Search NASA⌕ Search

SEARCH · Search NASA

Results for “randomized methods”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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

At least 1,153 records · Page 64
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Controlling radioisotope proportions when randomly sampling from Dirichlet distributions in PyRIID

As machine learning models for radioisotope quantification become more powerful, likewise the need for high-quality synthetic training data grows as well. For problem spaces that involve estimating the relative isotopic proportions of various sources in gamma spectra it is necessary to generate training data that accurately represents the variance of proportions encountered. In this report, we aim to provide guidance on how to target a desired variance of proportions which are randomly when using the PyRIID Seed Mixer, which samples from a Dirichlet distribution. We provide a method for properly parameterizing the Dirichlet distribution in order to maintain a constant variance across an arbitrary number of dimensions, where each dimension represents a distinct source template being mixed. We demonstrate that our method successfully parameterizes the Dirichlet distribution to target a specific variance of proportions, provided that several conditions are met. This allows us to follow a principled technique for controlling how random mixture proportions are generated which are then used downstream in the synthesis process to produce the final, noisy gamma spectra.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
NASA NTRS1993

On the apparent insignificance of the randomness of flexible joints on large space truss dynamics

Deployable periodic large space structures have been shown to exhibit high dynamic sensitivity to period-breaking imperfections and uncertainties. These can be brought on by manufacturing or assembly errors, structural imperfections, as well as nonlinear and/or nonconservative joint behavior. In addition, the necessity of precise pointing and position capability can require the consideration of these usually negligible and unknown parametric uncertainties and their effect on the overall dynamic response of large space structures. This work describes the use of a new design approach for the global dynamic solution of beam-like periodic space structures possessing parametric uncertainties. Specifically, the effect of random flexible joints on the free vibrations of simply-supported periodic large space trusses is considered. The formulation is a hybrid approach in terms of an extended Timoshenko beam continuum model, Monte Carlo simulation scheme, and first-order perturbation methods. The mean and mean-square response statistics for a variety of free random vibration problems are derived for various input random joint stiffness probability distributions. The results of this effort show that, although joint flexibility has a substantial effect on the modal dynamic response of periodic large space trusses, the effect of any reasonable uncertainty or randomness associated with these joint flexibilities is insignificant.

Koch, R. M.↗
DOE OSTI2018

Device-scale CFD study for mass transfer coefficient and effective mass transfer area in packed column

Carbon dioxide (CO 2 ) is a significant contributor of greenhouse gases emissions responsible for global warming. The fossil-fueled power plant is one of the major sources of CO 2 emission worldwide and needs to be mitigated. Post-combustion carbon capture by chemical absorption can be a promising technology to reduce CO 2 emission from power plants. The absorption is carried out via countercurrent flow in the packed columns. Although monoethanolamine (MEA) is a popular solvent, it has also limitations such as high volatility of solvent, corrosivity that leads to product degradation and damage to apparatuses, low CO 2 loading capacity, etc. Therefore, alternate solvents are being explored as prospective materials for carbon capture. A water-lean solvent, CO 2 binding organic liquids (CO2BOL), is currently being developed as a prospective material in carbon capture at the Pacific Northwest National Laboratory. Since the development of the material is at the primitive stage, it needs extensive investigation before industrial deployments. Hydrodynamics and mass transfer phenomena in the packed column using CO 2 BOL is unknown. Computational fluid dynamics (CFD) can be a useful tool to understand these phenomena in the packed column. Before employing the extensive CFD investigations, one needs to carefully validate CFD predictions for accurate flow predictions. This report demonstrates the validation of multiphase flow simulations against the experimental results for both structured and random packed columns. The reactive multiphase flow simulations using volume of fluid method were conducted at a wide range of liquid loads and contacts angles. The effective area is one of the critical factors dictating interphase mass transfer due to chemical reactions. Accordingly, CFD-predicted effective areas were compared with the experimental results of Tsai et al. [1] for Mellapak 250.Y structured packing. Both results matched well at different liquid loads in the presence of fast chemical kinetics using caustic solvents. Further, reactive multiphase flow simulations were also conducted in pall ring random packing to compute effective areas in the presence of fast chemical reactions. The CFD-predicted effective area compared well with correlation developed by Song et al. [2]. With successful validations, the current simulation framework can be further used for prediction of hydrodynamics and mass transfer phenomena using CO 2 BOL solvents in the packed column. Subsequently, it would be helpful in design optimization of a packed column. In addition to validation of the CFD model, this report also explains the mechanism for enhanced liquid side mass transfer for a structured packed column as compared to the wetted wall column.

20 FOSSIL-FUELED POWER PLANTS↗
DOE OSTI2017

Graph Sampling for Visual Analytics

Effectively visualizing large graphs and capturing the statistical properties are two challenging tasks. To aid in these two tasks, many sampling approaches for graph simplification have been proposed, falling into three categories: node sampling, edge sampling, and traversal-based sampling. It is still unknown which approach is the best. We evaluate commonly used graph sampling methods through a combined visual and statistical comparison of graphs sampled at various rates. We conduct our evaluation on three graph models: random graphs, small-world graphs, and scale-free graphs. Initial results indicate that the effectiveness of a sampling method is dependent on the graph model, the size of the graph, and the desired statistical property. This benchmark study can be used as a guideline in choosing the appropriate method for a particular graph sampling task, and the results presented can be incorporated into graph visualization and analysis tools.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2020

Amino Acid Encoding for Deep Learning Applications

Background: The number of applications of deep learning algorithms in bioinformatics is increasing as they usually achieve superior performance over classical approaches, especially, when bigger training datasets are available. In deep learning applications, discrete data, e.g. words or n-grams in language, or amino acids or nucleotides in bioinformatics, are generally represented as a continuous vector through an embedding matrix. Recently, learning this embedding matrix directly from the data as part of the continuous iteration of the model to optimize the target prediction – a process called ‘end-to-end learning’ – has led to state-of-the-art results in many fields. Although usage of embeddings is well described in the bioinformatics literature, the potential of end-to-end learning for single amino acids, as compared to more classical manually-curated encoding strategies, has not been systematically addressed. To this end, we compared classical encoding matrices, namely one-hot, VHSE8 and BLOSUM62, to end-to-end learning of amino acid embeddings for two different prediction tasks using three widely used architectures, namely recurrent neural networks (RNN), convolutional neural networks (CNN), and the hybrid CNN-RNN. Results: By using different deep learning architectures, we show that end-to-end learning is on par with classical encodings for embeddings of the same dimension even when limited training data is available, and might allow for a reduction in the embedding dimension without performance loss, which is critical when deploying the models to devices with limited computational capacities. We found that the embedding dimension is a major factor in controlling the model performance. Surprisingly, we observed that deep learning models are capable of learning from random vectors of appropriate dimension. Conclusion: Our study shows that end-to-end learning is a flexible and powerful method for amino acid encoding. Further, due to the flexibility of deep learning systems, amino acid encoding schemes should be benchmarked against random vectors of the same dimension to disentangle the information content provided by the encoding scheme from the distinguishability effect provided by the scheme.

Deep-learning↗
NASA NTRS2020

Amino Acid Encoding for Deep Learning Applications

Background: The number of applications of deep learning algorithms in bioinformatics is increasing as they usually achieve superior performance over classical approaches, especially, when bigger training datasets are available. In deep learning applications, discrete data, e.g. words or n-grams in language, or amino acids or nucleotides in bioinformatics, are generally represented as a continuous vector through an embedding matrix. Recently, learning this embedding matrix directly from the data as part of the continuous iteration of the model to optimize the target prediction – a process called ‘end-to-end learning’ – has led to state-ofthe-art results in many fields. Although usage of embeddings is well described in the bioinformatics literature, the potential of end-to-end learning for single amino acids, as compared to more classical manually-curated encoding strategies, has not been systematically addressed. To this end, we compared classical encoding matrices, namely one-hot, VHSE8 and BLOSUM62, to end-to-end learning of amino acid embeddings for two different prediction tasks using three widely used architectures, namely recurrent neural networks (RNN), convolutional neural networks (CNN), and the hybrid CNN-RNN. Results: By using different deep learning architectures, we show that end-to-end learning is on par with classical encodings for embeddings of the same dimension even when limited training data is available, and might allow for a reduction in the embedding dimension without performance loss, which is critical when deploying the models to devices with limited computational capacities. We found that the embedding dimension is a major factor in controlling the model performance. Surprisingly, we observed that deep learning models are capable of learning from random vectors of appropriate dimension. Conclusion: Our study shows that end-to-end learning is a flexible and powerful method for amino acid encoding. Further, due to the flexibility of deep learning systems, amino acid encoding schemes should be benchmarked against random vectors of the same dimension to disentangle the information content provided by the encoding scheme from the distinguishability effect provided by the scheme.

Hesham ElAbd↗
NASA NTRS2014

The Effect of General Statistical Fiber Misalignment on Predicted Damage Initiation in Composites

A micromechanical method is employed for the prediction of unidirectional composites in which the fiber orientation can possess various statistical misalignment distributions. The method relies on the probability-weighted averaging of the appropriate concentration tensor, which is established by the micromechanical procedure. This approach provides access to the local field quantities throughout the constituents, from which initiation of damage in the composite can be predicted. In contrast, a typical macromechanical procedure can determine the effective composite elastic properties in the presence of statistical fiber misalignment, but cannot provide the local fields. Fully random fiber distribution is presented as a special case using the proposed micromechanical method. Results are given that illustrate the effects of various amounts of fiber misalignment in terms of the standard deviations of in-plane and out-of-plane misalignment angles, where normal distributions have been employed. Damage initiation envelopes, local fields, effective moduli, and strengths are predicted for polymer and ceramic matrix composites with given normal distributions of misalignment angles, as well as fully random fiber orientation.

Micromechanical↗
DOE OSTI2018

Secure true random number generation using 1.5-T transistor flash memory

This disclosure relates generally to physically unclonable function (PUF) circuitry along with methods of generating numbers. In one embodiment, the PUF circuitry includes a memory, a memory control circuitry, and whitening circuitry. To reduce or eliminate the systematic bias from the array, whitening circuitry is configured to generate a random number comprising random number bits in response to the memory control circuit implementing at least one sequence of memory cycles on the array of the memory cells in the memory. The whitening circuitry is configured to provide the random number bits of the random number based on the variable bit states stored by the array of the memory cells. On average the whitening circuitry is configured to provide approximately half of the random number bits in the first bit state and half of random number bits in a second bit state.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Detecting hidden transient events in noisy nonlinear time-series

The information impulse function (IIF), running Variance, and local Hölder Exponent are three conceptually different time-series evaluation techniques. These techniques examine time-series for local changes in information content, statistical variation, and point-wise smoothness, respectively. Using simulated data emulating a randomly excited nonlinear dynamical system, this study interrogates the utility of each method to correctly differentiate a transient event from the background while simultaneously locating it in time. Computational experiments are designed and conducted to evaluate the efficacy of each technique by varying pulse size, time location, and noise level in time-series. Our findings reveal that, in most cases, the first instance of a transient event is more easily observed with the information-based approach of IIF than with the Variance and local Hölder Exponent methods. While our study highlights the unique strengths of each technique, the results suggest that very robust and reliable event detection for nonlinear systems producing noisy time-series data can be obtained by incorporating the IIF into the analysis.

42 ENGINEERING↗
NASA NTRS1981

Radiative transfer theory for passive microwave remote sensing of random media

The radiative transfer theory is applied to a random medium for the case of passive remote sensing. A general numerical solution with Gaussian quadrature method is studied. For a medium with laminar structure or a medium with cylindrical structure, closed form solutions are obtained by solving the radiative transfer equations of a two-layer random medium. Numerical results for the general case and the two limiting cases are compared.

Chuang, S. L.↗
NASA NTRS2015

Optimization of Microphone Locations for Acoustic Liner Impedance Eduction

Two impedance eduction methods are explored for use with data acquired in the NASA Langley Grazing Flow Impedance Tube. The first is an indirect method based on the convected Helmholtz equation, and the second is a direct method based on the Kumaresan and Tufts algorithm. Synthesized no-flow data, with random jitter to represent measurement error, are used to evaluate a number of possible microphone locations. Statistical approaches are used to evaluate the suitability of each set of microphone locations. Given the computational resources required, small sample statistics are employed for the indirect method. Since the direct method is much less computationally intensive, a Monte Carlo approach is employed to gather its statistics. A comparison of results achieved with full and reduced sets of microphone locations is used to determine which sets of microphone locations are acceptable. For the indirect method, each array that includes microphones in all three regions (upstream and downstream hard wall sections, and liner test section) provides acceptable results, even when as few as eight microphones are employed. The best arrays employ microphones well away from the leading and trailing edges of the liner. The direct method is constrained to use microphones opposite the liner. Although a number of arrays are acceptable, the optimum set employs 14 microphones positioned well away from the leading and trailing edges of the liner. The selected sets of microphone locations are also evaluated with data measured for ceramic tubular and perforate-over-honeycomb liners at three flow conditions (Mach 0.0, 0.3, and 0.5). They compare favorably with results attained using all 53 microphone locations. Although different optimum microphone locations are selected for the two impedance eduction methods, there is significant overlap. Thus, the union of these two microphone arrays is preferred, as it supports usage of both methods. This array contains 3 microphones in the upstream hard wall section, 14 microphones opposite the liner, and 3 microphones in the downstream hard wall section.

Jones, M. G.↗
DOE OSTI2018

A sub-sampled approach to extremely low-dose STEM

The inpainting of deliberately and randomly sub-sampled images offers a potential means to image specimens at a high resolution and under extremely low-dose conditions (≤1 e - /Å 2 ) using a scanning transmission electron microscope. We show that deliberate sub-sampling acquires images at least an order of magnitude faster than conventional low-dose methods for an equivalent electron dose. More importantly, when adaptive sub-sampling is implemented to acquire the images, there is a significant increase in the resolution and sensitivity which accompanies the increase in imaging speed. Lastly, we demonstrate the potential of this method for beam sensitive materials and in-situ observations by experimentally imaging the node distribution in a metal-organic framework.

47 OTHER INSTRUMENTATION↗
DOE OSTI2019

Determining velocity of detonation using high-resolution time domain reflectometry

Two methods are presented to make measurements of the length of a coaxial cable with submillimeter and submicrosecond accuracy. If the cable is destroyed by the passage of a strong shock, for instance, from the detonation of an explosive, a measure of the phase velocity of the detonation (VOD) can be obtained with high accuracy (≈0.25%). The first method introduces a series of short Gaussian electrical pulses (<300 ps) into the cable that reflect off the cable end and are compared to the time of the original outgoing pulse by using a fast digitizer (100 gigasamples per second). By curve fitting a Gaussian form to the outgoing and reflected pulses, a measurement of the cable length can be made with subsample accuracy. So, as long as the pulses do not overlap, a large number may be present in the cable at any time allowing pulse repetition rates of 40–135 MHz depending on the cable length. Averaging allows accurate VOD measurements to be made that are comparable with the best traditional techniques. The second method replaces the pulser with a wide-spectrum (0.1–10 GHz) amplified random noise source. A similar setup and digitizer are used before autocorrelating short (80–160 ns) subsamples of the total waveform to measure the reflected delay time. That is, a smaller amplitude version of the original waveform appears time shifted by twice the transit time in the cable and this time is estimated via autocorrelation. The advantage of this method is that a new subsample needs only to be selected from a time increment in which new spectral information is present for a new measure of the length to be made. In this manner, many more length estimates can be averaged to establish the VOD than with the Gaussian approach. As a result, a detailed description of the implementation of both methods is presented together with an example of the type of data collected and limitations found.

47 OTHER INSTRUMENTATION↗
NASA NTRSDate not supplied

Investigating the Effects of Exposure to Blue-Enriched Light or Peppermint Odor on Alertness, Mood, and Performance Upon Awakening from Deep Sleep at Night

Introduction: Sleep inertia refers the transient neurobehavioral impairments experienced immediately after waking from sleep. This period of reduced alertness and performance poses a significant safety risk to on-call workers who may be required to perform a safety-critical task immediately after waking (e.g., emergency services, health care, and military). In these operations, the need for a rapid return to full alertness is critical to mission safety and success. Several factors may exacerbate sleep inertia, resulting in greater impairment upon waking, including: waking from deep sleep, (i.e., slow wave sleep, SWS), waking at night, and waking following prior sleep loss. Awakenings under these conditions are common for on-call and extended shift workers who may need to perform safety-critical tasks soon after waking from unprotected sleep opportunities. Therefore, there is a need for evidence-based reactive countermeasures (i.e., used upon waking) to the cognitive consequences sleep inertia. Specifically, countermeasures that can rapidly restore alertness and performance immediately following sleep. A recent review of the literature on reactive countermeasures highlighted several research gaps and promising candidates for further investigation. The review also emphasized the need for countermeasures that are operationally viable and readily deployed in occupational settings. This study aims to address the identified gaps and limitations by assessing the efficacy of exposure to two known acute alerting stimuli - blue-enriched light and peppermint odor - to improve cognitive performance, alertness, and mood immediately after waking from SWS at night. Materials and Methods: Twelve participants completed a two-week within-subject, randomized, cross-over intervention study including two in-laboratory overnight visits. During each experimental week, the subjects experienced one intervention (light or peppermint) and a control condition upon awakening from SWS at night. The presentation order of the two conditions (intervention or control) at wake-up and the order of intervention (light or peppermint) by week was randomized by sex. Prior to each in-laboratory visit, participants maintained a sleep schedule of 8.5 h for 5 nights and 5 h for one night. Compliance with this sleep schedule was confirmed by actigraphy. In the laboratory, participants went to bed at their habitual bedtime and were monitored by standard polysomnography. After at least five minutes of continuous SWS, participants were awoken and exposed, in a randomized order, to either the control or intervention condition. During the hour after awakening from SWS (at 2, 17, 32, and 47 minutes after waking), participants completed a battery of tasks including a 5-minute psychomotor vigilance task (PVT), a subjective scale of alertness (Karolinska Sleepiness Scale, KSS), and visual analogue scales (VAS) of mood. Following this sleep inertia measurement period, all lights were turned off and participants were allowed to return to sleep. They were then awoken again from their subsequent SWS period and exposed to the alternative condition (control or intervention). Following this second awakening, participants were allowed to sleep until their habitual wake time and were then released from the laboratory. Participants then followed the at-home sleep schedule and returned to the laboratory for the second intervention (light or peppermint) following the procedures described above. The light intervention involved exposure to a blue-enriched light canvas illuminated for 1 hour at a distance of ~56 cm from the participant (~200 lux and ~60 melanopic lux at angle of gaze). For the peppermint intervention, peppermint oil was pipetted onto a mask, and participants inhaled the odor with the mask covering the nose and mouth for 1 minute. The control condition for both weeks involved a dim, red ambient light (<1 lux). An odorless mask, without any oil pipetted onto the mask, was also worn in the peppermint control condition. Results: Compared to the control condition, participants exposed to blue-enriched light had fewer PVT lapses (χ2 = 5.285, p = .022), reported feeling more alert (KSS: F1,77 = 4.955, p = .029; VASalert: F1,77 = 8.226, p = .005), and had improved mood (VAScheerful: F1,77 = 8.615, p = .004; VASdepressed: F1,77 = 4.649, p = .034; VASlethargic: F1,77 = 5.652, p = .020). Exposure to peppermint oil did not improve any outcome measures on any of the tasks compared to control condition (p > .05). Conclusions: We found that participants had fewer lapses of attention upon awakening when exposed to blue-enriched light compared to dim, red light. In addition, participants reported feeling more alert, more cheerful, less depressed, and less lethargic in the blue-enriched light condition. Brief exposure to a peppermint odor, however, did not appear to improve performance, alertness, or mood under the experimental conditions. Our null results in the peppermint condition may have been due to methodological limitations such as the duration and method of administration. Given the need to mitigate the potential impact of sleep inertia on safety-critical tasks in on-call operations, our findings suggest that blue-enriched light exposure upon awakening may help to improve performance and alertness during the sleep inertia period following awakening from deep, nocturnal sleep. We are currently exploring the potential mechanisms for the effect of light on cognitive performance upon awakening as well as investigating its application in real-world settings to explore the translational efficacy of this countermeasure to occupational environments. Continued exploration into light and other reactive countermeasures, and potentially their combination, is needed in order to provide evidence-based guidance on effective sleep inertia countermeasures to improve the alertness and performance of those required to perform safety-critical tasks soon after waking.

sleep inertia↗
DOE OSTI2019

Uncertainty quantification methodology for hyperbolic systems with application to blood flow in arteries

We present a Stochastic Finite Volume - ADER (SFV-ADER) methodology for Uncertainty Quantification (UQ) in the general framework of systems of hyperbolic balance laws with uncertainty in parameters. The resulting method is weakly intrusive, meaning that deterministic solvers need minor modifications to include random contributions, and has no theoretical accuracy barrier. An illustration of the second-order version for the viscous Burgers equation with uncertain viscosity coefficient is first given in detail, under different deterministic initial conditions; the attainment of the theoretically expected convergence rate is demonstrated empirically; results and features of our scheme are discussed. We then extend the SFV-ADER method to a non-linear hyperbolic system with source terms that models one-dimensional blood flow in arteries, assuming uncertainties in a set of parameters of the problem. Results are then compared with measurements in a well-defined 1:1 experimental replica of the network of largest arteries in the human systemic circulation. For a 99%-confidence level, severe influence of parameter fluctuations is seen in pressure profiles, while secondary effects on flow rate profiles become visible, particularly in terminal branches. In conclusion, results suggest that the proposed methodology could successfully be applied to other physical problems involving hyperbolic balance laws.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2021

Utilizing physics-based input features within a machine learning model to predict wind speed forecasting error

Machine learning is quickly becoming a commonly used technique for wind speed and power forecasting. Many machine learning methods utilize exogenous variables as input features, but there remains the question of which atmospheric variables are most beneficial for forecasting, especially in handling non-linearities that lead to forecasting error. This question is addressed via creation of a hybrid model that utilizes an autoregressive integrated moving-average (ARIMA) model to make an initial wind speed forecast followed by a random forest model that attempts to predict the ARIMA forecasting error using knowledge of exogenous atmospheric variables. Variables conveying information about atmospheric stability and turbulence as well as inertial forcing are found to be useful in dealing with non-linear error prediction. Streamwise wind speed, time of day, turbulence intensity, turbulent heat flux, vertical velocity, and wind direction are found to be particularly useful when used in unison for hourly and 3 h timescales. The prediction accuracy of the developed ARIMA–random forest hybrid model is compared to that of the persistence and bias-corrected ARIMA models. The ARIMA–random forest model is shown to improve upon the latter commonly employed modeling methods, reducing hourly forecasting error by up to 5 % below that of the bias-corrected ARIMA model and achieving an R 2 value of 0.84 with true wind speed.

17 WIND ENERGY↗
NASA NTRS1982

Investigation of continuously traversing microphone system for mode measurement

The continuously Traversing Microphone System consists of a data acquisition and processing method for obtaining the modal coefficients of the discrete, coherent acoustic field in a fan inlet duct. The system would be used in fan rigs or full scale engine installations where present measurement methods, because of the excessive number of microphones and long test times required, are not feasible. The purpose of the investigation reported here was to develop a method for defining modal structure by means of a continuously traversing microphone system and to perform an evaluation of the method, based upon analytical studies and computer simulated tests. A variety of system parameters were examined, and the effects of deviations from ideal were explored. Effects of traverse speed, digitizing rate, run time, roundoff error, calibration errors, and random noise background level were determined. For constant fan operating speed, the sensitivity of the method to normal errors and deviations was determined to be acceptable. Good recovery of mode coefficients was attainable. Fluctuating fan speed conditions received special attention, and it was concluded that by employing suitable time delay procedures, satisfactory information on mode coefficients can be obtained under realistic conditions. A plan for further development involving fan rig tests was prepared.

Cicon, D. E.↗
NASA NTRS1994

Fatigue reliability method with in-service inspections

The first reliability method (FORM) has traditionally been applied in a fatigue reliability setting to one inspection interval at a time, so that the random distribution of crack lengths must be recharacterized following each inspection. The FORM presented here allow each analysis to span several inspection periods without explicit characterization of the crack length distribution upon each inspection. The method thereby preserves the attractive feature of FORM in that relatively few realizations in the random variable space need to be considered. Examples are given which show that the present methodology gives estimates which are in good agreement with Monte Carlo simulations and is efficient even for complex components.

Harkness, H. H.↗