1 
To what extent is your data assimilation scheme designed to find the posterior mean, the posterior mode or something else?Hodyss, Daniel, Bishop, Craig H., Morzfeld, Matthias 30 September 2016 (has links)
Recently there has been a surge in interest in coupling ensemblebased data assimilation methods with variational methods (commonly referred to as 4DVar). Here we discuss a number of important differences between ensemblebased and variational methods that ought to be considered when attempting to fuse these methods. We note that the Best Linear Unbiased Estimate (BLUE) of the posterior mean over a data assimilation window can only be delivered by data assimilation schemes that utilise the 4dimensional (4D) forecast covariance of a prior distribution of nonlinear forecasts across the data assimilation window. An ensemble Kalman smoother (EnKS) may be viewed as a BLUE approximating data assimilation scheme. In contrast, we use the dual form of 4DVar to show that the most likely nonlinear trajectory corresponding to the posterior mode across a data assimilation window can only be delivered by data assimilation schemes that create counterparts of the 4D prior forecast covariance using a tangent linear model. Since 4DVar schemes have the required structural framework to identify posterior modes, in contrast to the EnKS, they may be viewed as mode approximating data assimilation schemes. Hence, when aspects of the EnKS and 4DVar data assimilation schemes are blended together in a hybrid, one would like to be able to understand how such changes would affect the modeor meanfinding abilities of the data assimilation schemes. This article helps build such understanding using a series of simple examples. We argue that this understanding has important implications to both the interpretation of the hybrid state estimates and to their design.

2 
Dynamical aspects of atmospheric data assimilation in the tropicsŽagar, Nedjeljka January 2004 (has links)
<p>A faithful depiction of the tropical atmosphere requires threedimensional sets of observations. Despite the increasing amount of observations presently available, these will hardly ever encompass the entire atmosphere and, in addition, observations have errors. Additional (background) information will always be required to complete the picture. Valuable added information comes from the physical laws governing the flow, usually mediated via a numerical weather prediction (NWP) model. These models are, however, never going to be errorfree, why a reliable estimate of their errors poses a real challenge since the whole truth will never be within our grasp. </p><p>The present thesis addresses the question of improving the analysis procedures for NWP in the tropics. Improvements are sought by addressing the following issues:</p><p> the efficiency of the internal model adjustment, </p><p> the potential of the reliable backgrounderror information, as compared to observations,</p><p> the impact of a new, spaceborne lineofsight wind measurements, and</p><p> the usefulness of multivariate relationships for data assimilation in the tropics.</p><p>Most NWP assimilation schemes are effectively univariate near the equator. In this thesis, a multivariate formulation of the variational data assimilation in the tropics has been developed. The proposed backgrounderror model supports the masswind coupling based on convectivelycoupled equatorial waves. The resulting assimilation model produces balanced analysis increments and hereby increases the efficiency of all types of observations.</p><p>Idealized adjustment and multivariate analysis experiments highlight the importance of direct wind measurements in the tropics. In particular, the presented results confirm the superiority of wind observations compared to mass data, in spite of the exact multivariate relationships available from the background information. The internal model adjustment is also more efficient for wind observations than for mass data. </p><p>In accordance with these findings, new satellite wind observations are expected to contribute towards the improvement of NWP and climate modeling in the tropics. Although incomplete, the new windfield information has the potential to reduce uncertainties in the tropical dynamical fields, if used together with the existing satellite massfield measurements.</p><p>The results obtained by applying the new backgrounderror representation to the tropical shortrange forecast errors of a stateofart NWP model suggest that achieving useful tropical multivariate relationships may be feasible within an operational NWP environment.</p>

3 
Cálculo de integrais de trajetória em mecânica estatística e teoria de campos através de técnicas variacionais / Calculation path integrals statistical mechanics field theory variational techniquesAragão, Cristiane Moura Lima de 06 June 2002 (has links)
Estendemos para a teria de campos o método variacional de Kleinert. Este método foi primeiramente usado na mecânica quântica e fornece uma expansão em cumulantes convergente. Sua extensão para a teoria de campos não é trivial devido às divergências ultravioletas que aparecem quando a dimensão do espaço é maior que 2. Devido a estas divergências, a teoria deve ser regularizada e normalizada. Além das dificuldades usuais associadas com a renormalização, devemos decidir se calculamos o valor ótimo do parâmetro variacional antes ou depois da renormalização. Nesta tese abordamos o problema da renormalização do potencial efetivo variacional. Primeiramente, mostramos que o potencial efetivo variacional em temperatura zero coincide com o \"potencial efetivo pósgaussiano\" introduzido por Stancu e Stevenson. Em seguida, apresentamos um esquema de renormalização que permite que renormalizemos a teoria antes de calcular o parâmetro variacional ótimo. Usando este esquema mostramos que o potencial efetivo usual, calculado em ordem 1loop, pode ser obtido a partir do esquema variacional de Kleinert inteirando uma única vez a equação que determina o parâmetro variacional. Para o potencial efetivo em ordem 2loops esta aproximação não é tão boa. A renormalização da teoria antes do cálculo do parâmetro variacional permite que estudemos o potencial efetivo variacional numericamente e de forma nãoperturbativa, como foi feito por Kleinert para a mecânica quântica. / We have extended the Kleinert variational technique to field theory. This method was first used in quantum mechanics and provides a convergent cumulate expansion that is extremely accurate. Its extension to field theory is nontrivial because of the ultraviolet divergences that appear when the space dimension is greater than 2. Due to these divergences the theory has to be regularized and renormalized. In addition to the usual difficulties associated with renormalization, one has to decide whether one calculates the optimum value of the variational parameter before or after renormalization. In this thesis we deal with the renormalization of the variational effective potential. Firstly, we show that the zero temperature regularized variational potential coincides with the postGaussian effective potential introduced by Stancu and Stenvenson. Secondly, we present a renormalization scheme that enables one to renormalize the theory before calculating the optimum variational parameter. Using this scheme we show that the usual 1loop effective potential can be obtained from the Kleinert variational scheme by interacting only once the equation that determines the variational parameter. In this sense, the 1loop expansion is contained within the variational scheme. For the 2loop effective potential the same approximation is not so good. The renormalization of the theory before the calculation of the variational parameter allows one to study the variational effective potential numerically and in a nonpertubative way, as it was done in quantum mechanics by Kleinert.

4 
Estudo de alguns problemas elípticos para o operador biharmônico / Study of some elliptic biharmonic problemsPimenta, Marcos Tadeu de Oliveira 09 May 2011 (has links)
Nesse trabalho estudamos questões de existência, multiplicidade e concentração de soluções de uma classe de problemas elípticos biharmônicos. Nos três primeiros capítulos são utilizados métodos variacionais para estudar a existência, multiplicidade e comportamento assintótico das soluções fracas nãotriviais de equações de Schrödinger estacionárias biharmônicas com diferentes hipóteses sobre o potencial e sobre a nãolinearidade. No último capítulo, o método de decomposição em cones duais é empregado para obter a existência de três soluções (positiva, negativa e nodal) para uma equação biharmônica / In this work we study some problems on existence, multiplicity and concentration of solutions of biharmonic elliptic equtions. In the first three chapters, variational methods are used to study the existence, multiplicity and the asymptotic behavior of weak nontrivial solutions of stationary Schrödinger biharmonic equations under certain assumptions on the potential function and the nonlinearity. In the last chapter we use variational methods again and also the dual decomposition method to get existence of positive, negative and signchanging solutions for a biharmonic equation

5 
Dynamical aspects of atmospheric data assimilation in the tropicsŽagar, Nedjeljka January 2004 (has links)
A faithful depiction of the tropical atmosphere requires threedimensional sets of observations. Despite the increasing amount of observations presently available, these will hardly ever encompass the entire atmosphere and, in addition, observations have errors. Additional (background) information will always be required to complete the picture. Valuable added information comes from the physical laws governing the flow, usually mediated via a numerical weather prediction (NWP) model. These models are, however, never going to be errorfree, why a reliable estimate of their errors poses a real challenge since the whole truth will never be within our grasp. The present thesis addresses the question of improving the analysis procedures for NWP in the tropics. Improvements are sought by addressing the following issues:  the efficiency of the internal model adjustment,  the potential of the reliable backgrounderror information, as compared to observations,  the impact of a new, spaceborne lineofsight wind measurements, and  the usefulness of multivariate relationships for data assimilation in the tropics. Most NWP assimilation schemes are effectively univariate near the equator. In this thesis, a multivariate formulation of the variational data assimilation in the tropics has been developed. The proposed backgrounderror model supports the masswind coupling based on convectivelycoupled equatorial waves. The resulting assimilation model produces balanced analysis increments and hereby increases the efficiency of all types of observations. Idealized adjustment and multivariate analysis experiments highlight the importance of direct wind measurements in the tropics. In particular, the presented results confirm the superiority of wind observations compared to mass data, in spite of the exact multivariate relationships available from the background information. The internal model adjustment is also more efficient for wind observations than for mass data. In accordance with these findings, new satellite wind observations are expected to contribute towards the improvement of NWP and climate modeling in the tropics. Although incomplete, the new windfield information has the potential to reduce uncertainties in the tropical dynamical fields, if used together with the existing satellite massfield measurements. The results obtained by applying the new backgrounderror representation to the tropical shortrange forecast errors of a stateofart NWP model suggest that achieving useful tropical multivariate relationships may be feasible within an operational NWP environment.

6 
Estudo de alguns problemas elípticos para o operador biharmônico / Study of some elliptic biharmonic problemsMarcos Tadeu de Oliveira Pimenta 09 May 2011 (has links)
Nesse trabalho estudamos questões de existência, multiplicidade e concentração de soluções de uma classe de problemas elípticos biharmônicos. Nos três primeiros capítulos são utilizados métodos variacionais para estudar a existência, multiplicidade e comportamento assintótico das soluções fracas nãotriviais de equações de Schrödinger estacionárias biharmônicas com diferentes hipóteses sobre o potencial e sobre a nãolinearidade. No último capítulo, o método de decomposição em cones duais é empregado para obter a existência de três soluções (positiva, negativa e nodal) para uma equação biharmônica / In this work we study some problems on existence, multiplicity and concentration of solutions of biharmonic elliptic equtions. In the first three chapters, variational methods are used to study the existence, multiplicity and the asymptotic behavior of weak nontrivial solutions of stationary Schrödinger biharmonic equations under certain assumptions on the potential function and the nonlinearity. In the last chapter we use variational methods again and also the dual decomposition method to get existence of positive, negative and signchanging solutions for a biharmonic equation

7 
Cálculo de integrais de trajetória em mecânica estatística e teoria de campos através de técnicas variacionais / Calculation path integrals statistical mechanics field theory variational techniquesCristiane Moura Lima de Aragão 06 June 2002 (has links)
Estendemos para a teria de campos o método variacional de Kleinert. Este método foi primeiramente usado na mecânica quântica e fornece uma expansão em cumulantes convergente. Sua extensão para a teoria de campos não é trivial devido às divergências ultravioletas que aparecem quando a dimensão do espaço é maior que 2. Devido a estas divergências, a teoria deve ser regularizada e normalizada. Além das dificuldades usuais associadas com a renormalização, devemos decidir se calculamos o valor ótimo do parâmetro variacional antes ou depois da renormalização. Nesta tese abordamos o problema da renormalização do potencial efetivo variacional. Primeiramente, mostramos que o potencial efetivo variacional em temperatura zero coincide com o \"potencial efetivo pósgaussiano\" introduzido por Stancu e Stevenson. Em seguida, apresentamos um esquema de renormalização que permite que renormalizemos a teoria antes de calcular o parâmetro variacional ótimo. Usando este esquema mostramos que o potencial efetivo usual, calculado em ordem 1loop, pode ser obtido a partir do esquema variacional de Kleinert inteirando uma única vez a equação que determina o parâmetro variacional. Para o potencial efetivo em ordem 2loops esta aproximação não é tão boa. A renormalização da teoria antes do cálculo do parâmetro variacional permite que estudemos o potencial efetivo variacional numericamente e de forma nãoperturbativa, como foi feito por Kleinert para a mecânica quântica. / We have extended the Kleinert variational technique to field theory. This method was first used in quantum mechanics and provides a convergent cumulate expansion that is extremely accurate. Its extension to field theory is nontrivial because of the ultraviolet divergences that appear when the space dimension is greater than 2. Due to these divergences the theory has to be regularized and renormalized. In addition to the usual difficulties associated with renormalization, one has to decide whether one calculates the optimum value of the variational parameter before or after renormalization. In this thesis we deal with the renormalization of the variational effective potential. Firstly, we show that the zero temperature regularized variational potential coincides with the postGaussian effective potential introduced by Stancu and Stenvenson. Secondly, we present a renormalization scheme that enables one to renormalize the theory before calculating the optimum variational parameter. Using this scheme we show that the usual 1loop effective potential can be obtained from the Kleinert variational scheme by interacting only once the equation that determines the variational parameter. In this sense, the 1loop expansion is contained within the variational scheme. For the 2loop effective potential the same approximation is not so good. The renormalization of the theory before the calculation of the variational parameter allows one to study the variational effective potential numerically and in a nonpertubative way, as it was done in quantum mechanics by Kleinert.

8 
Machine Learning and Field Inversion approaches to DataDriven Turbulence ModelingMichelen Strofer, Carlos Alejandro 27 April 2021 (has links)
There still is a practical need for improved closure models for the Reynoldsaveraged NavierStokes (RANS) equations. This dissertation explores two different approaches for using experimental data to provide improved closure for the Reynolds stress tensor field. The first approach uses machine learning to learn a general closure model from data. A novel framework is developed to train deep neural networks using experimental velocity and pressure measurements. The sensitivity of the RANS equations to the Reynolds stress, required for gradientbased training, is obtained by means of both variational and ensemble methods. The second approach is to infer the Reynolds stress field for a flow of interest from limited velocity or pressure measurements of the same flow. Here, this field inversion is done using a Monte Carlo Bayesian procedure and the focus is on improving the inference by enforcing known physical constraints on the inferred Reynolds stress field. To this end, a method for enforcing boundary conditions on the inferred field is presented. The two datadriven approaches explored and improved upon here demonstrate the potential for improved practical RANS predictions. / Doctor of Philosophy / The Reynoldsaveraged NavierStokes (RANS) equations are widely used to simulate fluid flows in engineering applications despite their known inaccuracy in many flows of practical interest. The uncertainty in the RANS equations is known to stem from the Reynolds stress tensor for which no universally applicable turbulence model exists. The computational cost of more accurate methods for fluid flow simulation, however, means RANS simulations will likely continue to be a major tool in engineering applications and there is still a need for improved RANS turbulence modeling. This dissertation explores two different approaches to use available experimental data to improve RANS predictions by improving the uncertain Reynolds stress tensor field. The first approach is using machine learning to learn a datadriven turbulence model from a set of training data. This model can then be applied to predict new flows in place of traditional turbulence models. To this end, this dissertation presents a novel framework for training deep neural networks using experimental measurements of velocity and pressure. When using velocity and pressure data, gradientbased training of the neural network requires the sensitivity of the RANS equations to the learned Reynolds stress. Two different methods, the continuous adjoint and ensemble approximation, are used to obtain the required sensitivity. The second approach explored in this dissertation is field inversion, whereby available data for a flow of interest is used to infer a Reynolds stress field that leads to improved RANS solutions for that same flow. Here, the field inversion is done via the ensemble Kalman inversion (EKI), a Monte Carlo Bayesian procedure, and the focus is on improving the inference by enforcing known physical constraints on the inferred Reynolds stress field. To this end, a method for enforcing boundary conditions on the inferred field is presented. While further development is needed, the two datadriven approaches explored and improved upon here demonstrate the potential for improved practical RANS predictions.

9 
Constructing and solving variational image registration problemsCahill, Nathan D. January 2009 (has links)
Nonrigid image registration has received much attention in the medical imaging and computer vision research communities, because it enables a wide variety of applications. Feature tracking, segmentation, classification, temporal image differencing, tumour growth estimation, and pharmacokinetic modeling are examples of the many tasks that are enhanced by the use of aligned imagery. Over the years, the medical imaging and computer vision communties have developed and refined image registration techniques in parallel, often based on similar assumptions or underlying paradigms. This thesis focuses on variational registration, which comprises a subset of nonrigid image registration. It is divided into chapters that are based on fundamental aspects of the variational registration problem: image dissimilarity measures, changing overlap regions, regularizers, and computational solution strategies. Key contributions include the development of local versions of standard dissimilarity measures, the handling of changing overlap regions in a manner that is insensitive to the amount of noninteresting background information, the combination of two standard taxonomies of regularizers, and the generalization of solution techniques based on Fourier methods and the Demons algorithm for use with many regularizers. To illustrate and validate the various contributions, two sets of example imagery are used: 3D CT, MR, and PET images of the brain as well as 3D CT images of lung cancer patients.

10 
On the regularity of holonomically constrained minimisers in the calculus of variationsHopper, Christopher Peter January 2014 (has links)
This thesis concerns the regularity of holonomic minimisers of variational integrals in the context of direct methods in the calculus of variations. Specifically, we consider Sobolev mappings from a bounded domain into a connected compact Riemannian manifold without boundary, to which such mappings are said to be holonomically constrained. For a general class of strictly quasiconvex integral functionals, we give a direct proof of local C<sup>1,α</sup>Hölder continuity, for some 0 < α < 1, of holonomic minimisers off a relatively closed 'singular set' of Lebesgue measure zero. Crucially, the proof constructs comparison maps using the universal covering of the target manifold, the lifting of Sobolev mappings to the covering space and the connectedness of the covering space. A certain tangential Aharmonic approximation lemma obtained directly using a Lipschitz approximation argument is also given. In the context of holonomic minimisers of regular variational integrals, we also provide bounds on the Hausdorff dimension of the singular set by generalising a variational difference quotient method to the holonomically constrained case with critical growth. The results are analogous to energyminimising harmonic maps into compact manifolds, however in this case the proof does not use a monotonicity formula. We discuss several applications to variational problems in condensed matter physics, in particular those concerning the superfluidity of liquid helium3 and nematic liquid crystals. In these problems, the class of mappings are constrained to an orbit of 'broken symmetries' or 'manifold of internal states', which correspond to a subgroup of residual symmetries.

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