Krylov subspace approximations for the exponential Euler method: Error estimates and the harmonic Ritz approximant

We study Krylov subspace methods for approximating the matrix-function vector product φ(tA)b where φ(z) = [exp(z) - 1]/z. This product arises in the numerical integration of large stiff systems of differential equations by the Exponential Euler Method, where A is the Jacobian matrix of the system. Recently, this method has found application in the simulation of transport phenomena in porous media within mathematical models of wood drying and groundwater flow. We develop an a posteriori upper bound on the Krylov subspace approximation error and provide a new interpretation of a previously published error estimate. This leads to an alternative Krylov approximation to φ(tA)b, the so-called Harmonic Ritz approximant, which we find does not exhibit oscillatory behaviour of the residual error.

Cephalometric analysis of prediction tracings: A comparison of three different methods

The purpose of this study was to compare-using cephalometric analysis (McNamara, and Legan and Burstone)-prediction tracings performed using three different methods, that is, manual and using the Dentofacial Planner Plus and Dolphin Image computer programs, with postoperative outcomes. Pre- and postoperative (6 months after surgery) lateral cephalometric radiographs were selected from 25 long-faced patients treated with combined surgery. Prediction tracings were made with each method and compared cephalometrically with the postoperative results. This protocol was repeated once more for method error evaluation. Statistical analysis was made by ANOVA and the Tukey test. The results showed superior predictability when the manual method was applied (50% similarity to postoperative results), followed by Dentofacial Planner Plus (31.2%) and Dolphin Image (18.8%). The experimental condition suggests that the manual method provides greater accuracy, although the predictability of the digital methods proved quite satisfactory. © 2013 World Federation of Orthodontists.

Estudo tridimensional da oclusão normal na população brasileira

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Análise do emprego do cálculo amostral e do erro do método em pesquisas científicas publicadas na literatura ortodôntica nacional e internacional

Esta investigação tem o objetivo de avaliar, quantitativamente, com que frequência os pesquisadores da ciência ortodôntica têm empregado o cálculo amostral e a análise do erro do método em pesquisas publicadas no Brasil e nos Estados Unidos.

Angulação dos caninos em indivíduos portadores de má oclusão de Classe I e de Classe III: análise comparativa através de um novo método utilizando imagens digitalizadas

OBJETIVO: determinar as angulações mesiodistais das coroas dos caninos em indivíduos portadores de má oclusão de Classe III, comparando-os a indivíduos Classe I. MÉTODOS: foram empregadas medidas tomadas em fotografias digitalizadas de modelos de gesso e transportadas para um programa gráfico para leitura das medidas (Image Tool). Tais procedimentos foram repetidos para avaliação do erro do método casual (fórmula de Dahlberg) e para a análise da reprodutibilidade através da Correlação intraclasse. A amostra constituiu-se de 57 pacientes com dentição permanente completa e não tratados ortodonticamente, dividida em dois grupos, de acordo com a má oclusão apresentada: o grupo I foi constituído por 33 pacientes portadores de má oclusão de Classe I, sendo 16 do sexo masculino e 17 do feminino, com média de idades de 27 anos; o grupo II era representado por 24 pacientes portadores de má oclusão de Classe III, 20 do sexo masculino e 4 do feminino, com média de idades de 22 anos. RESULTADOS: o erro casual mostrou-se com uma variação de 1,54 a 1,96 graus para a angulação dos caninos. A análise estatística revelou que o método apresenta uma excelente reprodutibilidade (p<0,01). Os resultados obtidos na angulação da coroa dos caninos não mostraram diferença estatisticamente significativa entre os caninos superiores nos grupos Classe I e Classe III, embora esse dente mostrasse, em média, uma angulação 2 graus maior nos indivíduos Classe III. Entretanto, para os caninos inferiores, foi observada uma diferença estatisticamente significativa em ambos os lados (p=0,0009 e p=0,0074) entre os grupos Classe I e Classe III. Os pacientes Classe III apresentaram uma menor angulação nos caninos inferiores em comparação aos pacientes Classe I, tendendo a acompanhar a compensação natural dos incisivos, descrita rotineiramente na literatura. CONCLUSÃO: os resultados permitem concluir que as compensações dentárias, frequentemente observadas na literatura para a região de incisivos, se estendem também à angulação dos caninos, principalmente no que se refere à arcada inferior.

Cephalometric evaluation of the predictability of bimaxillary surgical-orthodontic treatment outcomes in long face pattern patients: a retrospective study

OBJECTIVE: The aim of this study was to compare by means of McNamara as well as Legan and Burstone's cephalometric analyses, both manual and digitized (by Dentofacial Planner Plus and Dolphin Image software) prediction tracings to post-surgical results. METHODS: Pre and post-surgical teleradiographs (6 months) of 25 long face patients subjected to combined orthognathic surgery were selected. Manual and computerized prediction tracings of each patient were performed and cephalometrically compared to post-surgical outcomes. This protocol was repeated in order to evaluate the method error and statistical evaluation was conducted by means of analysis of variance and Tukey's test. RESULTS: A higher frequency of cephalometric variables, which were not statistically different from the actual post-surgical results for the manual method, was observed. It was followed by DFPlus and Dolphin software; in which similar cephalometric values for most variables were observed. CONCLUSION: It was concluded that the manual method seemed more reliable, although the predictability of the evaluated methods (computerized and manual) proved to be reasonably satisfactory and similar.

Dahlberg formula: a novel approach for its evaluation

INTRODUCTION: The accurate evaluation of error of measurement (EM) is extremely important as in growth studies as in clinical research, since there are usually quantitatively small changes. In any study it is important to evaluate the EM to validate the results and, consequently, the conclusions. Because of its extreme simplicity, the Dahlberg formula is largely used worldwide, mainly in cephalometric studies. OBJECTIVES: (I) To elucidate the formula proposed by Dahlberg in 1940, evaluating it by comparison with linear regression analysis; (II) To propose a simple methodology to analyze the results, which provides statistical elements to assist researchers in obtaining a consistent evaluation of the EM. METHODS: We applied linear regression analysis, hypothesis tests on its parameters and a formula involving the standard deviation of error of measurement and the measured values. RESULTS AND CONCLUSION: we introduced an error coefficient, which is a proportion related to the scale of observed values. This provides new parameters to facilitate the evaluation of the impact of random errors in the research final results.

Mesh-free approximations via the error reproducing kernel method and applications to nonlinear systems developing shocks

Error estimates for the error reproducing kernel method (ERKM) are provided. The ERKM is a mesh-free functional approximation scheme [A. Shaw, D. Roy, A NURBS-based error reproducing kernel method with applications in solid mechanics, Computational Mechanics (2006), to appear (available online)], wherein a targeted function and its derivatives are first approximated via non-uniform rational B-splines (NURBS) basis function. Errors in the NURBS approximation are then reproduced via a family of non-NURBS basis functions, constructed using a polynomial reproduction condition, and added to the NURBS approximation of the function obtained in the first step. In addition to the derivation of error estimates, convergence studies are undertaken for a couple of test boundary value problems with known exact solutions. The ERKM is next applied to a one-dimensional Burgers equation where, time evolution leads to a breakdown of the continuous solution and the appearance of a shock. Many available mesh-free schemes appear to be unable to capture this shock without numerical instability. However, given that any desired order of continuity is achievable through NURBS approximations, the ERKM can even accurately approximate functions with discontinuous derivatives. Moreover, due to the variation diminishing property of NURBS, it has advantages in representing sharp changes in gradients. This paper is focused on demonstrating this ability of ERKM via some numerical examples. Comparisons of some of the results with those via the standard form of the reproducing kernel particle method (RKPM) demonstrate the relative numerical advantages and accuracy of the ERKM.

Projection Error Propagation-based Regularization (PEPR) method for resistivity reconstruction in Electrical Impedance Tomography (EIT)

A novel Projection Error Propagation-based Regularization (PEPR) method is proposed to improve the image quality in Electrical Impedance Tomography (EIT). PEPR method defines the regularization parameter as a function of the projection error developed by difference between experimental measurements and calculated data. The regularization parameter in the reconstruction algorithm gets modified automatically according to the noise level in measured data and ill-posedness of the Hessian matrix. Resistivity imaging of practical phantoms in a Model Based Iterative Image Reconstruction (MoBIIR) algorithm as well as with Electrical Impedance Diffuse Optical Reconstruction Software (EIDORS) with PEPR. The effect of PEPR method is also studied with phantoms with different configurations and with different current injection methods. All the resistivity images reconstructed with PEPR method are compared with the single step regularization (STR) and Modified Levenberg Regularization (LMR) techniques. The results show that, the PEPR technique reduces the projection error and solution error in each iterations both for simulated and experimental data in both the algorithms and improves the reconstructed images with better contrast to noise ratio (CNR), percentage of contrast recovery (PCR), coefficient of contrast (COC) and diametric resistivity profile (DRP). (C) 2013 Elsevier Ltd. All rights reserved.

A Reliable Residual Based A Posteriori Error Estimator for a Quadratic Finite Element Method for the Elliptic Obstacle Problem

A residual based a posteriori error estimator is derived for a quadratic finite element method (FEM) for the elliptic obstacle problem. The error estimator involves various residuals consisting of the data of the problem, discrete solution and a Lagrange multiplier related to the obstacle constraint. The choice of the discrete Lagrange multiplier yields an error estimator that is comparable with the error estimator in the case of linear FEM. Further, an a priori error estimate is derived to show that the discrete Lagrange multiplier converges at the same rate as that of the discrete solution of the obstacle problem. The numerical experiments of adaptive FEM show optimal order convergence. This demonstrates that the quadratic FEM for obstacle problem exhibits optimal performance.