Inverse Problems for multiple invariant curves

Planar polynomial vector fields which admit invariant algebraic curves, Darboux integrating factors or Darboux first integrals are of special interest. In the present paper we solve the inverse problem for invariant algebraic curves with a given multiplicity and for integrating factors, under generic assumptions regarding the (multiple) invariant algebraic curves involved. In particular we prove, in this generic scenario, that the existence of a Darboux integrating factor implies Darboux integrability. Furthermore we construct examples where the genericity assumption does not hold and indicate that the situation is different for these.

Efficient total variation algorithm for fetal brain MRI reconstruction.

Fetal MRI reconstruction aims at finding a high-resolution image given a small set of low-resolution images. It is usually modeled as an inverse problem where the regularization term plays a central role in the reconstruction quality. Literature has considered several regularization terms s.a. Dirichlet/Laplacian energy, Total Variation (TV)- based energies and more recently non-local means. Although TV energies are quite attractive because of their ability in edge preservation, standard explicit steepest gradient techniques have been applied to optimize fetal-based TV energies. The main contribution of this work lies in the introduction of a well-posed TV algorithm from the point of view of convex optimization. Specifically, our proposed TV optimization algorithm or fetal reconstruction is optimal w.r.t. the asymptotic and iterative convergence speeds O(1/n2) and O(1/√ε), while existing techniques are in O(1/n2) and O(1/√ε). We apply our algorithm to (1) clinical newborn data, considered as ground truth, and (2) clinical fetal acquisitions. Our algorithm compares favorably with the literature in terms of speed and accuracy.

Efficient total variation algorithm for fetal MRI reconstruction

Fetal MRI reconstruction aims at finding a high-resolution image given a small set of low-resolution images. It is usually modeled as an inverse problem where the regularization term plays a central role in the reconstruction quality. Literature has considered several regularization terms s.a. Dirichlet/Laplacian energy [1], Total Variation (TV)based energies [2,3] and more recently non-local means [4]. Although TV energies are quite attractive because of their ability in edge preservation, standard explicit steepest gradient techniques have been applied to optimize fetal-based TV energies. The main contribution of this work lies in the introduction of a well-posed TV algorithm from the point of view of convex optimization. Specifically, our proposed TV optimization algorithm for fetal reconstruction is optimal w.r.t. the asymptotic and iterative convergence speeds O(1/n(2)) and O(1/root epsilon), while existing techniques are in O(1/n) and O(1/epsilon). We apply our algorithm to (1) clinical newborn data, considered as ground truth, and (2) clinical fetal acquisitions. Our algorithm compares favorably with the literature in terms of speed and accuracy.

Heat Source Estimation with the Conjugate Gradient Method in Inverse Linear Diffusive Problems

In this work, we present the solution of a class of linear inverse heat conduction problems for the estimation of unknown heat source terms, with no prior information of the functional forms of timewise and spatial dependence of the source strength, using the conjugate gradient method with an adjoint problem. After describing the mathematical formulation of a general direct problem and the procedure for the solution of the inverse problem, we show applications to three transient heat transfer problems: a one-dimensional cylindrical problem; a two-dimensional cylindrical problem; and a one-dimensional problem with two plates.

On-site calibration of a phase fraction meter by an inverse technique

The formal calibration procedure of a phase fraction meter is based on registering the outputs resulting from imposed phase fractions at known flow regimes. This can be straightforwardly done in laboratory conditions, but is rarely the case in industrial conditions, and particularly for on-site applications. Thus, there is a clear need for less restrictive calibration methods regarding to the prior knowledge of the complete set of inlet conditions. A new procedure is proposed in this work for the on-site construction of the calibration curve from total flown mass values of the homogeneous dispersed phase. The solution is obtained by minimizing a convenient error functional, assembled with data from redundant tests to handle the intrinsic ill-conditioned nature of the problem. Numerical simulations performed for increasing error levels demonstrate that acceptable calibration curves can be reconstructed, even from total mass measured within a precision of up to 2%. Consequently, the method can readily be applied, especially in on-site calibration problems in which classical procedures fail due to the impossibility of having a strict control of all the input/output parameters.

Conventional and Reciprocal Approaches to the Forward and Inverse Problems of Electroencephalography

Le problème inverse en électroencéphalographie (EEG) est la localisation de sources de courant dans le cerveau utilisant les potentiels de surface sur le cuir chevelu générés par ces sources. Une solution inverse implique typiquement de multiples calculs de potentiels de surface sur le cuir chevelu, soit le problème direct en EEG. Pour résoudre le problème direct, des modèles sont requis à la fois pour la configuration de source sous-jacente, soit le modèle de source, et pour les tissues environnants, soit le modèle de la tête. Cette thèse traite deux approches bien distinctes pour la résolution du problème direct et inverse en EEG en utilisant la méthode des éléments de frontières (BEM): l’approche conventionnelle et l’approche réciproque. L’approche conventionnelle pour le problème direct comporte le calcul des potentiels de surface en partant de sources de courant dipolaires. D’un autre côté, l’approche réciproque détermine d’abord le champ électrique aux sites des sources dipolaires quand les électrodes de surfaces sont utilisées pour injecter et retirer un courant unitaire. Le produit scalaire de ce champ électrique avec les sources dipolaires donne ensuite les potentiels de surface. L’approche réciproque promet un nombre d’avantages par rapport à l’approche conventionnelle dont la possibilité d’augmenter la précision des potentiels de surface et de réduire les exigences informatiques pour les solutions inverses. Dans cette thèse, les équations BEM pour les approches conventionnelle et réciproque sont développées en utilisant une formulation courante, la méthode des résidus pondérés. La réalisation numérique des deux approches pour le problème direct est décrite pour un seul modèle de source dipolaire. Un modèle de tête de trois sphères concentriques pour lequel des solutions analytiques sont disponibles est utilisé. Les potentiels de surfaces sont calculés aux centroïdes ou aux sommets des éléments de discrétisation BEM utilisés. La performance des approches conventionnelle et réciproque pour le problème direct est évaluée pour des dipôles radiaux et tangentiels d’excentricité variable et deux valeurs très différentes pour la conductivité du crâne. On détermine ensuite si les avantages potentiels de l’approche réciproquesuggérés par les simulations du problème direct peuvent êtres exploités pour donner des solutions inverses plus précises. Des solutions inverses à un seul dipôle sont obtenues en utilisant la minimisation par méthode du simplexe pour à la fois l’approche conventionnelle et réciproque, chacun avec des versions aux centroïdes et aux sommets. Encore une fois, les simulations numériques sont effectuées sur un modèle à trois sphères concentriques pour des dipôles radiaux et tangentiels d’excentricité variable. La précision des solutions inverses des deux approches est comparée pour les deux conductivités différentes du crâne, et leurs sensibilités relatives aux erreurs de conductivité du crâne et au bruit sont évaluées. Tandis que l’approche conventionnelle aux sommets donne les solutions directes les plus précises pour une conductivité du crâne supposément plus réaliste, les deux approches, conventionnelle et réciproque, produisent de grandes erreurs dans les potentiels du cuir chevelu pour des dipôles très excentriques. Les approches réciproques produisent le moins de variations en précision des solutions directes pour différentes valeurs de conductivité du crâne. En termes de solutions inverses pour un seul dipôle, les approches conventionnelle et réciproque sont de précision semblable. Les erreurs de localisation sont petites, même pour des dipôles très excentriques qui produisent des grandes erreurs dans les potentiels du cuir chevelu, à cause de la nature non linéaire des solutions inverses pour un dipôle. Les deux approches se sont démontrées également robustes aux erreurs de conductivité du crâne quand du bruit est présent. Finalement, un modèle plus réaliste de la tête est obtenu en utilisant des images par resonace magnétique (IRM) à partir desquelles les surfaces du cuir chevelu, du crâne et du cerveau/liquide céphalorachidien (LCR) sont extraites. Les deux approches sont validées sur ce type de modèle en utilisant des véritables potentiels évoqués somatosensoriels enregistrés à la suite de stimulation du nerf médian chez des sujets sains. La précision des solutions inverses pour les approches conventionnelle et réciproque et leurs variantes, en les comparant à des sites anatomiques connus sur IRM, est encore une fois évaluée pour les deux conductivités différentes du crâne. Leurs avantages et inconvénients incluant leurs exigences informatiques sont également évalués. Encore une fois, les approches conventionnelle et réciproque produisent des petites erreurs de position dipolaire. En effet, les erreurs de position pour des solutions inverses à un seul dipôle sont robustes de manière inhérente au manque de précision dans les solutions directes, mais dépendent de l’activité superposée d’autres sources neurales. Contrairement aux attentes, les approches réciproques n’améliorent pas la précision des positions dipolaires comparativement aux approches conventionnelles. Cependant, des exigences informatiques réduites en temps et en espace sont les avantages principaux des approches réciproques. Ce type de localisation est potentiellement utile dans la planification d’interventions neurochirurgicales, par exemple, chez des patients souffrant d’épilepsie focale réfractaire qui ont souvent déjà fait un EEG et IRM.

Certified Rapid Solution of Parametrized Linear Elliptic Equations: Application to Parameter Estimation

We present a technique for the rapid and reliable evaluation of linear-functional output of elliptic partial differential equations with affine parameter dependence. The essential components are (i) rapidly uniformly convergent reduced-basis approximations — Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N (optimally) selected points in parameter space; (ii) a posteriori error estimation — relaxations of the residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs; and (iii) offline/online computational procedures — stratagems that exploit affine parameter dependence to de-couple the generation and projection stages of the approximation process. The operation count for the online stage — in which, given a new parameter value, we calculate the output and associated error bound — depends only on N (typically small) and the parametric complexity of the problem. The method is thus ideally suited to the many-query and real-time contexts. In this paper, based on the technique we develop a robust inverse computational method for very fast solution of inverse problems characterized by parametrized partial differential equations. The essential ideas are in three-fold: first, we apply the technique to the forward problem for the rapid certified evaluation of PDE input-output relations and associated rigorous error bounds; second, we incorporate the reduced-basis approximation and error bounds into the inverse problem formulation; and third, rather than regularize the goodness-of-fit objective, we may instead identify all (or almost all, in the probabilistic sense) system configurations consistent with the available experimental data — well-posedness is reflected in a bounded "possibility region" that furthermore shrinks as the experimental error is decreased.

L1-regularisation for ill-posed problems in variational data assimilation

We consider four-dimensional variational data assimilation (4DVar) and show that it can be interpreted as Tikhonov or L2-regularisation, a widely used method for solving ill-posed inverse problems. It is known from image restoration and geophysical problems that an alternative regularisation, namely L1-norm regularisation, recovers sharp edges better than L2-norm regularisation. We apply this idea to 4DVar for problems where shocks and model error are present and give two examples which show that L1-norm regularisation performs much better than the standard L2-norm regularisation in 4DVar.

Resolution of sharp fronts in the presence of model error in variational data assimilation

We show that the four-dimensional variational data assimilation method (4DVar) can be interpreted as a form of Tikhonov regularization, a very familiar method for solving ill-posed inverse problems. It is known from image restoration problems that L1-norm penalty regularization recovers sharp edges in the image more accurately than Tikhonov, or L2-norm, penalty regularization. We apply this idea from stationary inverse problems to 4DVar, a dynamical inverse problem, and give examples for an L1-norm penalty approach and a mixed total variation (TV) L1–L2-norm penalty approach. For problems with model error where sharp fronts are present and the background and observation error covariances are known, the mixed TV L1–L2-norm penalty performs better than either the L1-norm method or the strong constraint 4DVar (L2-norm)method. A strength of the mixed TV L1–L2-norm regularization is that in the case where a simplified form of the background error covariance matrix is used it produces a much more accurate analysis than 4DVar. The method thus has the potential in numerical weather prediction to overcome operational problems with poorly tuned background error covariance matrices.

Parâmetro de regularização em problemas inversos: estudo numérico com a transformada de Radon

In general, an inverse problem corresponds to find a value of an element x in a suitable vector space, given a vector y measuring it, in some sense. When we discretize the problem, it usually boils down to solve an equation system f(x) = y, where f : U Rm ! Rn represents the step function in any domain U of the appropriate Rm. As a general rule, we arrive to an ill-posed problem. The resolution of inverse problems has been widely researched along the last decades, because many problems in science and industry consist in determining unknowns that we try to know, by observing its effects under certain indirect measures. Our general subject of this dissertation is the choice of Tykhonov´s regulaziration parameter of a poorly conditioned linear problem, as we are going to discuss on chapter 1 of this dissertation, focusing on the three most popular methods in nowadays literature of the area. Our more specific focus in this dissertation consists in the simulations reported on chapter 2, aiming to compare the performance of the three methods in the recuperation of images measured with the Radon transform, perturbed by the addition of gaussian i.i.d. noise. We choosed a difference operator as regularizer of the problem. The contribution we try to make, in this dissertation, mainly consists on the discussion of numerical simulations we execute, as is exposed in Chapter 2. We understand that the meaning of this dissertation lays much more on the questions which it raises than on saying something definitive about the subject. Partly, for beeing based on numerical experiments with no new mathematical results associated to it, partly for being about numerical experiments made with a single operator. On the other hand, we got some observations which seemed to us interesting on the simulations performed, considered the literature of the area. In special, we highlight observations we resume, at the conclusion of this work, about the different vocations of methods like GCV and L-curve and, also, about the optimal parameters tendency observed in the L-curve method of grouping themselves in a small gap, strongly correlated with the behavior of the generalized singular value decomposition curve of the involved operators, under reasonably broad regularity conditions in the images to be recovered

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

Estimativa da profundidade do embasamento na bacia potiguar usando inversão gravimétrica

The gravity inversion method is a mathematic process that can be used to estimate the basement relief of a sedimentary basin. However, the inverse problem in potential-field methods has neither a unique nor a stable solution, so additional information (other than gravity measurements) must be supplied by the interpreter to transform this problem into a well-posed one. This dissertation presents the application of a gravity inversion method to estimate the basement relief of the onshore Potiguar Basin. The density contrast between sediments and basament is assumed to be known and constant. The proposed methodology consists of discretizing the sedimentary layer into a grid of rectangular juxtaposed prisms whose thicknesses correspond to the depth to basement which is the parameter to be estimated. To stabilize the inversion I introduce constraints in accordance with the known geologic information. The method minimizes an objective function of the model that requires not only the model to be smooth and close to the seismic-derived model, which is used as a reference model, but also to honor well-log constraints. The latter are introduced through the use of logarithmic barrier terms in the objective function. The inversion process was applied in order to simulate different phases during the exploration development of a basin. The methodology consisted in applying the gravity inversion in distinct scenarios: the first one used only gravity data and a plain reference model; the second scenario was divided in two cases, we incorporated either borehole logs information or seismic model into the process. Finally I incorporated the basement depth generated by seismic interpretation into the inversion as a reference model and imposed depth constraint from boreholes using the primal logarithmic barrier method. As a result, the estimation of the basement relief in every scenario has satisfactorily reproduced the basin framework, and the incorporation of the constraints led to improve depth basement definition. The joint use of surface gravity data, seismic imaging and borehole logging information makes the process more robust and allows an improvement in the estimate, providing a result closer to the actual basement relief. In addition, I would like to remark that the result obtained in the first scenario already has provided a very coherent basement relief when compared to the known basin framework. This is significant information, when comparing the differences in the costs and environment impact related to gravimetric and seismic surveys and also the well drillings

The STATFLUX code: a statistical method for calculation of flow and set of parameters, based on the Multiple-Compartment Biokinetical Model

The code STATFLUX, implementing a new and simple statistical procedure for the calculation of transfer coefficients in radionuclide transport to animals and plants, is proposed. The method is based on the general multiple-compartment model, which uses a system of linear equations involving geometrical volume considerations. Flow parameters were estimated by employing two different least-squares procedures: Derivative and Gauss-Marquardt methods, with the available experimental data of radionuclide concentrations as the input functions of time. The solution of the inverse problem, which relates a given set of flow parameter with the time evolution of concentration functions, is achieved via a Monte Carlo Simulation procedure.Program summaryTitle of program: STATFLUXCatalogue identifier: ADYS_v1_0Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADYS_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions: noneComputer for which the program is designed and others on which it has been tested: Micro-computer with Intel Pentium III, 3.0 GHzInstallation: Laboratory of Linear Accelerator, Department of Experimental Physics, University of São Paulo, BrazilOperating system: Windows 2000 and Windows XPProgramming language used: Fortran-77 as implemented in Microsoft Fortran 4.0. NOTE: Microsoft Fortran includes non-standard features which are used in this program. Standard Fortran compilers such as, g77, f77, ifort and NAG95, are not able to compile the code and therefore it has not been possible for the CPC Program Library to test the program.Memory, required to execute with typical data: 8 Mbytes of RAM memory and 100 MB of Hard disk memoryNo. of bits in a word: 16No. of lines in distributed program, including test data, etc.: 6912No. of bytes in distributed Program, including test data, etc.: 229 541Distribution format: tar.gzNature of the physical problem: the investigation of transport mechanisms for radioactive substances, through environmental pathways, is very important for radiological protection of populations. One such pathway, associated with the food chain, is the grass-animal-man sequence. The distribution of trace elements in humans and laboratory animals has been intensively studied over the past 60 years [R.C. Pendlenton, C.W. Mays, R.D. Lloyd, A.L. Brooks, Differential accumulation of iodine-131 from local fallout in people and milk, Health Phys. 9 (1963) 1253-1262]. In addition, investigations on the incidence of cancer in humans, and a possible causal relationship to radioactive fallout, have been undertaken [E.S. Weiss, M.L. Rallison, W.T. London, W.T. Carlyle Thompson, Thyroid nodularity in southwestern Utah school children exposed to fallout radiation, Amer. J. Public Health 61 (1971) 241-249; M.L. Rallison, B.M. Dobyns, F.R. Keating, J.E. Rall, F.H. Tyler, Thyroid diseases in children, Amer. J. Med. 56 (1974) 457-463; J.L. Lyon, M.R. Klauber, J.W. Gardner, K.S. Udall, Childhood leukemia associated with fallout from nuclear testing, N. Engl. J. Med. 300 (1979) 397-402]. From the pathways of entry of radionuclides in the human (or animal) body, ingestion is the most important because it is closely related to life-long alimentary (or dietary) habits. Those radionuclides which are able to enter the living cells by either metabolic or other processes give rise to localized doses which can be very high. The evaluation of these internally localized doses is of paramount importance for the assessment of radiobiological risks and radiological protection. The time behavior of trace concentration in organs is the principal input for prediction of internal doses after acute or chronic exposure. The General Multiple-Compartment Model (GMCM) is the powerful and more accepted method for biokinetical studies, which allows the calculation of concentration of trace elements in organs as a function of time, when the flow parameters of the model are known. However, few biokinetics data exist in the literature, and the determination of flow and transfer parameters by statistical fitting for each system is an open problem.Restriction on the complexity of the problem: This version of the code works with the constant volume approximation, which is valid for many situations where the biological half-live of a trace is lower than the volume rise time. Another restriction is related to the central flux model. The model considered in the code assumes that exist one central compartment (e.g., blood), that connect the flow with all compartments, and the flow between other compartments is not included.Typical running time: Depends on the choice for calculations. Using the Derivative Method the time is very short (a few minutes) for any number of compartments considered. When the Gauss-Marquardt iterative method is used the calculation time can be approximately 5-6 hours when similar to 15 compartments are considered. (C) 2006 Elsevier B.V. All rights reserved.

Formulação e implementação de um pacote para interpretar mapas magnéticos utilizando os polinômios de Walsh

O presente trabalho consiste na formulação de uma metodologia para interpretação automática de dados de campo magnético. Desta forma, a sua utilização tornará possível a determinação das fronteiras e magnetização de cada corpo. Na base desta metodologia foram utilizadas as características de variações abruptas de magnetização dos corpos. Estas variações laterais abruptas serão representadas por polinômios descontínuos conhecidos como polinômios de Walsh. Neste trabalho, muitos conceitos novos foram desenvolvidos na aplicação dos polinômios de Walsh para resolver problemas de inversão de dados aeromagnéticos. Dentre os novos aspectos considerados, podemos citar. (i) - O desenvolvimento de um algoritmo ótimo para gerar um jôgo dos polinômios "quase-ortogonais" baseados na distribuição de magnetização de Walsh. (ii) - O uso da metodologia damped least squares para estabilizar a solução inversa. (iii) - Uma investigação dos problemas da não-invariância, inerentes quando se usa os polinômios de Walsh. (iv) - Uma investigação da escolha da ordem dos polinômios, tomando-se em conta as limitações de resolução e o comportamento dos autovalores. Utilizando estas características dos corpos magnetizados é possível formular o problema direto, ou seja, a magnetização dos corpos obedece a distribuição de Walsh. É também possível formular o problema inverso, na qual a magnetização geradora do campo observado obedece a série de Walsh. Antes da utilização do método é necessária uma primeira estimativa da localização das fontes magnéticas. Foi escolhida uma metodologia desenvolvida por LOURES (1991), que tem como base a equação homogênea de Euler e cujas exigências necessárias à sua utilização é o conhecimento do campo magnético e suas derivadas. Para testar a metodologia com dados reais foi escolhida uma região localizada na bacia sedimentar do Alto Amazonas. Os dados foram obtidos a partir do levantamento aeromagnético realizado pela PETROBRÁS.

Image Reconstruction Using Interval Simulated Annealing in Electrical Impedance Tomography

Electrical impedance tomography (EIT) is an imaging technique that attempts to reconstruct the impedance distribution inside an object from the impedance between electrodes placed on the object surface. The EIT reconstruction problem can be approached as a nonlinear nonconvex optimization problem in which one tries to maximize the matching between a simulated impedance problem and the observed data. This nonlinear optimization problem is often ill-posed, and not very suited to methods that evaluate derivatives of the objective function. It may be approached by simulated annealing (SA), but at a large computational cost due to the expensive evaluation process of the objective function, which involves a full simulation of the impedance problem at each iteration. A variation of SA is proposed in which the objective function is evaluated only partially, while ensuring boundaries on the behavior of the modified algorithm.