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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.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2011

An approach to assessing stochastic radiogenic risk in medical imaging

Purpose: This letter suggests a formalism, the medical effective dose (MED), that is suitable for assessing stochastic radiogenic risks in diagnostic medical procedures. Methods: The MED is derived from radiobiological and probabilistic first principals, including: (1) The independence of radiation-induced biological effects in neighboring voxels at low doses; (2) the linear no-threshold assumption for stochastic radiation injury (although other dose-response relationships could be incorporated, instead); (3) the best human radiation dose-response data currently available; and (4) the built-in possibility that the carcinogenic risk to an irradiated organ may depend on its volume. The MED involves a dose-risk summation over irradiated voxels at high spatial resolution; it reduces to the traditional effective dose when every organ is irradiated uniformly and when the dependence of risk on organ volumes is ignored. Standard relative-risk tissue weighting factors can be used with the MED approach until more refined data become available. Results: The MED is intended for clinical and phantom dosimetry, and it provides an estimate of overall relative radiogenic stochastic risk for any given dose distribution. A result of the MED derivation is that the stochastic risk may increase with the volume of tissue (i.e., the number of cells) irradiated, a feature that can be activated when forthcoming radiobiological research warrants it. In this regard, the MED resembles neither the standard effective dose (E) nor the CT dose index (CTDI), but it is somewhat like the CT dose-length product (DLP). Conclusions: The MED is a novel, probabilistically and biologically based means of estimating stochastic-risk-weighted doses associated with medical imaging. Built in, ab initio, is the ability to link radiogenic risk to organ volume and other clinical factors. It is straightforward to implement when medical dose distributions are available, provided that one is content, for the time being, to accept the relative tissue weighting factors published by the International Commission of Radiological Protection (ICRP). It requires no new radiobiological data and avoids major problems encountered by the E, CTDI, and CT-E formalisms. It makes possible relative inter-patient dosimetry, and also realistic intercomparisons of stochastic risks from different protocols that yield images of comparable quality.

61 RADIATION PROTECTION AND DOSIMETRY↗
DOE OSTI2016

SU-F-T-182: A Stochastic Approach to Daily QA Tolerances On Spot Properties for Proton Pencil Beam Scanning

Purpose: Proton pencil beam scanning is used clinically across the United States. There are no current guidelines on tolerances for daily QA specific to pencil beam scanning, specifically related to the individual spot properties (spot width). Using a stochastic method to determine tolerances has the potential to optimize tolerances on individual spots and decrease the number of false positive failures in daily QA. Individual and global spot tolerances were evaluated. Methods: As part of daily QA for proton pencil beam scanning, a field of 16 spots (corresponding to 8 energies) is measured using an array of ion chambers (Matrixx, IBA). Each individual spot is fit to two Gaussian functions (x,y). The spot width (σ) in × and y are recorded (32 parameters). Results from the daily QA were retrospectively analyzed for 100 days of data. The deviations of the spot widths were histogrammed and fit to a Gaussian function. The stochastic spot tolerance was taken to be the mean ± 3σ. Using these results, tolerances were developed and tested against known deviations in spot width. Results: The individual spot tolerances derived with the stochastic method decreased in 30/32 instances. Using the previous tolerances (± 20% width), the daily QA would have detected 0/20 days of the deviation. Using a tolerance of any 6 spots failing the stochastic tolerance, 18/20 days of the deviation would have been detected. Conclusion: Using a stochastic method we have been able to decrease daily tolerances on the spot widths for 30/32 spot widths measured. The stochastic tolerances can lead to detection of deviations that previously would have been picked up on monthly QA and missed by daily QA. This method could be easily extended for evaluation of other QA parameters in proton spot scanning.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2017

An assessment of flexibility reserves in stochastic modeling at multiple timescales

Due to recent technological achievements, stochastic optimization, which inherently captures the uncertainty of intermittent resources, is being used to capture the variability and uncertainty of wind and solar resources. However, due to persistent computational limitations, it is not practical to consider all possible variable generation scenarios. As a result, a reduced number of most likely scenarios is usually considered. While this helps reduce the computational burden, it also leaves the system operator vulnerable to some risk. In order to address this issue, this paper aims at providing insight into using an explicit reserve requirement in a stochastic modeling framework in order to provide system operators with greater confidence in stochastic dispatch solutions. This is accomplished by simulating a modified version of the IEEE 118 bus system in a fully stochastic, multi-timescale framework with flexibility reserve requirements. Results show that utilizing a stochastic flexibility reserve requirement within the stochastic modeling framework offers the most reliability benefit.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2017

Analytical Assessment for Transient Stability Under Stochastic Continuous Disturbances

Here, with the growing integration of renewable power generation, plug-in electric vehicles, and other sources of uncertainty, increasing stochastic continuous disturbances are brought to power systems. The impact of stochastic continuous disturbances on power system transient stability attracts significant attention. To address this problem, this paper proposes an analytical assessment method for transient stability of multi-machine power systems under stochastic continuous disturbances. In the proposed method, a probability measure of transient stability is presented and analytically solved by stochastic averaging. Compared with the conventional method (Monte Carlo simulation), the proposed method is many orders of magnitude faster, which makes it very attractive in practice when many plans for transient stability must be compared or when transient stability must be analyzed quickly. Also, it is found that the evolution of system energy over time is almost a simple diffusion process by the proposed method, which explains the impact mechanism of stochastic continuous disturbances on transient stability in theory.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2023

Probabilistic-learning-based stochastic surrogate model from small incomplete datasets for nonlinear dynamical systems

We consider a high-dimensional nonlinear computational model of a dynamical system, parameterized by a vector-valued control parameter, in the presence of uncertainties represented by an uncontrolled parameter modeled by a vector-valued random variable, and possibly with stochastic excitation. The objective is to construct a statistical surrogate model where the input is any deterministic value of the control parameter, and the output is a vector-valued observation of the computational model, which is a random vector whose probability measure is updated using a target dataset. To construct this statistical surrogate model, the stochastic response of the computational model must be built, which is a vector-valued time-discretized stochastic process in high dimension, depending on the control parameter. It is assumed that the computational cost of a single evaluation of the deterministic model is high. For the probabilistic updating, we consider a subset of the components of the observation of the computational model, defined as the “identification observation” of the computational model, for which a small target dataset is available. Therefore, the target dataset is associated with partial observability, corresponding to an incomplete data case. Given a prior probability model of the random control and uncontrolled parameters, a training dataset is constructed, consisting of realizations of the random triplet composed of the stochastic response, the random identification observation, and the random control parameter. Since the computational cost of a single evaluation of the deterministic model is assumed to be large, the training dataset is also of small size. The main challenges in this problem are the high dimensionality, partial observability leading to incomplete data in the target dataset for the identification observation of the computational model (which is not sufficient to identify the computational stochastic responses), and the availability of a small training dataset. To address these challenges, we propose a methodology based on statistical methods for constructing necessary reduced representations, direct probabilistic learning under constraints using probabilistic learning on manifolds (PLoM) constrained by the target dataset, and the use of a weak formulation of the Fourier transform of probability measures. Statistical conditioning is also employed to explore the learned dataset. The constructed predictive statistical surrogate model can be implemented in the context of online computation. Here, we apply this approach to a problem of nonlinear stochastic dynamics in high dimensions within the framework of deformable solids mechanics.

Engineering↗
DOE OSTI2017

Stochastic Formulation of the Resolution of Identity: Application to Second Order Møller–Plesset Perturbation Theory

A stochastic orbital approach to the resolution of identity (RI) approximation for 4-index electron repulsion integrals (ERIs) is presented. The stochastic RI-ERIs are then applied to second order Møller–Plesset perturbation theory (MP2) utilizing a multiple stochastic orbital approach. The introduction of multiple stochastic orbitals results in an O(N AO 3 ) scaling for both the stochastic RI-ERIs and stochastic RI-MP2, NAO being the number of basis functions. For a range of water clusters we demonstrate that this method exhibits a small prefactor and observed scalings of O(N e 2.4 ) for total energies and O(N e 3.1 ) for forces (N e being the number of correlated electrons), outperforming MP2 for clusters with as few as 21 water molecules.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Optimization and Evaluation of Stochastic Unified Convection Using Single-Column Model Simulations at Multiple Observation Sites

We extend the previously developed stochastic unified convection scheme (UNICON) for shallow convection to deep convection by parameterizing the impact of mesoscale organized flow on updraft properties. The extended stochastic UNICON parameterizes thermodynamic properties of updrafts at the near-surface as a multivariate Gaussian distribution, where the variances of the distribution are the summation of variances from non-organized turbulence and mesoscale organized flow. The distribution of updraft radius is parameterized as a power-law distribution with a scale break which is parameterized as a linear function of the strength of mesoscale organized flow. The proposed parameterization is validated using a series of large-eddy simulations of deep convection. The free parameters introduced in the formulation of stochastic UNICON are optimized using 10 cases of single-column model simulations over the ocean. Stochastic UNICON with the optimized parameters significantly reduces the biases of thermodynamic profiles and surface precipitation rates simulated in the original UNICON for tropical convection cases. The simulation of the variation in anomalies of temperature and moisture associated with the Madden-Julian oscillation is also improved. The overall improvements in simulated thermodynamic profiles are found to be due to the increased heating and drying tendencies by convective processes in stochastic UNICON. An additional simulation of an idealized deep convection case shows that stochastic UNICON produces enhanced cloud variabilities with dependency on updraft radius, indicating its ability to represent the coexistence of shallow and deep convection.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2019

Energy window stochastic density functional theory

We report that linear scaling density functional theory is important for understanding electronic structure properties of nanometer scale systems. Recently developed stochastic density functional theory can achieve linear or even sublinear scaling for various electronic properties without relying on the sparsity of the density matrix. The basic idea relies on projecting stochastic orbitals onto the occupied space by expanding the Fermi-Dirac operator and repeating this for N χ stochastic orbitals. Often, a large number of stochastic orbitals are required to reduce the statistical fluctuations (which scale as N$-1/2\atop{χ}$) below a tolerable threshold. In this work, we introduce a new stochastic density functional theory that can efficiently reduce the statistical fluctuations for certain observable and can also be integrated with an embedded fragmentation scheme. The approach is based on dividing the occupied space into energy windows and projecting the stochastic orbitals with a single expansion onto all windows simultaneously. This allows for a significant reduction of the noise as illustrated for bulk silicon with a large supercell. Finally, we also provide theoretical analysis to rationalize why the noise can be reduced only for a certain class of ground state properties, such as the forces and electron density.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2017

Energy Efficient Stochastic-Based Deep Spiking Neural Networks for Sparse Datasets

With large deep neural networks (DNNs) neces- sary to solve complex and data-intensive problems, energy efficiency is a key bottleneck for effectively deploying DL in the real world. Deep spiking NNs have gained much research at- tention recently due to the interest in building biological neural networks and the availability of neuromorphic platforms, which can be orders of magnitude more energy efficient compared to CPUs and GPUs. Although spiking NNs have proven to be an efficient technique for solving many machine learning and computer vision problems, to the best of our knowledge, this is the first attempt to adapt spiking NNs to sparse datasets. In this paper, we study the behaviour of spiking NNs in handling NLP datasets and the sparsity in their data representation. Then, we propose a novel framework for spiking NN using the concept of stochastic computing. Specifically, instead of generating spike trains with firing rates proportional to the intensity of each value in the feature set separately, the whole feature set is treated as a distribution function and a stochastic spiking train that follow this distribution is generated. This framework reduces the connectivity between NN layers from O(N) to O(logN). Also, it encodes input data differently and make suitable to handle sparse datasets. Finally, the framework achieves high energy efficiency since it uses Integrate and Fire neurons same as conventional spiking NNs. The results show that our proposed stochastic-based SNN achieves nearly the same accuracy as the original DNN on MNIST dataset, and it has better performance than state- of-the-art SNN. Besides that stochastic-based SNN is energy efficient, where the fully connected DNN, the conventional SNN, and the data normalized SNN consume 38.24, 1.83, and 1.85 -times more energy than the stochastic-based SNN, respectively. For sparse datasets, including IMDb and In-House clinical datasets, stochastic-based SNN achieves performance comparable to that of the conventional DNN. However, the conventional spiking NN has a significant decline in classification accuracy.

Alawad, Mohammed↗
DOE OSTI2016

Inverse Subspace Iteration for Spectral Stochastic Finite Element Methods

We study random eigenvalue problems in the context of spectral stochastic finite elements. In particular, given a parameter-dependent, symmetric positive-definite matrix operator, we explore the performance of algorithms for computing its eigenvalues and eigenvectors represented using polynomial chaos expansions. We formulate a version of stochastic inverse subspace iteration, which is based on the stochastic Galerkin finite element method, and we compare its accuracy with that of Monte Carlo and stochastic collocation methods. The coefficients of the eigenvalue expansions are computed from a stochastic Rayleigh quotient. Our approach allows the computation of interior eigenvalues by deflation methods, and we can also compute the coefficients of multiple eigenvectors using a stochastic variant of the modified Gram-Schmidt process. The effectiveness of the methods is illustrated by numerical experiments on benchmark problems arising from vibration analysis.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks

One of the open problems in scientific computing is the long-time integration of nonlinear stochastic partial differential equations (SPDEs), especially with arbitrary initial data. We address this problem by taking advantage of recent advances in scientific machine learning and the spectral dynamically orthogonal (DO) and borthogonal (BO) methods for representing stochastic processes. The recently introduced DO/BO methods reduce the SPDE to solving a system of deterministic PDEs and a system of stochastic ordinary differential equations. Specifically, we propose two new physics-informed neural networks (PINNs) for solving time-dependent SPDEs, namely the neural network (NN)-DO/BO methods. The proposed methods incorporate the DO/BO constraints into the loss function (along with the modal decomposition of the SPDE) with an implicit form instead of generating explicit expressions for the temporal derivatives of the DO/BO modes. Hence, the NN-DO/BO methods can overcome some of the drawbacks of the original DO/BO methods. For example, we do not need the assumption that the covariance matrix of the random coefficients is invertible as in the original DO method, and we can remove the assumption of no eigenvalue crossing as in the original BO method. Moreover, the NN-DO/BO methods can be used to solve time-dependent stochastic inverse problems with the same formulation and same computational complexity as for forward problems. Furthermore, we demonstrate the capability of the proposed methods via several numerical examples, namely: (1) A linear stochastic advection equation with deterministic initial condition: we obtain good results with the proposed methods, while the original DO/BO methods cannot be applied directly in this case. (2) Long-time integration of the stochastic Burgers' equation: we show the good performance of NN-DO/BO methods, especially the effectiveness of the NN-BO approach for such problems with many eigenvalue crossings during the whole time evolution, while the original BO method fails. (3) Nonlinear reaction diffusion equation: we consider both the forward problem and the inverse problems, including very noisy initial point values, to investigate the flexibility of the NN-DO/BO methods in handling inverse and mixed type problems. Taken together, these simulation results demonstrate that the NN-DO/BO methods can be employed to effectively quantify uncertainty propagation in a wide range of physical problems, but future work should address the efficiency issue of PINNs for forward problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

Spatial Stochastic Dynamics Enable Robust Cell Polarization

Although cell polarity is an essential feature of living cells, it is far from being well-understood. Using a combination of computational modeling and biological experiments we closely examine an important prototype of cell polarity: the pheromone-induced formation of the yeast polarisome. Focusing on the role of noise and spatial heterogeneity, we develop and investigate two mechanistic spatial models of polarisome formation, one deterministic and the other stochastic, and compare the contrasting predictions of these two models against experimental phenotypes of wild-type and mutant cells. We find that the stochastic model can more robustly reproduce two fundamental characteristics observed in wild-type cells: a highly polarized phenotype via a mechanism that we refer to as spatial stochastic amplification, and the ability of the polarisome to track a moving pheromone input. Moreover, we find that only the stochastic model can simultaneously reproduce these characteristics of the wild-type phenotype and the multi-polarisome phenotype of a deletion mutant of the scaffolding protein Spa2. Significantly, our analysis also demonstrates that higher levels of stochastic noise results in increased robustness of polarization to parameter variation. Furthermore, our work suggests a novel role for a polarisome protein in the stabilization of actin cables. These findings elucidate the intricate role of spatial stochastic effects in cell polarity, giving support to a cellular model where noise and spatial heterogeneity combine to achieve robust biological function.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2012

Detection and Characterization of Underground Facilities by Stochastic Inversion and Modeling of Data from the New Generation of Synthetic Aperture Satellites (Final Report)

Many clandestine development and production activities can be conducted underground to evade surveillance. The purpose of the study reported here was to develop a technique to detect underground facilities by broad-area search and then to characterize the facilities by inversion of the collected data. This would enable constraints to be placed on the types of activities that would be feasible at each underground site, providing a basis the design of targeted surveillance and analysis for more complete characterization. Excavation of underground cavities causes deformation in the host material and overburden that produces displacements at the ground surface. Such displacements are often measurable by a variety of surveying or geodetic techniques. One measurement technique, Interferometric Synthetic Aperture Radar (InSAR), uses data from satellite-borne (or airborne) synthetic aperture radars (SARs) and so is ideal for detecting and measuring surface displacements in denied access regions. Depending on the radar frequency and the acquisition mode and the surface conditions, displacement maps derived from SAR interferograms can provide millimeter- to centimeter-level measurement accuracy on regional and local scales at spatial resolution of {approx}1-10 m. Relatively low-resolution ({approx}20 m, say) maps covering large regions can be used for broad-area detection, while finer resolutions ({approx}1 m) can be used to image details of displacement fields over targeted small areas. Surface displacements are generally expected to be largest during or a relatively short time after active excavation, but, depending on the material properties, measurable displacement may continue at a decreasing rate for a considerable time after completion. For a given excavated volume in a given geological setting, the amplitude of the surface displacements decreases as the depth of excavation increases, while the area of the discernable displacement pattern increases. Therefore, the ability to detect evidence for an underground facility using InSAR depends on the displacement sensitivity and spatial resolution of the interferogram, as well as on the size and depth of the facility and the time since its completion. The methodology development described in this report focuses on the exploitation of synthetic aperture radar data that are available commercially from a number of satellite missions. Development of the method involves three components: (1) Evaluation of the capability of InSAR to detect and characterize underground facilities ; (2) inversion of InSAR data to infer the location, depth, shape and volume of a subsurface facility; and (3) evaluation and selection of suitable geomechanical forward models to use in the inversion. We adapted LLNL's general-purpose Bayesian Markov Chain-Monte Carlo procedure, the 'Stochastic Engine' (SE), to carry out inversions to characterize subsurface void geometries. The SE performs forward simulations for a large number of trial source models to identify the set of models that are consistent with the observations and prior constraints. The inverse solution produced by this kind of stochastic method is a posterior probability density function (pdf) over alternative models, which forms an appropriate input to risk-based decision analyses to evaluate subsequent response strategies. One major advantage of a stochastic inversion approach is its ability to deal with complex, non-linear forward models employing empirical, analytical or numerical methods. However, while a geomechanical model must incorporate adequate physics to enable sufficiently accurate prediction of surface displacements, it must also be computationally fast enough to render the large number of forward realizations needed in stochastic inversion feasible. This latter requirement prompted us first to investigate computationally efficient empirical relations and closed-form analytical solutions. However, our evaluation revealed severe limitations in the ability of existing empirical and analytical forms to predict deformations from underground cavities with an accuracy consistent with the potential resolution and precision of InSAR data. We followed two approaches to overcoming these limitations. The first was to develop a new analytical solution for a 3D cavity excavated in an elastic half-space. The second was to adapt a fast parallelized finite element method to the SE and evaluate the feasibility of using in the stochastic inversion. To date we have demonstrated the ability of InSAR to detect underground facilities and measure the associated surface displacements by mapping surface deformations that track the excavation of the Los Angeles Metro system. The Stochastic Engine implementation has been completed and undergone functional testing.

58 GEOSCIENCES↗
DOE OSTI2012

Large-scale Stochastic Optimization for Unit Commitment and Economic Dispatch

Our FY 2012 LDRD research on stochastic optimization with its application in power grid planning and operations has advanced the state-of-the-art of this area in the following aspects: (1) Deploying stochastic unit commitment on large real-world applications, (2) Stochastic optimization coupled with real-time production cost simulation (interleaved), and (3) Real-time stochastic optimization with rolling-horizon look-ahead. In conjunction, a number of computational experiments have been performed to evaluate the capability of the latest stochastic optimization software and explore opportunities for future research. In what follows, we discuss each of these contribution areas, including the results of our computational experiments and their implications on the scalability and solvability of large-scale stochastic optimization models.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2012

Large-Scale Stochastic Optimization for Unit Commitment and Economic Dispatch

Our FY 2012 LDRD research on stochastic optimization with its application in power grid planning and operations has advanced the state-of-the-art of this area in the following aspects: (1) Deploying stochastic unit commitment on large real-world applications, (2) Stochastic optimization coupled with real-time production costsimulation (interleaved), and (3) Real-time stochastic optimization with rolling-horizon look-ahead. In conjunction, a number of computational experiments have been performed to evaluate the capability of the latest stochastic optimization software and explore opportunities for future research. In what follows, we discuss each of these contribution areas, including the results of our computational experiments and their implications on the scalability and solvability of large-scale stochastic optimization models.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2023

Latency considerations for stochastic optimizers in variational quantum algorithms

Variational quantum algorithms, which have risen to prominence in the noisy intermediate-scale quantum setting, require the implementation of a stochastic optimizer on classical hardware. To date, most research has employed algorithms based on the stochastic gradient iteration as the stochastic classical optimizer. In this work we propose instead using stochastic optimization algorithms that yield stochastic processes emulating the dynamics of classical deterministic algorithms. This approach results in methods with theoretically superior worst-case iteration complexities, at the expense of greater per-iteration sample (shot) complexities. We investigate this trade-off both theoretically and empirically and conclude that preferences for a choice of stochastic optimizer should explicitly depend on a function of both latency and shot execution times.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

Temporal Coarse Graining for Classical Stochastic Noise in Quantum Systems

Simulations of quantum systems with Hamiltonian classical stochastic noise can be challenging when the noise exhibits temporal correlations over a multitude of time scales, such as for 1/f noise in solid-state quantum information processors. Here we present an approach for simulating Hamiltonian classical stochastic noise that performs temporal coarse-graining by effectively integrating out the high-frequency components of the noise. We focus on the case where the stochastic noise can be expressed as a sum of Ornstein-Uhlenbeck processes. Temporal coarse-graining is then achieved by conditioning the stochastic process on a coarse realization of the noise, expressing the conditioned stochastic process in terms of a sum of smooth, deterministic functions and bridge processes with boundaries fixed at zero, and performing the ensemble average over the bridge processes. For Ornstein-Uhlenbeck processes, the deterministic components capture all dependence on the coarse realization, and the stochastic bridge processes are not only independent but taken from the same distribution with correlators that can be expressed analytically, allowing the associated noise propagators to be precomputed once for all simulations. This combination of noise trajectories on a coarse time grid and ensemble averaging over bridge processes has practical advantages, such as a simple concatenation rule, that we highlight with numerical examples.

Albash, Tameem [Sandia National Lab. (SNL-NM), Alb↗
NASA NTRS1990

Stochastic performance robustness of aircraft control systems

Stochastic robustness, a simple technique used to estimate the robustness of linear, time-invariant systems, is applied to a twin-jet transport aircraft control system. Concepts behind stochastic stability robustness are extended to stochastic performance robustness. Stochastic performance robustness measures based on classical design specifications and measures specific to aircraft handling qualities are introduced. Confidence intervals for both individual stochastic robustness measures and for comparing two measures are presented. The application of stochastic performance robustness, the use of confidence intervals, and tradeoffs between performance objectives are demonstrated by means of the twin-jet aircraft example.

Stengel, Robert F.↗