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NASA NTRS2006

Realistic Covariance Prediction for the Earth Science Constellation

Routine satellite operations for the Earth Science Constellation (ESC) include collision risk assessment between members of the constellation and other orbiting space objects. One component of the risk assessment process is computing the collision probability between two space objects. The collision probability is computed using Monte Carlo techniques as well as by numerically integrating relative state probability density functions. Each algorithm takes as inputs state vector and state vector uncertainty information for both objects. The state vector uncertainty information is expressed in terms of a covariance matrix. The collision probability computation is only as good as the inputs. Therefore, to obtain a collision calculation that is a useful decision-making metric, realistic covariance matrices must be used as inputs to the calculation. This paper describes the process used by the NASA/Goddard Space Flight Center's Earth Science Mission Operations Project to generate realistic covariance predictions for three of the Earth Science Constellation satellites: Aqua, Aura and Terra.

Duncan, Matthew↗

NASA NTRS2020

Risk and Performance Assessment of Generic Mission Architectures: Showcasing the Artemis Mission

A has initiated a strong push to return face. In this work, we astronaut assess performance and risk for proposed mission architectures using a new Mission Architecture Risk Assessment (MARA) tool. The MARA tool can produce statistics about the availability of components and overall performance of the mission considering potential failures of any of its components. In a Monte Carlo approach, the tool repeats the mission simulation multiple times while a random generator lets modules fail according to their failure rates. The results provide statistically meaningful insights into the overall performance of the chosen architecture. A given mission architecture can be freely replicated in the tool, with the mission timeline and basic characteristics of employed mission modules (habitats, rovers, power generation units, etc.) specified in a configuration file. Crucially, failure rates for each module need to be known or estimated. The tool performs an event-driven simulation of the mission and accounts for random failure events. Failed modules can be repaired, which takes crew time but restores operations. In addition to tracking individual modules, MARA can assess the availability of predefined functions throughout the mission. For instance, the function of resource collection would require a rover to collect the resources, a power generation unit to charge the rover, and a resource processing module. Together, the modules that are required for a given function are called a functional group. Similarly, we can assess how much crew time is available to achieve a mission benefit (e.g. research, building a base, etc) as opposed to spending crew time on repairs. Here we employ the method on the proposed NASA Artemis mission. Artemis aims to return United States astronauts to the lunar surface by 2024. Results provide insights into mission failure probabilities, up- and downtime for individual modules and crew-time resources spent on the repair of failed modules. The tool also allows us to tweak the mission architecture in order to find setups that produce more favorable mission performance. As such, the tool can be an aid in improving the mission architect abling cost-benefit analysis for mission improvement.

Rumpf, Clemens M.↗

NASA NTRS1989

Development of a funding, cost, and spending model for satellite projects

The need for a predictive budget/funging model is obvious. The current models used by the Resource Analysis Office (RAO) are used to predict the total costs of satellite projects. An effort to extend the modeling capabilities from total budget analysis to total budget and budget outlays over time analysis was conducted. A statistical based and data driven methodology was used to derive and develop the model. Th budget data for the last 18 GSFC-sponsored satellite projects were analyzed and used to build a funding model which would describe the historical spending patterns. This raw data consisted of dollars spent in that specific year and their 1989 dollar equivalent. This data was converted to the standard format used by the RAO group and placed in a database. A simple statistical analysis was performed to calculate the gross statistics associated with project length and project cost ant the conditional statistics on project length and project cost. The modeling approach used is derived form the theory of embedded statistics which states that properly analyzed data will produce the underlying generating function. The process of funding large scale projects over extended periods of time is described by Life Cycle Cost Models (LCCM). The data was analyzed to find a model in the generic form of a LCCM. The model developed is based on a Weibull function whose parameters are found by both nonlinear optimization and nonlinear regression. In order to use this model it is necessary to transform the problem from a dollar/time space to a percentage of total budget/time space. This transformation is equivalent to moving to a probability space. By using the basic rules of probability, the validity of both the optimization and the regression steps are insured. This statistically significant model is then integrated and inverted. The resulting output represents a project schedule which relates the amount of money spent to the percentage of project completion.

Johnson, Jesse P.↗

NASA NTRS1994

Center for modeling of turbulence and transition: Research briefs, 1993

This research brief contains the progress reports of the research staff of the Center for Modeling of Turbulence and Transition (CMOTT) from June 1992 to July 1993. It is also an annual report to the Institute for Computational Mechanics in Propulsion located at Ohio Aerospace Institute and NASA Lewis Research Center. The main objectives of the research activities at CMOTT are to develop, validate, and implement turbulence and transition models for flows of interest in propulsion systems. Currently, our research covers eddy viscosity one- and two-equation models, Reynolds-stress algebraic equation models, Reynolds-stress transport equation models, nonequilibrium multiple-scale models, bypass transition models, joint scalar probability density function models, and Renormalization Group Theory and Direct Interaction Approximation methods. Some numerical simulations (LES and DNS) have also been carried out to support the development of turbulence modeling. Last year was CMOTT's third year in operation. During this period, in addition to the above mentioned research, CMOTT has also hosted the following programs: an eighteen-hour short course on 'Turbulence--Fundamentals and Computational Modeling (Part I)' given by CMOTT at the NASA Lewis Research Center; a productive summer visitor research program that has generated many encouraging results; collaborative programs with industry customers to help improve their turbulent flow calculations for propulsion system designs; a biweekly CMOTT seminar series with speakers from within and without the NASA Lewis Research Center including foreign speakers. In addition, CMOTT members have been actively involved in the national and international turbulence research activities. The current CMOTT roster and organization are listed in Appendix A. Listed in Appendix B are the abstracts of the biweekly CMOTT seminar. Appendix C lists the papers contributed by CMOTT members.

Liou, William W.↗

NASA NTRS1978

Variation of fan tone steadiness for several inflow conditions

An amplitude probability density function analysis technique for quantifying the degree of fan noise tone steadiness has been applied to data from a fan tested under a variety of inflow conditions. The test conditions included typical static operation, inflow control by a honeycomb/screen device and forward velocity in a wind tunnel simulating flight. The ratio of mean square sinusoidal-to-random signal content in the fundamental and second harmonic tones was found to vary by more than an order-of-magnitude. Some implications of these results concerning the nature of fan noise generation mechanisms are discussed.

Balombin, J. R.↗

NASA NTRS1978

Variation of fan tone steadiness for several inflow conditions

Balombin, J. R.↗

NASA NTRS1986

Spatial variations of ocean wave directional spectra from the Seasat synthetic aperture radar

Seasat synthetic aperture radar ocean wave spectra for a 900-km pass are analyzed and interpreted in the context of both their probable generation sources and their surface current and bathymetric modifiers. Systematic vector wavenumber variations of several times the standard error of determination (about 1.5 percent in magnitude and 0.9 deg in direction) occur along the entire 900-km pass. The large-scale spatial variation of a 200-m swell system can be accurately accounted for as a result of dispersion from a distant storm. The more local variations are qualitatively well correlated in position with known currents and bathymetry but show systematic biases that appear partly due to an environmentally dependent instrument transfer function in the regions of high current and highest sea state. There is also substantial evidence that a large angular deviation in the center of the pass is the result of a mesoscale eddy just to the east.

Beal, R. C.↗

NASA NTRS2013

Estimated Probability of a Cervical Spine Injury During an ISS Mission

Introduction: The Integrated Medical Model (IMM) utilizes historical data, cohort data, and external simulations as input factors to provide estimates of crew health, resource utilization and mission outcomes. The Cervical Spine Injury Module (CSIM) is an external simulation designed to provide the IMM with parameter estimates for 1) a probability distribution function (PDF) of the incidence rate, 2) the mean incidence rate, and 3) the standard deviation associated with the mean resulting from injury/trauma of the neck. Methods: An injury mechanism based on an idealized low-velocity blunt impact to the superior posterior thorax of an ISS crewmember was used as the simulated mission environment. As a result of this impact, the cervical spine is inertially loaded from the mass of the head producing an extension-flexion motion deforming the soft tissues of the neck. A multibody biomechanical model was developed to estimate the kinematic and dynamic response of the head-neck system from a prescribed acceleration profile. Logistic regression was performed on a dataset containing AIS1 soft tissue neck injuries from rear-end automobile collisions with published Neck Injury Criterion values producing an injury transfer function (ITF). An injury event scenario (IES) was constructed such that crew 1 is moving through a primary or standard translation path transferring large volume equipment impacting stationary crew 2. The incidence rate for this IES was estimated from in-flight data and used to calculate the probability of occurrence. The uncertainty in the model input factors were estimated from representative datasets and expressed in terms of probability distributions. A Monte Carlo Method utilizing simple random sampling was employed to propagate both aleatory and epistemic uncertain factors. Scatterplots and partial correlation coefficients (PCC) were generated to determine input factor sensitivity. CSIM was developed in the SimMechanics/Simulink environment with a Monte Carlo wrapper (MATLAB) used to integrate the components of the module. Results: The probability of generating an AIS1 soft tissue neck injury from the extension/flexion motion induced by a low-velocity blunt impact to the superior posterior thorax was fitted with a lognormal PDF with mean 0.26409, standard deviation 0.11353, standard error of mean 0.00114, and 95% confidence interval [0.26186, 0.26631]. Combining the probability of an AIS1 injury with the probability of IES occurrence was fitted with a Johnson SI PDF with mean 0.02772, standard deviation 0.02012, standard error of mean 0.00020, and 95% confidence interval [0.02733, 0.02812]. The input factor sensitivity analysis in descending order was IES incidence rate, ITF regression coefficient 1, impactor initial velocity, ITF regression coefficient 2, and all others (equipment mass, crew 1 body mass, crew 2 body mass) insignificant. Verification and Validation (V&V): The IMM V&V, based upon NASA STD 7009, was implemented which included an assessment of the data sets used to build CSIM. The documentation maintained includes source code comments and a technical report. The software code and documentation is under Subversion configuration management. Kinematic validation was performed by comparing the biomechanical model output to established corridors.

Brooker, John E.↗

NASA NTRS2021

Compact Augmented Spark Igniters for Liquid Rocket Engines

The objective of this dissertation is to characterize augmented spark impinging (torch) igniters. Torch igniters inject propellants into a small prechamber where a spark igniter imparts the activation energy to initiate combustion. The flame stabilizes within the prechamber and exits into the main combustion chamber where it ignites the main flow directly or by mixing with a coaxial flow to create a secondary flame. The underlying phenomena that promote reliable ignition are multifaceted and required a broad scope of investigations. Initial efforts focused on spark discharges across an annular geometry. Gaseous propellent projects the electrical arc toward the prechamber, where the arc and the thermal energy is imparted to the surrounding fluid then meets with a combustible mixture. Discharges are parametrically examined against pressure, spark-gap size, and exciter types to emulate transient engine startup. The energy imparted to the flow, approximate velocity, and distance traveled by the exhaust plume are determined. Spark quenching (no energy deposition) and its effects on perceived versus actual measurements are shown. Results outline the prechamber target zone where a combustible mixture is necessary for ignition. Next, three-dimensional, time-accurate, and nonreacting computational fluid dynamics simulations assess effects of geometric and mass flow differences on the size, location, and composition of a combustible mixture. Fluid dynamic phenomena are elucidated. Correlations between nondimensional variables are found. Results are coupled with a conservation-law analysis to produce novel objective functions. Lastly, a full-scale, modular torch igniter is tested. Injector sizes, mixture ratios, and injector momentum ratios are systematically varied to map ignition probability and ignition delay. Results show the importance of local mixture ratios, provide injector momentum ratios guidelines, and demonstrate reliable ignition of core mixture ratios typically outside flammability limits. The deliverable of this project is not a torch igniter or derivative thereof. Instead, this dissertation produces novel objective-functions for igniter design. These equations are functions of variables that may be constrained by engine-level requirements, physics, and experimental results from this work. They provide a foundation for future igniter design and will minimize iterative processes from the design cycle for next-generation ignition hardware.

Torch Igniter↗

NASA NTRS2000

The Next Generation MOD: A Microchip Amino Acid Analyzer for Detecting Extraterrestrial Life

The MOD (Mars Organic Detector) instrument which has selected for the definition phase of the BEDS package on the 2005 Mars Explorer Program spacecraft is designed to simply detect the presence of amino acids in Martian surface samples at a sensitivity of a few parts per billion (ppb). An additional important aspect of amino acid analyses of Martian samples is identifying and quantifying which compounds are present, and also distinguishing those produced abiotically from those synthesized by either extinct or extant life. Amino acid homochirality provides an unambiguous way of distinguishing between abiotic vs. biotic origins. Proteins made up of mixed D- and L-amino acids would not likely have been efficient catalysts in early organisms because they could not fold into bioactive configurations such as the a-helix. However, enzymes made up of all D-amino acids function just as well as those made up of only L-amino acids, but the two enzymes use the opposite stereoisomeric substrates. There are no biochemical reasons why L-amino acids would be favored over Damino acids. On Earth, the use of only L-amino acids in proteins by life is probably simply a matter of chance. We assume that if proteins and enzymes were a component of extinct or extant life on Mars, then amino acid homochirality would have been a requirement. However, the possibility that Martian life was (or is) based on D-amino acids would be equal to that based on L-amino acids. The detection of a nonracemic mixture of amino acids in a Martian sample would be strong evidence for the presence of an extinct or extant biota on Mars. The finding of an excess of D-amino acids would provide irrefutable evidence of unique Martian life that could not have been derived from seeding the planet with terrestrial life (or the seeding of the Earth with Martian life). In contrast, the presence of racemic amino acids, along with non-protein amino acids such as alpha-aminoisobutyric acid and isovaline, would be indicative of an abiotic origin, although we have to consider the possibility that the racemic amino acids were generated from the racemization of biotically produced amino acids.

Mathies, R. A.↗

NASA NTRS2007

Assessment of Immune Status, Latent Viral Reactivation and Stress during Long Duration Bed Rest as an Analog for Spaceflight

As logistical access for in-flight space research becomes more limited, the use of ground based spaceflight analogs for life science studies will increase. These studies are particularly important as NASA progresses towards the Lunar and eventually Mars missions outlined in the 2005 Vision for Space Exploration. Countermeasures must be developed to mitigate the clinical risks associated with exploration class space missions. In an effort to coordinate studies across multiple disciplines, NASA has selected 90-day bed rest as the analog of choice, and initiated the Flight Analogs Project to implement research studies with or without the evaluation of countermeasures. Although bed rest is not the analog of choice to evaluate spaceflight-associated immune dysfunction, a standard Immune Assessment was developed for subjects participating in the 90-day bed best studies. The Immune Assessment consists of: leukocyte subset distribution, T cell functional responses, intracellular cytokine production profiles, latent viral reactivation, virus specific T cell levels, virus specific T cell function, stress hormone levels and a behavioral assessment using stress questionnaires. The purpose of the assessment during the initial studies (without countermeasure) is to establish control data against which future studies (with countermeasure) will be evaluated. It is believed that some of the countermeasures planned to be evaluated in future studies, such as exercise, pharmacologic intervention or nutritional supplementation, have the ability to impact immune function. Therefore immunity will likely be monitored during those studies. The data generated during the first three control studies showed that the subjects in general did not display altered peripheral leukocyte subsets, constitutive immune activation, significant latent viral reactivation (EBV, VZV) or altered T cell function. Interestingly, for some subjects the level of constitutively activated T cells (CD8+/CD69+) and virus-specific T cells (CMV and EBV) both decreased during the studies. This likely reflects the isolation of the subjects (from an immunological perspective) and absence of everyday subclinical challenges to the immune system. Cortisol levels (plasma and saliva) did not vary significantly during the studies. This probably reflects a lack of physiological stress during the study and the stress of readaptation to the 1xG environment at R+1. These data demonstrate the absence of significant immune alteration during 90-day bed rest, and establish control data against which future studies (including countermeasures) may be compared.

Crucian, Brian E.↗

NASA NTRS1984

A stochastic model for photon noise induced by charged particles in multiplier phototubes of the space telescope fine guidance sensors

The Space Telescope (ST) is subjected to charged particle strikes in its space environment. ST's onboard fine guidance sensors utilize multiplier phototubes (PMT) for attitude determination. These tubes, when subjected to charged particle strikes, generate spurious photons in the form of Cerenkov radiation and fluorescence which give rise to unwanted disturbances in the pointing of the telescope. A stochastic model for the number of these spurious photons which strike the photocathode of the multiplier phototube which in turn produce the unwanted photon noise are presented. The model is applicable to both galactic cosmic rays and charged particles trapped in the Earth's radiation belts. The model which was programmed allows for easy adaption to a wide range of particles and different parameters for the phototube of the multiplier. The probability density functions for photons noise caused by protons, alpha particles, and carbon nuclei were using thousands of simulated strikes. These distributions are used as part of an overall ST dynamics simulation. The sensitivity of the density function to changes in the window parameters was also investigated.

Howell, L. W.↗

NASA NTRS1986

Stochastic model for photon noise induced by charged particles in multiplier phototubes of the Hubble Space Telescope fine guidance sensors

The Space Telescope (ST) is subjected to charged particle strikes in its space environment. ST's onboard fine guidance sensors utilize multiplier phototubes (PMT) for attitude determination. These tubes, when subjected to charged particle strikes, generate spurious photons in the form of Cerenkov radiation and fluorescence which give rise to unwanted disturbances in the pointing of the telescope. A stochastic model for the number of these spurious photons which strike the photocathodes of the multiplier phototube which in turn produce the unwanted photon noise are presented. The model is applicable to both galactic cosmic rays and charged particles trapped in the earth's radiation belts. The model which was programmed allows for easy adaption to a wide range of particles and different parameters for the phototube of the multiplier. The probability density functions for photons noise caused by protons, alpha particles, and carbon nuclei were using thousands of simulated strikes. These distributions are used as part of an overall ST dynamics simulation. The sensitivity of the density function to changes in the window parameters was also investigated.

Howell, L. W.↗

NASA NTRS1995

A count probability cookbok: Spurious effects and the scaling model

We study the errors brought by finite volume effects and dilution effects on the practical determination of the count probability distribution function P(sub N)(n,l), which is the probability of having N objects in a cell of volume l cubed for a set of average number density n. Dilution effects are particularly revelant to the so-called sparse sampling strategy. This work is mainly done in the framework of the Bailan & Schaeffer scaling model, which assumes that the Q-body correlation functions obey the scaling relation Xi(sub Q)(lambda r(sub l),....lambda r(sub Q) = lambda(exp -(Q-1)gamma) Xi(sub Q)(r(sub 1),....r(sub Q)). We use three synthetic samples as references to perform our analysis: a fractal generated by a Rayleigh-Levy random walk with approximately 3 x 10(exp 4) objects, a sample dominated by a spherical power-law cluster with approximately 3 x 10(exp 4) objects and a cold dark matter (CDM) universe involving approximately 3 x 10(exp 5) matter particles.

Colombi, S.↗

NASA NTRS2015

Nearly a Decade of CALIPSO Observations of Asian and Saharan Dust Properties Near Source and Transport Regions

The lidar on the Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) mission, makes robust measurements of dust and has generated a length of record that is significant both seasonally and inter-annually. We exploit this record to determine a multi-year climatology of the properties of Asian and Saharan dust, in particular seasonal optical depths, layer frequencies, and layer heights of dust gridded in accordance with the Level 3 data products protocol, between 2006-2015. The data are screened using standard CALIPSO quality assurance flags, cloud aerosol discrimination (CAD) scores, overlying features and layer properties. To evaluate the effects of transport on the morphology, vertical extent and size of the dust layers, we compare probability distribution functions of the layer integrated volume depolarization ratios, geometric depths and integrated attenuated color ratios near the source to the same distributions in the far field or transport region. CALIPSO is collaboration between NASA and Centre National D'études Spatiales (CNES), was launched in April 2006 to provide vertically resolved measurements of cloud and aerosol distributions. The primary instrument on the CALIPSO satellite is the Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP), a near-nadir viewing two-wavelength polarization-sensitive instrument. The unique nature of CALIOP measurements make it quite challenging to validate backscatter profiles, aerosol type, and cloud phase, all of which are used to retrieve extinction and optical depth. To evaluate the uncertainty in the lidar ratios, we compare the values computed from dust layers overlying opaque water clouds, considered nominal, with the constant lidar ratio value used in the CALIOP algorithms for dust. We also explore the effects of noise on the CALIOP retrievals at daytime by comparing the distributions of the properties at daytime to the nighttime distributions.

Omar, Ali H.↗

NASA NTRS1975

Wave theory of turbulence in compressible media (acoustic theory of turbulence)

The generation and the transmission of sound in turbulent flows are treated as one of the several aspects of wave propagation in turbulence. Fluid fluctuations are decomposed into orthogonal Fourier components, with five interacting modes of wave propagation: two vorticity modes, one entropy mode, and two acoustic modes. Wave interactions, governed by the inhomogeneous and nonlinear terms of the perturbed Navier-Stokes equations, are modeled by random functions which give the rates of change of wave amplitudes equal to the averaged interaction terms. The statistical framework adopted is a quantum-like formulation in terms of complex distribution functions. The spatial probability distributions are given by the squares of the absolute values of the complex characteristic functions. This formulation results in nonlinear diffusion-type transport equations for the probability densities of the five modes of wave propagation.

Kentzer, C. P.↗

NASA NTRS2004

Scatter in Carbon/Silicon Carbide (C/SiC) Composites Quantified

Carbon-fiber-reinforced silicon carbide matrix (C/SiC) composites processed by chemical vapor infiltration are candidate materials for aerospace thermal structures. Carbon fibers can retain properties at very high temperatures, but they are known to have poor oxidation resistance in adverse, high-temperature environments. Nevertheless, the combination of CVI-SiC matrix with higher stiffness and oxidation resistance, the interfacial coating, and additional surface-seal coating provides the necessary protection to the carbon fibers, and makes the material viable for high-temperature space applications operating under harsh environments. Furthermore, C/SiC composites, like other ceramic matrix composites (CMCs), exhibit graceful non-catastrophic failure because of various inherent energy dissipating mechanisms. The material exhibits nonlinearity in deformation even at very low stress levels. This is the result of the severe matrix microcracking present in the as processed composite because of large differences between the coefficients of thermal expansion of the fiber and the matrix. Utilization of these advanced composites in next generation space vehicles will require innovative structural configurations, updated materials, and refined analyses. Structural safety issues for these vehicles are in direct competition with performance and cost. One would have to quantify the uncertainties associated with the design using formal probabilistic methods. Specifically four fundamental aspects on which analyses are based-- (1) loading conditions, (2) material behavior, (3) geometrical configurations, and (4) structural connections between the composite components and baseline structure--are stochastic in nature. A direct way to formally account for uncertainties is to develop probabilistic structural analysis methods where all participating variables are described by appropriate probability density functions. The present work, however, focuses on analyzing the stochastic material behavior of these advanced composites using formal probabilistic analysis methods. Often, some of the desirable property characteristics that allow composites to offer advantages over conventional structural materials (like tailoring of composite properties) and the complexity are in fact responsible for their greater statistical variability and the requirements for more characterization tests. Composite properties are anisotropic as well, having different properties in different directions. This means that characterization of a property such as stiffness--which will vary greatly depending on the orientation of the fiber relative to the direction of the testing--must be repeated for several different directions and loading conditions. The fabrication process for composites also introduces statistical variations in properties and geometry. A composite part is produced in a number of steps, each of which introduces statistical variability. The matrix is usually produced from a combination of raw materials; and the fiber, which has its own set of properties, is often coated or surface treated, introducing yet another source of variability.

Murthy, Pappu L. N.↗

NASA NTRS2006

Exploring the Connection Between Sampling Problems in Bayesian Inference and Statistical Mechanics

The Bayesian and statistical mechanical communities often share the same objective in their work - estimating and integrating probability distribution functions (pdfs) describing stochastic systems, models or processes. Frequently, these pdfs are complex functions of random variables exhibiting multiple, well separated local minima. Conventional strategies for sampling such pdfs are inefficient, sometimes leading to an apparent non-ergodic behavior. Several recently developed techniques for handling this problem have been successfully applied in statistical mechanics. In the multicanonical and Wang-Landau Monte Carlo (MC) methods, the correct pdfs are recovered from uniform sampling of the parameter space by iteratively establishing proper weighting factors connecting these distributions. Trivial generalizations allow for sampling from any chosen pdf. The closely related transition matrix method relies on estimating transition probabilities between different states. All these methods proved to generate estimates of pdfs with high statistical accuracy. In another MC technique, parallel tempering, several random walks, each corresponding to a different value of a parameter (e.g. "temperature"), are generated and occasionally exchanged using the Metropolis criterion. This method can be considered as a statistically correct version of simulated annealing. An alternative approach is to represent the set of independent variables as a Hamiltonian system. Considerab!e progress has been made in understanding how to ensure that the system obeys the equipartition theorem or, equivalently, that coupling between the variables is correctly described. Then a host of techniques developed for dynamical systems can be used. Among them, probably the most powerful is the Adaptive Biasing Force method, in which thermodynamic integration and biased sampling are combined to yield very efficient estimates of pdfs. The third class of methods deals with transitions between states described by rate constants. These problems are isomorphic with chemical kinetics problems. Recently, several efficient techniques for this purpose have been developed based on the approach originally proposed by Gillespie. Although the utility of the techniques mentioned above for Bayesian problems has not been determined, further research along these lines is warranted

Pohorille, Andrew↗