signals stepped the RGMC across its dynamic range of performance using a radiometer that sampled four noise references; analysis indicates that the RGMC successfully modulated the receiver gain with an external signal. Calibration algorithms applied to four noise references demonstrate the RGMC produced ensemble data sets of the external signal.

Jones, Hollis↗

NASA NTRS1987

Quantitative analysis of the reconstruction performance of interpolants

The analysis presented provides a quantitative measure of the reconstruction or interpolation performance of linear, shift-invariant interpolants. The performance criterion is the mean square error of the difference between the sampled and reconstructed functions. The analysis is applicable to reconstruction algorithms used in image processing and to many types of splines used in numerical analysis and computer graphics. When formulated in the frequency domain, the mean square error clearly separates the contribution of the interpolation method from the contribution of the sampled data. The equations provide a rational basis for selecting an optimal interpolant; that is, one which minimizes the mean square error. The analysis has been applied to a selection of frequently used data splines and reconstruction algorithms: parametric cubic and quintic Hermite splines, exponential and nu splines (including the special case of the cubic spline), parametric cubic convolution, Keys' fourth-order cubic, and a cubic with a discontinuous first derivative. The emphasis in this paper is on the image-dependent case in which no a priori knowledge of the frequency spectrum of the sampled function is assumed.

Lansing, Donald L.↗

NASA NTRSDate not supplied

Training and Validation of Spectral Gap Filling Algorithm for Cpf-Ceres Intercalibration

The Climate Absolute Radiance and Refractivity Observatory (CLARREO) Pathfinder (CPF) mission is set to launch an SI-traceable reflective solar (RS) spectrometer aboard the International Space Station to measure Earth-reflected solar radiation with a radiometric uncertainty of 0.3% (k=1). The CPF intercalibration team has devised a cutting-edge methodology to accurately transfer the benchmark CPF calibration reference to the shortwave (SW) channel (200-5000 nm) of the Clouds and the Earth’s Radiant Energy System (CERES) instrument. The spectral range of CPF measurements spans from 350-2300 nm, while the CERES SW channel measures the Earth-reflected broadband solar radiances between 200 nm to 5 μm. To conduct precise CPF-CERES intercalibration analysis, the CPF-like spectral radiances outside the CPF spectral range need to be estimated to match the CERES SW spectral range. In response, the team has developed a fast algorithm that leverages spectrally redundant information within the CPF-measured portion through principal component analysis (PCA) and utilizes pre-established spectral correlation relationships among wavelengths to extend the CPF spectrum below 350 nm and above 2300 nm. Our results show that the algorithm achieves excellent accuracy in generating the missing energy in the UV and IR portions. The RMS error in the UV region is less than 4.5x10-3 W/m2/sr/nm, while in the IR region, it is smaller than 8x10-5 W/m2/sr/nm. Our methodology was validated using measured EMIT radiance data, which covers the spectral range from 0.381 μm to 2.493 μm. We employed EMIT radiances within the wavelength range of 0.43 – 2.25 μm to generate radiances for both the shorter wavelength range (0.381 – 0.43 μm) and longer wavelength range (2.25 – 2.493 μm). The generated radiances agree very well with the measured EMIT radiances. The standard deviation in the integrated broadband radiances was about 0.1%, and the bias is less than 0.004% for over 1.5 million EMIT measured samples. These statistics show that the spectral gap filling algorithm is robust and effective in substantially reducing the spectral difference-induced uncertainty in the CPF-CERES intercalibration samples.

Qiguang Yang↗

NASA NTRS1981

A family of solution algorithms for nonlinear structural analysis based on relaxation equations

A family of hierarchical algorithms for nonlinear structural equations are presented. The algorithms are based on the Davidenko-Branin type homotopy and shown to yield consistent hierarchical perturbation equations. The algorithms appear to be particularly suitable to problems involving bifurcation and limit point calculations. An important by-product of the algorithms is that it provides a systematic and economical means for computing the stepsize at each iteration stage when a Newton-like method is employed to solve the systems of equations. Some sample problems are provided to illustrate the characteristics of the algorithms.

Park, K. C.↗

NASA NTRS2014

Exact and Approximate Probabilistic Symbolic Execution

Probabilistic software analysis seeks to quantify the likelihood of reaching a target event under uncertain environments. Recent approaches compute probabilities of execution paths using symbolic execution, but do not support nondeterminism. Nondeterminism arises naturally when no suitable probabilistic model can capture a program behavior, e.g., for multithreading or distributed systems. In this work, we propose a technique, based on symbolic execution, to synthesize schedulers that resolve nondeterminism to maximize the probability of reaching a target event. To scale to large systems, we also introduce approximate algorithms to search for good schedulers, speeding up established random sampling and reinforcement learning results through the quantification of path probabilities based on symbolic execution. We implemented the techniques in Symbolic PathFinder and evaluated them on nondeterministic Java programs. We show that our algorithms significantly improve upon a state-of- the-art statistical model checking algorithm, originally developed for Markov Decision Processes.

Symbolic Execution↗

NASA NTRS2017

Spectral Characterization of Analog Samples in Anticipation of OSIRIS-REx's Arrival at Bennu

NASA's Origins, Spectral Interpretation, Resource Identification, and Security-Regolith Explorer (OSIRIS-REx) mission successfully launched on September 8th, 2016. During its rendezvous with near-Earth asteroid (101955) Bennu beginning in 2018, OSIRIS-REx will characterize the asteroid's physical, mineralogical, and chemical properties in an effort to globally map the properties of Bennu, a primitive carbonaceous asteroid, and choose a sampling location [e.g. 1]. In preparation for these observations, we spectrally characterized a suite of analog samples across visible, near- and thermal-infrared wavelengths and used these in initial tests of phase detection and abundance determination software algorithms. Here we present the thermal infrared laboratory measurements of the analog sample suite measured under asteroidlike conditions, which are relevant to the interpretation of spectroscopic observations by the OSIRIS-REx Thermal Emission Spectrometer (OTES) [2, 3]. This suite of laboratory measurements of asteroid analogs under asteroid-like conditions is the first of their kind.

Donaldson Hanna, K. L.↗

NASA NTRS1980

Adaptive/learning control of large space structures - System identification techniques

Techniques developed for the control of aircraft under changing operating conditions are used to develop a learning control system structure for a multi-configuration, flexible space vehicle. A configuration identification subsystem that is to be used with a learning algorithm and a memory and control process subsystem is developed. Adaptive gain adjustments can be achieved by this learning approach without prestoring of large blocks of parameter data and without dither signal inputs which will be suppressed during operations for which they are not compatible. The Space Shuttle Solar Electric Propulsion (SEP) experiment is used as a sample problem for the testing of adaptive/learning control system algorithms.

Thau, F. E.↗

NASA NTRS1980

Improved local linearization algorithm for solving the quaternion equations

The objective of this paper is to develop a new and more accurate local linearization algorithm for numerically solving sets of linear time-varying differential equations. Of special interest is the application of this algorithm to the quaternion rate equations. The results are compared, both analytically and experimentally, with previous results using local linearization methods. The new algorithm requires approximately one-third more calculations per step than the previously developed local linearization algorithm; however, this disadvantage could be reduced by using parallel implementation. For some cases the new algorithm yields significant improvement in accuracy, even with an enlarged sampling interval. The reverse is true in other cases. The errors depend on the values of angular velocity, angular acceleration, and integration step size. One important result is that for the worst case the new algorithm can guarantee eigenvalues nearer the region of stability than can the previously developed algorithm.

Yen, K.↗

NASA NTRS1993

Coherent lidar design and performance verification

The verification of LAWS beam alignment in space can be achieved by a measurement of heterodyne efficiency using the surface return. The crucial element is a direct detection signal that can be identified for each surface return. This should be satisfied for LAWS but will not be satisfied for descoped LAWS. The performance of algorithms for velocity estimation can be described with two basic parameters: the number of coherently detected photo-electrons per estimate and the number of independent signal samples per estimate. The average error of spectral domain velocity estimation algorithms are bounded by a new periodogram Cramer-Rao Bound. Comparison of the periodogram CRB with the exact CRB indicates a factor of two improvement in velocity accuracy is possible using non-spectral domain estimators. This improvement has been demonstrated with a maximum-likelihood estimator. The comparison of velocity estimation algorithms for 2 and 10 micron coherent lidar was performed by assuming all the system design parameters are fixed and the signal statistics are dominated by a 1 m/s rms wind fluctuation over the range gate. The beam alignment requirements for 2 micron are much more severe than for a 10 micron lidar. The effects of the random backscattered field on estimating the alignment error is a major problem for space based lidar operation, especially if the heterodyne efficiency cannot be estimated. For LAWS, the biggest science payoff would result from a short transmitted pulse, on the order of 0.5 microseconds instead of 3 microseconds. The numerically errors for simulation of laser propagation in the atmosphere have been determined as a joint project with the University of California, San Diego. Useful scaling laws were obtained for Kolmogorov atmospheric refractive turbulence and an atmospheric refractive turbulence characterized with an inner scale. This permits verification of the simulation procedure which is essential for the evaluation of the effects of refractive turbulence on coherent Doppler lidar systems. The analysis of 2 micron Doppler lidar data from Coherent Technologies, Inc. (CTI) has demonstrated many of the advantages of doppler lidar measurements of boundary layer winds. The effects of wind shear and wind turbulence over the pulse volume are probably the dominant source of the reduced performance. The effects of wind shear and wind turbulence on the statistical description of doppler lidar data has been derived and calculated.

Frehlich, Rod↗

NASA NTRS1995

A Hybrid Parallel Preconditioning Algorithm For CFD

A new hybrid preconditioning algorithm will be presented which combines the favorable attributes of incomplete lower-upper (ILU) factorization with the favorable attributes of the approximate inverse method recently advocated by numerous researchers. The quality of the preconditioner is adjustable and can be increased at the cost of additional computation while at the same time the storage required is roughly constant and approximately equal to the storage required for the original matrix. In addition, the preconditioning algorithm suggests an efficient and natural parallel implementation with reduced communication. Sample calculations will be presented for the numerical solution of multi-dimensional advection-diffusion equations. The matrix solver has also been embedded into a Newton algorithm for solving the nonlinear Euler and Navier-Stokes equations governing compressible flow. The full paper will show numerous examples in CFD to demonstrate the efficiency and robustness of the method.

Barth,Timothy J.↗