Algebraic Error Research Articles

Slender-body approximations for advection–diffusion problems

We consider steady advection–diffusion about a slender body of revolution at arbitrary $O(1)$ Péclet numbers ($\mathit{Pe}$). The transported scalar attenuates at large distances and is governed by axisymmetric (either Dirichlet or Neumann) data prescribed at the body boundary. The advecting field is assumed to be an axisymmetric Stokes flow approaching a uniform stream at large distances and satisfying impermeability at the boundary; otherwise, the interfacial distribution of tangential velocity is assumed to be arbitrary, irrotational and no-slip Stokes flows being particular cases. Employing the method of matched asymptotic expansions, we develop a systematic scheme for the calculation of the scalar concentration in increasing powers of $\ln ^{-1}(1/{\it\epsilon})$, ${\it\epsilon}\ll 1$ being the body’s characteristic thickness to length ratio. The leading term in the inner expansion coincides with the pure diffusion case, the second term depends nonlinearly on the magnitude of the far-field stream and higher-order terms depend on the boundary distribution of tangential velocity. In the special case of irrotational flow and Neumann boundary conditions the logarithmic expansion terminates, leaving an algebraic error in ${\it\epsilon}$. The general formulae developed can be directly applied to numerous physical scenarios. We here consider the classical problem of forced heat convection from an isothermal body, finding a two-term expansion for $\mathit{Nu}({\it\epsilon},\mathit{Pe})/\mathit{Nu}({\it\epsilon},0)$, the ratio of the Nusselt number to its value at $\mathit{Pe}=0$. This ratio is insensitive to the particle shape at the asymptotic orders considered; at moderately large $\mathit{Pe}$ ($\ll {\it\epsilon}^{-1}$) its deviation from unity is $O[\ln (\mathit{Pe})/\text{ln}\,(1/{\it\epsilon})]$, marking the poor effectiveness of advection about slender bodies. The expansion is compared with a numerical computation in the case of a prolate spheroid in both irrotational and no-slip Stokes flows.

Strict bounding of quantities of interest in computations based on domain decomposition

This paper deals with bounding the error on the estimation of quantities of interest obtained by finite element and domain decomposition methods. The proposed bounds are written in order to separate the two errors involved in the resolution of reference and adjoint problems : on the one hand the discretization error due to the finite element method and on the other hand the algebraic error due to the use of the iterative solver. Beside practical considerations on the parallel computation of the bounds, it is shown that the interface conformity can be slightly relaxed so that local enrichment or refinement are possible in the subdomains bearing singularities or quantities of interest which simplifies the improvement of the estimation. Academic assessments are given on 2D static linear mechanic problems.

Performance and Difficulties of Students in Formulating and Solving Quadratic Equations with One Unknown

This study attempts to investigate the performance of tenth-grade students in solving quadratic equations with one unknown, using symbolic equation and word-problem representations. The participants were 217 tenth-grade students, from three different public high schools. Data was collected through an open-ended questionnaire comprising eight symbolic equations and four word problems; furthermore, semi-structured interviews were conducted with sixteen of the students. In the data analysis, the percentage of the students’ correct, incorrect, blank, and incomplete responses was determined to obtain an overview of student performance in solving symbolic equations and word problems. In addition, the students’ written responses and interview data were qualitatively analyzed to determine the nature of the students’ difficulties in formulating and solving quadratic equations. The findings revealed that although students have difficulties in solving both symbolic quadratic equations and quadratic word problems, they performed better in the context of symbolic equations compared with word problems. Student difficulties in solving symbolic problems were mainly associated with arithmetic and algebraic manipulation errors. In the word problems, however, students had difficulties comprehending the context and were therefore unable to formulate the equation to be solved. Thus, it is concluded that the differences in the structural properties of the symbolic equations and word problem representations affected student performance in formulating and solving quadratic equations with one unknown.

A Posteriori Estimates for a Natural Neumann–Neumann Domain Decomposition Algorithm on a Unilateral Contact Problem

In this paper we present an error estimator for unilateral contact problems solved by a Neumann---Neumann Domain Decomposition algorithm. This error estimator takes into account both the spatial error due to the finite element discretization and the algebraic error due to the domain decomposition algorithm. To differentiate specifically the contribution of these two error sources to the global error, two quantities are introduced: a discretization error indicator and an algebraic error indicator. The effectivity indices and the convergence of both the global error estimator and the error indicators are shown on several examples.

Persistent and Pernicious Errors in Algebraic Problem Solving

Students hold many misconceptions as they transition from arithmetic to algebraic thinking, and these misconceptions can hinder their performance and learning in the subject. To identify the errors in Algebra I which are most persistent and pernicious in terms of predicting student difficulty on standardized test items, the present study assessed algebraic misconceptions using an in-depth error analysis on algebra students’ problem solving efforts at different points in the school year. Results indicate that different types of errors become more prominent with different content at different points in the year, and that there are certain types of errors that, when made during different levels of content are indicative of math achievement difficulties. Recommendations for the necessity and timing of intervention on particular errors are discussed.

An a posteriori-based, fully adaptive algorithm with adaptive stopping criteria and mesh refinement for thermal multiphase compositional flows in porous media

In this work we develop an a posteriori-based adaptive algorithm for thermal multiphase compositional flows in porous media. The key ingredients are fully computable a posteriori error estimates, bounding the dual norm of the residual supplemented by a nonconformity evaluation term. The theory hinges on assumptions that allow the application to variety of discretization methods. The estimators are then elaborated to estimate separately the space, time, linearization, and algebraic errors. This additional information is used to formulate a fully adaptive algorithm including adaptive stopping criteria for iterative solvers as well as refinement/derefinement criteria for both the time step and the mesh size. Numerical validation is provided on an industrial case study in the context of oil-recovery based on the steam-assisted gravity drainage procedure. Implicit cell-centered finite volumes with phase-upwind and two-point discretization of the diffusive fluxes are considered. It is shown that significant gains in computational cost can be achieved in this example, without hindering the quality of the results as measured by quantities of engineering interest.

A hybridizable discontinuous Galerkin method for fractional diffusion problems

We study the use of the hybridizable discontinuous Galerkin (HDG) method for numerically solving fractional diffusion equations of order $$-\alpha $$-? with $$-1<\alpha <0$$-1<?<0. For exact time-marching, we derive optimal algebraic error estimates assuming that the exact solution is sufficiently regular. Thus, if for each time $$t \in [0,T]$$t?[0,T] the approximations are taken to be piecewise polynomials of degree $$k\ge 0$$k?0 on the spatial domain $$\varOmega $$Ω, the approximations to $$u$$u in the $$L_\infty \bigr (0,T;L_2(\varOmega )\bigr )$$L?(0,T?L2(Ω))-norm and to $$\nabla u$$?u in the $$L_\infty \bigr (0,T;\mathbf{L}_2(\varOmega )\bigr )$$L?(0,T?L2(Ω))-norm are proven to converge with the rate $$h^{k+1}$$hk+1, where $$h$$h is the maximum diameter of the elements of the mesh. Moreover, for $$k\ge 1$$k?1 and quasi-uniform meshes, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for $$u$$u converging with a rate of $$\sqrt{\log (T h^{-2/(\alpha +1)})}\, \,h^{k+2}$$log(Th-2/(?+1))hk+2.

Algebraic and Discretization Error Estimation by Equilibrated Fluxes for Discontinuous Galerkin Methods on Nonmatching Grids

We derive a posteriori error estimates for the discontinuous Galerkin method applied to the Poisson equation. We allow for a variable polynomial degree and simplicial meshes with hanging nodes and propose an approach allowing for simple (nonconforming) flux reconstructions in such a setting. We take into account the algebraic error stemming from the inexact solution of the associated linear systems and propose local stopping criteria for iterative algebraic solvers. An algebraic error flux reconstruction is introduced in this respect. Guaranteed reliability and local efficiency are proven. We next propose an adaptive strategy combining both adaptive mesh refinement and adaptive stopping criteria. At last, we detail a form of the estimates avoiding any practical reconstruction of a flux and only working with the approximate solution, which simplifies greatly their evaluation. Numerical experiments illustrate a tight control of the overall error, good prediction of the distribution of both the discretization and algebraic error components, and efficiency of the adaptive strategy.

Determining the Numeracy and Algebra Errors of Students in a Two-year Vocational School

The goal of this study was to determine the mathematics achievement level in basic numeracy and algebra concepts of students in a two-year program in a technical vocational school of higher education and determine the errors that they make in these topics. The researcher developed a diagnostic mathematics achievement test related to numeracy and basic algebra concepts by determining the common errors in numeracy and algebra through a review of the literature. The 20-question test was administered to 130 students who were continuing their education in industrial electronic and electronic communication programs of a technical vocational school of higher education in a middle sized university in the western part of Turkey. The mean students’ achievement level of mathematics was 6.31, which was a quite low score, out of a possible 20 points. The results of the study point out that there is a lack of understanding of the concepts of numbers and algebra. Analysis of the errors that students exhibit provide a valuable insight in to the nature of mathematics learning that many vocational education students have carried with them into higher education from their earlier learning experiences.

A posteriori error estimates, stopping criteria, and adaptivity for multiphase compositional Darcy flows in porous media

In this paper we derive a posteriori error estimates for the compositional model of multiphase Darcy flow in porous media, consisting of a system of strongly coupled nonlinear unsteady partial differential and algebraic equations. We show how to control the dual norm of the residual augmented by a nonconformity evaluation term by fully computable estimators. We then decompose the estimators into the space, time, linearization, and algebraic error components. This allows to formulate criteria for stopping the iterative algebraic solver and the iterative linearization solver when the corresponding error components do not affect significantly the overall error. Moreover, the spatial and temporal error components can be balanced by time step and space mesh adaptation. Our analysis applies to a broad class of standard numerical methods, and is independent of the linearization and of the iterative algebraic solvers employed. We exemplify it for the two-point finite volume method with fully implicit Euler time stepping, the Newton linearization, and the GMRes algebraic solver. Numerical results on two real-life reservoir engineering examples confirm that significant computational gains can be achieved thanks to our adaptive stopping criteria, already on fixed meshes, without any noticeable loss of precision.

Quantitative Stochastic Homogenization of Elliptic Equations in Nondivergence Form

We introduce a new method for studying stochastic homogenization of elliptic equations in nondivergence form. The main application is an algebraic error estimate, asserting that deviations from the homogenized limit are at most proportional to a power of the microscopic length scale, assuming a finite range of dependence. The results are new even for linear equations. The arguments rely on a new geometric quantity which is controlled in part by adapting elements of the regularity theory for the Monge-Amp\`ere equation.

Guaranteed lower bounds for eigenvalues

This paper introduces fully computable two-sided bounds on the eigenvalues of the Laplace operator on arbitrarily coarse meshes based on some approximation of the corresponding eigenfunction in the nonconforming Crouzeix-Raviart finite element space plus some postprocessing. The efficiency of the guaranteed error bounds involves the global mesh-size and is proven for the large class of graded meshes. Numerical examples demonstrate the reliability of the guaranteed error control even with an inexact solve of the algebraic eigenvalue problem. This motivates an adaptive algorithm which monitors the discretisation error, the maximal mesh-size, and the algebraic eigenvalue error. The accuracy of the guaranteed eigenvalue bounds is surprisingly high with efficiency indices as small as 1.4.

Verification of Galois field based circuits by formal reasoning based on computational algebraic geometry

Algebraic error correcting codes (ECC) are widely used to implement reliability features in modern servers and systems and pose a formidable verification challenge. We present a novel methodology and techniques for provably correct design of ECC logics. The methodology is comprised of a design specification method that directly exposes the ECC algorithm's underlying math to a verification layer, encapsulated in a tool "BLUEVERI", which establishes the correctness of the design conclusively by using an apparatus of computational algebraic geometry (Buchberger's algorithm for Grobner basis construction). We present results from its application to example circuits to demonstrate the effectiveness of the approach. The methodology has been successfully applied to prove correctness of large error correcting circuits on IBM's POWER systems to protect memory storage and processor to memory communication, as well as a host of smaller error correcting circuits.

Distribution of the discretization and algebraic error in numerical solution of partial differential equations

In the adaptive numerical solution of partial differential equations, local mesh refinement is used together with a posteriori error analysis in order to equilibrate the discretization error distribution over the domain. Since the discretized algebraic problems are not solved exactly, a natural question is whether the spatial distribution of the algebraic error is analogous to the spatial distribution of the discretization error. The main goal of this paper is to illustrate using standard boundary value model problems that this may not hold. On the contrary, the algebraic error can have large local components which can significantly dominate the total error in some parts of the domain. The illustrated phenomenon is of general significance and it is not restricted to some particular problems or dimensions. To our knowledge, the discrepancy between the spatial distribution of the discretization and algebraic errors has not been studied in detail elsewhere.

Suboptimal Solutions to the Algebraic-Error Line Triangulation

Line triangulation, a foundational problem in computer vision, is to estimate the 3D line position from a set of measured image lines with known camera projection matrices. Aiming to improve the triangulation's efficiency, in this work, two algorithms are proposed to find suboptimal solutions under the algebraic-error optimality criterion of the Plucker line coordinates. In these proposed algorithms, the algebraic-error optimality criterion is reformulated by the transformation of the Klein constraint. By relaxing the quadratic unit norm constraint to six linear constraints, six new single-quadric-constraint optimality criteria are constructed in the new formulation, whose optimal solutions can be obtained by solving polynomial equations. Moreover, we prove that the minimum algebraic error of either the first three or the last three of the six new criteria is not more than $\sqrt{3}$ times of that of the original algebraic-error optimality criterion. Thus, with three new criteria and all the six criteria, suboptimal solutions under the algebraic error minimization and the geometric error minimization are obtained. Experimental results show the effectiveness of our proposed algorithms.

Internal Error Propagation in Explicit Runge--Kutta Methods

In practical computation with Runge--Kutta methods, the stage equations are not satisfied exactly, due to roundoff errors, algebraic solver errors, and so forth. We show by example that propagation of such errors within a single step can have catastrophic effects for otherwise practical and well-known methods. We perform a general analysis of internal error propagation, emphasizing that it depends significantly on how the method is implemented. We show that for a fixed method, essentially any set of internal stability polynomials can be obtained by modifying the implementation details. We provide bounds on the internal error amplification constants for some classes of methods with many stages, including strong stability preserving methods and extrapolation methods. These results are used to prove error bounds in the presence of roundoff or other internal errors.

A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows

This paper develops a general abstract framework for a posteriori estimates for immiscible incompressible two-phase flows in porous media. We measure the error by the dual norm of the residual and, for mathematical correctness, employ the concept of global and complementary pressures in the analysis. Our estimators allow to estimate separately the different error components, namely, the spatial discretization error, the temporal discretization error, the linearization error, the iterative coupling error, and the algebraic solver error. We propose an adaptive algorithm wherein the different iterative procedures (iterative linearization, iterative coupling, iterative solution of linear systems) are stopped when the corresponding errors do not affect significantly the overall error and wherein the spatial and temporal errors are equilibrated. Consequently, important computational savings can be achieved while guaranteeing a user-given precision. The developed framework covers fully implicit, implicit pressure–explicit saturation, or iterative coupling formulations; conforming spatial discretization schemes such as the vertex-centered finite volume method or the finite element method and nonconforming spatial discretization schemes such as the cell-centered finite volume method, the mixed finite element method, or the discontinuous Galerkin method; linearizations such as the Newton or the fixed-point one; and general linear solvers. Numerical experiments for a model problem are presented to illustrate the theoretical results. Only by stopping timely the linear and nonlinear solvers, speedups by a factor between 10 and 20 in terms of the number of total linear solver iterations are achieved.

Interplay between discretization and algebraic computation in adaptive numerical solutionof elliptic PDE problems

AbstractThe Adaptive Finite Element Method (AFEM) for approximating solutions of PDE boundary value and eigenvalue problems is a numerical scheme that automatically and iteratively adapts the finite element space until a sufficiently accurate approximate solution is found. The adaptation process is based on a posteriori error estimators, and at each step of this process an algebraic problem (linear or nonlinear algebraic system or eigenvalue problem) has to be solved. In practical computations the solution of the algebraic problem cannot be obtained exactly. As a consequence, the algebraic error should be incorporated in the context of the AFEM and its a posteriori error estimators. The goal of this paper is to survey some existing approaches in the AFEM context that consider the interplay between the finite element discretization and the algebraic computation. We believe that a better understanding of this interplay is of great importance for the future development in the area of numerically solving large‐scale real‐world motivated problems. (© 2013 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow

In this paper we derive an a posteriori error estimate for the numerical approximation of the solution of a system modeling the flow of two incompressible and immiscible fluids in a porous medium. We take into account the capillary pressure, which leads to a coupled system of two equations: parabolic and elliptic. The parabolic equation may become degenerate, i.e., the nonlinear diffusion coefficient may vanish over regions that are not known a priori. We first show that, under appropriate assumptions, the energy-type norm differences between the exact and the approximate nonwetting phase saturations, the global pressures, and the Kirchhoff transforms of the nonwetting phase saturations can be bounded by the dual norm of the residuals. We then bound the dual norm of the residuals by fully computable a posteriori estimators. Our analysis covers a large class of conforming, vertex-centered finite volume-type discretizations with fully implicit time stepping. As an example, we focus here on two approaches: a ``mathematical'' scheme derived from the weak formulation, and a phase-by-phase upstream weighting ``engineering'' scheme. Finally, we show how the different error components, namely the space discretization error, the time discretization error, the linearization error, the algebraic solver error, and the quadrature error can be distinguished and used for making the calculations efficient.

Indirect Estimates of Total Fertility Rate Using Child Woman/Ratio: A Comparison with the Bogue-Palmore Method

Indirect estimation methodologies of the total fertility rate (TFR) have a long history within demography and have provided important techniques applied demographers can use when data is sparse or lacking. However new methodologies for approximating the total fertility rate have not been proposed in nearly 30 years. This study presents a novel method for indirectly approximating the total fertility rate using an algebraic rearrangement of the general fertility rate (GFR) through the known relationship between GFR and TFR. It then compares the proposed method to the well-known Bogue-Palmore method. These methods are compared in 196 countries and include overall errors as well as characteristics of the countries that contribute to fertility behavior. Additionally, these methods were compared geographically to find any geographical patterns. We find this novel method is not only simpler than the Bogue-Palmore method, requiring fewer data inputs, but also has reduced algebraic and absolute errors when compared with the Bogue-Palmore method and specifically outperforms the Bogue-Palmore method in developing countries. We find that our novel method may be useful estimation procedure for demographers.