Probabilistic behavior in perceiving the environment and take critical decisions have an inevitable role in human life. A decision is concerned with a choice among the available alternatives and is always subject to unknown elements concerning the future. The lack of complete data, insufficient scientific, behavioral, and industry development and of course defects in measurement methods, affect the reliability of an action’s outcome. Thus, having a proper estimation of this reliability or uncertainty could be very advantageous particularly when an individual or generally a subject is faced with a high risk. With the fact that there are always uncertainty elements whose values are unknown and these enter into a processes through multiple sources, it has been a primary challenge to design an efficient representation of confidence objectively. With the aim of addressing this problem, a variety of researches have been conducted to introduce frameworks in metrology of uncertainty quantification that are comprehensive enough and have transferability into different areas. Moreover, it’s also a challenging task to define a proper index that reflects more aspects of the problem and measurement process. With significant advances in Artificial Intelligence in the past decade, one of the key elements, in order to ease human life by giving more control to machines, is to heed the uncertainty estimation for a prediction. With a focus on measurement aspects, this thesis attends to demonstrate how a different measurement index affects the quality of evaluated predictive uncertainty of neural networks. Finally, we propose a novel index that shows uncertainty values with the same or higher quality than existing methods which emphasizes the benefits of having a proper measurement index in managing the risk of the outcome from a classification model.

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This study has two parts. In the first part we develop a computational approach to the solution of an inverse modelling problem concerning the material properties of electrolytes used in Lithium-ion batteries. The dependence of the diffusion coefficient and the transference number on the concentration of Lithium ions is reconstructed based on the concentration data obtained from an in-situ NMR imaging experiment. This experiment is modelled by a system of 1D time-dependent Partial Differential Equations (PDE) describing the evolution of the concentration of Lithium ions with prescribed initial concentration and fluxes at the boundary. The material properties that appear in this model are reconstructed by solving a variational optimization problem in which the least-square error between the experimental and simulated concentration values is minimized. The uncertainty of the reconstruction is characterized by assuming that the material properties are random variables and their probability distribution estimated using a novel combination of Monte-Carlo approach and Bayesian statistics. In the second part of this study, we carefully analyze a number of secondary effects such as ion pairing and dendrite growth that may influence the estimation of the material properties and develop mathematical models to include these effects. We then use reconstructions of material properties based on inverse modelling along with their uncertainty estimates as a framework to validate or invalidate the models. The significance of certain secondary effects is assessed based on the influence they have on the reconstructed material properties. / Thesis / Doctor of Philosophy (PhD)

This dissertation addresses model form uncertainty quantification, non-deterministic simulations, and sensitivity analysis of the results of these simulations, with a focus on application to analysis of unmanned aircraft systems. The model form uncertainty quantification utilizes equation error to estimate the error between an identified model and flight test results. The errors are then related to aircraft states, and prediction intervals are calculated. This method for model form uncertainty quantification results in uncertainty bounds that vary with the aircraft state, narrower where consistent information has been collected and wider where data are not available. Non-deterministic simulations can then be performed to provide uncertainty quantified estimates of the system performance. The model form uncertainties could be time varying, so multiple sampling methods were considered. The two methods utilized were a fixed uncertainty level and a rate bounded variation in the uncertainty level. For analysis using fixed uncertainty level, the corner points of the model form uncertainty were sampled, providing reduced computational time. The second model better represents the uncertainty but requires significantly more simulations to sample the uncertainty. The uncertainty quantified performance estimates are compared to estimates based on flight tests to check the accuracy of the results. Sensitivity analysis is performed on the uncertainty quantified performance estimates to provide information on which of the model form uncertainties contribute most to the uncertainty in the performance estimates. The proposed method uses the results from the fixed uncertainty level analysis that utilizes the corner points of the model form uncertainties. The sensitivity of each parameter is estimated based on corner values of all the other uncertain parameters. This results in a range of possible sensitivities for each parameter dependent on the true value of the other parameters. / Ph. D.

This paper presents a framework for tuning, validating, and quantifying uncertainties for flight loads. The flight loads are computed using a Nastran doublet lattice model and are validated using measured data from a flight loads survey for a Cessna Model 525B business jet equipped with Tamarack® Aerospace Group’s active winglet modification, ATLAS® (Active Technology Load Alleviation System). ATLAS® allows for significant aerodynamic improvements to be realized by reducing loads to below the values of the original, unmodified airplane. Flight loads are measured using calibrated strain gages and are used to tune and validate a Nastran doublet-lattice flight loads model. Methods used to tune and validate the model include uncertainty quantification of the Nastran model form and lead to an uncertainty quantified model which can be used to estimate flight loads at any given flight condition within the operating envelope of the airplane. The methods presented herein improve the efficiency of the loads process and reduce conservatism in design loads through improved prediction techniques. Regression techniques and uncertainty quantification methods are presented to more accurately assess the complexities in comparing models to flight test results. / Master of Science / This paper presents a process for correlating analytical airplane loads models to flight test data and validating the results. The flight loads are computed using Nastran, a structural modeling tool coupled with an aerodynamic loads solver. The flight loads models are correlated to flight test data and are validated using measured data from a flight loads survey for a Cessna Model 525B business jet equipped with Tamarack ® Aerospace Group’s active winglet modification, ATLAS ® (Active Technology Load Alleviation System). ATLAS ® allows for significant aerodynamic improvements and efficiency gains to be realized by reducing loads to below the values of the original, unmodified airplane. Flight loads are measured using a series of strain gage sensors mounted on the wing. These sensors are calibrated to measure aerodynamic loads and are used to tune and validate the Nastran flight loads model. Methods used to tune and validate the model include quantification of error and uncertainties in the model. These efforts lead to a substantially increased understanding of the model limitations and uncertainties, which is especially valuable at the corners of the operating envelope of the airplane. The methods presented herein improve the efficiency of the loads process and reduce conservatism in design loads through improved prediction techniques. The results provide a greater amount of guidance for decision making throughout the design and certification of a load alleviation system and similar airplane aerodynamic improvements.

In this thesis a method for verification, validation and uncertainty quantification (VV&UQ) has been tested and evaluated on a fuel transfer application in the fuel rig currently used at Saab. A simplified model has been developed for the limited part of the fuel system in the rig that is affected in the transfer, and VV&UQ has been performed on this model. The scope for the thesis has been to investigate if and how simulation models can be used for certification of the fuel system in a fighter aircraft. The VV&UQ-analysis was performed with the limitation that no probability distributions for uncertainties were considered. Instead, all uncertainties were described using intervals (so called epistemic uncertainties). Simulations were performed on five different operating points in terms of fuel flow to the engine with five different initial conditions for each, resulting in 25 different operating modes. For each of the 25 cases, the VV&UQ resulted in a minimum and maximum limit for how much fuel that could be transferred. 6 cases were chosen for validation measurements and the resulting amount of fuel transferred ended up between the corresponding epistemic intervals. Performing VV&UQ is a time demanding and computationally heavy task, which quickly grows as the model becomes more complex. Our conclusion is that a pilot study is necessary, where time and costs are evaluated, before choosing to use a simulation model and perform VV&UQ for certification. Further investigation of different methods for increasing confidence in simulation models is also needed, for which VV&UQ is one suitable option.

This thesis examines the quantification of uncertainty in real-world engineering systems using the multilevel Monte Carlo method. It is often infeasible to use the traditional Monte Carlo method to investigate the impact of uncertainty because computationally it can be prohibitively expensive for complex systems. Therefore, the newer multilevel method is investigated and the cost of this method is analysed in the finite element framework. The Monte Carlo and multilevel Monte Carlo methods are compared for two prototypical examples: structural vibrations and buoyancy driven flows through porous media. In the first example, the impact of random mass density is quantified for structural vibration problems in several dimensions using the multilevel Monte Carlo method. Comparable eigenvalues and energy density approximations are found for the traditional Monte Carlo method and the multilevel Monte Carlo method, but for certain problems the expectation and variance of the quantities of interest can be computed over 100 times faster using the multilevel Monte Carlo method. It is also tractable to use the multilevel method for three dimensional structures, where the traditional Monte Carlo method is often prohibitively expensive. In the second example, the impact of uncertainty in buoyancy driven flows through porous media is quantified using the multilevel Monte Carlo method. Again, comparable results are obtained from the two methods for diffusion dominated flows and the multilevel method is orders of magnitude cheaper. The finite element models for this investigation are formulated carefully to ensure that spurious numerical artefacts are not added to the solution and are compared to an analytical model describing the long term sequestration of CO2 in the presence of a background flow. Additional cost reductions are achieved by solving the individual independent samples in parallel using the new podS library. This library schedules the Monte Carlo and multilevel Monte Carlo methods in parallel across different computer architectures for the two examples considered in this thesis. Nearly linear cost reductions are obtained as the number of processes is increased.

The main goal of this thesis is to investigate the frequentist asymptotic properties of nonparametric Bayesian procedures in inverse problems and the Gaussian white noise model. In the first part, we study the frequentist posterior contraction rate of nonparametric Bayesian procedures in linear inverse problems in both the mildly and severely ill-posed cases. This rate provides a quantitative measure of the quality of statistical estimation of the procedure. A theorem is proved in a general Hilbert space setting under approximation-theoretic assumptions on the prior. The result is applied to non-conjugate priors, notably sieve and wavelet series priors, as well as in the conjugate setting. In the mildly ill-posed setting, minimax optimal rates are obtained, with sieve priors being rate adaptive over Sobolev classes. In the severely ill-posed setting, oversmoothing the prior yields minimax rates. Previously established results in the conjugate setting are obtained using this method. Examples of applications include deconvolution, recovering the initial condition in the heat equation and the Radon transform. In the second part of this thesis, we investigate Bernstein--von Mises type results for adaptive nonparametric Bayesian procedures in both the Gaussian white noise model and the mildly ill-posed inverse setting. The Bernstein--von Mises theorem details the asymptotic behaviour of the posterior distribution and provides a frequentist justification for the Bayesian approach to uncertainty quantification. We establish weak Bernstein--von Mises theorems in both a Hilbert space and multiscale setting, which have applications in $L^2$ and $L^\infty$ respectively. This provides a theoretical justification for plug-in procedures, for example the use of certain credible sets for sufficiently smooth linear functionals. We use this general approach to construct optimal frequentist confidence sets using a Bayesian approach. We also provide simulations to numerically illustrate our approach and obtain a visual representation of the different geometries involved.

Nearly every decision must be made with a degree of uncertainty regarding the outcome. Decision making based on modeling and simulation predictions needs to incorporate and aggregate uncertain evidence. To validate multiscale simulation models, it may be necessary to consider evidence collected at a length scale that is different from the one at which a model predicts. In addition, traditional methods of uncertainty analysis do not distinguish between two types of uncertainty: uncertainty due to inherently random inputs, and uncertainty due to lack of information about the inputs. This thesis examines and applies a Bayesian approach for model parameter validation that uses generalized interval probability to separate these two types of uncertainty. A generalized interval Bayes’ rule (GIBR) is used to combine the evidence and update belief in the validity of parameters. The sensitivity of completeness and soundness for interval range estimation in GIBR is investigated. Several approaches to represent complete ignorance of probabilities’ values are tested. The result from the GIBR method is verified using Monte Carlo simulations. The method is first applied to validate the parameter set for a molecular dynamics simulation of defect formation due to radiation. Evidence is supplied by the comparison with physical experiments. Because the simulation includes variables whose effects are not directly observable, an expanded form of GIBR is implemented to incorporate the uncertainty associated with measurement in belief update. In a second example, the proposed method is applied to combining the evidence from two models of crystal plasticity at different length scales.