Allal Guessab

Sharp Integral Inequalities of the Hermite-Hadamard Type

We consider a family of two-point quadrature formulae and establish sharp estimates for the remainders under various regularity conditions. Improved forms of certain integral inequalities due to Hermite and Hadamard, Iyengar, Milovanovi? and Pecari?, and others are obtained as special cases. Our results can also be interpreted as analogues to a theorem of Ostrowski on the deviation of a function from its averages. Furthermore, we establish a generalization of a result of Fink concerning Lp estimates for the remainder of the trapezoidal rule and present the best constants in the error bounds.

Convexity results and sharp error estimates in approximate multivariate integration

jjAn interesting property of the midpoint rule and the trapezoidal rule, which is expressed by the so-called Hermite-Hadamard inequalities, is that they provide one-sided approximations to the integral of a convex function. We establish multivariate analogues of the Hermite-Hadamard inequalities and obtain access to multivariate integration formulae via convexity, in analogy to the univariate case. In particular, for simplices of arbitrary dimension, we present two families of integration formulae which both contain a multivariate analogue of the midpoint rule and the trapezoidal rule as boundary cases. The first family also includes a multivariate analogue of a Maclaurin formula and of the two-point Gaussian quadrature formula; the second family includes a multivariate analogue of a formula by P. C. Hammer and of Simpsons rule. In both families, we trace out those formulae which satisfy a Hermite-Hadamard inequality. As an immediate consequence of the latter, we obtain sharp error estimates for twice continuously differentiable functions.

A fast algorithm for Gaussian type quadrature formulae with mixed boundary conditions and some lumped mass spectral approximations

After studying Gaussian type quadrature formulae with mixed boundary conditions, we suggest a fast algorithm for computing their nodes and weights. It is shown that the latter are computed in the same manner as in the theory of the classical Gauss quadrature formulae. In fact, all nodes and weights are again computed as eigenvalues and eigenvectors of a real symmetric tridiagonal matrix. Hence, we can adapt existing procedures for generating such quadrature formulae. Comparative results with various methods now in use are given. In the second part of this paper, new algorithms for spectral approximations for second-order elliptic problems are derived. The key to the efficiency of our algorithms is to find an appropriate spectral approximation by using the most accurate quadrature formula, which takes the boundary conditions into account in such a way that the resulting discrete system has a diagonal mass matrix. Hence, our algorithms can be used to introduce explicit resolutions for the time-dependent problems. This is the so-called lumped mass method. The performance of the approach is illustrated with several numerical examples in one and two space dimensions.

Construction of positive definite cubature formulae and approximation of functions via Voronoi tessellations

Let Ω ⊂ ℝd be a compact convex set of positive measure. In a recent paper, we established a definiteness theory for cubature formulae of order two on Ω. Here we study extremal properties of those positive definite formulae that can be generated by a centroidal Voronoi tessellation of Ω. In this connection we come across a class of operators of the form

Approximations of differentiable convex functions on arbitrary convex polytopes

L_n[f](\boldsymbol{x}):= \sum_{i=1}^n \phi_i(\boldsymbol{x})(f(\boldsymbol{y}_i) + \langle\nabla f(\boldsymbol{y}_i), \boldsymbol{x}-\boldsymbol{y}_i\rangle)

Increasing the approximation order of spline quasi-interpolants

, where

Generalized barycentric coordinates and approximations of convex functions on arbitrary convex polytopes

\boldsymbol{y}_1,\dots, \boldsymbol{y}_n

Construction techniques for multivariate modified quasi-interpolants with high approximation order

are distinct points in Ω and {ϕ1, ..., ϕn} is a partition of unity on Ω. We present best possible pointwise error estimates and describe operators Ln with a smallest constant in an Lp error estimate for 1 ≤ p < ∞ . For a generalization, we introduce a new type of Voronoi tessellation in terms of a twice continuously differentiable and strictly convex function f. It allows us to describe a best operator Ln for approximating f by Ln[f] with respect to the Lp norm.

A new family of extended Gauss quadratures with an interior interval constraint

Abstract Let X n ≔ { x i } i = 0 n be a given set of ( n + 1 ) pairwise distinct points in R d (called nodes or sample points), let P = conv ( X n ) , let f be a convex function with Lipschitz continuous gradient on P and λ ≔ { λ i } i = 0 n be a set of barycentric coordinates with respect to the point set X n . We analyze the error estimate between f and its barycentric approximation: B n [ f ] ( x ) = ∑ i = 0 n λ i ( x ) f ( x i ) , ( x ∈ P ) and present the best possible pointwise error estimates of f . Additionally, we describe the optimal barycentric coordinates that provide the best operator B n for approximating f by B n [ f ] . We show that the set of (linear finite element) barycentric coordinates generated by the Delaunay triangulation gives access to efficient algorithms for computing optimal approximations. Finally, numerical examples are used to show the success of the method.

Symmetrization, convexity and applications

Abstract In this paper, we show how by a very simple modification of bivariate spline discrete quasi-interpolants, we can construct a new class of quasi-interpolants which have remarkable properties such as high order of regularity and polynomial reproduction. More precisely, given a spline discrete quasi-interpolation operator Q d , which is exact on the space P m of polynomials of total degree at most m , we first propose a general method to determine a new differential quasi-interpolation operator Q r D which is exact on P m + r . Q r D uses the values of the function to be approximated at the points involved in the linear functional defining Q d as well as the partial derivatives up to the order r at the same points. From this result, we then construct and study a first order differential quasi-interpolant based on the C 1 cubic B-spline on the equilateral triangulation with a hexagonal support. When the derivatives are not available or extremely expensive to compute, we approximate them by appropriate finite differences to derive new discrete quasi-interpolants Q d . We estimate with small constants the quasi-interpolation errors f − Q r D [ f ] and f − Q d [ f ] in the infinity norm. Finally, numerical examples are used to analyze the performance of the method.