Nonlinear error dynamics for cycled data assimilation methods

We investigate the error dynamics for cycled data assimilation systems, such that the inverse problem of state determination is solved at tk, k = 1, 2, 3, ..., with a first guess given by the state propagated via a dynamical system model from time tk − 1 to time tk. In particular, for nonlinear dynamical systems that are Lipschitz continuous with respect to their initial states, we provide deterministic estimates for the development of the error ||ek|| := ||x(a)k − x(t)k|| between the estimated state x(a) and the true state x(t) over time. Clearly, observation error of size δ > 0 leads to an estimation error in every assimilation step. These errors can accumulate, if they are not (a) controlled in the reconstruction and (b) damped by the dynamical system under consideration. A data assimilation method is called stable, if the error in the estimate is bounded in time by some constant C. The key task of this work is to provide estimates for the error ||ek||, depending on the size δ of the observation error, the reconstruction operator Rα, the observation operator H and the Lipschitz constants K(1) and K(2) on the lower and higher modes of controlling the damping behaviour of the dynamics. We show that systems can be stabilized by choosing α sufficiently small, but the bound C will then depend on the data error δ in the form c||Rα||δ with some constant c. Since ||Rα|| → ∞ for α → 0, the constant might be large. Numerical examples for this behaviour in the nonlinear case are provided using a (low-dimensional) Lorenz '63 system.

Variational data assimilation for very large environmental problems

Variational data assimilation is commonly used in environmental forecasting to estimate the current state of the system from a model forecast and observational data. The assimilation problem can be written simply in the form of a nonlinear least squares optimization problem. However the practical solution of the problem in large systems requires many careful choices to be made in the implementation. In this article we present the theory of variational data assimilation and then discuss in detail how it is implemented in practice. Current solutions and open questions are discussed.

Time-periodic perturbation of a Lienard equation with an unbounded homoclinic loop

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Incorporação do vínculo de suavidade no ajuste de histórico de reservatórios de petróleo

The history match procedure in an oil reservoir is of paramount importance in order to obtain a characterization of the reservoir parameters (statics and dynamics) that implicates in a predict production more perfected. Throughout this process one can find reservoir model parameters which are able to reproduce the behaviour of a real reservoir.Thus, this reservoir model may be used to predict production and can aid the oil file management. During the history match procedure the reservoir model parameters are modified and for every new set of reservoir model parameters found, a fluid flow simulation is performed so that it is possible to evaluate weather or not this new set of parameters reproduces the observations in the actual reservoir. The reservoir is said to be matched when the discrepancies between the model predictions and the observations of the real reservoir are below a certain tolerance. The determination of the model parameters via history matching requires the minimisation of an objective function (difference between the observed and simulated productions according to a chosen norm) in a parameter space populated by many local minima. In other words, more than one set of reservoir model parameters fits the observation. With respect to the non-uniqueness of the solution, the inverse problem associated to history match is ill-posed. In order to reduce this ambiguity, it is necessary to incorporate a priori information and constraints in the model reservoir parameters to be determined. In this dissertation, the regularization of the inverse problem associated to the history match was performed via the introduction of a smoothness constraint in the following parameter: permeability and porosity. This constraint has geological bias of asserting that these two properties smoothly vary in space. In this sense, it is necessary to find the right relative weight of this constrain in the objective function that stabilizes the inversion and yet, introduces minimum bias. A sequential search method called COMPLEX was used to find the reservoir model parameters that best reproduce the observations of a semi-synthetic model. This method does not require the usage of derivatives when searching for the minimum of the objective function. Here, it is shown that the judicious introduction of the smoothness constraint in the objective function formulation reduces the associated ambiguity and introduces minimum bias in the estimates of permeability and porosity of the semi-synthetic reservoir model

A General Solution of the Fokker-Planck Equation

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Poço quadrado quântico finito e método de fatorização

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

ACCELERATING ITERATIVE ALGORITHMS FOR SYMMETRICAL LINEAR COMPLEMENTARITY-PROBLEMS

We propose a method for accelerating iterative algorithms for solving symmetric linear complementarity problems. The method consists in performing a one-dimensional optimization in the direction generated by a splitting method even for non-descent directions. We give strong convergence proofs and present numerical experiments that justify using this acceleration.

Numerical simulation of viscous three-dimensional flow over aerospace vehicles using Euler/boundary-layer zonal approach

This paper presents a viscous three-dimensional simulations coupling Euler and boundary layer codes for calculating flows over arbitrary surfaces. The governing equations are written in a general non orthogonal coordinate system. The Levy-Lees transformation generalized to three-dimensional flows is utilized. The inviscid properties are obtained from the Euler equations using the Beam and Warming implicit approximate factorization scheme. The resulting equations are discretized and approximated by a two-point fmitedifference numerical scheme. The code developed is validated and applied to the simulation of the flowfield over aerospace vehicle configurations. The results present good correlation with the available data.

Optimal trajectory and solution of the inverse kinematics of a robotic manipulator by genetic algorithms

This paper presents the generation of optimal trajectories by genetic algorithms (GA) for a planar robotic manipulator. The implemented GA considers a multi-objective function that minimizes the end-effector positioning error together with the joints angular displacement and it solves the inverse kinematics problem for the trajectory. Computer simulations results are presented to illustrate this implementation and show the efficiency of the used methodology producing soft trajectories with low computing cost. © 2011 Springer-Verlag Berlin Heidelberg.

O formalismo supersimétrico e suas aplicações em mecânica quântica

Pós-graduação em Biofísica Molecular - IBILCE

Detecção de falhas estruturais usando sensores e atuadores piezelétricos e algoritmos genéticos

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Análise de métodos numéricos de diferenças finitas para solução da equação de Poisson em domínios irregulares

Pós-graduação em Matematica Aplicada e Computacional - FCT

Estudo da cinética de precipitação de Ag na liga Cu-7% Al-4% Ag

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Aplicação de deconvolução homomórfica a dados sísmicos

Um registro sísmico é frequentemente representado como a convolução de um pulso-fonte com a resposta do meio ao impulso, relacionada ao caminho da propagação. O processo de separação destes dois componentes da convolução é denominado deconvolução. Existe uma variedade de aproximações para o desenvolvimento de uma deconvolução. Uma das mais comuns é o uso da filtragem linear inversa, ou seja, o processamento do sinal composto, através de um filtro linear, cuja resposta de frequência é a recíproca da transformada de Fourier de um dos componentes do sinal. Obviamente, a fim de usarmos a filtragem inversa, tais componentes devem ser conhecidas ou estimadas. Neste trabalho, tratamos da aplicação a sinais sísmicos, de uma técnica de deconvolução não linear, proposta por Oppenheim (1965), a qual utiliza a teoria de uma classe de sistemas não lineares, que satisfazem um princípio generalizado de superposição, denominados de sistemas homomórficos. Tais sistemas são particularmente úteis na separação de sinais que estão combinados através da operação de convolução. O algoritmo da deconvolução homomórfica transforma o processo de convolução em uma superposição aditiva de seus componentes, com o resultado de que partes simples podem ser separadas mais facilmente. Esta classe de técnicas de filtragem representa uma generalização dos problemas de filtragem linear. O presente método oferece a considerável vantagem de que não é necessário fazer qualquer suposição prévia sobre a natureza do pulso sísmico fonte, ou da resposta do meio ao impulso, não requerendo assim, as considerações usuais de que o pulso seja de fase-mínima e que a distribuição dos impulsos seja aleatória, embora a qualidade dos resultados obtidos pela análise homomórfica seja muito sensível à razão sinal/ruído, como demonstrado.

Solution to bilharmonic equation with vanishing potential

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)