Abstract: Logic trees are widely used in probabilistic seismic hazard analysis as a tool to capture the epistemic uncertainty associated with the seismogenic sources and the ground-motion prediction models used in estimating the hazard. Combining two or more ground-motion relations within a logic tree will generally require several conversions to be made, because there are several definitions available for both the predicted ground-motion parameters and the explanatory parameters within the predictive ground-motion relations. Procedures for making conversions for each of these factors are presented, using a suite of predictive equations in current use for illustration. The sensitivity of the resulting ground-motion models to these conversions is shown to be pronounced for some of the parameters, especially the measure of source-to-site distance, highlighting the need to take into account any incompatibilities among the selected equations. Procedures are also presented for assigning weights to the branches in the ground-motion section of the logic tree in a transparent fashion, considering both intrinsic merits of the individual equations and their degree of applicability to the particular application.

Abstract: This paper discusses two stochastic approaches to computing the propagation of uncertainty in numerical simulations: polynomial chaos and stochastic collocation. Chebyshev polynomials are used in both cases for the conventional, deterministic portion of the discretization in physical space. For the stochastic parameters, polynomial chaos utilizes a Galerkin approximation based upon expansions in Hermite polynomials, whereas stochastic collocation rests upon a novel transformation between the stochastic space and an artificial space. In our present implementation of stochastic collocation, Legendre interpolating polynomials are employed. These methods are discussed in the specific context of a quasi-one-dimensional nozzle flow with uncertainty in inlet conditions and nozzle shape. It is shown that both stochastic approaches efficiently handle uncertainty propagation. Furthermore, these approaches enable computation of statistical moments of arbitrary order in a much more effective way than other usual techniques such as the Monte Carlo simulation or perturbation methods. The numerical results indicate that the stochastic collocation method is substantially more efficient than the full Galerkin, polynomial chaos method. Moreover, the stochastic collocation method extends readily to highly nonlinear equations. An important application is to the stochastic Riemann problem, which is of particular interest for spectral discontinuous Galerkin methods.

Abstract: In the current practice of probabilistic seismic hazard analysis (PSHA) using logic trees, it is common to use the mean hazard curve to determine ground motions for engineering design. We present the case against the use of the mean hazard curve and explain why this practice should be discontinued and, where necessary, removed from regulations. The identification and quantification of uncertainties is integral to modern seismic hazard analysis. In probabilistic seismic hazard studies, the variability of the earthquake magnitude, earthquake location, and ground motion level (expressed as the number of logarithmic standard deviations above the logarithmic mean) are considered explicitly in the computation of the hazard. In major seismic hazard projects, the scientific uncertainty in the models of the distributions of earthquake magnitude, location, and ground motion are also considered using logic trees (Kulkarni et al. 1984, Coppersmith and Youngs 1986, Reiter 1990, Bommer et al. 2005). The inherent variability considered directly in the hazard computation is called the aleatory variability, and the scientific uncertainty in the models of the earthquake occurrence and ground motion is called the epistemic uncertainty. The terms randomness and uncertainty have also been used for aleatory variability and epistemic uncertainty, respectively; however, the former terms are now commonly used interchangeably. As a result, they are often mixed up when used in hazard analysis. The terms ‘‘aleatory variability’’ and ‘‘epistemic uncertainty’’ are used to provide an unambiguous terminology. This is not simply semantics: distinguishing between the two types of uncertainty is fundamental to the way that they are dealt with in the hazard calculations and how uncertainty is handled in decision making on the basis of the hazard analysis. In application, the key difference is that aleatory variability leads to the shape of the hazard curve and the epistemic uncertainty leads to alternative hazard curves. There is no dilemma regarding the inclusion of the aleatory variability in the hazard calculations, particularly the variability associated with ground-motion prediction equations: a ‘‘hazard curve’’ calculated using only median values from the equations and neglecting the standard deviation has little meaning and cannot be considered a genuine hazard curve. The hazard analyst does, however, have control over the branches of the logic tree and the weights assigned to these, and hence over the degree to which epistemic uncer

Abstract: Logic trees have become a popular tool in seismic hazard studies. Commonly, the models corresponding to the end branches of the complete logic tree in a probabalistic seismic hazard analysis (psha) are treated separately until the final calculation of the set of hazard curves. This comes at the price that information regarding sensitivities and uncertainties in the ground-motion sections of the logic tree are only obtainable after disaggregation. Furthermore, from this end-branch model perspective even the designers of the logic tree cannot directly tell what ground-motion scenarios most likely would result from their logic trees for a given earthquake at a particular distance, nor how uncertain these scenarios might be or how they would be affected by the choices of the hazard analyst. On the other hand, all this information is already implicitly present in the logic tree. Therefore, with the ground-motion perspective that we propose in the present article, we treat the ground-motion sections of a complete logic tree for seismic hazard as a single composite model representing the complete state-of-knowledge-and-belief of a particular analyst on ground motion in a particular target region. We implement this view by resampling the ground-motion models represented in the ground-motion sections of the logic tree by Monte Carlo simulation (separately for the median values and the sigma values) and then recombining the sets of simulated values in proportion to their logic-tree branch weights. The quantiles of this resampled composite model provide the hazard analyst and the decision maker with a simple, clear, and quantitative representation of the overall physical meaning of the ground-motion section of a logic tree and the accompanying epistemic uncertainty. Quantiles of the composite model also provide an easy way to analyze the sensitivities and uncertainties related to a given logic-tree model. We illustrate this for a composite ground-motion model for central Europe. Further potential fields of applications are seen wherever individual best estimates of ground motion have to be derived from a set of candidate models, for example, for hazard maps, sensitivity studies, or for modeling scenario earthquakes.

Abstract: This study compares two techniques for uncertainty quantification in chemistry computations, one based on sensitivity analysis and error propagation, and the other on stochastic analysis using polynomial chaos techniques. The two constructions are studied in the context of H2O2 ignition under supercritical-water conditions. They are compared in terms of their prediction of uncertainty in species concentrations and the sensitivity of selected species concentrations to given parameters. The formulation is extended to one-dimensional reacting-flow simulations. The computations are used to study sensitivities to both reaction rate pre-exponentials and enthalpies, and to examine how this information must be evaluated in light of known, inherent parametric uncertainties in simulation parameters. The results indicate that polynomial chaos methods provide similar first-order information to conventional sensitivity analysis, while preserving higher-order information that is needed for accurate uncertainty quantification and for assigning confidence intervals on sensitivity coefficients. These higher-order effects can be significant, as the analysis reveals substantial uncertainties in the sensitivity coefficients themselves. © 2005 Wiley Periodicals, Inc. Int J Chem Kinet 37: 368–382, 2005

Abstract: The network-level infrastructure management problem involves selecting and scheduling maintenance, repair, and rehabilitation (MR&R) activities on networks of infrastructure facilities so as to maintain the level of service provided by the network in a cost-effective manner. This problem is frequently formulated as a Markov decision problem (MDP) solved via linear programming (LP). The conditions of facilities are represented by elements of discrete condition rating sets, and transition probabilities are employed to describe deterioration processes. Epistemic and parametric uncertainties not considered within the standard MDP/LP framework are associated with the transition probabilities used in infrastructure management optimization routines. This paper contrasts the expected costs incurred when model uncertainty is ignored with those incurred when this uncertainty is explicitly considered using robust optimization. A case study involving a network-level pavement management MDP/LP problem demonstrates how explicitly considering uncertainty may limit worst-case MR&R expenditures. The methods and results can also be used to identify the costs of uncertainty in transition probability matrices used in infrastructure management systems.

Abstract: Multi-objective evolutionary algorithms (MOEAs) have proven to be a powerful tool for global optimization purposes of deterministic problem functions. Yet, in many real-world problems, uncertainty about the correctness of the system model and environmental factors does not allow to determine clear objective values. Stochastic sampling as applied in noisy EAs neglects that this so-called epistemic uncertainty is not an inherent property of the system and cannot be reduced by sampling methods. Therefore, some extensions for MOEAs to handle epistemic uncertainty in objective functions are proposed. The extensions are generic and applicable to most common MOEAs. A density measure for uncertain objectives is proposed to maintain diversity in the nondominated set. The approach is demonstrated to the reliability optimization problem, where uncertain component failure rates are usual and exhaustive tests are often not possible due to time and budget reasons.

Abstract: The network-level infrastructure management problem involves selecting and scheduling Maintenance, Repair, and Rehabilitation (MR&R) activities on networks of infrastructure facilities so as to maintain the level of service provided by the network in a cost-effective manner. This problem is frequently formulated as a Markov Decision Problem (MDP) solved via Linear Programming (LP). The conditions of facilities are represented by elements of discrete condition rating sets, and transition probabilities are employed to describe deterioration processes. Epistemic and parametric uncertainties not considered within the standard MDP/LP framework are associated with the transition probabilities used in infrastructure management optimization routines. This paper contrasts the expected costs incurred when model uncertainty is ignored with those incurred when this uncertainty is explicitly considered using Robust Optimization. A case study involving a network-level pavement management MDP/LP problem demonstrates how explicitly considering uncertainty may limit worst case MR&R expenditures. The methods and results can also be used to identify the costs of uncertainty in transition probability matrices used in infrastructure management systems.

Abstract: Safety systems are important components of high-consequence systems that are intended to prevent the unintended operation of the system and thus the potentially significant negative consequences that could result from such an operation. This presentation investigates and illustrates formal procedures for assessing the uncertainty in the probability that a safety system will fail to operate as intended in an accident environment. Probability theory and evidence theory are introduced as possible mathematical structures for the representation of the epistemic uncertainty associated with the performance of safety systems, and a representation of this type is illustrated with a hypothetical safety system involving one weak link and one strong link that is exposed to a high temperature fire environment. Topics considered include (1) the nature of diffuse uncertainty information involving a system and its environment, (2) the conversion of diffuse uncertainty information into the mathematical structures associated with probability theory and evidence theory, and (3) the propagation of these uncertainty structures through a model for a safety system to obtain representations in the context of probability theory and evidence theory of the uncertainty in the probability that the safety system will fail to operate as intended. The results suggest that evidence theory provides a potentially valuable representational tool for the display of the implications of significant epistemic uncertainty in inputs to complex analyses.

Abstract: Geological formations are ubiquitously heterogeneous, and the equations that govern flow and transport in such formations can be treated as stochastic partial differential equations. The Monte Carlo method is a straightforward approach for simulating flow in heterogeneous porous media; an alternative based on the moment-equation approach has been developed in the last two decades to reduce the high computational expense required by the Monte Carlo method. However, the computational cost of the moment-equation approach is still high. For example, to solve head covariance up to first order in terms of $\sigma_Y^2$, the variance of log hydraulic conductivity Y = ln Ks, it is required to solve sets of linear algebraic equations with N unknowns for 2N times (N being the number of grid nodes). The cost is even higher if higher-order approximations are needed. Zhang and Lu [J. Comput. Phys., 194 (2004), pp. 773--794] developed a new approach to evaluate high-order moments (fourth order for mean head in terms of $\sigma_Y$, and third order for head variances in terms of $\sigma_Y^2$) of flow quantities based on the combination of Karhunen--Loeve decomposition and perturbation methods. In this study, we systematically investigate the computational efficiency and solution accuracy of three approaches: Monte Carlo simulations, the conventional moment-equation (CME) approach, and the moment-equation approach based on Karhunen--Loeve decomposition (KLME). It is evident that the computational cost for the KLME approach is significantly lower than those required by the Monte Carlo and CME approaches. More importantly, while the computational costs (in terms of the number of times for solving linear algebraic equations with N unknowns) for the CME approach depend on the number of grid nodes, the cost for the KLME approach is independent of the number of grid nodes. This makes it possible to apply the KLME method to solve more realistic large-scale flow problems.

Abstract: Rapid Prototyping (RP) is the process of building three-dimensional objects, in layers, using additive manufacturing. Rapid Manufacturing (RM) is the use of RP technologies to manufacture end-use, or finished, products. At small lot sizes, such as with customized products, traditional manufacturing technologies become infeasible due to the high costs of tooling and setup. RM offers the opportunity to produce these customized products economically. Coupled with the customization opportunities afforded by RM is a certain degree of uncertainty. This uncertainty is mainly attributed to the lack of information known about what the customer’s specific requirements and preferences are at the time of production. In this paper, we present an overall method for selection of a RM technology under the geometric uncertainty inherent to mass customization. Specifically, we define the types of uncertainty inherent to RM (epistemic), propose a method to account for this uncertainty in a selection process (interval analysis), and propose a method to select a technology under uncertainty (Hurwicz selection criterion). We illustrate our method with an example on the selection of an RM technology to produce custom caster wheels.Copyright © 2005 by ASME

Abstract: Metamodels have been used with success in many areas of engineering for decades but only recently in the field of structural dynamics. A metamodel is a fast running surrogate that is typically used to aid an analyst or test engineer in the fast and efficient exploration of the design space. Response surface metamodels are used in this work to perform parameter identification of a simple five degree of freedom system, motivated by their low training requirements and ease of use. In structural dynamics applications, response surface metamodels have been utilized in a forward sense, for activities such as sensitivity analysis or uncertainty quantification. In this study a polynomial response surface model is developed, relating system parameters to measurable output features. Once this relationship is established, the response surface is used in an inverse sense to identify system parameters from measured output features. A design of experiments is utilized to choose points, representing a fraction of the full design space of interest, for fitting the response surface metamodel. Two parameters commonly used to characterize damage in a structural system, stiffness and damping, are identified. First changes are identified and located with success in a linear 5DOF system. Then parameter identification is attempted with a nonlinear 5DOF system and limited success is achieved. This work will demonstrate that use of response surface metamodels in an inverse sense shows promise for use in system parameter identification for both linear and weakly nonlinear systems and that the method has potential for use in damage identification applications.

Abstract: Asset management systems help public works agencies decide when and how to maintain and rehabilitate infrastructure facilities in a cost effective manner Many sources of error, some difficult to quantify, can limit the ability of asset management systems to accurately predict how built systems will deteriorate This paper introduces the use of robust optimization to deal with epistemic uncertainty The Hurwicz criterion is employed to ensure management policies are never ‘too conservative’ An efficient solution algorithm is developed to solve robust counterparts of the asset management problem A case study demonstrates how the consideration of uncertainty alters optimal management policies and shows how the proposed approach may reduce maintenance and rehabilitation (M&R) expenditures

Abstract: [1] In the recent past the nonlinear prediction (NLP) method, initially developed in the context of nonlinear time series analysis, has been successfully applied to river flow deterministic forecasting In this work a probabilistic approach to the NLP method is proposed, which allows one to estimate the probability distribution of the predicted discharge values and to quantify the total uncertainty related to the forecast An ensemble technique is also proposed in order to optimize the choice of the parameter values and to provide robustness to the model calibration The probabilistic NLP method is applied to a river flow time series, giving results that confirm the effectiveness and reliability of the proposed approach

Abstract: In order to improve the robustness of vibroacoustic numerical predictions, one introduces a model of random uncertainties. The random uncertainty modelling relies on a nonparametric approach providing random system realizations with a maximum entropy. This approach only requires a few uncertainty parameters but takes into account data errors as well as model errors. It appears to be well adapted to study the variability of structural-acoustic systems; the implementation of the method for this class of problem is presented here for the first time. Practically, the paper deals with a classical low frequency vibroacoustic modelling such as used for booming noise predictions. The application of the nonparametric approach to vehicle uncertainties modelling shows the sensitivity of the vibroacoustic frequency responses to structural and cavity uncertainties as well as coupling interface uncertainties. Flexible parts appear to be more sensitive to random uncertainties than stiff parts. The sensitivity of the structural modes to structural random uncertainties is also shown in a stochastic MAC table.

Abstract: The propagation of parameter uncertainty through a model to obtain the uncertain vibration response is becoming more practical for industrial scale finite element models due to the increase in computing power available. In some cases the parametric uncertainty may be measured directly, for example the thickness of a panel. However the parameters for joint models (for example) must be estimated from measurements using the techniques of finite element model updating. In these cases the techniques of model updating must be extended to allow for uncertainty quantification from a series of measurements on nominally identically structures. The validation of these methods requires laboratory experiments where the uncertain parameter is measured directly and also estimated by updating. This paper outlines the results of experiments that may be used for this purpose, namely a moving mass on a free-free beam and a copper pipe with uncertain internal pressure. The data from these experiments will be freely available on the associated website at Bristol.

Abstract: Successful uncertainty quantification requires many reservoir models matching field production data and time lapse seismic. We have been able to automatically generate nearly 50 history matched models for a GOM oil field using the Neighbourhood Approximation stochastic sampling algorithm. This allows us to produce uncertainty forecasts for various production scenarios. Geostatistical simulation constrained by well log, core, and seismic data is applied for building geological and reservoir models. The parameters of the geostatistical simulation (channel directions, channel dimensions, variagram parameters, etc.) are considered as uncertain parameters. Additionally, end points of relative permeability curves, dependencies of compressibility factors and permeability on effective stress, transmissibility multipliers across faults, and water aquifer size are varied in the history matching process. A misfit function is selected to quantify history match of model results with field measurements of water cut, gas-oil ratio, and reservoir pressure in production wells in different moments of time. Trends and variances of the observation data are determined and incorporated in the objective function. The position of the water-oil contact at the beginning of the field development and after three years of production is estimated from time lapse seismic. Differences in the water-oil contact positions determined from the reservoir simulation and time lapse seismic are quantified and incorporated in the objective function. The reservoir model has been run nearly 2400 times in the history matching process. The Neighbourhood Algorithm is applied for the selection of the values of the history matching parameters in each run. The high quality of the history match is demonstrated, An ensemble appraising procedure based on a Bayesian framework is used to determine probability distributions of the history matching parameters and to assign probabilities to the simulation models (runs). About 50 models with the highest probabilities (which cover 99% of the cumulative probability range) are selected for the predictions. A general tool has been developed for the definition of statistical parameters of production predictions (mean values, confidence interval, etc) and their changes in time. Uncertainty estimations for the base case predictions and several production scenarios are demonstrated.

Abstract: Analytical models are used in engineering science to simulate a host of physical phenomena ranging from the constitutive behavior of materials to manufacturing processes to the static and dynamic performance of components and systems under a variety of environmental conditions. Numerical simulations are increasingly used to replace prototype testing in the development of new products, and must be relied upon when testing is impractical or impossible. The predictive accuracy of a model has been defined as the accuracy of model simulations, or predictions, under conditions for which the model has not been subjected to direct experimental verification. This paper addresses a "top-down" method for predictive accuracy assessment based on the statistical analysis of direct comparisons between physical observations and corresponding model predictions for generically similar sets of analysis-test data. Unlike traditional "bottom-up" methods based on the propagation of only parametric uncertainty through a model, this approach captures all sources of uncertainty represented in the generic database, i.e., "total uncertainty." This generic uncertainty model will include experimental uncertainty to the extent that it has not been separately quantified on the basis of replicate tests and removed from the estimate of total uncertainty, and model form uncertainty to the extent that various models are included in the database. The generic uncertainty model can be propagated by various means through a particular model belonging to the same generic category, whether or not the model has been included in the generic database, to assess the predictive accuracy of the model based on past experience. In this way, the predictive accuracy of future models, e.g. future models of systems not yet built or tested, may be assessed. A practical example is given.

Abstract: Seismic hazard assessments provide quantitative evaluations of the nature of ground shaking at a specified location that could be induced by future earthquakes. Such evaluations serve to inform engineering decisions about the location and design of new projects and the safety of existing structures. In order to provide the engineers and planners with complete information on which to base their decisions, the assessments must identify and, to the extent that is possible, quantify the associated uncertainties. There are major uncertainties associated with both the seismicity model and the ground-motion model in any seismic hazard assessment, but the uncertainties associated with the latter will generally have the larger impact on the results. Uncertainties in ground-motion prediction equations can be characterized as aleatory variability and epistemic uncertainty; the former can be directly integrated into the hazard calculations, although for very low annual exceedance frequencies it can become necessary to impose physical limits on the distribution of residuals. Epistemic uncertainty can be handled using logic-tree formulations. Combining several ground-motion prediction equations in a logic tree often requires adjustments to be made to compensate for the use of different parameter definitions; without these adjustments, the epistemic uncertainty can be grossly over- or underestimated. However, the adjustments themselves, which are often empirically derived, carry their own uncertainty and this must be included in the analyses.

Abstract: Structural Health Monitoring (SHM) systems that report in real-time a flight vehicle's condition are central to meeting the goals of increasing flight vehicle safety and reliability, while reducing operating and maintenance costs. The structural response of flight vehicles is inherently random, requiring deterministic finite element analyses to be augmented with uncertainty quantification methods to compute the response statistics and structural damage probability. To detect damage with maximum probability, sensors must be placed optimally. This requires combining a probabilistic finite element method (FEM) with damage detection algorithms and optimization techniques. This study develops a methodology to integrate the above disciplines into a sensor placement optimization (SPO) methodology for SHM systems under uncertainty. To achieve this, the structural component under consideration is analyzed via FEM and uncertainty of model input quantities is included in the analysis via random processes and fields. In the next two steps probabilistic FEM analyses are performed to determine the model output variability and using these results, damage detection procedures such as feature extraction and state classification are applied to assess the current structural state of the component. Repeating these two steps using both healthy and damaged structural models helps quantify the reliability of a given sensor layout. Finally, SPO is achieved to maximize the reliability of damage detection. The sensor layout design of a thermal protection system component is used as a numerical example.

Abstract: Quantifying uncertainty in production forecasts is critical to making good reservoir management decisions, particularly for many current investment opportunities that require intensive technology and large investments, and that may have marginal profitability indicators. Reservoir studies are conducted to support decision making, but reservoir management decisions must often be made before completion of these studies. This paper presents a new approach to reservoir studies that combines production forecasting with history matching. The approach provides preliminary production forecasts much earlier in reservoir studies. More importantly, the approach provides estimates of uncertainty associated with the forecasts. This is accomplished by using the mismatch of history match runs to weight corresponding forecast runs. We illustrate application of the method to the 8-Sand reservoir in the Green Canyon 18 field, Gulf of Mexico. We observed that, as the accuracy of the model increased during the histo...

Abstract: Many reliability-based design optimization (RBDO) techniques have been developed to obtain a reliable design accurately and efficiently in the presence of uncertainties. Probability theory has primarily been used for uncertainty analysis in RBDO. However, in a situation in which information of uncertain variables is imprecise and incomplete, probabilistic techniques are not appropriate to describe the propagation of uncertainty. In this work, evidence theory, also called Dempster-Shafer theory, is employed to handle imprecise uncertain variables as an alternative to classical probability theory. Since the uncertainty quantification method of evidence theory is not compatible with prevailing probabilistic RBDO techniques, we propose an efficient optimization strategy by defining a plausibility function in a trustregion. The procedure for handling imprecise uncertain variables is presented on a practical large-scale structural RBDO problem.

Abstract: Accurate modeling of flow in oil/gas reservoirs requires a detailed description of reservoir properties such as permeability and porosity. However, such reservoirs are inherently heterogeneous and exhibit a high degree of spatial variability in medium properties. Significant spatial heterogeneity and a limited number of measurements lead to uncertainty in characterization of reservoir properties and thus, to uncertainty in predicting flow in the reservoirs. As a result, the equations that govern flow in such reservoirs are treated as stochastic partial differential equations. The current industrial practice is to tackle the problem of uncertainty quantification by Monte Carlo simulations (MCS). This entails generating a large number of equally likely random realizations of the reservoir fields with parameter statistics derived from sampling, solving deterministic flow equations for each realization, and post-processing the results over all realizations to obtain sample moments of the solution. This approach has the advantages of applying to a broad range of both linear and nonlinear flow problems, but has a number of potential drawbacks. To properly resolve high frequency space-time fluctuations in random parameters, it is necessary to employ fine numerical grids in space-time. Therefore, the computation effort for each realization is usually large, especially for largescale reservoirs. As a result, a detailed assessment of the uncertainty associated with flow performance predictions is rarely performed. In this work, we develop an accurate yet efficient approach for solving flow problems in heterogeneous reservoirs. We do so by obtaining higher-order solutions of the prediction and the associated uncertainty of reservoir flow quantities using the moment-equation approach based on Karhunen-Loeve decomposition (KLME). The KLME approach is developed on the basis of the Karhunen-Loeve (KL) decomposition, polynomial expansion, and perturbation methods. We conduct Monte Carlo simulations and compare MCS results against different orders of approximations from the KLME method. The three-dimensional computational examples demonstrate that this KLME method is computationally more efficient than both Monte Carlo simulations and the conventional momentequation method. The KLME approach allows us to evaluate higher-order terms that are needed for the highly heterogeneous reservoirs. In addition, just like the Monte Carlo method the KLME approach can be implemented with existing simulators in a straightforward manner and are inherently parallel. The efficiency of the KLME method makes it possible to simulate fluid flow in large-scale heterogeneous reservoirs.

Abstract: Sensitivity analysis is critically important to numerous analysis algorithms, including large scale optimization, uncertainty quantification,reduced order modeling, and error estimation. Our research focused on developing tools, algorithms and standard interfaces to facilitate the implementation of sensitivity type analysis into existing code and equally important, the work was focused on ways to increase the visibility of sensitivity analysis. We attempt to accomplish the first objective through the development of hybrid automatic differentiation tools, standard linear algebra interfaces for numerical algorithms, time domain decomposition algorithms and two level Newton methods. We attempt to accomplish the second goal by presenting the results of several case studies in which direct sensitivities and adjoint methods have been effectively applied, in addition to an investigation of h-p adaptivity using adjoint based a posteriori error estimation. A mathematical overview is provided of direct sensitivities and adjoint methods for both steady state and transient simulations. Two case studies are presented to demonstrate the utility of these methods. A direct sensitivity method is implemented to solve a source inversion problem for steady state internal flows subject to convection diffusion. Real time performance is achieved using novel decomposition into offline and online calculations. Adjoint methods are used to reconstruct initialmore » conditions of a contamination event in an external flow. We demonstrate an adjoint based transient solution. In addition, we investigated time domain decomposition algorithms in an attempt to improve the efficiency of transient simulations. Because derivative calculations are at the root of sensitivity calculations, we have developed hybrid automatic differentiation methods and implemented this approach for shape optimization for gas dynamics using the Euler equations. The hybrid automatic differentiation method was applied to a first order approximation of the Euler equations and used as a preconditioner. In comparison to other methods, the AD preconditioner showed better convergence behavior. Our ultimate target is to perform shape optimization and hp adaptivity using adjoint formulations in the Premo compressible fluid flow simulator. A mathematical formulation for mixed-level simulation algorithms has been developed where different physics interact at potentially different spatial resolutions in a single domain. To minimize the implementation effort, explicit solution methods can be considered, however, implicit methods are preferred if computational efficiency is of high priority. We present the use of a partial elimination nonlinear solver technique to solve these mixed level problems and show how these formulation are closely coupled to intrusive optimization approaches and sensitivity analyses. Production codes are typically not designed for sensitivity analysis or large scale optimization. The implementation of our optimization libraries into multiple production simulation codes in which each code has their own linear algebra interface becomes an intractable problem. In an attempt to streamline this task, we have developed a standard interface between the numerical algorithm (such as optimization) and the underlying linear algebra. These interfaces (TSFCore and TSFCoreNonlin) have been adopted by the Trilinos framework and the goal is to promote the use of these interfaces especially with new developments. Finally, an adjoint based a posteriori error estimator has been developed for discontinuous Galerkin discretization of Poisson's equation. The goal is to investigate other ways to leverage the adjoint calculations and we show how the convergence of the forward problem can be improved by adapting the grid using adjoint-based error estimates. Error estimation is usually conducted with continuous adjoints but if discrete adjoints are available it may be possible to reuse the discrete version for error estimation. We investigate the advantages and disadvantages of continuous and discrete adjoints through a simple example.« less