Journal articles: 'A posteriori error bound'

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Liu, Jie, Tian Xia, and Wei Jiang. "A Posteriori Error Estimates with Computable Upper Bound for the Nonconforming RotatedQ1Finite Element Approximation of the Eigenvalue Problems." Mathematical Problems in Engineering 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/891278.

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This paper discusses the nonconforming rotatedQ1finite element computable upper bound a posteriori error estimate of the boundary value problem established by M. Ainsworth and obtains efficient computable upper bound a posteriori error indicators for the eigenvalue problem associated with the boundary value problem. We extend the a posteriori error estimate to the Steklov eigenvalue problem and also derive efficient computable upper bound a posteriori error indicators. Finally, through numerical experiments, we verify the validity of the a posteriori error estimate of the boundary value problem; meanwhile, the numerical results show that the a posteriori error indicators of the eigenvalue problem and the Steklov eigenvalue problem are effective.

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Antonopoulou, Dimitra, and Michael Plexousakis. "A posteriori analysis for space-time, discontinuous in time Galerkin approximations for parabolic equations in a variable domain." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 2 (March 2019): 523–49. http://dx.doi.org/10.1051/m2an/2018059.

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This paper presents an a posteriori error analysis for the discontinuous in time space–time scheme proposed by Jamet for the heat equation in multi-dimensional, non-cylindrical domains Jamet (SIAM J. Numer. Anal. 15 (1978) 913–928). Using a Clément-type interpolant, we prove abstract a posteriori error bounds for the numerical error. Furthermore, in the case of two-dimensional spatial domains we transform the problem into an equivalent one, of parabolic type, with space-time dependent coefficients but posed on a cylindrical domain. We formulate a discontinuous in time space–time scheme and prove a posteriori error bounds of optimal order. The a priori estimates of Evans (American Mathematical Society (1998)) for general parabolic initial and boundary value problems are used in the derivation of the upper bound. Our lower bound coincides with that of Picasso (Comput. Meth. Appl. Mech. Eng. 167 (1998) 223–237), proposed for adaptive, Runge-Kutta finite element methods for linear parabolic problems. Our theoretical results are verified by numerical experiments.

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Cochez-Dhondt, Sarah, and Serge Nicaise. "A Posteriori Error Estimators Based on Equilibrated Fluxes." Computational Methods in Applied Mathematics 10, no. 1 (2010): 49–68. http://dx.doi.org/10.2478/cmam-2010-0002.

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Abstract We consider the conforming of finite element approximations of reactiondiffusion problems. We propose new a posteriori error estimators based on H(div)- conforming finite elements and equilibrated fluxes. It is shown that these estimators give rise to an upper bound where the constant is one in front of the indicator, up to higher order terms. Lower bounds can also be established with constants depending on the shape regularity of the mesh and the local variation of the coefficients. We further analyze the convergence of an adaptive algorithm. The reliability and efficiency of the proposed estimators are confirmed by various numerical tests.

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Mandic, Danilo P., and Jonathon A. Chambers. "Relationships Between the A Priori and A Posteriori Errors in Nonlinear Adaptive Neural Filters." Neural Computation 12, no. 6 (June 1, 2000): 1285–92. http://dx.doi.org/10.1162/089976600300015358.

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The lower bounds for the a posteriori prediction error of a nonlinear predictor realized as a neural network are provided. These are obtained for a priori adaptation and a posteriori error networks with sigmoid nonlinearities trained by gradient-descent learning algorithms. A contractivity condition is imposed on a nonlinear activation function of a neuron so that the a posteriori prediction error is smaller in magnitude than the corresponding a priori one. Furthermore, an upper bound is imposed on the learning rate η so that the approach is feasible. The analysis is undertaken for both feedforward and recurrent nonlinear predictors realized as neural networks.

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CREUSÉ, E., S. NICAISE, and G. KUNERT. "A POSTERIORI ERROR ESTIMATION FOR THE STOKES PROBLEM: ANISOTROPIC AND ISOTROPIC DISCRETIZATIONS." Mathematical Models and Methods in Applied Sciences 14, no. 09 (September 2004): 1297–341. http://dx.doi.org/10.1142/s0218202504003635.

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The paper presents a posteriori error estimators for the stationary Stokes problem. We consider anisotropic finite element discretizations (i.e. elements with very large aspect ratio) where conventional, isotropic error estimators fail. Our analysis covers two- and three-dimensional domains, conforming and non-conforming discretizations as well as different elements. This large variety of settings requires different approaches and results in different estimators. Furthermore many examples of finite element pairs that are covered by the analysis are presented. Lower and upper error bounds form the main result with minimal assumptions on the elements. The lower error bound is uniform with respect to the mesh anisotropy with the exception of nonconforming 3D discretizations made of pentahedra or hexahedra. The upper error bound depends on a proper alignment of the anisotropy of the mesh which is a common feature of anisotropic error estimation. In the special case of isotropic meshes, the results simplify, and upper and lower error bounds hold unconditionally. Some of the corresponding results seem to be novel (in particular for 3D domains), and cover element pairs of practical importance. The numerical experiments confirm the theoretical predictions and show the usefulness of the anisotropic error estimators.

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KNEZEVIC, DAVID J., NGOC-CUONG NGUYEN, and ANTHONY T. PATERA. "REDUCED BASIS APPROXIMATION ANDA POSTERIORIERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS." Mathematical Models and Methods in Applied Sciences 21, no. 07 (July 2011): 1415–42. http://dx.doi.org/10.1142/s0218202511005441.

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In this paper we present reduced basis (RB) approximations and associated rigorous a posteriori error bounds for the parametrized unsteady Boussinesq equations. The essential ingredients are Galerkin projection onto a low-dimensional space associated with a smooth parametric manifold — to provide dimension reduction; an efficient proper orthogonal decomposition–Greedy sampling method for identification of optimal and numerically stable approximations — to yield rapid convergence; accurate (online) calculation of the solution-dependent stability factor by the successive constraint method — to quantify the growth of perturbations/residuals in time; rigorous a posteriori bounds for the errors in the RB approximation and associated outputs — to provide certainty in our predictions; and an offline–online computational decomposition strategy for our RB approximation and associated error bound — to minimize marginal cost and hence achieve high performance in the real-time and many-query contexts. The method is applied to a transient natural convection problem in a two-dimensional "complex" enclosure — a square with a small rectangle cutout — parametrized by Grashof number and orientation with respect to gravity. Numerical results indicate that the RB approximation converges rapidly and that furthermore the (inexpensive) rigorous a posteriori error bounds remain practicable for parameter domains and final times of physical interest.

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Korneev, V. G. "A Posteriori Error Control at Numerical Solution of Plate Bending Problem." Applied Mechanics and Materials 725-726 (January 2015): 674–80. http://dx.doi.org/10.4028/www.scientific.net/amm.725-726.674.

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The classical approach to a posteriori error control considered in this paper bears on the counter variational principles of Lagrange and Castigliano. Its efficient implementation for problems of mechanics of solids assumes obtaining equilibrated stresses/resultants which, at the same time, are sufficiently close to the exact values. Besides, it is important that computation of the error bound with the use of such stresses/resultants would be cheap in respect to the arithmetic work. Following these guide lines, we expand the preceding results for elliptic partial differential equations and theory elasticity equations upon the problem of thin plate bending. We obtain guaranteed a posteriori error bounds of simple forms for solutions by the finite element method and discuss the algorithms of linear complexity for their computation. The approach of the paper also allows to improve some known general a posteriori estimates by means of arbitrary not equilibrated stress fields.

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Buffa, Annalisa, and Eduardo M. Garau. "A posteriori error estimators for hierarchical B-spline discretizations." Mathematical Models and Methods in Applied Sciences 28, no. 08 (July 2018): 1453–80. http://dx.doi.org/10.1142/s0218202518500392.

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In this paper, we develop a function-based a posteriori error estimators for the solution of linear second-order elliptic problems considering hierarchical spline spaces for the Galerkin discretization. We obtain a global upper bound for the energy error over arbitrary hierarchical mesh configurations which simplifies the implementation of adaptive refinement strategies. The theory hinges on some weighted Poincaré-type inequalities where the B-spline basis functions are the weights appearing in the norms. Such inequalities are derived following the lines in [A. Veeser and R. Verfürth, Explicit upper bounds for dual norms of residuals, SIAM J. Numer. Anal. 47 (2009) 2387–2405], where the case of standard finite elements is considered. Additionally, we present numerical experiments that show the efficiency of the error estimators independently of the degree of the splines used for the discretization, together with an adaptive algorithm guided by these local estimators that yields optimal meshes and rates of convergence, exhibiting an excellent performance.

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Sabawi, Younis A. "Posteriori Error bound For Fullydiscrete Semilinear Parabolic Integro-Differential equations." Journal of Physics: Conference Series 1999, no. 1 (September 1, 2021): 012085. http://dx.doi.org/10.1088/1742-6596/1999/1/012085.

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GREPL, MARTIN A. "CERTIFIED REDUCED BASIS METHODS FOR NONAFFINE LINEAR TIME-VARYING AND NONLINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS." Mathematical Models and Methods in Applied Sciences 22, no. 03 (March 2012): 1150015. http://dx.doi.org/10.1142/s0218202511500151.

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We present reduced basis approximations and associated a posteriori error bounds for parabolic partial differential equations involving (i) a nonaffine dependence on the parameter and (ii ) a nonlinear dependence on the field variable. The method employs the Empirical Interpolation Method in order to construct "affine" coefficient-function approximations of the "nonaffine" (or nonlinear) parametrized functions. We consider linear time-invariant as well as linear time-varying nonaffine functions and introduce a new sampling approach to generate the function approximation space for the latter case. Our a posteriori error bounds take both error contributions explicitly into account — the error introduced by the reduced basis approximation and the error induced by the coefficient function interpolation. We show that these bounds are rigorous upper bounds for the approximation error under certain conditions on the function interpolation, thus addressing the demand for certainty of the approximation. As regards efficiency, we develop an offline–online computational procedure for the calculation of the reduced basis approximation and associated error bound. The method is thus ideally suited for the many-query or real-time contexts. Numerical results are presented to confirm and test our approach.

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Brown, Judith A., and Joseph E. Bishop. "Quantifying the Impact of Material-Model Error on Macroscale Quantities-of-Interest Using Multiscale a Posteriori Error-Estimation Techniques." MRS Advances 1, no. 40 (2016): 2789–94. http://dx.doi.org/10.1557/adv.2016.524.

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ABSTRACTAn a posteriori error-estimation framework is introduced to quantify and reduce modeling errors resulting from approximating complex mesoscale material behavior with a simpler macroscale model. Such errors may be prevalent when modeling welds and additively manufactured structures, where spatial variations and material textures may be present in the microstructure. We consider a case where a <100> fiber texture develops in the longitudinal scanning direction of a weld. Transversely isotropic elastic properties are obtained through homogenization of a microstructural model with this texture and are considered the reference weld properties within the error-estimation framework. Conversely, isotropic elastic properties are considered approximate weld properties since they contain no representation of texture. Errors introduced by using isotropic material properties to represent a weld are assessed through a quantified error bound in the elastic regime. An adaptive error reduction scheme is used to determine the optimal spatial variation of the isotropic weld properties to reduce the error bound.

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Hoppe, Ronald H. W., and Youri Iliash. "An equilibrated a posteriori error estimator for an Interior Penalty Discontinuous Galerkin approximation of the p-Laplace problem." Russian Journal of Numerical Analysis and Mathematical Modelling 36, no. 6 (December 1, 2021): 313–36. http://dx.doi.org/10.1515/rnam-2021-0026.

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Abstract We are concerned with an Interior Penalty Discontinuous Galerkin (IPDG) approximation of the p-Laplace equation and an equilibrated a posteriori error estimator. The IPDG method can be derived from a discretization of the associated minimization problem involving appropriately defined reconstruction operators. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in the broken W 1,p norm and relies on the construction of an equilibrated flux in terms of a numerical flux function associated with the mixed formulation of the IPDG approximation. The relationship with a residual-type a posteriori error estimator is established as well. Numerical results illustrate the performance of both estimators.

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LAFOREST, M. "MECHANISMS FOR ERROR PROPAGATION AND CANCELLATION IN GLIMM'S SCHEME WITHOUT RAREFACTIONS." Journal of Hyperbolic Differential Equations 04, no. 03 (September 2007): 501–31. http://dx.doi.org/10.1142/s0219891607001239.

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We derive an a posteriori error bound for Glimm's approximate solutions to convex scalar conservation laws containing only shock waves. Using Liu's wave-tracing method, we show that the L1 norm of the error is bounded by a sum of residuals containing independent contributions from each wave in the approximate solution. We introduce a framework, similar to the method of characteristics, for the analysis of the local errors generated by wave interactions. The analysis allows for explicit cancellation among the errors created by a single wave and for error propagation along discontinuities.

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Plum, M., and Ch Wieners. "Numerical Enclosures for Variational Inequalities." Computational Methods in Applied Mathematics 7, no. 4 (2007): 376–88. http://dx.doi.org/10.2478/cmam-2007-0023.

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AbstractWe present a new method for proving the existence of a unique solution of variational inequalities within guaranteed close error bounds to a numerical approximation. The method is derived for a specific model problem featuring most of the difficulties of perfect plasticity. We introduce a finite element method for the computation of admissible primal and dual solutions which a posteriori guarantees the existence of a unique solution (by the verification of the safe load condition) and which allows determination of a guaranteed error bound. Finally, we present explicit existence results and error bounds in some significant specific configurations.

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Petković, Miodrag S., Snežana Ilić, and Ivan Petković. "A posteriori error bound methods for the inclusion of polynomial zeros." Journal of Computational and Applied Mathematics 208, no. 2 (November 2007): 316–30. http://dx.doi.org/10.1016/j.cam.2006.09.014.

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Ciarlet, Patrick, and Martin Vohralík. "Localization of global norms and robust a posteriori error control for transmission problems with sign-changing coefficients." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 5 (September 2018): 2037–64. http://dx.doi.org/10.1051/m2an/2018034.

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We present a posteriori error analysis of diffusion problems where the diffusion tensor is not necessarily symmetric and positive definite and can in particular change its sign. We first identify the correct intrinsic error norm for such problems, covering both conforming and nonconforming approximations. It combines a dual (residual) norm together with the distance to the correct functional space. Importantly, we show the equivalence of both these quantities defined globally over the entire computational domain with the Hilbertian sums of their localizations over patches of elements. In this framework, we then design a posteriori estimators which deliver simultaneously guaranteed error upper bound, global and local error lower bounds, and robustness with respect to the (sign-changing) diffusion tensor. Robustness with respect to the approximation polynomial degree is achieved as well. The estimators are given in a unified setting covering at once conforming, nonconforming, mixed, and discontinuous Galerkin finite element discretizations in two or three space dimensions. Numerical results illustrate the theoretical developments.

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Bartels, Sören, and Marijo Milicevic. "Primal-dual gap estimators for a posteriori error analysis of nonsmooth minimization problems." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 5 (July 28, 2020): 1635–60. http://dx.doi.org/10.1051/m2an/2019074.

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The primal-dual gap is a natural upper bound for the energy error and, for uniformly convex minimization problems, also for the error in the energy norm. This feature can be used to construct reliable primal-dual gap error estimators for which the constant in the reliability estimate equals one for the energy error and equals the uniform convexity constant for the error in the energy norm. In particular, it defines a reliable upper bound for any functions that are feasible for the primal and the associated dual problem. The abstract a posteriori error estimate based on the primal-dual gap is provided in this article, and the abstract theory is applied to the nonlinear Laplace problem and the Rudin–Osher–Fatemi image denoising problem. The discretization of the primal and dual problems with conforming, low-order finite element spaces is addressed. The primal-dual gap error estimator is used to define an adaptive finite element scheme and numerical experiments are presented, which illustrate the accurate, local mesh refinement in a neighborhood of the singularities, the reliability of the primal-dual gap error estimator and the moderate overestimation of the error.

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Liu, Jia, Mingyu Zhang, Chaoyong Wang, Rongjun Chen, Xiaofeng An, and Yufei Wang. "Upper Bound on the Bit Error Probability of Systematic Binary Linear Codes via Their Weight Spectra." Discrete Dynamics in Nature and Society 2020 (January 29, 2020): 1–11. http://dx.doi.org/10.1155/2020/1469090.

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In this paper, upper bound on the probability of maximum a posteriori (MAP) decoding error for systematic binary linear codes over additive white Gaussian noise (AWGN) channels is proposed. The proposed bound on the bit error probability is derived with the framework of Gallager’s first bounding technique (GFBT), where the Gallager region is defined to be an irregular high-dimensional geometry by using a list decoding algorithm. The proposed bound on the bit error probability requires only the knowledge of weight spectra, which is helpful when the input-output weight enumerating function (IOWEF) is not available. Numerical results show that the proposed bound on the bit error probability matches well with the maximum-likelihood (ML) decoding simulation approach especially in the high signal-to-noise ratio (SNR) region, which is better than the recently proposed Ma bound.

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Smears, Iain, and Martin Vohralík. "Simple and robust equilibrated flux a posteriori estimates for singularly perturbed reaction–diffusion problems." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 6 (October 12, 2020): 1951–73. http://dx.doi.org/10.1051/m2an/2020034.

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We consider energy norm a posteriori error analysis of conforming finite element approximations of singularly perturbed reaction–diffusion problems on simplicial meshes in arbitrary space dimension. Using an equilibrated flux reconstruction, the proposed estimator gives a guaranteed global upper bound on the error without unknown constants, and local efficiency robust with respect to the mesh size and singular perturbation parameters. Whereas previous works on equilibrated flux estimators only considered lowest-order finite element approximations and achieved robustness through the use of boundary-layer adapted submeshes or via combination with residual-based estimators, the present methodology applies in a simple way to arbitrary-order approximations and does not request any submesh or estimators combination. The equilibrated flux is obtained via local reaction–diffusion problems with suitable weights (cut-off factors), and the guaranteed upper bound features the same weights. We prove that the inclusion of these weights is not only sufficient but also necessary for robustness of any flux equilibration estimate that does not employ submeshes or estimators combination, which shows that some of the flux equilibrations proposed in the past cannot be robust. To achieve the fully computable upper bound, we derive explicit bounds for some inverse inequality constants on a simplex, which may be of independent interest.

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Apushkinskaya, Darya, and Sergey Repin. "Functional A Posteriori Error Estimates for the Parabolic Obstacle Problem." Computational Methods in Applied Mathematics 22, no. 2 (January 23, 2022): 259–76. http://dx.doi.org/10.1515/cmam-2021-0156.

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Abstract The paper is concerned with functional-type a posteriori estimates for the initial boundary value problem for a parabolic partial differential equation with an obstacle. We deduce a guaranteed and computable bound of the distance between the exact minimizer and any function from the admissible (energy) class of functions. Applications to the analysis of modeling errors caused by data implification are discussed. An important case of time incremental approximations is specially studied. Numerical examples presented in the last section show how the estimates work in practice.

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Journal, Baghdad Science. "A Note on the Perturbation of arithmetic expressions." Baghdad Science Journal 13, no. 1 (March 6, 2016): 190–97. http://dx.doi.org/10.21123/bsj.13.1.190-197.

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In this paper we present the theoretical foundation of forward error analysis of numerical algorithms under;• Approximations in "built-in" functions.• Rounding errors in arithmetic floating-point operations.• Perturbations of data.The error analysis is based on linearization method. The fundamental tools of the forward error analysis are system of linear absolute and relative a prior and a posteriori error equations and associated condition numbers constituting optimal of possible cumulative round – off errors. The condition numbers enable simple general, quantitative bounds definitions of numerical stability. The theoretical results have been applied a Gaussian elimination, and have proved to be very effective means of both a priori and a posteriori error analysis.

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REPIN, SERGEY, and STEFAN SAUTER. "COMPUTABLE ESTIMATES OF THE MODELING ERROR RELATED TO KIRCHHOFF–LOVE PLATE MODEL." Analysis and Applications 08, no. 04 (October 2010): 409–28. http://dx.doi.org/10.1142/s0219530510001709.

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The Kirchhoff–Love plate model is widely used in the analysis of thin elastic plates. It is well known that Kirchhoff–Love solutions can be viewed as certain limits of displacements and stresses for elastic plates where the thickness tends to zero. In this note, we consider the problem from a different point of view and derive computable upper bounds of the difference between the exact three-dimensional solution and a solution computed by using the Kirchhoff–Love hypotheses. This estimate is valid for any value of the thickness parameter. In combination with a posteriori error estimates for approximation errors, this estimate allows the direct measurement of both, approximation and modeling errors, encompassed in a numerical solution of the Kirchhoff–Love model. We prove that the upper bound possesses necessary asymptotic properties and, therefore, does not deteriorate as the thickness tends to zero.

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Jawecki, Tobias, Winfried Auzinger, and Othmar Koch. "Computable upper error bounds for Krylov approximations to matrix exponentials and associated $${\varvec{\varphi }}$$-functions." BIT Numerical Mathematics 60, no. 1 (September 11, 2019): 157–97. http://dx.doi.org/10.1007/s10543-019-00771-6.

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Abstract An a posteriori estimate for the error of a standard Krylov approximation to the matrix exponential is derived. The estimate is based on the defect (residual) of the Krylov approximation and is proven to constitute a rigorous upper bound on the error, in contrast to existing asymptotical approximations. It can be computed economically in the underlying Krylov space. In view of time-stepping applications, assuming that the given matrix is scaled by a time step, it is shown that the bound is asymptotically correct (with an order related to the dimension of the Krylov space) for the time step tending to zero. This means that the deviation of the error estimate from the true error tends to zero faster than the error itself. Furthermore, this result is extended to Krylov approximations of $$\varphi $$φ-functions and to improved versions of such approximations. The accuracy of the derived bounds is demonstrated by examples and compared with different variants known from the literature, which are also investigated more closely. Alternative error bounds are tested on examples, in particular a version based on the concept of effective order. For the case where the matrix exponential is used in time integration algorithms, a step size selection strategy is proposed and illustrated by experiments.

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Jawecki, Tobias. "A study of defect-based error estimates for the Krylov approximation of φ-functions." Numerical Algorithms 90, no. 1 (November 8, 2021): 323–61. http://dx.doi.org/10.1007/s11075-021-01190-x.

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AbstractPrior recent work, devoted to the study of polynomial Krylov techniques for the approximation of the action of the matrix exponential etAv, is extended to the case of associated φ-functions (which occur within the class of exponential integrators). In particular, a posteriori error bounds and estimates, based on the notion of the defect (residual) of the Krylov approximation are considered. Computable error bounds and estimates are discussed and analyzed. This includes a new error bound which favorably compares to existing error bounds in specific cases. The accuracy of various error bounds is characterized in relation to corresponding Ritz values of A. Ritz values yield properties of the spectrum of A (specific properties are known a priori, e.g., for Hermitian or skew-Hermitian matrices) in relation to the actual starting vector v and can be computed. This gives theoretical results together with criteria to quantify the achieved accuracy on the fly. For other existing error estimates, the reliability and performance are studied by similar techniques. Effects of finite precision (floating point arithmetic) are also taken into account.

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Dib, Séréna, Vivette Girault, Frédéric Hecht, and Toni Sayah. "A posteriori error estimates for Darcy’s problem coupled with the heat equation." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 6 (November 2019): 2121–59. http://dx.doi.org/10.1051/m2an/2019049.

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This work derives a posteriori error estimates, in two and three dimensions, for the heat equation coupled with Darcy’s law by a nonlinear viscosity depending on the temperature. We introduce two variational formulations and discretize them by finite element methods. We prove optimal a posteriori errors with two types of computable error indicators. The first one is linked to the linearization and the second one to the discretization. Then we prove upper and lower error bounds under regularity assumptions on the solutions. Finally, numerical computations are performed to show the effectiveness of the error indicators.

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Langer, Ulrich, Sergey Repin, and Monika Wolfmayr. "Functional A Posteriori Error Estimates for Parabolic Time-Periodic Boundary Value Problems." Computational Methods in Applied Mathematics 15, no. 3 (July 1, 2015): 353–72. http://dx.doi.org/10.1515/cmam-2015-0012.

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AbstractThe paper is concerned with parabolic time-periodic boundary value problems which are of theoretical interest and arise in different practical applications. The multiharmonic finite element method is well adapted to this class of parabolic problems. We study properties of multiharmonic approximations and derive guaranteed and fully computable bounds of approximation errors. For this purpose, we use the functional a posteriori error estimation techniques earlier introduced by S. Repin. Numerical tests confirm the efficiency of the a posteriori error bounds derived.

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Braess, Dietrich, Astrid S. Pechstein, and Joachim Schöberl. "An equilibration-based a posteriori error bound for the biharmonic equation and two finite element methods." IMA Journal of Numerical Analysis 40, no. 2 (February 20, 2019): 951–75. http://dx.doi.org/10.1093/imanum/drz005.

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Abstract We develop an a posteriori error bound for the interior penalty discontinuous Galerkin approximation of the biharmonic equation with continuous finite elements. The error bound is based on the two-energies principle and requires the computation of an equilibrated moment tensor. The natural space for the moment tensor is that of symmetric tensor fields with continuous normal-normal components, and is well-known from the Hellan-Herrmann-Johnson mixed formulation. We propose a construction that is totally local. The procedure can also be applied to the original Hellan–Herrmann–Johnson formulation, which directly provides an equilibrated moment tensor.

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Lord, Natacha H., and Anthony J. Mulholland. "A dual weighted residual method applied to complex periodic gratings." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, no. 2160 (December 8, 2013): 20130176. http://dx.doi.org/10.1098/rspa.2013.0176.

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An extension of the dual weighted residual (DWR) method to the analysis of electromagnetic waves in a periodic diffraction grating is presented. Using the α ,0-quasi-periodic transformation, an upper bound for the a posteriori error estimate is derived. This is then used to solve adaptively the associated Helmholtz problem. The goal is to achieve an acceptable accuracy in the computed diffraction efficiency while keeping the computational mesh relatively coarse. Numerical results are presented to illustrate the advantage of using DWR over the global a posteriori error estimate approach. The application of the method in biomimetic, to address the complex diffraction geometry of the Morpho butterfly wing is also discussed.

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Bertrand, Fleurianne, Daniele Boffi, and Rolf Stenberg. "Asymptotically Exact A Posteriori Error Analysis for the Mixed Laplace Eigenvalue Problem." Computational Methods in Applied Mathematics 20, no. 2 (April 1, 2020): 215–25. http://dx.doi.org/10.1515/cmam-2019-0099.

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AbstractThis paper derives a posteriori error estimates for the mixed numerical approximation of the Laplace eigenvalue problem. We discuss a reconstruction in the standard {H_{0}^{1}}-conforming space for the primal variable of the mixed Laplace eigenvalue problem and compare it with analogous approaches present in the literature for the corresponding source problem. In the case of Raviart–Thomas finite elements of arbitrary polynomial degree, the resulting error estimator constitutes a guaranteed upper bound for the error and is shown to be local efficient. Our reconstruction is performed locally on a set of vertex patches.

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Sharipov, Kosnazar. "On the Recovery of Continuous Functions from Noisy Fourier Coefficients." Computational Methods in Applied Mathematics 11, no. 1 (2011): 75–82. http://dx.doi.org/10.2478/cmam-2011-0004.

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AbstractWe consider the classical ill-posed problem of the recovery of continuous functions from noisy Fourier coefficients. For the classes of functions given in terms of generalized smoothness, we present a priori and a posteriori regularization parameter choice realizing an order-optimal error bound.

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Yano, Masayuki, Anthony T. Patera, and Karsten Urban. "A space-time hp-interpolation-based certified reduced basis method for Burgers' equation." Mathematical Models and Methods in Applied Sciences 24, no. 09 (May 20, 2014): 1903–35. http://dx.doi.org/10.1142/s0218202514500110.

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We present a space-time interpolation-based certified reduced basis method for Burgers' equation over the spatial interval (0, 1) and the temporal interval (0, T] parametrized with respect to the Peclet number. We first introduce a Petrov–Galerkin space-time finite element discretization which enjoys a favorable inf–sup constant that decreases slowly with Peclet number and final time T. We then consider an hp interpolation-based space-time reduced basis approximation and associated Brezzi–Rappaz–Raviart a posteriori error bounds. We describe computational offline–online decomposition procedures for the three key ingredients of the error bounds: the dual norm of the residual, a lower bound for the inf–sup constant, and the space-time Sobolev embedding constant. Numerical results demonstrate that our space-time formulation provides improved stability constants compared to classical L2-error estimates; the error bounds remain sharp over a wide range of Peclet numbers and long integration times T, in marked contrast to the exponentially growing estimate of the classical formulation for high Peclet number cases.

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32

Chahlaoui, Y. "A posteriori error bounds for discrete balanced truncation." Linear Algebra and its Applications 436, no. 8 (April 2012): 2744–63. http://dx.doi.org/10.1016/j.laa.2011.07.025.

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Leisten, Rainer. "A posteriori error bounds in linear programming aggregation." Computers & Operations Research 24, no. 1 (January 1997): 1–16. http://dx.doi.org/10.1016/s0305-0548(96)00034-2.

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Giacomini, Matteo, Olivier Pantz, and Karim Trabelsi. "Certified Descent Algorithm for shape optimization driven by fully-computable a posteriori error estimators." ESAIM: Control, Optimisation and Calculus of Variations 23, no. 3 (April 28, 2017): 977–1001. http://dx.doi.org/10.1051/cocv/2016021.

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In this paper we introduce a novel certified shape optimization strategy – named Certified Descent Algorithm (CDA) – to account for the numerical error introduced by the Finite Element approximation of the shape gradient. We present a goal-oriented procedure to derive a certified upper bound of the error in the shape gradient and we construct a fully-computable, constant-free a posteriori error estimator inspired by the complementary energy principle. The resulting CDA is able to identify a genuine descent direction at each iteration and features a reliable stopping criterion. After validating the error estimator, some numerical simulations of the resulting certified shape optimization strategy are presented for the well-known inverse identification problem of Electrical Impedance Tomography.

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Prud’homme, C., D. V. Rovas, K. Veroy, L. Machiels, Y. Maday, A. T. Patera, and G. Turinici. "Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods." Journal of Fluids Engineering 124, no. 1 (November 2, 2001): 70–80. http://dx.doi.org/10.1115/1.1448332.

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We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced-basis approximations—Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation—relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage in which, given a new parameter value, we calculate the output of interest and associated error bound, depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.

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Ovchinnikov, George V., Denis Zorin, and Ivan V. Oseledets. "Robust regularization of topology optimization problems with a posteriori error estimators." Russian Journal of Numerical Analysis and Mathematical Modelling 34, no. 1 (February 25, 2019): 57–69. http://dx.doi.org/10.1515/rnam-2019-0005.

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Abstract Topological optimization finds a material density distribution minimizing a functional of the solution of a partial differential equation (PDE), subject to a set of constraints (typically, a bound on the volume or mass of the material). Using a finite elements discretization (FEM) of the PDE and functional we obtain an integer programming problem. Due to approximation error of the FEM discretization, optimization problem becomes mesh-depended and possess false, physically inadequate optimums, while functional value heavily depends on the fineness of discretization scheme used to compute it. To alleviate this problem, we propose regularization of given functional by error estimate of the FEM discretization. This regularization provides robustness of solutions and improves obtained functional values as well. While the idea is broadly applicable, in this paper we apply our method to the heat conduction optimization. Problems of this type are of practical importance in design of heat conduction channels, heat sinks and other types of heat guides.

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Yamamoto, Tetsuro. "A note on a posteriori error bound of zabrejko and nguen for zincenko's iteration." Numerical Functional Analysis and Optimization 9, no. 9-10 (January 1987): 987–94. http://dx.doi.org/10.1080/01630568708816270.

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GERNER, ANNA-LENA, and KAREN VEROY. "REDUCED BASISA POSTERIORIERROR BOUNDS FOR THE STOKES EQUATIONS IN PARAMETRIZED DOMAINS: A PENALTY APPROACH." Mathematical Models and Methods in Applied Sciences 21, no. 10 (October 2011): 2103–34. http://dx.doi.org/10.1142/s0218202511005672.

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We present reduced basis approximations and associated rigorous a posteriori error bounds for the Stokes equations in parametrized domains. The method, built upon the penalty formulation for saddle point problems, provides error bounds not only for the velocity but also for the pressure approximation, while simultaneously admitting affine geometric variations with relative ease. The essential ingredients are: (i) dimension reduction through Galerkin projection onto a low-dimensional reduced basis space; (ii) stable, good approximation of the pressure through supremizer-enrichment of the velocity reduced basis space; (iii) optimal and numerically stable approximations identified through an efficient greedy sampling method; (iv) certainty, through rigorous a posteriori bounds for the errors in the reduced basis approximation; and (v) efficiency, through an offline-online computational strategy. The method is applied to a flow problem in a two-dimensional channel with a (parametrized) rectangular obstacle. Numerical results show that the reduced basis approximation converges rapidly, the effectivities associated with the (inexpensive) rigorous a posteriori error bounds remain good even for reasonably small values of the penalty parameter, and that the effects of the penalty parameter are relatively benign.

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Chen, Chuanjun, Xin Zhao, and Yuanyuan Zhang. "A Posteriori Error Estimate for Finite Volume Element Method of the Second-Order Hyperbolic Equations." Mathematical Problems in Engineering 2015 (2015): 1–11. http://dx.doi.org/10.1155/2015/510241.

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We establish a posteriori error estimate for finite volume element method of a second-order hyperbolic equation. Residual-type a posteriori error estimator is derived. The computable upper and lower bounds on the error in theH1-norm are established. Numerical experiments are provided to illustrate the performance of the proposed estimator.

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Bertrand, Fleurianne, Marcel Moldenhauer, and Gerhard Starke. "A Posteriori Error Estimation for Planar Linear Elasticity by Stress Reconstruction." Computational Methods in Applied Mathematics 19, no. 3 (July 1, 2019): 663–79. http://dx.doi.org/10.1515/cmam-2018-0004.

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AbstractThe nonconforming triangular piecewise quadratic finite element space by Fortin and Soulie can be used for the displacement approximation and its combination with discontinuous piecewise linear pressure elements is known to constitute a stable combination for incompressible linear elasticity computations. In this contribution, we extend the stress reconstruction procedure and resulting guaranteed a posteriori error estimator developed by Ainsworth, Allendes, Barrenechea and Rankin [2] and by Kim [18] to linear elasticity. In order to get a guaranteed reliability bound with respect to the energy norm involving only known constants, two modifications are carried out: (i) the stress reconstruction in next-to-lowest order Raviart–Thomas spaces is modified in such a way that its anti-symmetric part vanishes in average on each element; (ii) the auxiliary conforming approximation is constructed under the constraint that its divergence coincides with the one for the nonconforming approximation. An important aspect of our construction is that all results hold uniformly in the incompressible limit. Global efficiency is also shown and the effectiveness is illustrated by adaptive computations involving different Lamé parameters including the incompressible limit case.

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Ali Hassan, Sarah, Caroline Japhet, Michel Kern, and Martin Vohralík. "A Posteriori Stopping Criteria for Optimized Schwarz Domain Decomposition Algorithms in Mixed Formulations." Computational Methods in Applied Mathematics 18, no. 3 (July 1, 2018): 495–519. http://dx.doi.org/10.1515/cmam-2018-0010.

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AbstractThis paper develops a posteriori estimates for domain decomposition methods with optimized Robin transmission conditions on the interface between subdomains. We choose to demonstrate the methodology for mixed formulations, with a lowest-order Raviart–Thomas–Nédélec discretization, often used for heterogeneous and anisotropic porous media diffusion problems. Our estimators allow to distinguish the spatial discretization and the domain decomposition error components. We propose an adaptive domain decomposition algorithm wherein the iterations are stopped when the domain decomposition error does not affect significantly the overall error. Two main goals are thus achieved. First, a guaranteed bound on the overall error is obtained at each step of the domain decomposition algorithm. Second, important savings in terms of the number of domain decomposition iterations can be realized. Numerical experiments illustrate the efficiency of our estimates and the performance of the adaptive stopping criteria.

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Samrowski, Tatiana. "Combined Error Estimates in the Case of Dimension Reduction." Computational Methods in Applied Mathematics 14, no. 1 (January 1, 2014): 113–34. http://dx.doi.org/10.1515/cmam-2013-0024.

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Abstract. We consider the stationary reaction-diffusion problem in a domain $\Omega \subset \mathbb {R}^3$ having the size along one coordinate direction essentially smaller than along the others. By an energy type argumentation, different simplified models of lower dimension can be deduced and solved numerically. For these models, we derive a guaranteed upper bound of the difference between the exact solution of the original problem and a three-dimensional reconstruction generated by the solution of a dimensionally reduced problem. This estimate of the total error is determined as the sum of discretization and modeling errors, which are both explicit and computable. The corresponding discretization errors are estimated by a posteriori estimates of the functional type. Modeling error majorants are also explicitly evaluated. Hence, a numerical strategy based on the balancing modeling and discretization errors can be derived in order to provide an economical way of getting an approximate solution with an a priori given accuracy. Numerical tests are presented and discussed.

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Giesselmann, Jan, and Tristan Pryer. "A posteriori analysis for dynamic model adaptation in convection-dominated problems." Mathematical Models and Methods in Applied Sciences 27, no. 13 (October 19, 2017): 2381–423. http://dx.doi.org/10.1142/s0218202517500476.

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In this work, we present an a posteriori error indicator for approximation schemes of Runge–Kutta–discontinuous–Galerkin type arising in applications of compressible fluid flows. The purpose of this indicator is not only for mesh adaptivity, we also make use of this to drive model adaptivity. This is perhaps where a costly complex model and a cheaper simple model are solved over different parts of the domain. The a posteriori bound we derive indicates the regions where the complex model can be relatively well approximated with the cheaper one. One such example which we choose to highlight is that of the Navier–Stokes–Fourier equations approximated by Euler’s equations.

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Eftang, Jens L., Martin A. Grepl, and Anthony T. Patera. "A posteriori error bounds for the empirical interpolation method." Comptes Rendus Mathematique 348, no. 9-10 (May 2010): 575–79. http://dx.doi.org/10.1016/j.crma.2010.03.004.

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Hala Raad and Mohammad Sabawi. "A Priori and a Posteriori Error Analysis for Generic Linear Elliptic Problems." Tikrit Journal of Pure Science 27, no. 2 (November 30, 2022): 57–64. http://dx.doi.org/10.25130/tjps.v27i2.68.

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In this paper, a priori error analysis has been examined for the continuous Galerkin finite element method which is used for solving a generic scalar and a generic system of linear elliptic equations. We derived optimal order a priori error bounds in (energy) norm utilising standard a priori error analysis techniques and tools. Also, a posteriori error analysis is investigated for a generic scalar linear elliptic equation and for a generic system of linear elliptic equations. We derived optimal residual-based a posteriori error estimates energy technique in norm

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Key, Kerry, and Chester Weiss. "Adaptive finite-element modeling using unstructured grids: The 2D magnetotelluric example." GEOPHYSICS 71, no. 6 (November 2006): G291—G299. http://dx.doi.org/10.1190/1.2348091.

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Existing numerical modeling techniques commonly used for electromagnetic (EM) exploration are bound by the limitations of approximating complex structures using a rectangular grid. A more flexible tool is the adaptive finite-element (FE) method using unstructured grids. Composed of irregular triangles, an unstructured grid can readily conform to complicated structural boundaries. To ensure numerical accuracy, adaptive refinement using an a posteriori error estimator is performed iteratively to refine the grid where solution accuracy is insufficient. Two recently developed asymptotically exact a posteriori error estimators are based on a superconvergent gradient recovery operator. The first relies solely on the normed difference between the recovered gradients and the piecewise constant FE gradients and is effective for lowering the global error in the FE solution. For many problems, an accurate solution is required only in a few discreteregions and a more efficient error estimator is possible by considering the local influence of errors from coarse elements elsewhere in the grid. The second error estimator accomplishes this by using weights determined from the solution to an appropriate dual problem to modify the first error estimator. Application of these methods for 2D magnetotelluric (MT) modeling reveals, as expected, that the dual weighted error estimator is far more efficient in achieving accurate MT responses. Refining about 15% of elements per iteration gives the fastest convergence rate. For a given refined grid, the solution error at higher frequencies varies in proportion to the skin depth, requiring refinement about every two decades of frequency. The transverse electric (TE) and transverse magnetic (TM) modes exhibit different field behavior, and refinement should consider the effects of both. An example resistivity model of seafloor bathymetry underlain by complex salt intrusions and dipping and faulted sedimentary layers illustrates the benefits of this new technique.

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Eigel, Martin, and Tatiana Samrowski. "Functional A Posteriori Error Estimation for Stationary Reaction-Convection-Diffusion Problems." Computational Methods in Applied Mathematics 14, no. 2 (April 1, 2014): 135–50. http://dx.doi.org/10.1515/cmam-2014-0005.

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Abstract. A functional type a posteriori error estimator for the finite element discretization of the stationary reaction-convection-diffusion equation is derived. In case of dominant convection, the solution for this class of problems typically exhibits boundary layers and shock-front like areas with steep gradients. This renders the accurate numerical solution very demanding and appropriate techniques for the adaptive resolution of regions with large approximation errors are crucial. Functional error estimators as derived here contain no mesh-dependent constants and provide guaranteed error bounds for any conforming approximation. To evaluate the error estimator, a minimization problem is solved which does not require any Galerkin orthogonality or any specific properties of the employed approximation space. Based on a set of numerical examples, we assess the performance of the new estimator. It is observed that it exhibits a good efficiency also with convection-dominated problem settings.

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PATERA, ANTHONY T., and EINAR M. RØNQUIST. "A GENERAL OUTPUT BOUND RESULT: APPLICATION TO DISCRETIZATION AND ITERATION ERROR ESTIMATION AND CONTROL." Mathematical Models and Methods in Applied Sciences 11, no. 04 (June 2001): 685–712. http://dx.doi.org/10.1142/s0218202501001057.

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We present a general adjoint procedure that, under certain hypotheses, provides inexpensive, rigorous, accurate, and constant-free lower and upper asymptotic bounds for the error in "outputs" which are linear functionals of solutions to linear (e.g. partial-differential or algebraic) equations. We describe two particular instantiations for which the necessary hypotheses can be readily verified. The first case — a re-interpretation of earlier work — assesses the error due to discretization: an implicit Neumann-subproblem finite element a posteriori technique applicable to general elliptic partial differential equations. The second case — new to this paper — assesses the error due to solution, in particular, incomplete iteration: a primal-dual preconditioned conjugate-gradient Lanczos method for symmetric positive-definite linear systems, in which the error bounds for the output serve as stopping criterion; numerical results are presented for additive-Schwarz domain-decomposition-preconditioned solution of a spectral element discretization of the Poisson equation in three space dimensions. In both instantiations, the computational savings are significant: since the error in the output of interest can be precisely quantified, very fine meshes, and extremely small residuals, are no longer required to ensure adequate accuracy; numerical uncertainty, though certainly not eliminated, is greatly reduced.

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Carstensen, Carsten, and Christian Merdon. "Computational Survey on A Posteriori Error Estimators for the Crouzeix–Raviart Nonconforming Finite Element Method for the Stokes Problem." Computational Methods in Applied Mathematics 14, no. 1 (January 1, 2014): 35–54. http://dx.doi.org/10.1515/cmam-2013-0021.

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Abstract. This survey compares different strategies for guaranteed error control for the lowest-order nonconforming Crouzeix–Raviart finite element method for the Stokes equations. The upper error bound involves the minimal distance of the computed piecewise gradient $\operatorname{D}_{\textup {NC}}u_{\textup {CR}}$ to the gradients of Sobolev functions with exact boundary conditions. Several improved suggestions for the cheap computation of such test functions compete in five benchmark examples. This paper provides numerical evidence that guaranteed error control of the nonconforming FEM is indeed possible for the Stokes equations with overall efficiency indices between 1 to 4 in the asymptotic range.

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Kurz, Stefan, Dirk Pauly, Dirk Praetorius, Sergey Repin, and Daniel Sebastian. "Functional a posteriori error estimates for boundary element methods." Numerische Mathematik 147, no. 4 (March 18, 2021): 937–66. http://dx.doi.org/10.1007/s00211-021-01188-6.

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AbstractFunctional error estimates are well-established tools for a posteriori error estimation and related adaptive mesh-refinement for the finite element method (FEM). The present work proposes a first functional error estimate for the boundary element method (BEM). One key feature is that the derived error estimates are independent of the BEM discretization and provide guaranteed lower and upper bounds for the unknown error. In particular, our analysis covers Galerkin BEM and the collocation method, what makes the approach of particular interest for scientific computations and engineering applications. Numerical experiments for the Laplace problem confirm the theoretical results.

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Journal articles on the topic 'A posteriori error bound'

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