Uncertainty Propagation Algorithm Research Articles

Reliability analysis of deep tunnels in spatially varying brittle rocks using interval and random field modelling

Rock properties are estimated using objective lab/in-situ testing and subjective judgements invoking different types of uncertainties, i.e., aleatory, and epistemic, along them, often indicated by varying information levels. This study presents a unified reliability method to integrate the spatial variability of inputs modelled via alternate uncertainty models (intervals and probability-boxes (p-boxes)) with those modelled as stochastic variables via probability distributions. The methodology employs advanced Karhunen–Loève decomposition to generate interval and random fields of inputs modelled via alternate and stochastic models, respectively. Input properties are allocated to the zones of the numerical model in Fast Lagrangian Analysis of Continua-2D (FLAC-2D) based on their spatial dependency and correlation functions through a developed MATLAB-FLAC coupled code. The methodology is demonstrated for a deep tunnel in Canada to be constructed along a massive rock prone to brittle failures. Intact rock properties are modelled as stochastic variables due to objective estimation, while Geological Strength Index (GSI) and deformation modulus are modelled using alternate models (interval and p-box, respectively) due to subjective and hybrid estimation (double uncertainty propagation algorithm). The results of the methodology are compared with those of traditional deterministic and random field methods. The methodology reduces the subjectivity invoked by including unavailable additional information (e.g., assuming probability distributions of inputs based on literature) and propagates the originally available information of inputs accurately. The final outputs are the p-boxes of response parameters (i.e., displacements and damage zone) instead of their fixed values (deterministic analysis) and probability distributions (traditional reliability analysis), indicating the propagation of both impreciseness and variability of inputs by the method. For this case study, the p-boxes of outputs were bounding their values/distributions estimated via traditional analyses, verifying the accuracy of the methodology. Further, the impreciseness in the outputs, highest in the damage zone extent, was due to imprecision in the estimated GSI.

Robust convex optimization for reentry glide trajectory using polynomial chaos

Aiming at the reentry glide trajectory design of hypersonic vehicles with uncertain parameters, a robust trajectory optimization method is proposed to enhance the anti-interference ability and process reliability. Firstly, a robust trajectory optimization model for re-entry gliding under uncertain conditions is constructed, which is transformed into a deterministic trajectory optimization problem with dimension expansion by using an uncertainty propagation algorithm based on the Gaussian quadrature and non-intrusive polynomial chaos. Then, the convex optimization technology is used to convexity and discretize the problem, and a deterministic trajectory optimization strategy based on the sequential convex optimization algorithm is designed to achieve a fast solution for the high-dimensional deterministic problem. Finally, the hypersonic glide reentry of the American X-33 is chosen as an example for the numerical simulation. The results show that comparing with the traditional deterministic trajectory optimization algorithm, the present method can effectively reduce the impact of random interference on the reentry glide trajectory and improve the reliability and robustness of the trajectory.

Exploiting nonlinear invariants and path constraints to achieve tighter reachable set enclosures using differential inequalities

This article presents a new method for computing sharp bounds on the solutions of nonlinear dynamic systems subject to uncertain initial conditions, parameters, and time-varying inputs. Such bounds are widely used in algorithms for uncertainty propagation, robust state estimation, system verification, global dynamic optimization, and more. Recently, it has been shown that bounds computed via differential inequalities can often be made much less conservative by exploiting state constraints that are known to hold for all trajectories of interest (e.g., path constraints that describe feasible trajectories in the context of dynamic optimization, or constraints that explicitly describe invariant sets containing all system trajectories). However, effective bounding algorithms of this type are currently only available for problems with linear constraints. Moreover, the theoretical results underlying these algorithms do not apply to constraints that depend on time-varying inputs and rely on assumptions that prove to be very restrictive for nonlinear constraints. This article contributes a new differential inequalities theorem that permits the use of a very general class of nonlinear state constraints. Moreover, a new algorithm is presented for efficiently exploiting nonlinear constraints to achieve tighter bounds. The proposed approach is shown to produce very sharp bounds for two challenging case studies.

Effect of delamination on the stochastic natural frequencies of composite laminates

The coupled effect of manufacturing uncertainty and a critical service-life damage condition (delamination) is investigated on the natural frequencies of laminated composite plates. In general, delamination is an unavoidable phenomenon in composite materials encountered often in real-life operating conditions. We have focused on the characterization of dynamic responses of composite plates considering source-uncertainty in the material and geometric properties along with various single and multiple delamination scenarios. A hybrid high dimensional model representation based uncertainty propagation algorithm coupled with layer-wise stochastic finite element model of composites is developed to achieve computational efficiency. The finite element formulation is based on Mindlin's theory considering transverse shear deformation. Numerical results are presented for the stochastic natural frequencies of delaminated composites along with a comprehensive deterministic analysis. Further, an inevitable effect of noise is induced in the surrogate based analysis to explore the effect of various errors and epistemic uncertainties involved with the system.

A laboratory information management system for the analysis of tritium (3H) in environmental waters

Accurate and precise measurements of low levels of tritium (3H) in environmental waters are difficult to attain due to complex steps of sample preparation, electrolytic enrichment, liquid scintillation decay counting, and extensive data processing. We present a Microsoft Access™ relational database application, TRIMS (Tritium Information Management System) to assist with sample and data processing of tritium analysis by managing the processes from sample registration and analysis to reporting and archiving. A complete uncertainty propagation algorithm ensures tritium results are reported with robust uncertainty metrics. TRIMS will help to increase laboratory productivity and improve the accuracy and precision of 3H assays. The software supports several enrichment protocols and LSC counter types. TRIMS is available for download at no cost from the IAEA at www.iaea.org/water.

A generalized multi-resolution expansion for uncertainty propagation with application to cardiovascular modeling

Computational models are used in a variety of fields to improve our understanding of complex physical phenomena. Recently, the realism of model predictions has been greatly enhanced by transitioning from deterministic to stochastic frameworks, where the effects of the intrinsic variability in parameters, loads, constitutive properties, model geometry and other quantities can be more naturally included. A general stochastic system may be characterized by a large number of arbitrarily distributed and correlated random inputs, and a limited support response with sharp gradients or event discontinuities. This motivates continued research into novel adaptive algorithms for uncertainty propagation, particularly those handling high dimensional, arbitrarily distributed random inputs and non-smooth stochastic responses.In this work, we generalize a previously proposed multi-resolution approach to uncertainty propagation to develop a method that improves computational efficiency, can handle arbitrarily distributed random inputs and non-smooth stochastic responses, and naturally facilitates adaptivity, i.e., the expansion coefficients encode information on solution refinement. Our approach relies on partitioning the stochastic space into elements that are subdivided along a single dimension, or, in other words, progressive refinements exhibiting a binary tree representation. We also show how these binary refinements are particularly effective in avoiding the exponential increase in the multi-resolution basis cardinality and significantly reduce the regression complexity for moderate to high dimensional random inputs. The performance of the approach is demonstrated through previously proposed uncertainty propagation benchmarks and stochastic multi-scale finite element simulations in cardiovascular flow.

Uncertainty propagation algorithm from the radiometric calibration to the restored earth observation radiance.

The uncertainty of the radiometric calibration affects the accuracy of the earth observation (EO) radiance restored from the remote sensing digital number (DN) data. However, it has not been intensively analyzed whether they are equivalent to each other. The algorithm to deduce the uncertainty of the restored EO radiance in the solar-reflective spectral range (400-2500 nm) along the uncertainty propagation chain in the radiometric calibration process is proposed. It was validated compared with the traditional calibration uncertainty algorithm through an example of calibrating an imaging spectrometer. The interval about the real EO radiance and the corresponding level of confidence was reported as a result, which shows the possibility to accurately expressing the quality of the restored EO radiance following the rules used in the field of metrology.

Introducing a Comprehensive Data Reduction and Uncertainty Propagation Algorithm for U‐Th Geochronometry with Extraction Chromatography and Isotope Dilution MC‐ICP‐MS

We present an open‐source algorithm in Mathematica application (Wolfram Research) with a transparent data reduction and Monte Carlo simulation of systematic and random uncertainties for U‐Th geochronometry by multi‐collector ICP‐MS. Uranium and thorium were quantitatively separated from matrix elements through a single U/TEVA extraction chromatography step. A rigorous calibrator‐sample bracketing routine was adopted using CRM‐112A and IRMM‐035 standard solutions, doped with an IRMM‐3636a 233U/236U ‘double‐spike’ to account for instrumental mass bias and deviations of measured isotope ratios from certified values. The mean of 234U/238U and 230Th/232Th in the standard solutions varied within 0.42 and 0.25‰ (permil) of certified ratios, respectively, and were consistent with literature values within uncertainties. Based on multiple dissolutions with lithium metaborate flux fusion, U and Th concentrations in USGS BCR‐2 CRM were updated to 1739 ± 2 and 5987 ± 50 ng g−1 (95% CI), respectively. The measurement reproducibility of our analytical technique was evaluated by analysing six aliquots of an in‐house reference material, prepared by homogenising a piece of speleothem (CC3A) from Cathedral Cave, Utah, which returned a mean age of 21483 ± 63 years (95% CI, 2.9‰). Replicate analysis of ten samples from CC3A was consistent with ages previously measured at the University of Minnesota by single‐collector ICP‐MS within uncertainties.

Research of Expert System in Nuclear Power Plant

With the rising of the nuclear power plant accident, making the knowledge base of knowledge express difficultly, the inference efficiency of the device system is decreasing as well, but the complexity of nuclear power plant has led to the uncertainty of knowledge. In order to solve these problems, the credibility factor and uncertainty propagation algorithm of the network model is applied to the knowledge representation and inference of expert system, and using the CLIPS implements the fault reasoning. Experiments shows that use of credibility factor and uncertainty propagation algorithm of the network model can effectively solve the uncertainty information and structured knowledge representation in the event of a failure, and inference efficiency of nuclear power plant is improved.

Autour de la théorie des possibilités quantitative

This article deals with logical and graphical representations of uncertain information in a quantitative possibilistic theory framework. We first provide a deep analysis of relationships between these two forms of representational frameworks. Then, we provide a novel algorithm for reasoning with quantitative possibilistic logic. The algorithm exploits the syntactic relations between quantitative possibilistic logic bases and penalty logic bases. We provide experimental results that compare the propagation algorithm developed for the possibilistic product-based networks and the inference algorithm developed in this paper. quantitative possibilistic theory, penalty logic, uncertainty propagation algorithm.

Algorithms for uncertainty propagation in transmission‐line matrix (TLM) method

AbstractThis paper presents new techniques coupling uncertainty propagation with the transmission‐line matrix method. This improves the quality of electromagnetic simulation. The reason is that in this case the solution provides not only the average field value but also the associated covariances. This is very useful when incomplete or uncertain information is inherent to the problem. The technique is much faster than regular Monte Carlo simulations, which are used in other methods. Copyright © 2007 John Wiley & Sons, Ltd.

Efficient Closed-Loop Production Optimization Under Uncertainty

This article, written by Technology Editor Dennis Denney, contains highlights of paper SPE 94241, “Efficient Closed-Loop Production Optimization Under Uncertainty,” by P. Sarma, L.J. Durlofsky, and K. Aziz, Stanford U., prepared for the 2005 SPE Europec/EAGE Annual Conference, Madrid, Spain, 13–16 June. This closed-loop approach for real-time production optimization consists of three key elements—adjoint models for efficient parameter and control-gradient calculation, polynomial chaos expansions for efficient uncertainty propagation, and Karhunen-Loeve (K-L) expansions and Bayesian inversion theory for efficient real-time model updating (i.e., history matching). The closed-loop procedure provides a substantial improvement in net present value (NPV) over the base case, and the results seem to be very close to those obtained when the reservoir description is known a priori. Introduction A primary goal of reservoir modeling and management is to enable decisions that maximize the production potential of the reservoir. Real-time model-based reservoir management, also known as the closed-loop approach, has generated significant interest. This method uses model-based optimization of reservoir performance under geological uncertainty while incorporating dynamic information in real time, which acts to reduce model uncertainty. For such schemes to be practical, several algorithmic advances are required. The full-length paper details algorithms for efficient closed-loop reservoir management that are based on the application of optimal-control theory, Bayesian inversion theory, K-L expansions, and polynomial chaos expansions. Closed-Loop Approach Fig. 1 shows the “System” box that represents the real system over which some cost function (e.g., NPV or cumulative oil produced) is to be optimized for real-time model-based reservoir management. The system consists of the reservoir, wells, and surface facilities. Here, a set of controls [well rates and bottomhole pressures (BHPs)] are used to maximize or minimize NPV or cumulative oil produced. The low-order-model box represents the approximate model of the system, which, in this case, is the simulation model of the reservoir and facilities. Because knowledge of the reservoir generally is uncertain, the simulation model and its output also are uncertain. The basic idea of the closed-loop approach is to perform an optimization step on the approximate model, apply the obtained optimal controls on the real reservoir, gather new data as they become available, and assimilate these data to reduce model uncertainty. This loop can be repeated many times over the life of the reservoir. The closed-loop approach for efficient real-time optimization consists of three key components: efficient optimization algorithms, efficient model-updating algorithms, and efficient algorithms for uncertainty propagation. In this work, an adjoint model is applied for the efficient calculation of gradients of the objective function with respect to the controls, which then are used by gradient-based optimization algorithms, such as sequential quadratic programming, for optimization. For the model-updating procedure, Bayesian inversion theory is used, and an efficient parameterization of the uncertain parameter field is applied using the K-L expansion. The K-L expansion essentially transforms the correlated input random field into a much smaller set of independent random variables, thereby enabling the use of standard gradient-based inversion algorithms and associated adjoints for very efficient model updating while maintaining the geological correlation.