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NASA NTRS2001

Monitoring Java Programs with Java PathExplorer

We present recent work on the development Java PathExplorer (JPAX), a tool for monitoring the execution of Java programs. JPAX can be used during program testing to gain increased information about program executions, and can potentially furthermore be applied during operation to survey safety critical systems. The tool facilitates automated instrumentation of a program's late code which will then omit events to an observer during its execution. The observer checks the events against user provided high level requirement specifications, for example temporal logic formulae, and against lower level error detection procedures, for example concurrency related such as deadlock and data race algorithms. High level requirement specifications together with their underlying logics are defined in the Maude rewriting logic, and then can either be directly checked using the Maude rewriting engine, or be first translated to efficient data structures and then checked in Java.

Havelund, Klaus↗

NASA NTRS2002

Program Instrumentation and Trace Analysis

Several attempts have been made recently to apply techniques such as model checking and theorem proving to the analysis of programs. This shall be seen as a current trend to analyze real software systems instead of just their designs. This includes our own effort to develop a model checker for Java, the Java PathFinder 1, one of the very first of its kind in 1998. However, model checking cannot handle very large programs without some kind of abstraction of the program. This paper describes a complementary scalable technique to handle such large programs. Our interest is turned on the observation part of the equation: How much information can be extracted about a program from observing a single execution trace? It is our intention to develop a technology that can be applied automatically and to large full-size applications, with minimal modification to the code. We present a tool, Java PathExplorer (JPaX), for exploring execution traces of Java programs. The tool prioritizes scalability for completeness, and is directed towards detecting errors in programs, not to prove correctness. One core element in JPaX is an instrumentation package that allows to instrument Java byte code files to log various events when executed. The instrumentation is driven by a user provided script that specifies what information to log. Examples of instructions that such a script can contain are: 'report name and arguments of all called methods defined in class C, together with a timestamp'; 'report all updates to all variables'; and 'report all acquisitions and releases of locks'. In more complex instructions one can specify that certain expressions should be evaluated and even that certain code should be executed under various conditions. The instrumentation package can hence be seen as implementing Aspect Oriented Programming for Java in the sense that one can add functionality to a Java program without explicitly changing the code of the original program, but one rather writes an aspect and compiles it into the original program using the instrumentation. Another core element of JPaX is an observation package that supports the analysis of the generated event stream. Two kinds of analysis are currently supported. In temporal analysis the execution trace is evaluated against formulae written in temporal logic. We have implemented a temporal logic evaluator on finite traces using the Maude rewriting system from SRI International, USA. Temporal logic is defined in Maude by giving its syntax as a signature and its semantics as rewrite equations. The resulting semantics is extremely efficient and can handle event streams of hundreds of millions events in few minutes. Furthermore, the implementation is very succinct. The second form of even stream analysis supported is error pattern analysis where an execution trace is analyzed using various error detection algorithms that can identify error-prone programming practices that may potentially lead to errors in some different executions. Two such algorithms focusing on concurrency errors have been implemented in JPaX, one for deadlocks and the other for data races. It is important to note, that a deadlock or data race potential does not need to occur in order for its potential to be detected with these algorithms. This is what makes them very scalable in practice. The data race algorithm implemented is the Eraser algorithm from Compaq, however adopted to Java. The tool is currently being applied to a code base for controlling a spacecraft by the developers of that software in order to evaluate its applicability.

Havelund, Klaus↗

NASA NTRS2002

A Rewriting-Based Approach to Trace Analysis

We present a rewriting-based algorithm for efficiently evaluating future time Linear Temporal Logic (LTL) formulae on finite execution traces online. While the standard models of LTL are infinite traces, finite traces appear naturally when testing and/or monitoring red applications that only run for limited time periods. The presented algorithm is implemented in the Maude executable specification language and essentially consists of a set of equations establishing an executable semantics of LTL using a simple formula transforming approach. The algorithm is further improved to build automata on-the-fly from formulae, using memoization. The result is a very efficient and small Maude program that can be used to monitor program executions. We furthermore present an alternative algorithm for synthesizing probably minimal observer finite state machines (or automata) from LTL formulae, which can be used to analyze execution traces without the need for a rewriting system, and can hence be used by observers written in conventional programming languages. The presented work is part of an ambitious runtime verification and monitoring project at NASA Ames, called PATHEXPLORER, and demonstrates that rewriting can be a tractable and attractive means for experimenting and implementing program monitoring logics.

Havelund, Klaus↗

NASA NTRS2000

Upper Eocene Spherules at ODP Site 1090B

Our two labs independently discovered upper Eocene microtektites and microkrystites at ODP Site 1090, a new South Atlantic locality near the Agulhus Ridge. This is a significant new data point for the strewn fields of these spherules, which were recently extended into the Atlantic sector of the Southern Ocean when they were reported at ODP Site 689 on the Maude Rise. The microtektites have been regarded as related to North American tektites and the microkrystites as belonging to the clinopyroxene-bearing (cpx) spherule strewn field. Initial reports indicate that Site 1090 contains a complete sequence of upper Eocene sediments composed of diatom and nannofossil oozes. The magneto- and bio-stratigraphy indicate that impact-age sediments should occur in core 30X of Hole 1090B. One of us (FTK) took 2 cc samples at 10 cm intervals over 600 cm of core for Ir analyses and the senior author (SL) took 3 cc samples at 20 cm intervals to search for spherules. Both studies proved successful and additional samples were obtained to confirm initial results and better define the Ir anomaly and spherule abundances. Peak Ir concentrations of 0.97 ng/g were found at 1090B-30X-5, 105-106cm and 0.78 ng/g at 115-116 cm. Anomalous Ir concentrations (greater than 0.1 ng/g) extend over about 100 cm of core. Preliminary results indicate that the excess Ir at this site is about 25 ng per sq cm. About 380 microtektites (>63 pm) and 2492 microkrystites (>63 pm) were recovered over a 1.8 m interval with a peak abundance of microtektites (106/gram) and microkrystites (562/gram) at 1090B-30X- 5, 114-115 cm. The largest microtektite is approximately 960 x 1140 micron in size. About 55 % are spherical, and the rest are disc, cylinder, dumbbell, teardrop, or fragments. Most of the microtektites are transparent colorless, but a few are transparent pale brown or green. Preliminary data indicate that the microtektites at Site 1090 have similar major oxide compositions to those at Site 689. About 50% of the cpx spherules are spherical, the rest are fragments. They range from yellowish translucent to dark opaque. Based on stratigraphic data, occurrence of a positive Ir anomaly, and similar appearance and major oxide compositions of the Site 1090 spherules to those at Site 689, we believe that the spherule layer(s) are the same at both sites. However, there are significant differences between these two sites. At Site 689 the peak abundance of the cpx spherules is slightly below that of the microtektites. We can distinguish no such separation at Site 1090, despite a somewhat higher sediment accumulation rates. Peak Ir concentrations are about four times higher at Site 1090, resulting in a similarly higher total flux of Ir for this site compared to Site 689 on the Maude Rise. This is generally consistent with the flux of cpx spherules (greater than 63 pm) which is estimated to be approximately -2100 per sq cm, about twice that of Site 689D (1040 sq cm) and with the cpx-spherule component being a principal carrier of the Ir signal. The number of microtektites per sq cm (greater than 63 micron) is nearly the same at both sites, approximately 300 in 1090B and 280 in 689B. A decrease in the greater than 63 micron size fraction (consisting primarily of diatoms and radiolaria) is coincident with the spherule layer at both sites suggesting that the impact(s) that produced the spherule layer(s) may have had an adverse affect on the marine plankton.

Liu, S.↗

NASA NTRS2011

Simulation and Verification of Synchronous Set Relations in Rewriting Logic

This paper presents a mathematical foundation and a rewriting logic infrastructure for the execution and property veri cation of synchronous set relations. The mathematical foundation is given in the language of abstract set relations. The infrastructure consists of an ordersorted rewrite theory in Maude, a rewriting logic system, that enables the synchronous execution of a set relation provided by the user. By using the infrastructure, existing algorithm veri cation techniques already available in Maude for traditional asynchronous rewriting, such as reachability analysis and model checking, are automatically available to synchronous set rewriting. The use of the infrastructure is illustrated with an executable operational semantics of a simple synchronous language and the veri cation of temporal properties of a synchronous system.

Rocha, Camilo↗

NASA NTRS1988

Mass extinctions in the deep sea

The character of mass extinctions can be assessed by studying extinction patterns of organisms, the fabric of the extinction, and assessing the environmental niche and mode of life of survivors. Deep-sea benthic foraminifera have been listed as little affected by the Cretaceous-Tertiary (K-T) mass extinction, but very few quantitative data are available. New data on deep-sea Late Maestrichtian-Eocene benthic foraminifera from Maud Rise (Antractica) indicate that about 10 percent of the species living at depths of 2000 to 2500 m had last appearances within 1 my of the Cretaceous-Tertiary (K-T) boundary, versus about 25 percent of species at 1000 to 1500 m. Many survivors from the Cretaceous became extinct in a period of global deep-sea benthic foraminiferal extinction at the end of the Paleocene, a time otherwise marked by very few extinctions. Preliminary conclusions suggest that the deep oceanic environment is essentially decoupled from the shallow marine and terrestrial environment, and that even major disturbances of one of these will not greatly affect the other. This gives deep-sea benthic faunas a good opportunity to recolonize shallow environments from greater depths and vice versa after massive extinctions. The decoupling means that data on deep-sea benthic boundary was caused by the environmental effects of asteriod impact or excessive volcanism. The benthic foraminiferal data strongly suggest, however, that the environmental results were strongest at the Earth's surface, and that there was no major disturbance of the deep ocean; this pattern might result both from excessive volcanism and from an impact on land.

Thomas, E.↗

NASA NTRS1990

Spatial cross-correlation of Antarctic Sea ice and seabed topography

A time series of derived sea ice concentrations as observed about Antarctica by the Nimbus-7 Scanning Multichannel Microwave Radiometer (SMMR) satellite in 1983 is considered. The degree of spatial cross correlation between these data and seabed topography is quantified. The approach is to implement a statistical image processing filter designed to extract local patterns of spatial cross correlation over the entire sea ice field as it undergoes daily changes. Throughout the sea ice, it was found that large scale variations in sea ice concentration correlate systematically with variations in the topography of the seabed. Generally speaking, high concentrations of sea ice occur over deep ocean, whereas areas of encavement, early dissipation and polynya formation develop over topographic features of high elevation. The latter was studied in detail with respect to the features Maud Rise, Astrid Ridge and the continental shelf in the Cosmonaut and Ross Seas. In each case, it is shown that an encavement in sea ice, a polynya, or both develops in the vicinity of the feature in question. As these results are quantified in terms of spatial cross correlation, a potential role is inferred for seabed topography in such fluctuations in the sea ice about Antarctica.

Deveaux, Richard D.↗

NASA NTRS1991

Fine structure of the late Eocene Ir anomaly in marine sediments

The Late Eocene Ir abundance profile in deep sea cores from Ocean Drilling Program Leg 113 Hole 689B on the Maude Rise off of Antarctica was studied with 410 samples continuously in 10 cm increments and measured with the Iridium Coincidence (ICS). The ICS was subsequently modified to measure 13 other elements simultaneously with the Ir. The abundance profiles of these elements were then determined in the Late Eocene Massignano section in central Italy with 250 samples (encompassing roughly 2 million years of accumulation) which were collected about every 5 cm in about 2 cm increments. These studies augmented a previous one (which included many elements) of deep sea cores from Deep Sea Drilling Project Site 592 on the Lord Howe Rise in the Tasman Sea between Australia and New Zealand. In the latter study, 50 samples (encompassing roughly 0.7 million years of accumulation) were collected continuously in 10 cm increments. The results from these studies are discussed.

Asaro, F.↗

NASA NTRS1989

Antarctic carbonaceous chondrites - New opportunities for research

An account is given of the types of carbonaceous meteorites available in the Antarctic collections of the U.S. and Japan. In the case of the collection for Victoria Land and Queen Maud Land, all known classes for meteorites except C1 are present; available pairing data, though limited, are indicative of the presence of many different falls. Thus far, attention has been focused on the largest meteorites. Most samples, however, are small.

Mcsween, Harry Y., Jr.↗

NASA NTRS1993

Geochemical evidences for two chondritic-like cometary or asteroidal impacts before and at the K/T boundary

A number of geological and palaeontological evidences support multiple impacts of cometary showers within a short time (approximately 1-3 Ma) and their connection with mass extinctions. Observations include clustered crater ages, stratigraphic horizons of impact ejecta closely spaced in time, and evidence for stepwise mass extinctions spanning intervals of 1-3 Ma. For the K/T boundary, three candidates, Popigai, Manson, and Yucatan, have been proposed as impact craters. Two distinct strata at the K/T boundary in western North America have been interpreted as evidence for two sequential impacts. If multiple impacts occurred within a time span of about 1 Ma then multiple Ir enrichments should be observed. DSDP Hole 577B on the Shatsky Plateau in the northern Pacific at K/T time is the first site. Samples contain approximately greater than 97 percent CaCO3, which exhibit clear chemical signals associated with asteroidal/cometary impact. Ir, Fe, and Cr data are presented. From the Th-normalized data, two satellite peaks below the major peak at 78 cm and 81 cm of 577B-1-4 are clearly shown. The major Ir peak (K/T boundary) is at 72 cm. Fe and Cr, from C1-like impactor ejecta fallout, also show two peaks at the same positions. For hole 738C on the southern Kerguelen Plateau, Ir values reach a peak concentration of 18 ppb in the clay layer at 96.0-96.2 cm in section 20R-5, and gradually tail off. In the sample 115 cm above the boundary, Ir concentrations have still not reached background levels. From the Ir peak downward to the lowermost sample analyzed at 102 cm, the Ir concentration is still as high as 1.7 ppb. From the Th-normalized data, we observe a small Ir/Th peak at 100-101 cm. Though this peak is within the error margin, the trend is clear. Fe and Cr exhibit the same pattern. The third case is Hole 690C on the Queen Maud Ridge. Again, the Ir/Th plot indicates the strong possibility of satellite peaks at approximately 52 cm. The main peak is at 39-40 cm. For the Stevns Klint K/T boundary layers, the stratification of trace elements appears threefold with peak concentrations in sublayers A1, A3, and B2 for different element groups, including Ir. C1 ratios for many siderophile elements found in combined layers III and IV, corresponding to layers A, B, C, and D, strongly support the impact hypothesis. Also, multiple Ir anomalies in the K/T section at Lattengebirge, Bavarian Alps are reported. Recent works on Ni-rich spinels and Ir at the K/T boundaries clearly establish cometary/asteroidal impacts at the K/T boundary. Lastly, cometary showers can explain the enhanced Ir contents over approximately a 1 Ma interval in Gubbio shales.

Liu, Y.-G.↗

NASA NTRS1995

Chemical studies of H chondrites. 6: Antarctic/non-Antarctic compositional differences revisited

We report data for the trace elements Au, Co, Sb, Ga, Rb, Ag, Se, Cs, Te, Zn, Cd, Bi, T1, and In (ordered by putative volatility during nebular condensation and accretion) determined by radiochemical neutron activation analysis of 14 additional H5 and H6 chondrite falls. Data for the 10 most volatile elements (Rb to In) treated by the multivariate techniques of linear discriminant analysis and logistic regression in these and 44 other falls are compared with those of 59 H4-6 chondrites from Antarctica. Various populations are tested by the multivariate techniques, using the previously developed method of randomization-simulation to assess significance levels. An earlier conclusion, based on fewer examples, that H4-6 chondrite falls are compositionally distinguishable from the Antarctic suite is verified by the additional data. This distinctiveness is highly significant because of the presence of samples from Victoria Land in the Antarctic population, which differ compositionally from falls beyond any reasonable doubt. However, it cannot be proven unequivocally that falls and Antarctic samples from Queen Maud Land are compositionally distinguishable. Trivial causes (e.g., analyst bias, weathering) cannot explain the Victoria Land (Antarctic)/non-Antarctic compositional difference for paradigmatic H4-6 chondrites. This seems to reflect a time-dependent variation of near-Earth meteoroid source regions differing in average thermal history.

Wolf, Stephen F.↗

NASA NTRS2001

Testing Linear Temporal Logic Formulae on Finite Execution Traces

We present an algorithm for efficiently testing Linear Temporal Logic (LTL) formulae on finite execution traces. The standard models of LTL are infinite traces, reflecting the behavior of reactive and concurrent systems which conceptually may be continuously alive. In most past applications of LTL. theorem provers and model checkers have been used to formally prove that down-scaled models satisfy such LTL specifications. Our goal is instead to use LTL for up-scaled testing of real software applications. Such tests correspond to analyzing the conformance of finite traces against LTL formulae. We first describe what it means for a finite trace to satisfy an LTL property. We then suggest an optimized algorithm based on transforming LTL formulae. The work is done using the Maude rewriting system. which turns out to provide a perfect notation and an efficient rewriting engine for performing these experiments.

Havelund, Klaus↗

NASA NTRS2001

Motion of Major Ice Shelf Fronts in Antarctica from Slant Range Analysis of Radar Altimeter Data, 1978 - 1998

Slant range analysis of radar altimeter data from the Seasat, Geosat, ERS-1 and ERS-2 databases are used to determine barrier location at particular times, and estimate barrier motion (km/yr) for major Antarctic ice shelves. The barrier locations, which are the seaward edges or fronts of floating ice shelves, advance with time as the ice flows from the grounded ice sheets and retreat whenever icebergs calve from the fronts. The analysis covers various multiyear intervals from 1978 to 1998, supplemented by barrier location maps produced elsewhere for 1977 and 1986. Barrier motion is estimated as the ratio between mean annual ice shelf area change for a particular interval, and the length of the discharge periphery. This value is positive if the barrier location progresses seaward, or negative if the barrier location regresses (break-back). Either positive or negative values are lower limit estimates because the method does not detect relatively small area changes due to calving or surge events. The findings are discussed in the context of the three ice shelves that lie in large embayments (the Filchner-Ronne, Amery, and Ross), and marginal ice shelves characterized by relatively short distances between main segments of grounding line and barrier (those in the Queen Maud Land sector between 10.1 deg. W and 32.5 deg. E, and the West and Shackleton ice shelves). All the ice shelves included in the study account for approximately three-fourths of the total ice shelf area of Antarctica, and discharge approximately two-thirds of the total grounded ice area.

Zwally, H. J.↗

NASA NTRS2001

Monitoring Programs Using Rewriting

We present a rewriting algorithm for efficiently testing future time Linear Temporal Logic (LTL) formulae on finite execution traces, The standard models of LTL are infinite traces, reflecting the behavior of reactive and concurrent systems which conceptually may be continuously alive in most past applications of LTL, theorem provers and model checkers have been used to formally prove that down-scaled models satisfy such LTL specifications. Our goal is instead to use LTL for up-scaled testing of real software applications, corresponding to analyzing the conformance of finite traces against LTL formulae. We first describe what it means for a finite trace to satisfy an LTL property end then suggest an optimized algorithm based on transforming LTL formulae. We use the Maude rewriting logic, which turns out to be a good notation and being supported by an efficient rewriting engine for performing these experiments. The work constitutes part of the Java PathExplorer (JPAX) project, the purpose of which is to develop a flexible tool for monitoring Java program executions.

Havelund, Klaus↗

NASA NTRS2001

Identification of Late Eocene Impact Deposits at ODP Site 1090

Anomalous concentrations of Ir have been found in upper Eocene sediments from Ocean Drilling Program (ODP) Hole 1090B. Clear and dark-colored spherules that are believed to be microtektites and clinopyroxene- bearing microkrystites, respectively, were found in the samples with highest Ir. The peak Ir concentration in Sample 177- 1090B-30X-5,105-106 cm (954 pg/g) and the net Ir fluence (14 ng/cm2) at this site are higher that at most other localities except for Caribbean site RC9-58. The Ir anomaly and impact debris are probably correlative with similar deposits found at ODP Site 689 on the Maude Rise and at other localities around the world.

Kyte, Frank T.↗

NASA NTRS2003

Reliability of CHAMP Anomaly Continuations

CHAMP is recording state-of-the-art magnetic and gravity field observations at altitudes ranging over roughly 300 - 550 km. However, anomaly continuation is severely limited by the non-uniqueness of the process and satellite anomaly errors. Indeed, our numerical anomaly simulations from satellite to airborne altitudes show that effective downward continuations of the CHAMP data are restricted to within approximately 50 km of the observation altitudes while upward continuations can be effective over a somewhat larger altitude range. The great unreliability of downward continuation requires that the satellite geopotential observations must be analyzed at satellite altitudes if the anomaly details are to be exploited most fully. Given current anomaly error levels, joint inversion of satellite and near- surface anomalies is the best approach for implementing satellite geopotential observations for subsurface studies. We demonstrate the power of this approach using a crustal model constrained by joint inversions of near-surface and satellite magnetic and gravity observations for Maude Rise, Antarctica, in the southwestern Indian Ocean. Our modeling suggests that the dominant satellite altitude magnetic anomalies are produced by crustal thickness variations and remanent magnetization of the normal polarity Cretaceous Quiet Zone.

vonFrese, Ralph R. B.↗

NASA NTRS2010

Study of the Microbial Diversity of a Newly Discovered East Antarctic Freshwater Lake, L27C, and of a Perennially Ice-Covered Lake Untersee

The microbial communities that reside within freshwater lakes of Schirmacher and Untersee Oases in East Antarctica must cope with extreme conditions that may include cold temperature, annual freeze-thaw cycles, exposure to UV radiation, especially during the austral summer months, low light beneath thick ice-cover, followed by seasonal darkness. The objective of this study was to assess the microbial biodiversity and distribution from samples taken from two freshwater lakes (L27C and Lake Untersee) that were collected during the Tawani 2008 International Antarctic Expedition that conducted research in this region of Antarctica. L27C is a small, previously unreported lake residing 2 km WNW of Maitri Station at Schirmacher Oasis. Biodiversity and distribution of microorganisms within the lake were studied using both culture-independent and culture-dependent methodologies based upon the analysis of eubacterial 16S rRNA gene sequences. Lake Untersee, a perennially ice-covered, ultra-oligotrophic, lake in the Otto-von-Gruber-Gebirge (Gruber Mountains) of central Dronning Maud Land was also sampled and the microbial diversity was analyzed by eubacterial 16S rRNA gene sequences derived from pure cultures. Direct culturing of water samples from each lake on separate R2A growth medium exhibited a variety of microorganisms including: Janthinobacterium, Hymenobacter, Sphingamonas, Subtercola, Deinococcus, Arthrobacter, Flavobacterium, Polaromonas, Rhodoferax and Duganella. The evaluation of samples from L27C through culture-independent methodology identified a rich microbial diversity consisting of six different phyla of bacteria. The culture-independent analysis also displayed the majority of bacteria (56%) belonged to the Class gamma-proteobacteria within the phylum Proteobacteria. Within the Class gamma-proteobacteria, Acinetobacter dominated (48%) the total microbial load. Overall, L27C exhibited 7 different phyla of bacteria and 20 different genera. Statistical analysis (Shannon-Weaver Diversity Index and Simpson Diversity Index) of the biodiversity of L27C displayed a moderately rich and diverse community. Investigations of the biodiversity and distribution of microorganisms in these lakes will help further our understanding of how the physical environment impact the structure and function within these microbially dominated ecosystems.

Huang, Jonathan P.↗

NASA NTRS2017

Reconfigurable Model Execution in the OpenMDAO Framework

NASA's OpenMDAO framework facilitates constructing complex models and computing their derivatives for multidisciplinary design optimization. Decomposing a model into components that follow a prescribed interface enables OpenMDAO to assemble multidisciplinary derivatives from the component derivatives using what amounts to the adjoint method, direct method, chain rule, global sensitivity equations, or any combination thereof, using the MAUD architecture. OpenMDAO also handles the distribution of processors among the disciplines by hierarchically grouping the components, and it automates the data transfer between components that are on different processors. These features have made OpenMDAO useful for applications in aircraft design, satellite design, wind turbine design, and aircraft engine design, among others. This paper presents new algorithms for OpenMDAO that enable reconfigurable model execution. This concept refers to dynamically changing, during execution, one or more of: the variable sizes, solution algorithm, parallel load balancing, or set of variables-i.e., adding and removing components, perhaps to switch to a higher-fidelity sub-model. Any component can reconfigure at any point, even when running in parallel with other components, and the reconfiguration algorithm presented here performs the synchronized updates to all other components that are affected. A reconfigurable software framework for multidisciplinary design optimization enables new adaptive solvers, adaptive parallelization, and new applications such as gradient-based optimization with overset flow solvers and adaptive mesh refinement. Benchmarking results demonstrate the time savings for reconfiguration compared to setting up the model again from scratch, which can be significant in large-scale problems. Additionally, the new reconfigurability feature is applied to a mission profile optimization problem for commercial aircraft where both the parametrization of the mission profile and the time discretization are adaptively refined, resulting in computational savings of roughly 10% and the elimination of oscillations in the optimized altitude profile.

Hwang, John T.↗