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All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Multigrid Reduction in Time for Chaotic and Hyperbolic Problems (Final Report)

The coming massive parallelism of exascale computing presents a pressing challenge for the many DOE simulations of time-dependent partial differential equations (PDEs), which typically use traditional sequential time stepping methods. Since this traditional approach is inherently serial, it presents a sequential bottleneck when moving to exascale computing, because future performance gains will come through greater concurrency, not faster clock speeds. Thus, the goal of this work is to research parallelism in time, i.e., methods that compute multiple time values simultaneously, not sequentially. The focus will be on hyperbolic and chaotic problems of interest to DOE, with the goal of enabling scalable simulations of time-dependent hyperbolic and chaotic problems on future architectures. The chosen methodology for solving these problems parallel-in-time is multigrid, because multigrid (when it works) is a powerful, optimal, and scalable solver for discretized PDEs. Multigrid is already commonly used in many DOE simulations for scalably and optimally solving space-only PDE problems. The areas of hyperbolic and chaotic problems are chosen because of their relevance to problems of programmatic interest to DOE. However, these problems are also well-known to be difficult for parallel-in-time methods, with the most common method, parareal, diverging in many cases. The current state of-the-art for parallel-in-time at LLNL is the multigrid reduction in time (MGRIT) XBraid package, which also struggles for such problems, while still showing some improvement over parareal. In summary, new methods are needed for an efficient parallel-in-time scheme for hyperbolic and chaotic problems, and this work shall research promising new multigrid methods in this area. In particular, this work shall continue researching the directions from the current collaboration with Dr. Falgout, which are laid out in the work Toward Parallel in Time for Chaotic Dynamical Systems and showed the first known results of a parallel-in-time speedup for a chaotic problem. This work outlines two key improvements to XBraid for chaotic problems, the so-called “theta” and “delta-correction” methods. Here, these two improvements will be further researched and improved (including with a new relaxation method inspired by on Least Squares Shadowing (LSS)) and explored for more complicated problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Experimental and Numerical Study on the Effect of Nitric Oxide on Autoignition and Knock in a Direct-Injection Spark-Ignition Engine

We report Nitric Oxide (NO) can significantly influence the autoignition reactivity and this can affect knock limits in conventional stoichiometric SI engines. Previous studies also revealed that the role of NO changes with fuel type. Fuels with high RON (Research Octane Number) and high Octane Sensitivity (S = RON – MON (Motor Octane Number)) exhibited monotonically retarding knocklimited combustion phasing (KL-CA50) with increasing NO. In contrast, for a high-RON, low-S fuel, the addition of NO initially resulted in a strongly retarded KL-CA50 but beyond the certain amount of NO, KL-CA50 advanced again. The current study focuses on same high-RON, low-S Alkylate fuel to better understand the mechanisms responsible for the reversal in the effect of NO on KLCA50 beyond a certain amount of NO. Experiments were conducted to measure the responses of KL-CA50 and trace-autoignition CA50, the latter being indicative of CA50 at which end-gas autoignition starts to become measurable from the apparent heat-release rate. Chemical-kinetics simulations were conducted to reveal the role of NO for end-gas autoignition, with a specific focus on sequential autoignition in a thermally stratified end-gas. The simulation results reveal that the magnitude of low-temperature heat release (LTHR) generally increases with NO. However, the relative importance of NO for enhancing LTHR diminishes when the LTHR inherent to a fuel’s chemistry is strong, such as at lower temperatures in a thermal boundary layer. This rendered more uniform LTHR within a hypothetical thermal boundary and led to a more sequential (i.e. slower) autoignition event. It was also revealed that a change in compression ratio influences the importance of intermediate-temperature heat release (ITHR) due to changes of the temperature-pressure history of the end-gas. Together with the condition where end-gas autoignition occurs more sequentially, the shorter time spent in LTHR and ITHR regime can counter the increase in autoignition reactivity at high NO levels and allow KLCA50 to advance

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
NASA NTRS1971

High data rate coding for the space station telemetry links.

Coding systems for high data rates were examined from the standpoint of potential application in space-station telemetry links. Approaches considered included convolutional codes with sequential, Viterbi, and cascaded-Viterbi decoding. It was concluded that a high-speed (40 Mbps) sequential decoding system best satisfies the requirements for the assumed growth potential and specified constraints. Trade-off studies leading to this conclusion are viewed, and some sequential (Fano) algorithm improvements are discussed, together with real-time simulation results.

Lumb, D. R.↗
NASA NTRS1973

Detection, mapping and estimation of rate of spread of grass fires from southern African ERTS-1 imagery

Sequential band-6 imagery of the Zambesi Basin of southern Africa recorded substantial changes in burn patterns resulting from late dry season grass fires. One example from northern Botswana, indicates that a fire consumed approximately 70 square miles of grassland over a 24-hour period. Another example from western Zambia indicates increased fire activity over a 19-day period. Other examples clearly define the area of widespread grass fires in Angola, Botswana, Rhodesia and Zambia. From the fire patterns visible on the sequential portions of the imagery, and the time intervals involved, the rates of spread of the fires are estimated and compared with estimates derived from experimental burning plots in Zambia and Canada. It is concluded that sequential ERTS-1 imagery, of the quality studied, clearly provides the information needed to detect and map grass fires and to monitor their rates of spread in this region during the late dry season.

Wightman, J. M.↗
NASA NTRS1974

Navigation of the Mariner 10 spacecraft to Venus and Mercury

Orbit determination techniques used during the highly successful flight of Mariner 10 to Venus and Mercury are presented. Comparisons are made between different data sets, different sets of parameters, and between a conventional least squares batch filter and a sequential batch filter and smoother that was designed for this mission. The sequential filter was able to account for small spacecraft forces that the batch filter was unable to handle effectively, and hence, contributed greatly to the mission success. The sequential filter and smoother design is given as well as results for each phase of the mission.-

Christensen, C. S.↗
NASA NTRS1978

A single user efficiency measure for evaluation of parallel or pipeline computer architectures

A precise statement of the relationship between sequential computation at one rate, parallel or pipeline computation at a much higher rate, the data movement rate between levels of memory, the fraction of inherently sequential operations or data that must be processed sequentially, the fraction of data to be moved that cannot be overlapped with computation, and the relative computational complexity of the algorithms for the two processes, scalar and vector, was developed. The relationship should be applied to the multirate processes that obtain in the employment of various new or proposed computer architectures for computational aerodynamics. The relationship, an efficiency measure that the single user of the computer system perceives, argues strongly in favor of separating scalar and vector processes, sometimes referred to as loosely coupled processes, to achieve optimum use of hardware.

Jones, W. P.↗
NASA NTRS1980

A generalized formulation of minimum shift keying modulation

The modulator structure consisting of a single-input two-output sequential transducer whose output sequences control the modulated signal in the manner s(t) = x(i)p(t) + y(i)q(t) is considered. Here, iT is less than or equal to t, which is less than iT + T, where T is the modulation bit duration and where x(i) and y(i) take values in /+1, -1, 0/. The behavior of a sequential transducer is characterized by means of a 'trellis'. It is shown that, provided only that the 'carriers' p(t) and q(t) have the same energy in every bit interval, the optimum demodulator for the additive white Gaussian noise channel need process the received signal over only two bit intervals in making each data bit decision. It is demonstrated that particular choices of the sequential transducer yield differential minimum shift keying and differential staggered quadriphase shift keying. Possible generalizations of this approach are discussed.

Massey, J. L.↗
NASA NTRS1981

Comparison between variable and fixed dwell-time PN acquisition algorithms

Pseudo noise (PN) spread spectrum systems require a very accurate alignment between the PN code epochs at the transmitter and receiver. This synchronism is typically established through a two-step algorithm, including a coarse synchronization procedure and a fine synchronization procedure. A standard approach for the coarse synchronization is a sequential search over all code phases. The measurement of the power in the filtered signal is used to either accept or reject the code phase under test as the phase of the received PN code. This acquisition strategy, called a single dwell-time system, has been analyzed by Holmes and Chen (1977). A synopsis of the field of sequential analysis as it applies to the PN acquisition problem is provided. From this, the implementation of the variable dwell time algorithm as a sequential probability ratio test is developed. The performance of this algorithm is compared to the optimum detection algorithm and to the fixed dwell-time system.

Braun, W. R.↗
NASA NTRS1984

Flat-plate solar array project process development area, process research of non-CZ silicon material

The program is designed to investigate the fabrication of solar cells on N-type base material by a simultaneous diffusion of N-type and P-type dopants to form an P(+)NN(+) structure. The results of simultaneous diffusion experiments are being compared to cells fabricated using sequential diffusion of dopants into N-base material in the same resistivity range. The process used for the fabrication of the simultaneously diffused P(+)NN(+) cells follows the standard Westinghouse baseline sequence for P-base material except that the two diffusion processes (boron and phosphorus) are replaced by a single diffusion step. All experiments are carried out on N-type dendritic web grown in the Westinghouse pre-pilot facility. The resistivities vary from 0.5 (UC OMEGA)cm to 5 (UC OMEGA)cm. The dopant sources used for both the simultaneous and sequential diffusion experiments are commercial metallorganic solutions with phosphorus or boron components. After these liquids are applied to the web surface, they are baked to form a hard glass which acts as a diffusion source at elevated temperatures. In experiments performed thus far, cells produced in sequential diffusion tests have properties essentially equal to the baseline N(+)PP(+) cells. However, the simultaneous diffusions have produced cells with much lower IV characteristics mainly due to cross-doping of the sources at the diffusion temperature. This cross-doping is due to the high vapor pressure phosphorus (applied as a metallorganic to the back surface) diffusion through the SiO2 mask and then acting as a diffusant source for the front surface.

Campbell, R. B.↗
NASA NTRS1985

Flow rate and pressure profiles for 1 to 4 axially alined orifice inlets

Choked flow rate and pressure profile data were taken on sequential axially alined inlets of the orifice type, with an orifice length-to-diameter ratio of 0.5. The configuration consisted of two to four inlets spaced at 0.66 and 32 orifice diameters apart. At a spacing of 32 diameters the reduced flow rate appeared to follow the simple power-law relation G(sub r)/G(sub r,1) = N (sup-b), where G(sub r,1) is the reduced flow rate for a single inlet, N is the number of inlets, and b, although temperature dependent, is approximately 0.4. At this spacing the instrumented orifices and spacers gave pressure profiles that dropped sharply at the entrance and partially recovered within each inlet, somewhat independent of N. At low inlet temperature jetting through the last orifice was common. At a spacing of 0.66 diameter fluid jetting through all N inlets was prevalent at low temperatures for each configuration studied, as indicated by the flat pressure profiles and flow rates that were nearlly identical to those for a single orifice inlet. A simplifying relation was developed between the friction loss parameters for flow through N sequential tubes and N sequential inlets. The predicted flow rates for N tubes were in reasonable agreement with the N inlet analysis and followed the simple power-law relation.

Hendricks, R. C.↗
NASA NTRS1983

Reduction of display artifacts by random sampling

The application of random-sampling techniques to remove visible artifacts (such as flicker, moire patterns, and paradoxical motion) introduced in TV-type displays by discrete sequential scanning is discussed and demonstrated. Sequential-scanning artifacts are described; the window of visibility defined in spatiotemporal frequency space by Watson and Ahumada (1982 and 1983) and Watson et al. (1983) is explained; the basic principles of random sampling are reviewed and illustrated by the case of the human retina; and it is proposed that the sampling artifacts can be replaced by random noise, which can then be shifted to frequency-space regions outside the window of visibility. Vertical sequential, single-random-sequence, and continuously renewed random-sequence plotting displays generating 128 points at update rates up to 130 Hz are applied to images of stationary and moving lines, and best results are obtained with the single random sequence for the stationary lines and with the renewed random sequence for the moving lines.

Ahumada, A. J., Jr.↗
NASA NTRS1985

The star-formation history of very young clusters

The popular idea that star formation has proceeded sequentially from lowest to highest mass members in open clusters is examined critically. For extremely young clusters, such as NGC 2264 and NGC 6530, this sequential hypothesis is a consequence of the assignment of pre-main-sequence contraction ages to all member stars. However, such ages yield a formation history which is implausible from a physical point of view, since the critical time for the onset of formation at any stellar mass is equal to the pre-main-sequence contraction time for that mass. Moreover, these ages are in conflict with the strong observational evidence that a substantial fraction of cluster members have already reached the main sequence. After reconsideration of the probable main-sequence members, the stellar ages in NGC 2264 and NGC 6530 are consistent with a variety of formation histories, and, in particular, with the view that all stellar masses form in approximately the same interval of time within a given cluster, i.e., that there is no mass-age correlation. A notion closely related to the sequential hypothesis, that the total star-formation rate increases exponentially with time, is subject to the same criticism.

Stahler, S. W.↗
NASA NTRS1986

Multigrid methods in structural mechanics

Although the application of multigrid methods to the equations of elasticity has been suggested, few such applications have been reported in the literature. In the present work, multigrid techniques are applied to the finite element analysis of a simply supported Bernoulli-Euler beam, and various aspects of the multigrid algorithm are studied and explained in detail. In this study, six grid levels were used to model half the beam. With linear prolongation and sequential ordering, the multigrid algorithm yielded results which were of machine accuracy with work equivalent to 200 standard Gauss-Seidel iterations on the fine grid. Also with linear prolongation and sequential ordering, the V(1,n) cycle with n greater than 2 yielded better convergence rates than the V(n,1) cycle. The restriction and prolongation operators were derived based on energy principles. Conserving energy during the inter-grid transfers required that the prolongation operator be the transpose of the restriction operator, and led to improved convergence rates. With energy-conserving prolongation and sequential ordering, the multigrid algorithm yielded results of machine accuracy with a work equivalent to 45 Gauss-Seidel iterations on the fine grid. The red-black ordering of relaxations yielded solutions of machine accuracy in a single V(1,1) cycle, which required work equivalent to about 4 iterations on the finest grid level.

Raju, I. S.↗
NASA NTRS1989

Multiplier-continuation algorthms for constrained optimization

Several path following algorithms based on the combination of three smooth penalty functions, the quadratic penalty for equality constraints and the quadratic loss and log barrier for inequality constraints, their modern counterparts, augmented Lagrangian or multiplier methods, sequential quadratic programming, and predictor-corrector continuation are described. In the first phase of this methodology, one minimizes the unconstrained or linearly constrained penalty function or augmented Lagrangian. A homotopy path generated from the functions is then followed to optimality using efficient predictor-corrector continuation methods. The continuation steps are asymptotic to those taken by sequential quadratic programming which can be used in the final steps. Numerical test results show the method to be efficient, robust, and a competitive alternative to sequential quadratic programming.

Lundberg, Bruce N.↗
NASA NTRS1989

Concurrent processing simulation of the space station

The development of a new capability for the time-domain simulation of multibody dynamic systems and its application to the study of a large angle rotational maneuvers of the Space Station is described. The effort was divided into three sequential tasks, which required significant advancements of the state-of-the art to accomplish. These were: (1) the development of an explicit mathematical model via symbol manipulation of a flexible, multibody dynamic system; (2) the development of a methodology for balancing the computational load of an explicit mathematical model for concurrent processing; and (3) the implementation and successful simulation of the above on a prototype Custom Architectured Parallel Processing System (CAPPS) containing eight processors. The throughput rate achieved by the CAPPS operating at only 70 percent efficiency, was 3.9 times greater than that obtained sequentially by the IBM 3090 supercomputer simulating the same problem. More significantly, analysis of the results leads to the conclusion that the relative cost effectiveness of concurrent vs. sequential digital computation will grow substantially as the computational load is increased. This is a welcomed development in an era when very complex and cumbersome mathematical models of large space vehicles must be used as substitutes for full scale testing which has become impractical.

Gluck, R.↗
NASA NTRS1992

Computational aspects of helicopter trim analysis and damping levels from Floquet theory

Helicopter trim settings of periodic initial state and control inputs are investigated for convergence of Newton iteration in computing the settings sequentially and in parallel. The trim analysis uses a shooting method and a weak version of two temporal finite element methods with displacement formulation and with mixed formulation of displacements and momenta. These three methods broadly represent two main approaches of trim analysis: adaptation of initial-value and finite element boundary-value codes to periodic boundary conditions, particularly for unstable and marginally stable systems. In each method, both the sequential and in-parallel schemes are used and the resulting nonlinear algebraic equations are solved by damped Newton iteration with an optimally selected damping parameter. The impact of damped Newton iteration, including earlier-observed divergence problems in trim analysis, is demonstrated by the maximum condition number of the Jacobian matrices of the iterative scheme and by virtual elimination of divergence. The advantages of the in-parallel scheme over the conventional sequential scheme are also demonstrated.

Gaonkar, Gopal H.↗
NASA NTRS1993

Helicopter trim analysis by shooting and finite element methods with optimally damped Newton iterations

Helicopter trim settings of periodic initial state and control inputs are investigated for convergence of Newton iteration in computing the settings sequentially and in parallel. The trim analysis uses a shooting method and a weak version of two temporal finite element methods with displacement formulation and with mixed formulation of displacements and momenta. These three methods broadly represent two main approaches of trim analysis: adaptation of initial-value and finite element boundary-value codes to periodic boundary conditions, particularly for unstable and marginally stable systems. In each method, both the sequential and in-parallel schemes are used, and the resulting nonlinear algebraic equations are solved by damped Newton iteration with an optimally selected damping parameter. The impact of damped Newton iteration, including earlier-observed divergence problems in trim analysis, is demonstrated by the maximum condition number of the Jacobian matrices of the iterative scheme and by virtual elimination of divergence. The advantages of the in-parallel scheme over the conventional sequential scheme are also demonstrated.

Achar, N. S.↗
NASA NTRS1993

Efficient partitioning and assignment on programs for multiprocessor execution

The general problem studied is that of segmenting or partitioning programs for distribution across a multiprocessor system. Efficient partitioning and the assignment of program elements are of great importance since the time consumed in this overhead activity may easily dominate the computation, effectively eliminating any gains made by the use of the parallelism. In this study, the partitioning of sequentially structured programs (written in FORTRAN) is evaluated. Heuristics, developed for similar applications are examined. Finally, a model for queueing networks with finite queues is developed which may be used to analyze multiprocessor system architectures with a shared memory approach to the problem of partitioning. The properties of sequentially written programs form obstacles to large scale (at the procedure or subroutine level) parallelization. Data dependencies of even the minutest nature, reflecting the sequential development of the program, severely limit parallelism. The design of heuristic algorithms is tied to the experience gained in the parallel splitting. Parallelism obtained through the physical separation of data has seen some success, especially at the data element level. Data parallelism on a grander scale requires models that accurately reflect the effects of blocking caused by finite queues. A model for the approximation of the performance of finite queueing networks is developed. This model makes use of the decomposition approach combined with the efficiency of product form solutions.

Standley, Hilda M.↗