Analytical and Numerical Methods

Abstract

Uncertainty accompanies our life processes and covers almost all fields of scientific studies. Two general categories of uncertainty, namely, aleatory uncertainty and epistemic uncertainty, exist in the world. While aleatory uncertainty refers to the inherent randomness in nature, derived from natural variability of the physical world (e.g., random show of a flipped coin), epistemic uncertainty origins from human’s lack of knowledge of the physical world, as well as ability of measuring and modeling the physical world (e.g., computation of the distance between two cities). Different kinds of uncertainty call for different handling methods. Aggarwal, Yu, Sarma, and Zhang et al. have made good surveys on uncertain database management based on the probability theory. This paper reviews multidisciplinary uncertainty processing activities in diverse fields. Beyond the dominant probability theory and fuzzy theory, we also review information-gap theory and recently derived uncertainty theory. Practices of these uncertainty handling theories in the domains of economics, engineering, ecology, and information sciences are also described. It is our hope that this study could provide insights to the database community on how uncertainty is managed in other disciplines, and further challenge and inspire database researchers to develop more advanced data management techniques and tools to cope with a variety of uncertainty issues in the real world.

keywords

Uncertainty management, probability theory, Dempster-Shafer theory, fuzzy theory, info-gap theory, probabilistic database, fuzzy database

Abstract

Linear regression quantifies the linear relationship between paired sets of input and output observations. The well known least-squares regression optimizes the performance criterion defined by the residual error, but is highly sensitive to uncertainties or perturbations in the observations. Robust least-squares algorithms have been developed to optimize the worst case performance for a given limit on the level of uncertainty, but they are applicable only when that limit is known. Herein, we present a robust-satisficing approach that maximizes the robustness to uncertainties in the observations, while satisficing a critical sub-optimal level of performance. The method emphasizes the trade-off between performance and robustness, which are inversely correlated. To resolve the resulting trade-off we introduce a new criterion, which assesses the consistency between the observations and the linear model. The proposed criterion determines a unique robust-satisficing regression and reveals the underlying level of uncertainty in the observations with only weak assumptions. These algorithms are demonstrated for the challenging application of linear regression to neural decoding for brain-machine interfaces. The model-consistent robust-satisfying regression provides superior performance for new observations under both similar and different conditions. Keywords: Linear regression, Robust regression, Regularization, Information-gap, Uncertainties, Brain machine interface.

Abstract

The aeronautics industry has benefited from the use of numerical models to supplement or replace the costly design-build-test paradigm. These models are often calibrated using experimental data to obtain optimal fidelity-to-data but compensating effects between calibration parameters can complicate the model selection process due to the non-uniqueness of the solution. One way to reduce this ambiguity is to include a robustness requirement to the selection criteria. In this study, the info-gap decision theory is used to represent the lack of knowledge resulting from compensating effects and a robustness analysis is performed to investigate the impact of uncertainty on both deterministic and stochastic fidelity metrics. The proposed methodology is illustrated on an academic example representing the dynamic response of a composite turbine blade.

Author keywords

Info-gap approach; Model calibration; Performance metric; Robust solution; Uncertainty.

Abstract

Purpose: Numerical models are being increasingly relied upon to evaluate wind turbine performance by simulating phenomena that are infeasible to measure experimentally. These numerical models, however, require a large number of input parameters that often need to be calibrated against available experiments. Owing to the unavoidable scarcity of experiments and inherent uncertainties in measurements, this calibration process may yield non-unique solutions, i.e. multiple sets of parameters may reproduce the available experiments with similar fidelity. The purpose of this paper is to study the trade-off between fidelity to measurements and the robustness of this fidelity to uncertainty in calibrated input parameters.

Design/methodology/approach: Here, fidelity is defined as the ability of the model to reproduce measurements and robustness is defined as the allowable variation in the input parameters with which the model maintains a predefined level of threshold fidelity. These two vital attributes of model predictiveness are evaluated in the development of a simplified finite element beam model of the CX-100 wind turbine blade.

Findings: Findings of this study show that calibrating the input parameters of a numerical model with the sole objective of improving fidelity to available measurements degrades the robustness of model predictions at both tested and untested settings. A more optimal model may be obtained by calibration methods considering both fidelity and robustness. Multi-criteria Decision Making further confirms the conclusion that the optimal model performance is achieved by maintaining a balance between fidelity and robustness during calibration.

Originality/value: Current methods for model calibration focus solely on fidelity while the authors focus on the trade-off between fidelity and robustness.

Keywords Uncertainty quantification, Validation, Experimental modal analysis, Prediction accuracy, Self-consistency, Test-analysis correlation

Abstract

Traditionally, model calibration is formulated as a single objective problem, where fidelity to measurements is maximized by adjusting model parameters. In such a formulation however, the model with best fidelity merely represents an optimum compromise between various forms of errors and uncertainties and thus, multiple calibrated models can be found to demonstrate comparable fidelity producing non-unique solutions. To alleviate this problem, the authors formulate model calibration as a multi-objective problem with two distinct objectives: fidelity and robustness. Herein, robustness is defined as the maximum allowable uncertainty in calibrating model parameters with which the model continues to yield acceptable agreement with measurements. The proposed approach is demonstrated through the calibration of a finite element model of a steel moment resisting frame.

Author keywords

Experiment-based model validation; Info-gap decision theory; Info-gap uncertainty model; Nondominated sorting genetic algorithm; Prediction looseness; Self-consistency

Abstract

In computer modeling, errors and uncertainties inevitably arise due to the mathematical idealization of physical processes stemming from insufficient knowledge regarding accurate model forms as well as the precise values of input parameters. While these errors and uncertainties are quantifiable, compensations between them can lead to multiple model forms and input parameter sets exhibiting a similar level of agreement with available experimental observations. Such nonuniqueness makes the selection of a single, best computer model (i.e. model form and values for its associate parameters) unjustifiable. Therefore, it becomes necessary to evaluate model performance based not only on the fidelity of the predictions to available experiments but also on a model’s ability to sustain such fidelity given the incompleteness of knowledge regarding the model itself, such an ability will herein be referred to as robustness. In this paper, the authors present a multiobjective approach to model calibration that accounts for not only the model’s fidelity to experiments but also its robustness to incomplete knowledge. With two conflicting objectives, the multi-objective model calibration results in a family of nondominated solutions exhibiting varying levels of fidelity and robustness effectively forming a Pareto front. The Pareto front solutions can be grouped depending on their nature of compromise between the two objectives, which can in turn help determine clusters in the parameter domain. The knowledge of these clusters can shed light on the nature of compensations as well as aid in the inference of uncertain input parameters. To demonstrate the feasibility and application of this new approach, we consider the computer model of a structural steel frame with uncertain connection stiffness parameters under static loading conditions. © The Society for Experimental Mechanics, Inc. 2015.

Author keywords

Info-gap decision theory; K-means clustering; Model calibration; Multi-objective optimization; Non-dominated sorting genetic algorithm (NSGA-II)

Abstract

The paper describes a new method based on the information-gap theory which enables an evaluation of worst case error predictions of the kNN method in the presence of a specified level of uncertainty in the data. There are presented concepts of a robustness and an opportunity of the kNN model and calculations of these concepts were performed for a simple 1-D data set and next, for a more complicated 6-D data set. In both cases the method worked correctly and enabled evaluation of the robustness and the opportunity for a given lowest acceptable quality rc or a windfall quality rw. The method enabled also choosing of the most robust kNN model for a given level of an uncertainty alpha.

Keywords

Data uncertainty; Function approximation; Information-gap theory; k-nearest neighbours method; Local regression

Abstract

In this paper, we consider newsboy problem under Knightian uncertainty. That is, we assume that the uncertainty of demand is Knightian uncertainty. We use info-gap uncertainty to model the uncertainty of the demand. The objective is to study the robustness of the optimal policy. We first assume that the price is exogenous and then the price is determined endogenously and is a decision variable. We study the optimal policies for the inventory control system with and without price setting under Knightian uncertainty and their robustness. We show that the relative robustness of the sole inventory system and the joint pricing and inventory control with multiplicative demand function to profit loss are the same and higher than that of the joint pricing and inventory control with additive demand function.

Keywords

Info-gap; Inventory control; Newsboy problem; Pricing

Abstract

The goal of this work is to propose a model calibration strategy for an industrial problem consisting in a MW class geared wind turbine power train subjected to uncertain loads. Lack of knowledge is commonplace in this kind of engineering system and a realistic model calibration cannot be performed without taking into account this type of uncertainty. The question at stake in this study is how to perform a robust predictive model of a dynamic system given that the excitations are poorly known. The uncertainty in the latter will be represented with an info-gap model. The tradeoff between fidelity to data and robustness to uncertainty is then investigated in order to maximize the robustness of the prediction error at a given horizon of uncertainty. This methodology is illustrated on a simple academic model and on a more complex engineering system representing a wind turbine geared power train.

Keywords

Load uncertainty; Model fidelity; Model updating; Robust calibration; Transient analysis; Wind turbine.

Abstract

Uncertainty quantification is an integral part of the model validation process and is important to take into account during the design of mechanical systems. Sources of uncertainty are diverse but generally fall into two categories: aleatory uncertainties due to random processes and epistemic uncertainty resulting from a lack of knowledge or erroneous assumptions.

This work focuses on the impact of uncertain levels of prestress on the behavior of solar arrays in their stowed configuration. In this context, snubbers are inserted between two adjacent panels to maintain contact and absorb vibrations during launch. However, under high excitation loads, a loss of contact between the two panels may occur.

This results in impacts that can cause extensive damages to fragile elements. In practice, the specific load configuration for which the separation of the two panels occurs is difficult to determine precisely since the exact level of prestress applied to the structure is unknown. An info-gap robustness analysis is applied to study the impact of this lack of knowledge on the prestress safety factor required to avoid loss of contact. The proposed methodology is illustrated using a simplified model of a solar array.

Abstract

This paper addresses the NASA Langley Multidisciplinary Uncertainty Quantification Challenge (MUQC) Problem, which is intended to pose challenges to the uncertainty quantification and robust design communities. The goals of the MUQC problem can be formulated into four main topics that are commonly encountered in the model development process: calibration, sensitivity analysis, uncertainty propagation, and robust design. Our analysis places a particular emphasis on the use of info-gap decision theory (IGDT) to address the goals of the MUQC problem. IGDT provides a convenient framework to treat epistemic uncertainty when using simulation models for decision-making. We utilize a robustness criterion, defined in the context of IGDT, to pursue calibration, uncertainty propagation, and robust design. Herein, our calibration utilizes IGDT to address the situation whereby traditional calibration techniques might result in non-unique results where different sets of calibration variables are able to replicate experiments with comparable fidelity. Uncertainty propagation is performed such that the worst-case and best-case performances of the model output are conditioned on the level of uncertainty that is permitted in the simulations. To pursue robust design, we utilize the robustness criterion to establish whether the amount of uncertainty tolerable in our optimized design is an improvement over the baseline design. We demonstrate that improving the robustness of the model requires different knowledge than improving performance of the model. The main conclusion is that IGDT provides a sound theoretical basis, and practical implementation, to meet the goals of the NASA MUQC problem without formulating simplifying assumptions.

Abstract

Several plausible modeling strategies are available to develop numerical models for simulating the dynamics of wind turbine blades. While the modeling strategy is typically selected according to expert judgment, the “best” modeling approach is unknown to the model developer. Thus, comparing plausible modeling strategies through a systematic and rigorous approach becomes necessary. This manuscript departs from the conventional approach that selects the model with the highest fidelity-to-data; and instead explores the trade-off between fidelity of model predictions to experiments and robustness of model predictions to model imprecision and inexactness. Exploring robustness in addition to fidelity lends credibility to the model, ensuring model predictions can be trusted even when lack-of-knowledge in the modeling assumptions and/or input parameters result in unforeseen errors and uncertainties. This concept is demonstrated on the CX-100 wind turbine blade in an experimental configuration with large masses added to load the blade in bending during vibration testing. The finite element model of the blade is built with shell elements and validated against experimental evidence, while the large masses are modeled according to two different, but plausible strategies using (i) a combination of point-mass and spring elements, and (ii) solid elements. These two modeling strategies are evaluated considering both the fidelity of the natural frequency predictions against experiments, and the robustness of the predicted natural frequencies to uncertainties in the input parameters. By considering robustness during model selection, the authors determine the extent to which prediction accuracy deteriorates as the lack-of-knowledge increases. The findings suggest the model with solid elements offers a higher degree of fidelity-to-data and robustness to uncertainties, thus providing a superior modeling strategy than the model with point masses and stiffening springs.

Keywords

Info-Gap decision theory; Model selection; Wind turbine blade; Model complexity; Prediction; Test-analysis correlation

Highlights

• Model selection is formulated in the context of info-gap decision theory.
• Competing finite element models of the CX-100 wind turbine blade are developed.
• Results demonstrate the trade-offs of accuracy and uncertainty.

Abstract

Mobile, unmanned, power and resource-rich devices, called actors, deployed within a Wireless Sensor Network (WSN) application area, enable faster response times to events. Due to cost constraints, only a few actors can be placed within a WSN application area. Determining which actor or set of actors should respond to an event is important, because the correct decision will increase the event response time and reduce energy expenditure. Since the mobile actors are widely dispersed over the application area, the actors’ accurate location and energy details will not always be available. In this paper, we show that using info-gap decision theory to choose the correct actors to respond to an event when uncertainty about an actor’s location and/or energy exists ensures that the actors chosen can adequately respond to the event. The robustness of the decision choice of the set of actor(s) assigned to respond to an event means that all chosen actor(s) have sufficient energy to respond to the event in real time.

Abstract

Wireless Sensor Networks (WSN) depend on remote human interaction. This slows the real time response to an event. Certain applications require rapid response to events detected by sensor nodes. The use of actors in a WSN enhances the real time response of the whole network. The coordination of actors to sensed events is thus vital for the efficient operation of the network. The goal is to optimise coordination between actors to determine which actors respond to an event message. We provide a model to determine which actors respond to an event based on Info-gap Decision Theory. We show that it is possible to choose some correct actors out of a set when faced with severe uncertainty about the environment.

Abstract

Structural health monitoring (SHM) systems provide real-time damage and performance information for civil, aerospace, and mechanical infrastructure through analysis of structural response measurements. The supervised learning methodology for data-driven SHM involves computation of low-dimensional, damage-sensitive features from raw measurement data that are then used in conjunction with machine learning algorithms to detect, classify, and quantify damage states. However, these systems often suffer from performance degradation in real-world applications due to varying operational and environmental conditions. Probabilistic approaches to robust SHM system design suffer from incomplete knowledge of all conditions a system will experience over its lifetime. Info-gap decision theory enables nonprobabilistic evaluation of the robustness of competing models and systems in a variety of decision making applications. Previous work employed info-gap models to handle feature uncertainty when selecting various components of a supervised learning system, namely features from a pre-selected family and classifiers. In this work, the info-gap framework is extended to robust feature design and classifier selection for general time series classification through an efficient, interval arithmetic implementation of an info-gap data model. Experimental results are presented for a damage type classification problem on a ball bearing in a rotating machine. The info-gap framework in conjunction with an evolutionary feature design system allows for fully automated design of a time series classifier to meet performance requirements under maximum allowable uncertainty. © 2014 SPIE.

Author keywords

Feature extraction; Info-gap decision theory; Robustness; Structural health monitoring; Time series classification