Optimal Approximation Order Research Articles

Optimal Lagrange interpolation by quartic [formula omitted] splines on triangulations

We develop a local Lagrange interpolation scheme for quartic C 1 splines on triangulations. Given an arbitrary triangulation Δ , we decompose Δ into pairs of neighboring triangles and add “diagonals” to some of these pairs. Only in exceptional cases, a few triangles are split. Based on this simple refinement of Δ , we describe an algorithm for constructing Lagrange interpolation points such that the interpolation method is local, stable and has optimal approximation order. The complexity for computing the interpolating splines is linear in the number of triangles. For the local Lagrange interpolation methods known in the literature, about half of the triangles have to be split.

On geometric interpolation by planar parametric polynomial curves

In this paper the problem of geometric interpolation of planar data by parametric polynomial curves is revisited. The conjecture that a parametric polynomial curve of degree < n can interpolate 2n given points in R 2 is confirmed for n < 5 under certain natural restrictions. This conclusion also implies the optimal asymptotic approximation order. More generally, the optimal order 2n can be achieved as soon as the interpolating curve exists.

A posteriori estimates for approximations of time-dependent Stokes equations

In this paper, we derive a posteriori error estimates for space-discrete approximations of the time-dependent Stokes equations. By using an appropriate Stokes reconstruction operator, we are able to write an auxiliary error equation, in pointwise form, that satisfies the exact divergence-free condition. Thus, standard energy estimates from partial differential equation theory can be applied directly, and yield a posteriori estimates that rely on available corresponding estimates for the stationary Stokes equation. Estimates of optimal order in L ∞ (L 2 ) and L ∞ (H 1 ) for the velocity are derived for finite-element and finite-volume approximations.

Theory and Application of Characteristic Finite Element Domain Decomposition Procedures for Coupled System of Dynamics of Fluids in Porous Media

For a coupled system of multiplayer dynamics of fluids in porous media, the characteristic finite element domain decomposition procedures applicable to parallel arithmetic are put forward. Techniques such as calculus of variations, domain decomposition, characteristic method, negative norm estimate, energy method and the theory of prior estimates are adopted. Optimal order estimates in L2 norm are derived for the error in the approximate solution.

Universal Algorithms for Learning Theory. Part II: Piecewise Polynomial Functions

This paper is concerned with estimating the regression function fρ in supervised learning by utilizing piecewise polynomial approximations on adaptively generated partitions. The main point of interest is algorithms that with high probability are optimal in terms of the least square error achieved for a given number m of observed data. In a previous paper [1], we have developed for each β > 0 an algorithm for piecewise constant approximation which is proven to provide such optimal order estimates with probability larger than 1- m-β. In this paper we consider the case of higher-degree polynomials. We show that for general probability measures ρ empirical least squares minimization will not provide optimal error estimates with high probability. We go further in identifying certain conditions on the probability measure ρ which will allow optimal estimates with high probability.

On geometric interpolation of circle-like curves

In this paper, geometric interpolation of certain circle-like curves by parametric polynomial curves is studied. It is shown that such an interpolating curve of degree n achieves the optimal approximation order 2 n, the fact already known for particular small values of n. Furthermore, numerical experiments suggest that the error decreases exponentially with growing n.

The upwind finite difference fractional steps method for nonlinear coupled systems

AbstractFor nonlinear coupled system of multilayer dynamics of fluids in porous media, a second‐order upwind finite‐difference fractional‐steps scheme applicable to parallel arithmetic are put forward, and two‐ and three‐dimensional schemes are used to form a complete set. Some techniques, such as calculus of variations, multiplicative commutation rule of difference operators, decomposition of high‐order difference operators, and prior estimates are adopted. Optimal order estimates in l2 norm are derived to determine the error in the approximate solution. This method has already been applied to the numerical simulation of migration‐accumulation of oil resources. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

Galerkin alternating-direction method for a kind of second-order quasi-linear hyperbolic problems

A kind of second-order quasi-linear hyperbolic equation is firstly transformed into a first-order system of equations, then the Galerkin alternating-direction procedure for the system is derived. The optimal order estimates in H 1 norm and L 2 norm of the procedure are obtained respectively by using the theory and techniques of priori estimate of differential equations. The numerical experiment is also given to support the theoretical analysis. Comparing the results of numerical example with the theoretical analysis, they are uniform.

Quadratic spline quasi-interpolants on Powell-Sabin partitions

In this paper we address the problem of constructing quasi-interpolants in the space of quadratic Powell-Sabin splines on nonuniform triangulations. Quasi-interpolants of optimal approximation order are proposed and numerical tests are presented.

Local quasi-interpolation by cubic C1 splines on type-6 tetrahedral partitions

We describe an approximating scheme based on cubic C1 splines on type-6 tetrahedral partitions using data on volumetric grids. The quasi-interpolating piecewise polynomials are directly determined by setting their Bernstein-Bezier coefficients to appropriate combinations of the data values. Hence, each polynomial piece of the approximating spline is immediately available from local portions of the data, without using prescribed derivatives at any point of the domain. The locality of the method and the uniform boundedness of the associated operator provide an error bound, which shows that the approach can be used to approximate and reconstruct trivariate functions. Simultaneously, we show that the derivatives of the quasi-interpolating splines yield nearly optimal approximation order. Numerical tests with up to 17 x 10 6 data sites show that the method can be used for efficient approximation.

Optimization on class of operator equations in the probabilistic case setting

In this paper, we introduce a problem of the optimization of approximate solutions of operator equations in the probabilistic case setting, and prove a general result which connects the relation between the optimal approximation order of operator equations with the asymptotic order of the probabilistic width. Moreover, using this result, we determine the exact orders on the optimal approximate solutions of multivariate Freldholm integral equations of the second kind with the kernels belonging to the multivariate Sobolev class with the mixed derivative in the probabilistic case setting.

Bernstein–Bézier representation and near-minimally normed discrete quasi-interpolation operators

We are concerned with the construction of bivariate box-spline discrete quasi-interpolants with small infinity norms and optimal approximation orders. They are defined by minimizing a sharp upper bound of the uniform norm which is derived from the Bernstein–Bézier representation of the corresponding fundamental function. We detail the construction of such quadratic and quartic quasi-interpolants.

The Upwind Finite Difference Fractional Steps Method for Nonlinear Coupled System of Dynamics of Fluids in Porous Media

For nonlinear coupled system of multilayer dynamics of fluids in porous media, the second order and first order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward, and two-dimensional and three-dimensional schemes are used to form a complete set. Some techniques, such as calculus of variations, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates, are adopted. Optimal order estimates in L2 norm are derived to determine the error in the second order approximate solution. This method has already been applied to the numerical simulation of migration-accumulation of oil resources.

An unfitted finite-element method for elliptic and parabolic interface problems

A finite-element discretization, independent of the location of the interface, is proposed and analysed for linear elliptic and parabolic interface problems. We establish error estimates of optimal order in the H 1 -norm and almost optimal order in the L 2 -norm for elliptic interface problems. An extension to parabolic interface problems is also discussed and an optimal error estimate in the L 2 (0, T; H1 (Ω))-norm and an almost optimal order estimate in the L 2 (0, T; L 2 (Ω))-norrn are derived for the spatially discrete scheme. A fully discrete scheme based on the backward Euler method is analysed and an optimal order error estimate in the L 2 (0, T; H I (Ω))-norm is derived. The interfaces are assumed to be of arbitrary shape and smooth for our purpose.

The modified method of upwind with finte difference fractional steps procedure for the numerical simulation and analysis of seawater intrusion

Abstract Numerical simulation and analysis of seawater intrusion is the mathematical basis for modern environmental science. Its mathematical model is the nonlinear coupled system of partial differential equations with initial-boudary problems. For a generic case of a three-dimensional bounded region, the modified method of upwind with finite difference fractional steps procedure is put forward. Optimal order estimates in L 2 norm are derived for the error in the approximation solution. The present method has been successfully used in predicting the consequences of seawater intrusion and protection projects. * Supported by the Major State Basic Research Development Program of China (Grant No. G19990328), National Tackling Key Problems Program (Grant No. 2002020094), National Natural Science Foundation of China (Grant Nos. 10271066, 10372052) and the Doctorate Foundation of the Ministry of Education of China (Grant No. 20030422047)

Numerical method and application for three-dimensional nonlinear system of dynamics of fluids in porous media

For the system of multilayer dynamics of fluids in porous media, the second order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward. Some techniques, such as calculus of variations, energy method, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates are adopted. Optimal order estimates are derived to determine the error in the second order approximate solution. These methods have already been applied to the numerical simulation of migration-accumulation of oil resources.

An octahedral [formula omitted] macro-element

A macro-element of smoothness C 2 is constructed on the split of an octahedron into eight tetrahedra. This new element complements those recently constructed on the Clough–Tocher and Worsey–Farin splits of a tetrahedron (cf. Alfeld, P., Schumaker, L.L., 2005a, 2005b). The octahedral element uses supersplines of degree thirteen, and provides optimal order of approximation of smooth functions.

Numerical integration with Taylor truncations for the quadrilateral and hexahedral finite elements

For general quadrilateral or hexahedral meshes, the finite-element methods require evaluation of integrals of rational functions, instead of traditional polynomials. It remains as a challenge in mathematics to show the traditional Gauss quadratures would ensure the correct order of approximation for the numerical integration in general. However, in the case of nested refinement, the refined quadrilaterals and hexahedra converge to parallelograms and parallelepipeds, respectively. Based on this observation, the rational functions of inverse Jacobians can be approximated by the Taylor expansion with truncation. Then the Gauss quadrature of exact order can be adopted for the resulting integrals of polynomials, retaining the optimal order approximation of the finite-element methods. A theoretic justification and some numerical verification are provided in the paper.

Characteristic‐mixed covolume methods for advection‐dominated diffusion problems

AbstractCharacteristic‐mixed covolume methods for time‐dependent advection‐dominated diffusion problems are developed and studied. The diffusion term in these problems is discretized using covolume methods applied to the mixed formulation of the problems on quadrilaterals, and the temporal differentiation and advection terms are treated by characteristic tracking schemes. Three characteristic tracking schemes are studied in the context of mixed covolume methods: the modified method of characteristics, the modified method of characteristics with adjusted advection, and the Eulerian–Lagrangian localized adjoint method. The proposed methods preserve the conceptual and computational merits of both characteristics‐based schemes and the mixed covolume methods. Existence and uniqueness of a solution to the discrete problem arising from the methods is shown. Stability and convergence properties of these methods are also obtained; unconditionally stable results and error estimates of optimal order are established. Copyright © 2006 John Wiley &amp; Sons, Ltd.

On the finite element method on quadrilateral meshes

The theoretical analysis of the finite element method is well established in the case of triangular or tetrahedral meshes. In this case optimal approximation properties have been proved in all reasonable functional norms. Several commercial codes use this method which is now in the common practice of engineering applications. On the other hand, the case of quadrilateral or hexahedral meshes, even if commonly used in the applications (sometimes it seems to be even more popular than the previous one), has not been studied in such deep detail, probably because it hides some insidious issues. For simplicity, we restrict our analysis to the two-dimensional case; three-dimensional analysis in some cases is a straightforward extension, in some others is more complicated. Indeed, we shall show that some commonly used quadrilateral finite elements present a lack of convergence when general regular meshes are used. The list of suboptimal elements includes serendipity (or trunk) elements, face elements, edge elements. A rigorous theory is presented, which gives necessary and sufficient conditions for optimal order approximation. Our theory is supported by several numerical experiments, which are taken from various engineering applications, ranging from elasticity and fluid dynamics to acoustics and electromagnetics. Example of suboptimal elements include: 8-node element for Poisson problem, Q 2 – P 1 Stokes element, face elements for acoustic problem, edge elements for electromagnetic problems.