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All sourcesNASA NTRSDOE OSTI
NASA NTRS1988

Fast Fourier Transform algorithm design and tradeoffs

The Fast Fourier Transform (FFT) is a mainstay of certain numerical techniques for solving fluid dynamics problems. The Connection Machine CM-2 is the target for an investigation into the design of multidimensional Single Instruction Stream/Multiple Data (SIMD) parallel FFT algorithms for high performance. Critical algorithm design issues are discussed, necessary machine performance measurements are identified and made, and the performance of the developed FFT programs are measured. Fast Fourier Transform programs are compared to the currently best Cray-2 FFT program.

Kamin, Ray A., III↗
NASA NTRS1986

On realizations of least-squares estimation and Kalman filtering by systolic arrays

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.

Chen, M. J.↗
NASA NTRS2021

Small UAV Flight Planning in Urban Environments

This work proposes a fast algorithm for generating obstacle-free and wind-efficient flight paths at a constant above-ground-level altitude in urban environments because a fast flight path planning algorithm is an essential function or service needed for enabling small unmanned aerial vehicle (sUAV) to operate in urban environments within Class G airspace. The proposed method first converts the 3D path planning problem to a 2D problem by constructing an obstacle map at a given above-ground-level altitude. A quad-tree decomposition is then used to build a search space in terms of obstacle occupancy and wind difference. The wind cost of traveling through each cell is defined based on energy consumption under various wind conditions. A repulsive potential is also adopted to make sure that the flight plans stay away from obstacles. The Theta* search algorithm, a variant of A* algorithm, is applied to mitigate the path angle change constraints introduced by grid-based graphs. With the Theta* and postsmoothing techniques, an obstacle-free, wind efficient, and constant above-ground-level flight plan can be quickly generated for sUAV operations in urban environments while meeting the lateral path angle constraints. The results showed that the path planning algorithm is efficient and can be finished within several seconds. With a proper choice of wind coefficient, the proposed path planning algorithm outperforms the multiple-shooting trajectory optimization method even in an obstacle-free environment. With the flexibility of incorporating other geo-related costs and computation efficiency, the proposed algorithm shows the potential for real-time flight path planning in complex urban environments.

Path planning↗
NASA NTRS1987

Approximating the linear quadratic optimal control law for hereditary systems with delays in the control

The fundamental control synthesis issue of establishing a priori convergence rates of approximation schemes for feedback controllers for a class of distributed parameter systems is addressed within the context of hereditary systems. Specifically, a factorization approach is presented for deriving approximations to the optimal feedback gains for the linear regulator-quadratic cost problem associated with time-varying functional differential equations with control delays. The approach is based on a discretization of the state penalty which leads to a simple structure for the feedback control law. General properties of the Volterra factors of Hilbert-Schmidt operators are then used to obtain convergence results for the controls, trajectories and feedback kernels. Two algorithms are derived from the basic approximation scheme, including a fast algorithm, in the time-invariant case. A numerical example is also considered.

Milman, Mark H.↗
NASA NTRS1988

Approximating the linear quadratic optimal control law for hereditary systems with delays in the control

The fundamental control synthesis issue of establishing a priori convergence rates of approximation schemes for feedback controllers for a class of distributed parameter systems is addressed within the context of hereditary schemes. Specifically, a factorization approach is presented for deriving approximations to the optimal feedback gains for the linear regulator-quadratic cost problem associated with time-varying functional differential equations with control delays. The approach is based on a discretization of the state penalty which leads to a simple structure for the feedback control law. General properties of the Volterra factors of Hilbert-Schmidt operators are then used to obtain convergence results for the controls, trajectories and feedback kernels. Two algorithms are derived from the basic approximation scheme, including a fast algorithm, in the time-invariant case. A numerical example is also considered.

Milman, Mark H.↗
NASA NTRS1993

Real-time dynamic simulation of the Cassini spacecraft using DARTS. Part 2: Parallel/vectorized real-time implementation

Part 1 of this paper presented the requirements for the real-time simulation of Cassini spacecraft along with some discussion of the DARTS algorithm. Here, in Part 2 we discuss the development and implementation of parallel/vectorized DARTS algorithm and architecture for real-time simulation. Development of the fast algorithms and architecture for real-time hardware-in-the-loop simulation of spacecraft dynamics is motivated by the fact that it represents a hard real-time problem, in the sense that the correctness of the simulation depends on both the numerical accuracy and the exact timing of the computation. For a given model fidelity, the computation should be computed within a predefined time period. Further reduction in computation time allows increasing the fidelity of the model (i.e., inclusion of more flexible modes) and the integration routine.

Fijany, A.↗
NASA NTRS1980

On the application of a fast polynomial transform and the Chinese remainder theorem to compute a two-dimensional convolution

A fast algorithm is developed to compute two dimensional convolutions of an array of d sub 1 X d sub 2 complex number points, where d sub 2 = 2(M) and d sub 1 = 2(m-r+) for some 1 or = r or = m. This algorithm requires fewer multiplications and about the same number of additions as the conventional fast fourier transform method for computing the two dimensional convolution. It also has the advantage that the operation of transposing the matrix of data can be avoided.

Truong, T. K.↗
NASA NTRS2015

Ultrafast High Accuracy PCRTM_SOLAR Model for Cloudy Atmosphere

An ultrafast high accuracy PCRTM_SOLAR model is developed based on PCA compression and principal component-based radiative transfer model (PCRTM). A fast algorithm for simulation of multi-scattering properties of cloud and/or aerosols is integrated into the fast infrared PCRTM. We completed radiance simulation and training for instruments, such as IASI, AIRS, CrIS, NASTI and SHIS, under diverse conditions. The new model is 5 orders faster than 52-stream DISORT with very high accuracy for cloudy sky radiative transfer simulation. It is suitable for hyperspectral remote data assimilation and cloudy sky retrievals.

Yang, Qiguang↗
NASA NTRS1992

Fast wavelet based algorithms for linear evolution equations

A class was devised of fast wavelet based algorithms for linear evolution equations whose coefficients are time independent. The method draws on the work of Beylkin, Coifman, and Rokhlin which they applied to general Calderon-Zygmund type integral operators. A modification of their idea is applied to linear hyperbolic and parabolic equations, with spatially varying coefficients. A significant speedup over standard methods is obtained when applied to hyperbolic equations in one space dimension and parabolic equations in multidimensions.

Engquist, Bjorn↗
NASA NTRS1996

Visualizing Vector Fields Using Line Integral Convolution and Dye Advection

We present local and global techniques to visualize three-dimensional vector field data. Using the Line Integral Convolution (LIC) method to image the global vector field, our new algorithm allows the user to introduce colored 'dye' into the vector field to highlight local flow features. A fast algorithm is proposed that quickly recomputes the dyed LIC images. In addition, we introduce volume rendering methods that can map the LIC texture on any contour surface and/or translucent region defined by additional scalar quantities, and can follow the advection of colored dye throughout the volume.

Shen, Han-Wei↗
NASA NTRS1997

Adaptation of a Fast Optimal Interpolation Algorithm to the Mapping of Oceangraphic Data

A fast, recently developed, multiscale optimal interpolation algorithm has been adapted to the mapping of hydrographic and other oceanographic data. This algorithm produces solution and error estimates which are consistent with those obtained from exact least squares methods, but at a small fraction of the computational cost. Problems whose solution would be completely impractical using exact least squares, that is, problems with tens or hundreds of thousands of measurements and estimation grid points, can easily be solved on a small workstation using the multiscale algorithm. In contrast to methods previously proposed for solving large least squares problems, our approach provides estimation error statistics while permitting long-range correlations, using all measurements, and permitting arbitrary measurement locations. The multiscale algorithm itself, published elsewhere, is not the focus of this paper. However, the algorithm requires statistical models having a very particular multiscale structure; it is the development of a class of multiscale statistical models, appropriate for oceanographic mapping problems, with which we concern ourselves in this paper. The approach is illustrated by mapping temperature in the northeastern Pacific. The number of hydrographic stations is kept deliberately small to show that multiscale and exact least squares results are comparable. A portion of the data were not used in the analysis; these data serve to test the multiscale estimates. A major advantage of the present approach is the ability to repeat the estimation procedure a large number of times for sensitivity studies, parameter estimation, and model testing. We have made available by anonymous Ftp a set of MATLAB-callable routines which implement the multiscale algorithm and the statistical models developed in this paper.

Menemenlis, Dimitris↗
NASA NTRS1989

A fast, preconditioned conjugate gradient Toeplitz solver

A simple factorization is given of an arbitrary hermitian, positive definite matrix in which the factors are well-conditioned, hermitian, and positive definite. In fact, given knowledge of the extreme eigenvalues of the original matrix A, an optimal improvement can be achieved, making the condition numbers of each of the two factors equal to the square root of the condition number of A. This technique is to applied to the solution of hermitian, positive definite Toeplitz systems. Large linear systems with hermitian, positive definite Toeplitz matrices arise in some signal processing applications. A stable fast algorithm is given for solving these systems that is based on the preconditioned conjugate gradient method. The algorithm exploits Toeplitz structure to reduce the cost of an iteration to O(n log n) by applying the fast Fourier Transform to compute matrix-vector products. Matrix factorization is used as a preconditioner.

Pan, Victor↗
NASA NTRS2009

High-Performance Algorithm for Solving the Diagnosis Problem

An improved method of model-based diagnosis of a complex engineering system is embodied in an algorithm that involves considerably less computation than do prior such algorithms. This method and algorithm are based largely on developments reported in several NASA Tech Briefs articles: The Complexity of the Diagnosis Problem (NPO-30315), Vol. 26, No. 4 (April 2002), page 20; Fast Algorithms for Model-Based Diagnosis (NPO-30582), Vol. 29, No. 3 (March 2005), page 69; Two Methods of Efficient Solution of the Hitting-Set Problem (NPO-30584), Vol. 29, No. 3 (March 2005), page 73; and Efficient Model-Based Diagnosis Engine (NPO-40544), on the following page. Some background information from the cited articles is prerequisite to a meaningful summary of the innovative aspects of the present method and algorithm. In model-based diagnosis, the function of each component and the relationships among all the components of the engineering system to be diagnosed are represented as a logical system denoted the system description (SD). Hence, the expected normal behavior of the engineering system is the set of logical consequences of the SD. Faulty components lead to inconsistencies between the observed behaviors of the system and the SD. Diagnosis the task of finding faulty components is reduced to finding those components, the abnormalities of which could explain all the inconsistencies. The solution of the diagnosis problem should be a minimal diagnosis, which is a minimal set of faulty components. The calculation of a minimal diagnosis is inherently a hard problem, the solution of which requires amounts of computation time and memory that increase exponentially with the number of components of the engineering system. Among the developments to reduce the computational burden, as reported in the cited articles, is the mapping of the diagnosis problem onto the integer-programming (IP) problem. This mapping makes it possible to utilize a variety of algorithms developed previously for IP to solve the diagnosis problem. In the IP approach, the diagnosis problem can be formulated as a linear integer optimization problem, which can be solved by use of well-developed integer-programming algorithms. This concludes the background information.

Fijany, Amir↗
NASA NTRS1996

Numerical Algorithms Based on Biorthogonal Wavelets

Wavelet bases are used to generate spaces of approximation for the resolution of bidimensional elliptic and parabolic problems. Under some specific hypotheses relating the properties of the wavelets to the order of the involved operators, it is shown that an approximate solution can be built. This approximation is then stable and converges towards the exact solution. It is designed such that fast algorithms involving biorthogonal multi resolution analyses can be used to resolve the corresponding numerical problems. Detailed algorithms are provided as well as the results of numerical tests on partial differential equations defined on the bidimensional torus.

Ponenti, Pj.↗
NASA NTRS1996

Numerical Algorithms Based on Biorthogonal Wavelets

Wavelet bases are used to generate spaces of approximation for the resolution of bidimensional elliptic and parabolic problems. Under some specific hypotheses relating the properties of the wavelets to the order of the involved operators, it is shown that an approximate solution can be built. This approximation is then stable and converges towards the exact solution. It is designed such that fast algorithms involving biorthogonal multi resolution analyses can be used to resolve the corresponding numerical problems. Detailed algorithms are provided as well as the results of numerical tests on partial differential equations defined on the bidimensional torus.

Ponenti, Pj.↗
NASA NTRS1990

Finding the complete path and weight enumerators of convolutional codes

A method for obtaining the complete path enumerator T(D, L, I) of a convolutional code is described. A system of algebraic equations is solved, using a new algorithm for computing determinants, to obtain T(D, L, I) for the (7,1/2) NASA standard code. Generating functions, derived from T(D, L, I) are used to upper bound Viterbi decoder error rates. This technique is currently feasible for constraint length K less than 10 codes. A practical, fast algorithm is presented for computing the leading nonzero coefficients of the generating functions used to bound the performance of constraint length K less than 20 codes. Code profiles with about 50 nonzero coefficients are obtained with this algorithm for the experimental K = 15, rate 1/4, code in the Galileo mission and for the proposed K = 15, rate 1/6, 2-dB code.

Onyszchuk, I.↗
NASA NTRS2000

IFSAR Simulation Using the Shooting and Bouncing Ray Technique

Interferometric Synthetic Aperture Radar (IFSAR) is a technique that allows an automated way to carry out terrain mapping. IFSAR is carried out by first generating a SAR image pair from two antennas that are spatially separated. The phase difference between the SAR image pair is proportional to the topography. After registering the SAR images, the difference in phase in each pixel is extracted to generate an interferogram. Since the phase can only be measured within 2pi radians, phase unwrapping is carried out to extract the absolute phase for each pixel that will be proportional to the local height. While IFSAR algorithm is typically applied to measurement data, it is useful to develop an IFSAR simulator to develop a better understanding of the IFSAR technique. The IFSAR simulator can be used in choosing system parameters, experimenting with processing procedures and mission planning. In this paper we will present an IFSAR simulation methodology to simulate the interferogram based on the shooting and bouncing ray (SBR) technique. SBR is a standard ray-tracing technique used to simulate scattering from large, complex targets. SBR is carried out by shooting rays at the target or scene. At the exit point of each ray, a ray-tube integration is done to find its contribution to the total field. A fast algorithm has been developed for the SBR for simulating SAR images of complex targets. In the IFSAR simulation, we build upon the fast SAR simulation technique. Given the antenna pair configuration, radar system parameters and the geometrical description of the scene, we first simulate two SAR images from each antenna. After post processing the two SAR images, we generate an interferogram. Phase unwrapping is then performed on the interferogram to arrive at the desired terrain map. We will present results from the SBR-based IFSAR simulator. The results will include terrain map reconstruction of urban environments. The reconstruction will be compared to the ground truth to examine the fidelity of the simulation. We will also investigate the effect of multi-bounce scattering in urban environments on phase unwrapping and reconstruction.

Houshmand, Bijan↗