Variational inverse problems integrate computational simulations of physical phenomena with physical measurements in an informational feedback control system. Control parameters of the computational model are optimized such that the simulation results fit the physical measurements.The solution procedure is computationally expensive since it involves running the simulation computer model (the emph{forward model}) and the associated emph {adjoint model} multiple times. In practice, our knowledge of the underlying physics is incomplete and hence the associated computer model is laden with emph {model errors}. Similarly, it is not possible to measure the physical quantities exactly and hence the measurements are associated with emph {data errors}. The errors in data and model adversely affect the inference solutions. This work develops methods to address the challenges posed by the computational costs and by the impact of data and model errors in solving variational inverse problems. Variational inverse problems of interest here are formulated as optimization problems constrained by partial differential equations (PDEs). The solution process requires multiple evaluations of the constraints, therefore multiple solutions of the associated PDE. To alleviate the computational costs we develop a parallel in time discretization algorithm based on a nonlinear optimization approach. Like in the emph{parareal} approach, the time interval is partitioned into subintervals, and local time integrations are carried out in parallel. Solution continuity equations across interval boundaries are added as constraints. All the computational steps - forward solutions, gradients, and Hessian-vector products - involve only ideally parallel computations and therefore are highly scalable. This work develops a systematic mathematical framework to compute the impact of data and model errors on the solution to the variational inverse problems. The computational algorithm makes use of first and second order adjoints and provides an a-posteriori error estimate for a quantity of interest defined on the inverse solution (i.e., an aspect of the inverse solution). We illustrate the estimation algorithm on a shallow water model and on the Weather Research and Forecast model. Presence of outliers in measurement data is common, and this negatively impacts the solution to variational inverse problems. The traditional approach, where the inverse problem is formulated as a minimization problem in $L_2$ norm, is especially sensitive to large data errors. To alleviate the impact of data outliers we propose to use robust norms such as the $L_1$ and Huber norm in data assimilation. This work develops a systematic mathematical framework to perform three and four dimensional variational data assimilation using $L_1$ and Huber norms. The power of this approach is demonstrated by solving data assimilation problems where measurements contain outliers. / Ph. D.

Multibody dynamics simulations are currently widely accepted as valuable means for dynamic performance analysis of mechanical systems. The evolution of theoretical and computational aspects of the multibody dynamics discipline make it conducive these days for other types of applications, in addition to pure simulations. One very important such application is design optimization for multibody systems. Sensitivity analysis of multibody system dynamics, which is performed before optimization or in parallel, is essential for optimization. Current sensitivity approaches have limitations in terms of efficiently performing sensitivity analysis for complex systems with respect to multiple design parameters. Thus, we bring new contributions to the state-of-the-art in analytical sensitivity approaches in this study. A direct differentiation method is developed for multibody dynamic models that employ Maggi's formulation. An adjoint variable method is developed for explicit and implicit first order Maggi's formulations, second order Maggi's formulation, and first and second order penalty formulations. The resulting sensitivities are employed to perform optimization of different multibody systems case studies. The collection of benchmark problems includes a five-bar mechanism, a full vehicle model, and a passive dynamic robot. The five-bar mechanism is used to test and validate the sensitivity approaches derived in this paper by comparing them with other sensitivity approaches. The full vehicle system is used to demonstrate the capability of the adjoint variable method based on the penalty formulation to perform sensitivity analysis and optimization for large and complex multibody systems with respect to multiple design parameters with high efficiency. In addition, a new multibody dynamics software library MBSVT (Multibody Systems at Virginia Tech) is developed in Fortran 2003, with forward kinematics and dynamics, sensitivity analysis, and optimization capabilities. Several different contact and friction models, which can be used to model point contact and surface contact, are developed and included in MBSVT. Finally, this study employs reference point coordinates and the penalty formulation to perform dynamic analysis for the passive dynamic robot, simplifying the modeling stage and making the robotic system more stable. The passive dynamic robot is also used to test and validate all the point contact and surface contact models developed in MBSVT. / Ph. D.

Excess phosphorus (P) that is transported into water bodies can cause water quality problems. A high potential for P delivery occurs when there is a high transport potential from erosion, runoff, and/or leaching coupled with high soil test P and/or high rate of fertilizer P application. A management tool is needed to identify those fields that have a high transport and source potential to deliver P to surface water. The Virginia P-Index is a mass-based tool that estimates the annual risk of delivery of P from a given field to surface water. Guidelines on P application rates are then given based on the level of risk. This is a new tool and additional research and testing are needed to determine the dependability and validity of the index. The overall goal of the research was to contribute to the continued development of the Virginia P-Index as an effective P management tool. A sensitivity analysis was completed to identify the parameters to which the P-Index was most sensitive under a range of conditions. In low erosion and runoff conditions, the P-Index was most sensitive to P management factors including application rate. As erosion and runoff potential increased, the P-Index was most sensitive to the erosion risk factors including soil loss. Under conditions with subsurface leaching, the P-Index was most sensitive to the subsurface leaching factors and Mehlich I soil test P. A stochastic analysis was also conducted to determine the effects of parameter variability. Variability of the P-Index output was greater as the risk of P delivery increased and this could affect management recommendations. A survey was completed to determine expert opinion as to the appropriateness of parameter estimation methods used in the Virginia P-Index. Thirty-eight surveys were returned, representing a diverse range of participants within and outside of Virginia. Comments from the respondents were used to evaluate the appropriateness of the parameter methods. All factors were determined to be appropriate given the state of the science. Estimation methods for the following factors were determined to be less appropriate than the other sub-factors by the survey respondents: soil texture/drainage class, subsurface dissolved reactive orthophosphate (DRP), runoff delivery, and sediment delivery. The Virginia P-Index was determined to be a well thought out management tool and implementation should identify fields with the greatest risk of P delivery to surface water. Recommendations for improvement were identified including a need for additional analysis and studies. / Master of Science

Numerical noise is a prevalent concern in many practical optimization problems. Convergence of gradient based optimization algorithms in the presence of numerical noise is not always assured. One way to improve optimization algorithm performance in the presence of numerical noise is to adjust the method of gradient computation. This study investigates the use of Continuous Sensitivity Equation (CSE) gradient approximations in the context of numerical noise and optimization. Three problems are considered: a problem with a system of ODE constraints, a single parameter flow problem constrained by the Navier-Stokes equations, and a multiple parameter flow problem constrained by the Navier-Stokes equations. All three problems use adaptive methods in the simulation of the constraint and are numerically noisy. Gradients for each problem are computed with both CSE and finite difference methods. The gradients are analyzed and compared. The two flow problems are optimized with a trust region optimization algorithm using both sets of gradient calculations. Optimization results are also compared, and the CSE gradient approximation yields impressive results for these examples. / Master of Science

An important part of an aerodynamic design process is optimizing designs to maximize quantities such as lift and the lift-to-drag ratio, in a process known as shape optimization. It is the goal of this thesis to develop and apply understanding of mixed finite element method and sensitivity analysis in a way that sets the foundation for shape optimization. The open-source Incompressible Flow Iterative Solution Software (IFISS) mixed finite element method toolbox for MATLAB developed by Silvester, Elman, and Ramage is used. Meshes are produced for a backward-facing step problem, using built-in tools from IFISS as well as the mesh generation software Gmsh, and grid convergence studies are performed for both sets of meshes along a sampled data line to ensure that the simulations converge asymptotically with increasing mesh resolution. As a preliminary study of sensitivity analysis, analytic sensitivities of velocity components along the backward-facing step data line to inflow velocity parameters are determined and verified using finite difference and complex step sensitivity values. The method is then applied to pressure drag calculated by integrating the pressure over the surface of a circular cylinder in a freestream flow, and verified and validated using published simulation data and experimental data. The sensitivity analysis study is extended to shape optimization, wherein the shape of a circular cylinder is altered and the sensitivities of the pressure drag coefficient to the changes in the cylinder shape are determined and verified. / Master of Science / When looking at designing an aircraft, it is important to consider the forces air flow exerts on the wings. The primary forces of interest for aerodynamic analysis are lift, which generally acts upward perpendicular to the flow of air, and drag, which opposes the motion of the wing through the air. Optimization is the process of developing a design in such a way that a specific quantity, such as lift or drag, is either maximized or minimized. Many methods exist of predicting the behavior of air flow, and various methods of optimization exist which take already existing predictive software and progressively alter the design to try to meet the minimized or maximized objective. This thesis outlines a multi-step effort to modify an open source software such that it could be used for design optimization.

This dissertation provides a complete mathematical framework to compute the sensitivities with respect to system parameters for any second order hybrid Ordinary Differential Equation (ODE) and rank 1 and 3 Differential Algebraic Equation (DAE) systems. The hybrid system is characterized by discontinuities in the velocity state variables due to an impulsive forces at the time of event. At the time of event, such system may also exhibit a change in the equations of motion or in the kinematic constraints. The analytical methodology that solves the sensitivities for hybrid systems is structured based on jumping conditions for both, the velocity state variables and the sensitivities matrix. The proposed analytical approach is then benchmarked against a known numerical method. The mathematical framework is extended to compute sensitivities of the states of the model and of the general cost functionals with respect to model parameters for both, unconstrained and constrained, hybrid mechanical systems. This dissertation emphasizes the penalty formulation for modeling constrained mechanical systems since this formalism has the advantage that it incorporates the kinematic constraints inside the equation of motion, thus easing the numerical integration, works well with redundant constraints, and avoids kinematic bifurcations. In addition, this dissertation provides a unified mathematical framework for performing the direct and the adjoint sensitivity analysis for general hybrid systems associated with general cost functions. The mathematical framework computes the jump sensitivity matrix of the direct sensitivities which is found by computing the Jacobian of the jump conditions with respect to sensitivities right before the event. The main idea is then to obtain the transpose of the jump sensitivity matrix to compute the jump conditions for the adjoint sensitivities. Finally, the methodology developed obtains the sensitivity matrix of cost functions with respect to parameters for general hybrid ODE systems. Such matrix is a key result for design analysis as it provides the parameters that affect the given cost functions the most. Such results could be applied to gradient based algorithms, control optimization, implicit time integration methods, deep learning, etc. / Ph. D. / A mechanical system is composed of many different parameters, like the length, weight and inertia of a body or the spring and damping constant of a suspension system. A variation of these constants can modify the motion a mechanical system. This dissertation provides a complete mathematical framework that aims at identifying the parameters that affect at most the motion of a mechanical system. Such system could be hybrid like the human body. Indeed, when walking the foot/ground impact causes an abrupt change of velocity of the foot, while the position of the foot remains the same. Such change makes the velocity of the human body to be discontinuous at such event, which makes the human body when walking a hybrid system. The same can be applied to a vehicle driving over a bump. The main result obtained from the mathematical framework is called the "sensitivity matrix". Such matrix is a key result for design analysis as it identifies the parameters that affect at most the motion of a mechanical system. Such results are very relevant and could be applied to different softwares with prebuilt gradient based algorithms, control optimization, implicit time integration methods, or deep learning, etc.

Maize is the most grown crop in the world. Each year, 5% of maize is lost due to a phenomenon known as stalk lodging (breakage of the stalk below the ear). One of the most promising solutions to stalk lodging is to design stalks with superior geometry to increase stalk strength. Researchers have developed a 3D parameterized maize stalk model, but these models take a long time to structurally analyze and are missing important material properties. This thesis addressed these problems by developing an automated package for analyzing the 3D parameterized maize stalk model, and by measuring the longitudinal shear modulus of both pith and rind stalk tissues. This thesis also identified the most influential geometric patterns in the 3D parameterized maize stalk model, which can be used to breed stronger maize. The results of this thesis are an increased understanding of the factors that influence stalk lodging, and geometric details for how stronger maize can be designed.

The main focus of this research is to use graph-theoretic formulations to develop an automated algorithm for the generation of sensitivity equations. The idea is to combine the benefits of direct differentiation with that of graph-theoretic formulation. The primary deliverable of this work is the developed software module which can derive the system equations and the sensitivity equations directly from the linear graph of the system. Sensitivity analysis refers to the study of changes in system behaviour brought about by the changes in model parameters. Due to the rapid increase in the sizes and complexities of the models being analyzed, it is important to extend the capabilities of the current tools of sensitivity analysis, and an automated, efficient, and accurate method for the generation of sensitivity equations is highly desirable. In this work, a graph-theoretic algorithm is developed to generate the sensitivity equations. In the current implementation, the proposed algorithm uses direct differentiation to generate sensitivity equations at the component level and graph-theoretic methods to assemble the equation fragments to form the sensitivity equations. This way certain amount of control can be established over the size and complexity of the generated sensitivity equations. The implementation of the algorithm is based on a commercial software package \verb MapleSim[Multibody] and can generate governing and sensitivity equations for multibody models created in MapleSim. In this thesis, the algorithm is tested on various mechanical, hydraulic, electro-chemical, multibody, and multi-domain systems. The generated sensitivity information are used to perform design optimization and parametric importance studies. The sensitivity results are validated using finite difference formulations. The results demonstrate that graph-theoretic sensitivity analysis is an automated, accurate, algorithmic method of generation for sensitivity equations, which enables the user to have some control over the form and complexity of the generated equations. The results show that the graph-theoretic method is more efficient than the finite difference approach. It is also demonstrated that the efficiency of the generated equations are at par or better than the equation obtained by direct differentiation.

A aplicação de modelos hidrológicos vem sendo bastante utilizada como apoio à tomada de decisão no planejamento dos recursos hídricos. Tendo em vista que os dados que servem de entrada para esses modelos estão sujeitos a erros diversos, o presente estudo teve o intuito de contribuir com o conhecimento do impacto desses erros no desempenho do modelo e na estimativa de seus parâmetros. O modelo analisado foi o IPH II fazendo uso do programa computacional WIN_IPH2. Entendendo que a avaliação da sensibilidade ainda é uma área que requer mais estudos, o presente trabalho é focado na utilização das análises de sensibilidade estática e dinâmica. Para isso foram geradas diversas séries temporais de dados de entradas do modelo hidrológico obtidas pela perturbação da série de dados observados. A perturbação foi representada por erros aleatórios (seguindo uma distribuição normal ou uniforme) ou sistemáticos incorporados ás séries temporais das variáveis: precipitação e evapotranspiração. Posteriormente, as análises de sensibilidade estática e dinâmica foram executadas. Para efetuar o acompanhamento da interferência dos erros, na eficiência do modelo, foi feita a avaliação dos resultados obtidos com a aplicação do modelo WIN_IPH2 para diferentes medidas de desempenho, e verificado o impacto dos erros nos dados de entrada no desempenho do modelo (sensibilidade estática) e no desempenho do modelo e na estimativa dos parâmetros (sensibilidade dinâmica). Na análise de sensibilidade estática verificou-se o decaimento mais acentuado da eficiência do modelo, em comparação com a análise de sensibilidade dinâmica, onde o modelo consegue contornar os erros nos dados de entrada com a alteração dos valores dos parâmetros. Por fim, o presente estudo confirmou as conclusões obtidas em estudos anteriores: Oudin et al. (2006), Andréassian et al. (2004), Kavetski et al. (2003). Além disso, o presente estudo apontou para outros fatores, na medida em que, observa-se junto à tendência do desempenho do modelo cair quando a intensidade do erro gerado é elevada, a importância de avaliar o possível comprometimento de dados em eventos extremos, uma vez que, nessa situação o desempenho do modelo passa a ser afetado de forma mais acentuada. / The hydrologic models had been used to support the decision making in water resources management. Since the input data of those models are subject to several kinds of errors, this study aimed to assess how this errors affect the model performance and the estimate of its parameters. The hydrologic model IPH II was used. Perceiving that the sensitivity analysis is still a field that requires further knowledge, this study was focused in the use of the dynamic and the static sensitivity procedures. In this sense, several time series of input data were obtained through the perturbations of an observed time serie. The perturbation was represented by the addition of random errors (with a normal or uniform distribution) or systematic errors to the observed time series of evapotranspiration and precipitation. Then, the static and dynamic sensibility analysis were performed. The effect of input data errors was assessed for several calibration processes of the IPH II using several performance measures. Thus, modification of the model performance (static sensitivity analysis) and model performance and parameter estimation (dynamic sensitivity analysis) were estimated. In the static sensitivity analysis it was found a most pronounced decay of the model efficiency in comparison with the dynamic sensitivity analysis, where the model can circumvent the errors in the input data with modification of the optimum parameter values. Finally, this study confirmed the conclusions of other previous studies as Oudin et al. (2006), Andréassian et al. (2004), Kavetski et al. (2003). In addition this study found other factors, as was observed that if the intensity of the error is high in an extreme event of precipitation, it reduced the model performance more than when it is low, in spite of the time series of errors have the same statistics.

A aplicação de modelos hidrológicos vem sendo bastante utilizada como apoio à tomada de decisão no planejamento dos recursos hídricos. Tendo em vista que os dados que servem de entrada para esses modelos estão sujeitos a erros diversos, o presente estudo teve o intuito de contribuir com o conhecimento do impacto desses erros no desempenho do modelo e na estimativa de seus parâmetros. O modelo analisado foi o IPH II fazendo uso do programa computacional WIN_IPH2. Entendendo que a avaliação da sensibilidade ainda é uma área que requer mais estudos, o presente trabalho é focado na utilização das análises de sensibilidade estática e dinâmica. Para isso foram geradas diversas séries temporais de dados de entradas do modelo hidrológico obtidas pela perturbação da série de dados observados. A perturbação foi representada por erros aleatórios (seguindo uma distribuição normal ou uniforme) ou sistemáticos incorporados ás séries temporais das variáveis: precipitação e evapotranspiração. Posteriormente, as análises de sensibilidade estática e dinâmica foram executadas. Para efetuar o acompanhamento da interferência dos erros, na eficiência do modelo, foi feita a avaliação dos resultados obtidos com a aplicação do modelo WIN_IPH2 para diferentes medidas de desempenho, e verificado o impacto dos erros nos dados de entrada no desempenho do modelo (sensibilidade estática) e no desempenho do modelo e na estimativa dos parâmetros (sensibilidade dinâmica). Na análise de sensibilidade estática verificou-se o decaimento mais acentuado da eficiência do modelo, em comparação com a análise de sensibilidade dinâmica, onde o modelo consegue contornar os erros nos dados de entrada com a alteração dos valores dos parâmetros. Por fim, o presente estudo confirmou as conclusões obtidas em estudos anteriores: Oudin et al. (2006), Andréassian et al. (2004), Kavetski et al. (2003). Além disso, o presente estudo apontou para outros fatores, na medida em que, observa-se junto à tendência do desempenho do modelo cair quando a intensidade do erro gerado é elevada, a importância de avaliar o possível comprometimento de dados em eventos extremos, uma vez que, nessa situação o desempenho do modelo passa a ser afetado de forma mais acentuada. / The hydrologic models had been used to support the decision making in water resources management. Since the input data of those models are subject to several kinds of errors, this study aimed to assess how this errors affect the model performance and the estimate of its parameters. The hydrologic model IPH II was used. Perceiving that the sensitivity analysis is still a field that requires further knowledge, this study was focused in the use of the dynamic and the static sensitivity procedures. In this sense, several time series of input data were obtained through the perturbations of an observed time serie. The perturbation was represented by the addition of random errors (with a normal or uniform distribution) or systematic errors to the observed time series of evapotranspiration and precipitation. Then, the static and dynamic sensibility analysis were performed. The effect of input data errors was assessed for several calibration processes of the IPH II using several performance measures. Thus, modification of the model performance (static sensitivity analysis) and model performance and parameter estimation (dynamic sensitivity analysis) were estimated. In the static sensitivity analysis it was found a most pronounced decay of the model efficiency in comparison with the dynamic sensitivity analysis, where the model can circumvent the errors in the input data with modification of the optimum parameter values. Finally, this study confirmed the conclusions of other previous studies as Oudin et al. (2006), Andréassian et al. (2004), Kavetski et al. (2003). In addition this study found other factors, as was observed that if the intensity of the error is high in an extreme event of precipitation, it reduced the model performance more than when it is low, in spite of the time series of errors have the same statistics.