An Ensemble Kalman Smoother for Nonlinear Dynamics

It is for mally proved that the general smoother for nonlinear dynamics can be for mulated as a sequential method, that is, obser vations can be assimilated sequentially during a for ward integration. The general filter can be derived from the smoother and it is shown that the general smoother and filter solutions at the final time become identical, as is expected from linear theor y. Then, a new smoother algorithm based on ensemble statistics is presented and examined in an example with the Lorenz equations. The new smoother can be computed as a sequential algorithm using only for ward-in-time model integrations. It bears a strong resemblance with the ensemble Kalman filter . The difference is that ever y time a new dataset is available during the for ward integration, an analysis is computed for all previous times up to this time. Thus, the first guess for the smoother is the ensemble Kalman filter solution, and the smoother estimate provides an improvement of this, as one would expect a smoother to do. The method is demonstrated in this paper in an intercomparison with the ensemble Kalman filter and the ensemble smoother introduced by van Leeuwen and Evensen, and it is shown to be superior in an application with the Lorenz equations. Finally , a discussion is given regarding the properties of the analysis schemes when strongly non-Gaussian distributions are used. It is shown that in these cases more sophisticated analysis schemes based on Bayesian statistics must be used.

A joint state and parameter estimation scheme for nonlinear dynamical systems

We present a novel algorithm for concurrent model state and parameter estimation in nonlinear dynamical systems. The new scheme uses ideas from three dimensional variational data assimilation (3D-Var) and the extended Kalman filter (EKF) together with the technique of state augmentation to estimate uncertain model parameters alongside the model state variables in a sequential filtering system. The method is relatively simple to implement and computationally inexpensive to run for large systems with relatively few parameters. We demonstrate the efficacy of the method via a series of identical twin experiments with three simple dynamical system models. The scheme is able to recover the parameter values to a good level of accuracy, even when observational data are noisy. We expect this new technique to be easily transferable to much larger models.

Cone-beam computerized tomographic, radiographic, and histologic evaluation of periapical repair in dogs` post-endodontic treatment

Objective. To evaluate the periapical repair after root canal treatment in the teeth of dogs using CT and conventional radiography and to compare these findings with the gold standard microscopic evaluation. Study design. The animals were divided into three groups according to endodontic treatment performed: Group 1, single-visit endodontic treatment in teeth without apical periodontitis; Group 2, single-visit endodontic treatment in teeth with apical periodontitis; and Group 3, endodontic treatment in teeth with apical periodontitis using calcium hydroxide as a root canal dressing. Group 4 consisted of teeth with apical periodontitis not submitted to root canal treatment and Group 5 consisted of healthy teeth without periapical disease. Radiographic, tomographic, and microscopic evaluations were performed by blind examiners. At 180 days experimental time, CT and radiographic measurements of periapical disease were compared with the gold standard microscopic measurement using intraclass correlation coefficient. Intergroup comparisons considering different methods of periapical lesions measurement or different clinical protocols of root canal treatment were performed by Kruskal Wallis test followed by Dunn. Integrity of lamina dura, presence of radiolucent areas, and presence of root resorption were analyzed by Fisher`s exact test. Results. There was discontinuity of the lamina dura and CPD in all teeth from Groups 2, 3, and 4 evaluated by tomography and radiography 45 days after CPD induction. Radiographically, 180 days after root canal treatment, there was no periapical lesion in teeth from Groups 1 and 3, different from groups 2 and 4 (p < .05). The highest reduction in the CPD size was observed on Group 3 (p < .05). According to the tomographic results, there was decrease of the size of the CPD on Group 3 but not on Groups 2 or 4. However, in all groups the periapical lesions presented larger mesio-distal extension if compared with radiography, both 45 days after CPD induction and 180 days after root canal treatment. At 180 days, CT measurements were closely related to microscopic results (ICC = 0.95) differently from radiographic evaluation (ICC = 0.86). Conclusion. CT Scan evaluation of periapical repair following root canal treatment provided similar information than that obtained by microscopic analysis, whereas radiographic evaluation underestimated the size do periapical lesion. (Oral Surg Oral Med Oral Pathol Oral Radiol Endod 2009; 108:796-805)

Accuracy of Periapical Radiography and Cone-Beam Computed Tomography Scans in Diagnosing Apical Periodontitis Using Histopathological Findings as a Gold Standard

Introduction: The aim of this study was to evaluate the accuracy of two imaging methods in diagnosing apical periodontitis (AP) using histopathological findings as a gold standard. Methods: The periapex of 83 treated or untreated roots of dogs` teeth was examined using periapical radiography (PR), cone-beam computed tomography (CBCT) scans, and histology. Sensitivity, specificity, predictive values, and accuracy of PR and CBCT diagnosis were calculated. Results: PR detected AP in 71% of roots, a CBCT scan detected AP in 84%, and AP was histologically diagnosed in 93% (p = 0.001). Overall, sensitivity was 0.77 and 0.91 for PR and CBCT, respectively. Specificity was 1 for both. Negative predictive value was 0.25 and 0.46 for PR and CBCT, respectively. Positive predictive value was 1 for both. Diagnostic accuracy (true positives + true negatives) was 0.78 and 0.92 for PR and CBCT (p = 0.028), respectively. Conclusion: A CBCT scan was more sensitive in detecting AP compared with PR, which was more likely to miss AP when it was still present. (J Endod 2009;35:1009-1012)

Outcome of Root Canal Treatment in Dogs Determined by Periapical Radiography and Cone-Beam Computed Tomography Scans

The purpose of this study was to compare the favorable outcome of root canal treatment determined by periapical radiographs (PRs) and cone beam computed tomography (CBCT) scans. Ninety-six roots of dogs` teeth were used to form four groups (n = 24). In group 1, root canal treatments were performed in healthy teeth. Root canals in groups 2 through 4 were infected until apical periodontitis (AP) was radiographically confirmed. Roots with AP were treated by one-visit therapy in group 2, by two-visit therapy in group 3, and left untreated in group 4. The radiolucent area in the PRs and the volume of CBCT-scanned periapical lesions were measured before and 6 months after the treatment. In groups 1, 2, and 3, a favorable outcome (lesions absent or reduced) was shown in 57 (79%) roots using PRs but only in 25 (35%) roots using CBCT scans (p = 0.0001). Unfavorable outcomes occurred more frequently after one-visit therapy than two-visit therapy when determined by CBCT scans (p = 0.023). (J Endod 2009; 35:723-726)

Heteroscedastic Nonlinear Regression Models

In this article, we present a generalization of the Bayesian methodology introduced by Cepeda and Gamerman (2001) for modeling variance heterogeneity in normal regression models where we have orthogonality between mean and variance parameters to the general case considering both linear and highly nonlinear regression models. Under the Bayesian paradigm, we use MCMC methods to simulate samples for the joint posterior distribution. We illustrate this algorithm considering a simulated data set and also considering a real data set related to school attendance rate for children in Colombia. Finally, we present some extensions of the proposed MCMC algorithm.

Exact Search Directions for Optimization of Linear and Nonlinear Models Based on Generalized Orthonormal Functions

A novel technique for selecting the poles of orthonormal basis functions (OBF) in Volterra models of any order is presented. It is well-known that the usual large number of parameters required to describe the Volterra kernels can be significantly reduced by representing each kernel using an appropriate basis of orthonormal functions. Such a representation results in the so-called OBF Volterra model, which has a Wiener structure consisting of a linear dynamic generated by the orthonormal basis followed by a nonlinear static mapping given by the Volterra polynomial series. Aiming at optimizing the poles that fully parameterize the orthonormal bases, the exact gradients of the outputs of the orthonormal filters with respect to their poles are computed analytically by using a back-propagation-through-time technique. The expressions relative to the Kautz basis and to generalized orthonormal bases of functions (GOBF) are addressed; the ones related to the Laguerre basis follow straightforwardly as a particular case. The main innovation here is that the dynamic nature of the OBF filters is fully considered in the gradient computations. These gradients provide exact search directions for optimizing the poles of a given orthonormal basis. Such search directions can, in turn, be used as part of an optimization procedure to locate the minimum of a cost-function that takes into account the error of estimation of the system output. The Levenberg-Marquardt algorithm is adopted here as the optimization procedure. Unlike previous related work, the proposed approach relies solely on input-output data measured from the system to be modeled, i.e., no information about the Volterra kernels is required. Examples are presented to illustrate the application of this approach to the modeling of dynamic systems, including a real magnetic levitation system with nonlinear oscillatory behavior.

Prediction of dynamical, nonlinear, and unstable process behavior

Process scheduling techniques consider the current load situation to allocate computing resources. Those techniques make approximations such as the average of communication, processing, and memory access to improve the process scheduling, although processes may present different behaviors during their whole execution. They may start with high communication requirements and later just processing. By discovering how processes behave over time, we believe it is possible to improve the resource allocation. This has motivated this paper which adopts chaos theory concepts and nonlinear prediction techniques in order to model and predict process behavior. Results confirm the radial basis function technique which presents good predictions and also low processing demands show what is essential in a real distributed environment.

MULTIPLICITY OF POSITIVE SOLUTIONS FOR A CLASS OF NONLINEAR SCHRODINGER EQUATIONS

This paper proves the multiplicity of positive solutions for the following class of quasilinear problems: {-epsilon(p)Delta(p)u+(lambda A(x) + 1)vertical bar u vertical bar(p-2)u = f(u), R(N) u(x)>0 in R(N), where Delta(p) is the p-Laplacian operator, N > p >= 2, lambda and epsilon are positive parameters, A is a nonnegative continuous function and f is a continuous function with subcritical growth. Here, we use variational methods to get multiplicity of positive solutions involving the Lusternick-Schnirelman category of intA(-1)(0) for all sufficiently large lambda and small epsilon.

Continuity of the dynamics in a localized large diffusion problem with nonlinear boundary conditions

This paper is concerned with singular perturbations in parabolic problems subjected to nonlinear Neumann boundary conditions. We consider the case for which the diffusion coefficient blows up in a subregion Omega(0) which is interior to the physical domain Omega subset of R(n). We prove, under natural assumptions, that the associated attractors behave continuously as the diffusion coefficient blows up locally uniformly in Omega(0) and converges uniformly to a continuous and positive function in Omega(1) = (Omega) over bar\Omega(0). (C) 2009 Elsevier Inc. All rights reserved.

Dynamics of a class of ODEs more general than almost periodic

A temporally global solution, if it exists, of a nonautonomous ordinary differential equation need not be periodic, almost periodic or almost automorphic when the forcing term is periodic, almost periodic or almost automorphic, respectively. An alternative class of functions extending periodic and almost periodic functions which has the property that a bounded temporally global solution solution of a nonautonomous ordinary differential equation belongs to this class when the forcing term does is introduced here. Specifically, the class of functions consists of uniformly continuous functions, defined on the real line and taking values in a Banach space, which have pre-compact ranges. Besides periodic and almost periodic functions, this class also includes many nonrecurrent functions. Assuming a hyperbolic structure for the unperturbed linear equation and certain properties for the linear and nonlinear parts, the existence of a special bounded entire solution, as well the existence of stable and unstable manifolds of this solution are established. Moreover, it is shown that this solution and these manifolds inherit the temporal behaviour of the vector field equation. In the stable case it is shown that this special solution is the pullback attractor of the system. A class of infinite dimensional examples involving a linear operator consisting of a time independent part which generates a C(0)-semigroup plus a small time dependent part is presented and applied to systems of coupled heat and beam equations. (C) 2010 Elsevier Ltd. All rights reserved.