Correlated uncertainty arithmetic with application to fusion neutronics

his thesis advances the idea of automatic and rigorous uncertainty propagation for computational science. The aim is to replace the deterministic arithmetic and logical operations composing a function or a computer program with their uncertain equivalents. In this thesis, uncertain computer variables are labelled uncertain numbers, which may be probability distributions, intervals, probability boxes, and possibility distributions. The individual models of uncertainty are surveyed in the context of imprecise probability theory, and their individual arithmetic described and developed, with new results presented in each arithmetic. The presented arithmetic framework allows random variables to be imprecisely characterised or partially defined. It is a common situation that input random variables are unknown or that only certain characteristics of the inputs are known. How uncertain numbers can be rigorously represented by a finite numerical discretisation is described. Further, it is shown how arithmetic operations are computed by numerical convolution, accounting for both the error from the input's discretisation and from the numerical integration, yielding guaranteed bounds on computed uncertain numbers. One of the central topics of this thesis is stochastic dependency. Considering complex dependencies amongst uncertain numbers is necessary, as it plays a key role in operations. An arithmetic operation between two uncertain numbers is a function not only of the input numbers, but also how they are correlated. This is often more important than the marginal information. In the presented arithmetic, dependencies between uncertain numbers may also be partially defined or missing entirely. A major proposition of this thesis are methods to propagate dependence information through functions alongside marginal information. The long-term goal is to solve probabilistic problems with partial knowledge about marginal distributions and dependencies using algorithms which were written deterministically. The developed arithmetic frameworks can be used individually, or may be combined into a larger uncertainty computing framework. We present an application of the developed method to a radiation transport algorithm for nuclear fusion neutronics problems

Uncertainty propagation in nonlinear systems.

This thesis examines the effects of uncertainty on a variety of different engineering systems. Uncertainty can be best described as a lack of knowledge for a particular system, and can come from a variety of different sources. Within this thesis the possibilistic branch of uncertainty quantification is used. A combination of simulated and real-life engineering systems are studied, covering some of the most popular types of computational models. An outline of various background topics is presented first, as these topics are all subsequently used within the thesis. The most important of these is the transformation method, a possibilistic uncertainty approach derived from fuzzy arithmetic. Most of the work here examines uncertain systems by implementing Ben-Haim's information gap theory. Uncertainty is deliberately introduced into the parameters of the various computational models to use the concept of “opportunity”. The basic rationale is that if some degree of tolerance can be accepted on a model prediction of a system, it is possible to obtain a lower value of prediction error than with a standard crisp-valued model. For the use of interval-valued computational models there is generally a trade-off to be made between minimising the prediction error of the model and minimising the range of predicted outputs, to reduce the tolerance on the solution. The studied models all use a “degree of uncertainty” parameter that allows any user to select the suitable trade-off level for their particular application. The thesis then concludes with a real-life engineering study, undertaken as a nine month placement on a European Union project entitled MADUSE. The work was done at Centro Ricerche Fiat, and examined the dynamic effects of uncertainties related to automotive spot welds. This study used both finite element modelling and experimental modal testing of manufactured specimens

A survey of probabilistic methods used in reliability, risk and uncertainty analysis: Analytical techniques 1

This report provides an introduction to the various probabilistic methods developed roughly between 1956--1985 for performing reliability or probabilistic uncertainty analysis on complex systems. This exposition does not include the traditional reliability methods (e.g. parallel-series systems, etc.) that might be found in the many reliability texts and reference materials (e.g. and 1977). Rather, the report centers on the relatively new, and certainly less well known across the engineering community, analytical techniques. Discussion of the analytical methods has been broken into two reports. This particular report is limited to those methods developed between 1956--1985. While a bit dated, methods described in the later portions of this report still dominate the literature and provide a necessary technical foundation for more current research. A second report (Analytical Techniques 2) addresses methods developed since 1985. The flow of this report roughly follows the historical development of the various methods so each new technique builds on the discussion of strengths and weaknesses of previous techniques. To facilitate the understanding of the various methods discussed, a simple 2-dimensional problem is used throughout the report. The problem is used for discussion purposes only; conclusions regarding the applicability and efficiency of particular methods are based on secondary analyses and a number of years of experience by the author. This document should be considered a living document in the sense that as new methods or variations of existing methods are developed, the document and references will be updated to reflect the current state of the literature as much as possible. For those scientists and engineers already familiar with these methods, the discussion will at times become rather obvious. However, the goal of this effort is to provide a common basis for future discussions and, as such, will hopefully be useful to those more intimate with probabilistic analysis and design techniques. There are clearly alternative methods of dealing with uncertainty (e.g. fuzzy set theory, possibility theory), but this discussion will be limited to those methods based on probability theory

Biclustering on expression data: A review

Biclustering has become a popular technique for the study of gene expression data, especially for discovering functionally related gene sets under different subsets of experimental conditions. Most of biclustering approaches use a measure or cost function that determines the quality of biclusters. In such cases, the development of both a suitable heuristics and a good measure for guiding the search are essential for discovering interesting biclusters in an expression matrix. Nevertheless, not all existing biclustering approaches base their search on evaluation measures for biclusters. There exists a diverse set of biclustering tools that follow different strategies and algorithmic concepts which guide the search towards meaningful results. In this paper we present a extensive survey of biclustering approaches, classifying them into two categories according to whether or not use evaluation metrics within the search method: biclustering algorithms based on evaluation measures and non metric-based biclustering algorithms. In both cases, they have been classified according to the type of meta-heuristics which they are based on.Ministerio de Economía y Competitividad TIN2011-2895

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

B

Distribution-free stochastic simulation methodology for model updating under hybrid uncertainties

In the real world, a significant challenge faced in the safe operation and maintenance of infrastructures is the lack of available information or data. This results in a large degree of uncertainty and the requirement for robust and efficient uncertainty quantification (UQ) tools in order to derive the most realistic estimates of the behavior of structures. While the probabilistic approach has long been utilized as an essential tool for the quantitative mathematical representation of uncertainty, a common criticism is that the approach often involves insubstantiated subjective assumptions because of the scarcity or imprecision of available information. To avoid the inclusion of subjectivity, the concepts of imprecise probabilities have been developed, and the distributional probability-box (p-box) has gained the most attention among various types of imprecise probability models since it can straightforwardly provide a clear separation between aleatory and epistemic uncertainty. This thesis concerns the realistic consideration and numerically efficient calibraiton and propagation of aleatory and epistemic uncertainties (hybrid uncertainties) based on the distributional p-box. The recent developments including the Bhattacharyya distance-based approximate Bayesian computation (ABC) and non-intrusive imprecise stochastic simulation (NISS) methods have strengthened the subjective assumption-free approach for uncertainty calibration and propagation. However, these methods based on the distributional p-box stand on the availability of the prior knowledge determining a specific distribution family for the p-box. The target of this thesis is hence to develop a distribution-free approach for the calibraiton and propagation of hybrid uncertainties, strengthening the subjective assumption-free UQ approach. To achieve the above target, this thesis presents five main developments to improve the Bhattacharyya distance-based ABC and NISS frameworks. The first development is on improving the scope of application and efficiency of the Bhattacharyya distance-based ABC. The dimension reduction procedure is proposed to evaluate the Bhattacharyya distance when the system under investigation is described by time-domain sequences. Moreover, the efficient Bayesian inference method within the Bayesian updating with structural reliability methods (BUS) framework is developed by combining BUS with the adaptive Kriging-based reliability method, namely AK-MCMC. The second development of the distribution-free stochastic model updating framework is based on the combined application of the staircase density functions and the Bhattacharyya distance. The staircase density functions can approximate a wide range of distributions arbitrarily close; hence the development achieved to perform the Bhattacharyya distance-based ABC without limiting hypotheses on the distribution families of the parameters having to be updated. The aforementioned two developments are then integrated in the third development to provide a solution to the latest edition (2019) of the NASA UQ challenge problem. The model updating tasks under very challenging condition, where prior information of aleatory parameters are extremely limited other than a common boundary, are successfully addressed based on the above distribution-free stochastic model updating framework. Moreover, the NISS approach that simplifies the high-dimensional optimization to a set of one-dimensional searching by a first-order high-dimensional model representation (HDMR) decomposition with respect to each design parameter is developed to efficiently solve the reliability-based design optimization tasks. This challenge, at the same time, elucidates the limitations of the current developments, hence the fourth development aims at addressing the limitation that the staircase density functions are designed for univariate random variables and cannot acount for the parameter dependencies. In order to calibrate the joint distribution of correlated parameters, the distribution-free stochastic model updating framework is extended by characterizing the aleatory parameters using the Gaussian copula functions having marginal distributions as the staircase density functions. This further strengthens the assumption-free approach for uncertainty calibration in which no prior information of the parameter dependencies is required. Finally, the fifth development of the distribution-free uncertainty propagation framework is based on another application of the staircase density functions to the NISS class of methods, and it is applied for efficiently solving the reliability analysis subproblem of the NASA UQ challenge 2019. The above five developments have successfully strengthened the assumption-free approach for both uncertainty calibration and propagation thanks to the nature of the staircase density functions approximating arbitrary distributions. The efficiency and effectiveness of those developments are sufficiently demonstrated upon the real-world applications including the NASA UQ challenge 2019

Risk as a tool in water resource management

Please read the abstract in the section 00front of this documentThesis (PhD (Water Utilisation))--University of Pretoria, 2005.Civil Engineeringunrestricte