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Error Estimator (error + estimator)
Kinds of Error Estimator Selected AbstractsA continuum mechanics-based framework for boundary and finite element mesh optimization in two dimensions for application in excavation analysisINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 4 2005Attila M. Zsáki Abstract The determination of the optimum excavation sequences in mining and civil engineering using numerical stress analysis procedures requires repeated solution of large models. Often such models contain much more complexity and geometric detail than required to arrive at an accurate stress analysis solution, especially considering our limited knowledge of rock mass properties. This paper develops an automated framework for estimating the effects of excavations at a region of interest, and optimizing the geometry used for stress analysis. It eliminates or simplifies the excavations in a model while maintaining the accuracy of analysis results. The framework can equally be applied to two-dimensional boundary and finite element models. The framework will have the largest impact for non-linear finite element analysis. It can significantly reduce computational times for such analysis by simplifying models. Error estimators are used in the framework to assess accuracy. The advantages of applying the framework are demonstrated on an excavation-sequencing scenario. Copyright © 2005 John Wiley & Sons, Ltd. [source] Rapid Exponential Convergence of Finite Element Estimates of the Effective Properties of Heterogeneous MaterialsADVANCED ENGINEERING MATERIALS, Issue 11 2007A. Gusev We develop and validate a general-purpose error estimator for the finite element solutions for the effective properties of heterogeneous materials. We show that the error should decrease exponentially upon increasing order of the polynomial interpolation. We use this finding to demonstrate the practical feasibility of reliable property predictions for a majority of particulate-morphology heterogeneous materials. [source] Numerical study of the effectivity index for an anisotropic error indicator based on Zienkiewicz,Zhu error estimatorINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 1 2003M. Picasso Abstract The framework of Formaggia and Perotto (Numerische Mathematik 2001; 89: 641,667) is considered to derive a new anisotropic error indicator for a Laplace problem in the energy norm. The matrix containing the error gradient is approached using a Zienkiewicz,Zhu error estimator. A numerical study of the effectivity index is proposed for anisotropic unstructured meshes, showing that our indicator is sharp. An anisotropic adaptive algorithm is implemented, aiming at controlling the estimated relative error. Copyright © 2003 John Wiley & Sons, Ltd. [source] Error estimation in a stochastic finite element method in electrokineticsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2010S. Clénet Abstract Input data to a numerical model are not necessarily well known. Uncertainties may exist both in material properties and in the geometry of the device. They can be due, for instance, to ageing or imperfections in the manufacturing process. Input data can be modelled as random variables leading to a stochastic model. In electromagnetism, this leads to solution of a stochastic partial differential equation system. The solution can be approximated by a linear combination of basis functions rising from the tensorial product of the basis functions used to discretize the space (nodal shape function for example) and basis functions used to discretize the random dimension (a polynomial chaos expansion for example). Some methods (SSFEM, collocation) have been proposed in the literature to calculate such approximation. The issue is then how to compare the different approaches in an objective way. One solution is to use an appropriate a posteriori numerical error estimator. In this paper, we present an error estimator based on the constitutive relation error in electrokinetics, which allows the calculation of the distance between an average solution and the unknown exact solution. The method of calculation of the error is detailed in this paper from two solutions that satisfy the two equilibrium equations. In an example, we compare two different approximations (Legendre and Hermite polynomial chaos expansions) for the random dimension using the proposed error estimator. In addition, we show how to choose the appropriate order for the polynomial chaos expansion for the proposed error estimator. Copyright © 2009 John Wiley & Sons, Ltd. [source] An a posteriori error estimator for the mimetic finite difference approximation of elliptic problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2008Lourenço Beirão da Veiga Abstract We present an a posteriori error indicator for the mimetic finite difference approximation of elliptic problems in the mixed form. We show that this estimator is reliable and efficient with respect to an energy-type error comprising both flux and pressure. Its performance is investigated by numerically solving the diffusion equation on computational domains with different shapes, different permeability tensors, and different types of computational meshes. Copyright © 2008 John Wiley & Sons, Ltd. [source] Adaptive mesh technique for thermal,metallurgical numerical simulation of arc welding processesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2008M. Hamide Abstract A major problem arising in finite element analysis of welding is the long computer times required for a complete three-dimensional analysis. In this study, an adaptative strategy for coupled thermometallurgical analysis of welding is proposed and applied in order to provide accurate results in a minimum computer time. The anisotropic adaptation procedure is controlled by a directional error estimator based on local interpolation error and recovery of the second derivatives of different fields involved in the finite element calculation. The methodology is applied to the simulation of a gas,tungsten-arc fusion line processed on a steel plate. The temperature field and the phase distributions during the welding process are analyzed by the FEM method showing the benefits of dynamic mesh adaptation. A significant increase in accuracy is obtained with a reduced computational effort. Copyright © 2007 John Wiley & Sons, Ltd. [source] Adaptive superposition of finite element meshes in non-linear transient solid mechanics problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2007Z. Yue Abstract An s-adaptive finite element procedure is developed for the transient analysis of 2-D solid mechanics problems with material non-linearity due to progressive damage. The resulting adaptive method simultaneously estimates and controls both the spatial error and temporal error within user-specified tolerances. The spatial error is quantified by the Zienkiewicz,Zhu error estimator and computed via superconvergent patch recovery, while the estimation of temporal error is based on the assumption of a linearly varying third-order time derivatives of the displacement field in conjunction with direct numerical time integration. The distinguishing characteristic of the s-adaptive procedure is the use of finite element mesh superposition (s-refinement) to provide spatial adaptivity. Mesh superposition proves to be particularly advantageous in computationally demanding non-linear transient problems since it is faster, simpler and more efficient than traditional h-refinement schemes. Numerical examples are provided to demonstrate the performance characteristics of the s-adaptive method for quasi-static and transient problems with material non-linearity. Copyright © 2007 John Wiley & Sons, Ltd. [source] A methodology for the formulation of error estimators for time integration in linear solid and structural dynamicsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2006I. Romero Abstract In this article, we present a novel methodology for the formulation of a posteriori error estimators applicable to time-stepping algorithms of the type commonly employed in solid and structural mechanics. The estimators constructed with the presented methodology are accurate and can be implemented very efficiently. More importantly, they provide reliable error estimations even in non-smooth problems where many standard estimators fail to capture the order of magnitude of the error. The proposed methodology is applied, as an illustrative example, to construct an error estimator for the Newmark method. Numerical examples of its performance and comparison with existing error estimators are presented. These examples verify the good accuracy and robustness predicted by the analysis. Copyright © 2005 John Wiley & Sons, Ltd. [source] Lower bound limit analysis with adaptive remeshingINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 14 2005Andrei V. Lyamin Abstract The objective of this work is to present an adaptive remeshing procedure for lower bound limit analysis with application to soil mechanics. Unlike conventional finite element meshes, a lower bound grid incorporates statically admissible stress discontinuities between adjacent elements. These discontinuities permit large stress jumps over an infinitesimal distance and reduce the number of elements needed to predict the collapse load accurately. In general, the role of the discontinuities is crucial as their arrangement and distribution has a dramatic influence on the accuracy of the lower bound solution (Limit Analysis and Soil Plasticity, 1975). To ensure that the discontinuities are positioned in an optimal manner requires an error estimator and mesh adaptation strategy which accounts for the presence of stress singularities in the computed stress field. Recently, Borges et al. (Int. J. Solids Struct. 2001; 38:1707,1720) presented an anisotropic mesh adaptation strategy for a mixed limit analysis formulation which used a directional error estimator. In the present work, this strategy has been tailored to suit a discontinuous lower bound formulation which employs the stresses and body forces as primary unknowns. The adapted mesh has a maximum density of discontinuities in the direction of the maximum rate of change in the stress field. For problems involving strong stress singularities in the boundary conditions (e.g. a strip footing), the automatic generation of discontinuity fans, centred on the singular points, has been implemented. The efficiency of the proposed technique is demonstrated by analysis of two classical soil mechanics problems; namely the bearing capacity of a rigid strip footing and the collapse of a vertical cut. Copyright © 2005 John Wiley & Sons, Ltd. [source] Smart element method I. The Zienkiewicz,Zhu feedbackINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2005Shaofan Li Abstract A new error control finite element formulation is developed and implemented based on the variational multiscale method, the inclusion theory in homogenization, and the Zienkiewicz,Zhu error estimator. By synthesizing variational multiscale method in computational mechanics, the equivalent eigenstrain principle in micromechanics, and the Zienkiewicz,Zhu error estimator in the finite element method (FEM), the new finite element formulation can automatically detect and subsequently homogenize its own discretization errors in a self-adaptive and a self-adjusting manner. It is the first finite element formulation that combines an optimal feedback mechanism and a precisely defined homogenization procedure to reduce its own discretization errors and hence to control numerical pollutions. The paper focuses on the following two issues: (1) how to combine a multiscale method with the existing finite element error estimate criterion through a feedback mechanism, and (2) convergence study. It has been shown that by combining the proposed variational multiscale homogenization method with the Zienkiewicz,Zhu error estimator a clear improvement can be made on the coarse scale computation. It is also shown that when the finite element mesh is refined, the solution obtained by the variational eigenstrain multiscale method will converge to the exact solution. Copyright © 2004 John Wiley & Sons, Ltd. [source] Numerical simulation of granular materials by an improved discrete element methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2005J. Fortin Abstract In this paper, we present an improved discrete element method based on the non-smooth contact dynamics and the bi-potential concept. The energy dissipated during the collisions is taken into account by means of restitution coefficients. The interaction between particles is modelled by Coulomb unilateral contact law with dry friction which is typically non-associated: during the contact, the sliding vector is not normal to the friction cone. The main feature of our algorithm is to overcome this difficulty by means of the bi-potential theory. It leads to an easy implement predictor,corrector scheme involving just an orthogonal projection onto the friction cone. Moreover the convergence test is based on an error estimator in constitutive law using the corner stone inequality of the bipotential. Then we present numerical simulations which show the robustness of our algorithm and the various possibilities of the software ,MULTICOR' developed with this approach. Copyright © 2004 John Wiley & Sons, Ltd. [source] An a posteriori error estimator for the p - and hp -versions of the finite element methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2005J. E. Tarancón Abstract An a posteriori error estimator is proposed in this paper for the p - and hp -versions of the finite element method in two-dimensional linear elastostatic problems. The local error estimator consists in an enhancement of an error indicator proposed by Bertóti and Szabó (Int. J. Numer. Meth. Engng. 1998; 42:561,587), which is based on the minimum complementary energy principle. In order to obtain the local error estimate, this error indicator is corrected by a factor which depends only on the polynomial degree of the element. The proposed error estimator shows a good effectivity index in meshes with uniform and non-uniform polynomial distributions, especially when the global error is estimated. Furthermore, the local error estimator is reliable enough to guide p - and hp -adaptive refinement strategies. Copyright © 2004 John Wiley & Sons, Ltd. [source] Voxel-based meshing and unit-cell analysis of textile compositesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 7 2003Hyung Joo Kim Abstract Unit-cell homogenization techniques are frequently used together with the finite element method to compute effective mechanical properties for a wide range of different composites and heterogeneous materials systems. For systems with very complicated material arrangements, mesh generation can be a considerable obstacle to usage of these techniques. In this work, pixel-based (2D) and voxel-based (3D) meshing concepts borrowed from image processing are thus developed and employed to construct the finite element models used in computing the micro-scale stress and strain fields in the composite. The potential advantage of these techniques is that generation of unit-cell models can be automated, thus requiring far less human time than traditional finite element models. Essential ideas and algorithms for implementation of proposed techniques are presented. In addition, a new error estimator based on sensitivity of virtual strain energy to mesh refinement is presented and applied. The computational costs and rate of convergence for the proposed methods are presented for three different mesh-refinement algorithms: uniform refinement; selective refinement based on material boundary resolution; and adaptive refinement based on error estimation. Copyright © 2003 John Wiley & Sons, Ltd. [source] On the use of anisotropic a posteriori error estimators for the adaptative solution of 3D inviscid compressible flowsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2009Y. Bourgault Abstract This paper describes the use of an a posteriori error estimator to control anisotropic mesh adaptation for computing inviscid compressible flows. The a posteriori error estimator and the coupling strategy with an anisotropic remesher are first introduced. The mesh adaptation is controlled by a single-parameter tolerance (TOL) in regions where the solution is regular, whereas a condition on the minimal element size hmin is enforced across solution discontinuities. This hmin condition is justified on the basis of an asymptotic analysis. The efficiency of the approach is tested with a supersonic flow over an aircraft. The evolution of a mesh adaptation/flow solution loop is shown, together with the influence of the parameters TOL and hmin. We verify numerically that the effect of varying hmin is concordant with the conclusions of the asymptotic analysis, giving hints on the selection of hmin with respect to TOL. Finally, we check that the results obtained with the a posteriori error estimator are at least as accurate as those obtained with anisotropic a priori error estimators. All the results presented can be obtained using a standard desktop computer, showing the efficiency of these adaptative methods. Copyright © 2008 John Wiley & Sons, Ltd. [source] A posteriori error estimation for convection dominated problems on anisotropic meshesMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2003Gerd Kunert Abstract A singularly perturbed convection,diffusion problem in two and three space dimensions is discretized using the streamline upwind Petrov Galerkin (SUPG) variant of the finite element method. The dominant convection frequently gives rise to solutions with layers; hence anisotropic finite elements can be applied advantageously. The main focus is on a posteriori energy norm error estimation that is robust in the perturbation parameter and with respect to the mesh anisotropy. A residual error estimator and a local problem error estimator are proposed and investigated. The analysis reveals that the upper error bound depends on the alignment of the anisotropies of the mesh and of the solution. Hence reliable error estimation is possible for suitable anisotropic meshes. The lower error bound depends on the problem data via a local mesh Peclet number. Thus efficient error estimation is achieved for small mesh Peclet numbers. Altogether, error estimation approaches for isotropic meshes are successfully extended to anisotropic elements. Several numerical experiments support the analysis. Copyright © 2003 John Wiley & Sons, Ltd. [source] Convergence of adaptive edge finite element methods for H(curl)-elliptic problemsNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2-3 2010Liuqiang Zhong Abstract The standard adaptive edge finite element method (AEFEM), using first/second family Nédélec edge elements with any order, for the three-dimensional H(curl)-elliptic problems with variable coefficients is shown to be convergent for the sum of the energy error and the scaled error estimator. The special treatment of the data oscillation and the interior node property are removed from the proof. Numerical experiments indicate that the adaptive meshes and the associated numerical complexity are quasi-optimal. Copyright © 2010 John Wiley & Sons, Ltd. [source] Equilibrated error estimators for discontinuous Galerkin methodsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2008Sarah Cochez-Dhondt Abstract We consider some diffusion problems in domains of ,d, d = 2 or 3 approximated by a discontinuous Galerkin method with polynomials of any degree. We propose a new a posteriori error estimator based on H(div)-conforming elements. It is shown that this estimator gives rise to an upper bound where the constant is one up to higher order terms. The lower bound is also established with a constant depending on the aspect ratio of the mesh, the dependence with respect to the coefficients being also traced. The reliability and efficiency of the proposed estimator is confirmed by some numerical tests. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 [source] A posteriori error estimator for expanded mixed hybrid methods,NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2007Dongho Kim Abstract In this article, we construct an a posteriori error estimator for expanded mixed hybrid finite-element methods for second-order elliptic problems. An a posteriori error analysis yields reliable and efficient estimate based on residuals. Several numerical examples are presented to show the effectivity of our error indicators. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 330,349, 2007 [source] On the subdomain-Galerkin/least squares method for 2- and 3-D mixed elliptic problems with reaction termsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2002Suh-Yuh Yang Abstract In this article we apply the subdomain-Galerkin/least squares method, which is first proposed by Chang and Gunzburger for first-order elliptic systems without reaction terms in the plane, to solve second-order non-selfadjoint elliptic problems in two- and three-dimensional bounded domains with triangular or tetrahedral regular triangulations. This method can be viewed as a combination of a direct cell vertex finite volume discretization step and an algebraic least-squares minimization step in which the pressure is approximated by piecewise linear elements and the flux by the lowest order Raviart-Thomas space. This combined approach has the advantages of both finite volume and least-squares methods. Among other things, the combined method is not subject to the Ladyzhenskaya-Babus,ka-Brezzi condition, and the resulting linear system is symmetric and positive definite. An optimal error estimate in the H1(,) × H(div; ,) norm is derived. An equivalent residual-type a posteriori error estimator is also given. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 738,751, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10030. [source] Two-sample Comparison Based on Prediction Error, with Applications to Candidate Gene Association StudiesANNALS OF HUMAN GENETICS, Issue 1 2007K. Yu Summary To take advantage of the increasingly available high-density SNP maps across the genome, various tests that compare multilocus genotypes or estimated haplotypes between cases and controls have been developed for candidate gene association studies. Here we view this two-sample testing problem from the perspective of supervised machine learning and propose a new association test. The approach adopts the flexible and easy-to-understand classification tree model as the learning machine, and uses the estimated prediction error of the resulting prediction rule as the test statistic. This procedure not only provides an association test but also generates a prediction rule that can be useful in understanding the mechanisms underlying complex disease. Under the set-up of a haplotype-based transmission/disequilibrium test (TDT) type of analysis, we find through simulation studies that the proposed procedure has the correct type I error rates and is robust to population stratification. The power of the proposed procedure is sensitive to the chosen prediction error estimator. Among commonly used prediction error estimators, the .632+ estimator results in a test that has the best overall performance. We also find that the test using the .632+ estimator is more powerful than the standard single-point TDT analysis, the Pearson's goodness-of-fit test based on estimated haplotype frequencies, and two haplotype-based global tests implemented in the genetic analysis package FBAT. To illustrate the application of the proposed method in population-based association studies, we use the procedure to study the association between non-Hodgkin lymphoma and the IL10 gene. [source] Guaranteed computable error bounds for conforming and nonconforming finite element analyses in planar elasticityINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2010Mark Ainsworth Abstract We obtain fully computable a posteriori error estimators for the energy norm of the error in second-order conforming and nonconforming finite element approximations in planar elasticity. These estimators are completely free of unknown constants and give a guaranteed numerical upper bound on the norm of the error. The estimators are shown to also provide local lower bounds, up to a constant and higher-order data oscillation terms. Numerical examples are presented illustrating the theory and confirming the effectiveness of the estimator. Copyright © 2009 John Wiley & Sons, Ltd. [source] A methodology for the formulation of error estimators for time integration in linear solid and structural dynamicsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2006I. Romero Abstract In this article, we present a novel methodology for the formulation of a posteriori error estimators applicable to time-stepping algorithms of the type commonly employed in solid and structural mechanics. The estimators constructed with the presented methodology are accurate and can be implemented very efficiently. More importantly, they provide reliable error estimations even in non-smooth problems where many standard estimators fail to capture the order of magnitude of the error. The proposed methodology is applied, as an illustrative example, to construct an error estimator for the Newmark method. Numerical examples of its performance and comparison with existing error estimators are presented. These examples verify the good accuracy and robustness predicted by the analysis. Copyright © 2005 John Wiley & Sons, Ltd. [source] Generalization of robustness test procedure for error estimators.INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2005Part I: formulation for patches near kinked boundaries Abstract In this part of paper we shall extend the formulation proposed by Babu,ka and co-workers for robustness patch test, for quality assessment of error estimators, to more general cases of patch locations especially in three-dimensional problems. This is performed first by finding an asymptotic finite element solution at interior parts of a problem with assumed smooth exact solution and then adding a correction part to obtain the solution near a kinked boundary irrespective of other boundary conditions at far ends of the domain. It has been shown that the solution corresponding to the correction part may be obtained in a spectral form by assuming a suitable proportionality relation between the nodal values of a mesh with repeatable pattern of macro-patches. Having found the asymptotic finite element solution, the performance of error estimators may be examined. Although in this paper we focus on the asymptotic behaviour of error estimators, the method described in this part may be used to obtain finite element solution for two/three-dimensional unbounded heat/elasticity problems with homogeneous differential equations. Some numerical results are presented to show the validity and performance of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd. [source] Generalization of robustness test procedure for error estimators.INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2005Part II: test results for error estimators using SPR Abstract In this part of the paper we shall use the formulation given in the first part to assess the quality of recovery-based error estimators using two recovery methods, i.e. superconvergent patch recovery (SPR) and recovery by equilibrium in patches (REP). The recovery methods have been shown to be asymptotically robust and superconvergent when applied to two-dimensional problems. In this study we shall examine the behaviour of the recovery methods on several three-dimensional mesh patterns for patches located either inside or at boundaries. This is performed by first finding an asymptotic finite element solution, irrespective of boundary conditions at far ends of the domain, and then applying the recovery methods. The test procedure near kinked boundaries is explained in a step-by-step manner. The results are given in a series of tables and figures for various cases of three-dimensional mesh patterns. It has been experienced that the full superconvergent property is generally lost due to presence of boundary layer solution and the definition of the recoveries near boundaries though the results of the robustness test is still within an acceptable range. Copyright © 2005 John Wiley & Sons, Ltd. [source] On the use of anisotropic a posteriori error estimators for the adaptative solution of 3D inviscid compressible flowsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2009Y. Bourgault Abstract This paper describes the use of an a posteriori error estimator to control anisotropic mesh adaptation for computing inviscid compressible flows. The a posteriori error estimator and the coupling strategy with an anisotropic remesher are first introduced. The mesh adaptation is controlled by a single-parameter tolerance (TOL) in regions where the solution is regular, whereas a condition on the minimal element size hmin is enforced across solution discontinuities. This hmin condition is justified on the basis of an asymptotic analysis. The efficiency of the approach is tested with a supersonic flow over an aircraft. The evolution of a mesh adaptation/flow solution loop is shown, together with the influence of the parameters TOL and hmin. We verify numerically that the effect of varying hmin is concordant with the conclusions of the asymptotic analysis, giving hints on the selection of hmin with respect to TOL. Finally, we check that the results obtained with the a posteriori error estimator are at least as accurate as those obtained with anisotropic a priori error estimators. All the results presented can be obtained using a standard desktop computer, showing the efficiency of these adaptative methods. Copyright © 2008 John Wiley & Sons, Ltd. [source] A least square extrapolation method for the a posteriori error estimate of the incompressible Navier Stokes problemINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2005M. Garbey Abstract A posteriori error estimators are fundamental tools for providing confidence in the numerical computation of PDEs. To date, the main theories of a posteriori estimators have been developed largely in the finite element framework, for either linear elliptic operators or non-linear PDEs in the absence of disparate length scales. On the other hand, there is a strong interest in using grid refinement combined with Richardson extrapolation to produce CFD solutions with improved accuracy and, therefore, a posteriori error estimates. But in practice, the effective order of a numerical method often depends on space location and is not uniform, rendering the Richardson extrapolation method unreliable. We have recently introduced (Garbey, 13th International Conference on Domain Decomposition, Barcelona, 2002; 379,386; Garbey and Shyy, J. Comput. Phys. 2003; 186:1,23) a new method which estimates the order of convergence of a computation as the solution of a least square minimization problem on the residual. This method, called least square extrapolation, introduces a framework facilitating multi-level extrapolation, improves accuracy and provides a posteriori error estimate. This method can accommodate different grid arrangements. The goal of this paper is to investigate the power and limits of this method via incompressible Navier Stokes flow computations. Copyright © 2005 John Wiley & Sons, Ltd. [source] Error analysis in cross-correlation of sky maps: application to the Integrated Sachs,Wolfe detectionMONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Issue 4 2007Anna Cabré ABSTRACT Constraining cosmological parameters from measurements of the Integrated Sachs,Wolfe effect requires developing robust and accurate methods for computing statistical errors in the cross-correlation between maps. This paper presents a detailed comparison of such error estimation applied to the case of cross-correlation of cosmic microwave background (CMB) and large-scale structure data. We compare theoretical models for error estimation with Monte Carlo simulations where both the galaxy and the CMB maps vary around a fiducial autocorrelation and cross-correlation model which agrees well with the current concordance , cold dark matter cosmology. Our analysis compares estimators both in harmonic and configuration (or real) space, quantifies the accuracy of the error analysis and discusses the impact of partial sky survey area and the choice of input fiducial model on dark energy constraints. We show that purely analytic approaches yield accurate errors even in surveys that cover only 10 per cent of the sky and that parameter constraints strongly depend on the fiducial model employed. Alternatively, we discuss the advantages and limitations of error estimators that can be directly applied to data. In particular, we show that errors and covariances from the jackknife method agree well with the theoretical approaches and simulations. We also introduce a novel method in real space that is computationally efficient and can be applied to real data and realistic survey geometries. Finally, we present a number of new findings and prescriptions that can be useful for analysis of real data and forecasts, and present a critical summary of the analyses done to date. [source] Equilibrated error estimators for discontinuous Galerkin methodsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2008Sarah Cochez-Dhondt Abstract We consider some diffusion problems in domains of ,d, d = 2 or 3 approximated by a discontinuous Galerkin method with polynomials of any degree. We propose a new a posteriori error estimator based on H(div)-conforming elements. It is shown that this estimator gives rise to an upper bound where the constant is one up to higher order terms. The lower bound is also established with a constant depending on the aspect ratio of the mesh, the dependence with respect to the coefficients being also traced. The reliability and efficiency of the proposed estimator is confirmed by some numerical tests. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 [source] Adaptive numerical solution of thick plates using first-order shear deformation theory.NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2003Part I: Error estimates Abstract A posteriori error estimation employing both a residual based estimator and a recovery based estimator is discussed. Interest is focused upon the application to Reissner-Mindlin type thick plates modeled using first-order shear deformation theory, and our investigation is limited to uniform meshes of bilinear quadrilateral elements. Numerical results for selected test problems are presented for the resulting error estimators and discussed. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 44,66, 2003 [source] Two-sample Comparison Based on Prediction Error, with Applications to Candidate Gene Association StudiesANNALS OF HUMAN GENETICS, Issue 1 2007K. Yu Summary To take advantage of the increasingly available high-density SNP maps across the genome, various tests that compare multilocus genotypes or estimated haplotypes between cases and controls have been developed for candidate gene association studies. Here we view this two-sample testing problem from the perspective of supervised machine learning and propose a new association test. The approach adopts the flexible and easy-to-understand classification tree model as the learning machine, and uses the estimated prediction error of the resulting prediction rule as the test statistic. This procedure not only provides an association test but also generates a prediction rule that can be useful in understanding the mechanisms underlying complex disease. Under the set-up of a haplotype-based transmission/disequilibrium test (TDT) type of analysis, we find through simulation studies that the proposed procedure has the correct type I error rates and is robust to population stratification. The power of the proposed procedure is sensitive to the chosen prediction error estimator. Among commonly used prediction error estimators, the .632+ estimator results in a test that has the best overall performance. We also find that the test using the .632+ estimator is more powerful than the standard single-point TDT analysis, the Pearson's goodness-of-fit test based on estimated haplotype frequencies, and two haplotype-based global tests implemented in the genetic analysis package FBAT. To illustrate the application of the proposed method in population-based association studies, we use the procedure to study the association between non-Hodgkin lymphoma and the IL10 gene. [source] | ||||||||||||||||