Discrete Orthogonal Projections Research Articles

A Theoretical Analysis for a New Finite Volume Scheme for a Linear Schrödinger Evolution Equation on General Nonconforming Spatial Meshes

In this article, we shall present in detail the results announced [5]. We consider a model of a linear time dependent Schrödinger equation with a time dependent potential. This model arises, for example, in underwater acoustics and has been studied by Akrivis and Dougalis [3]. We derive a new finite volume scheme on general nonconforming multidimensional spatial meshes introduced recently by Eymard et al. [18] for stationary anisotropic heterogeneous diffusion problems. The unknowns of this scheme are the values at the centre of the control volumes, at some internal interfaces, and at the mesh points of the time discretization. The discretization of the initial condition is performed using a discrete orthogonal projection. A new a priori estimate for the discrete problem is proved. Thanks to this a priori estimate for the discrete problem, we derive error estimates in discrete norms of and W 1, ∞(L 2). Moreover, we establish an error estimate for an approximation for the gradient, in a general framework. We prove that the convergence order is h 𝒟 + k, where h 𝒟 (resp. k) is the mesh size of the spatial (resp. time) discretization. This estimate is valid under the regularity assumption for the exact solution u. These error estimates are useful because they allow us to get error estimates for the approximations of the exact solution and its first derivatives. Results of the present work have been obtained by a comparison with an appropriately chosen auxiliary finite volume scheme.

Hybridisation of genetic algorithms and tabu search approach for reconstructing convex binary images from discrete orthogonal projections

In this paper, we consider a variant of the NP-complete problem of reconstructing HV-convex binary images from two orthogonal projections, noted by RCBIH, V. This variant is reformulated as a new integer programming problem. Since this problem is NP-complete, a new hybrid optimisation algorithm combining the techniques of genetic algorithms and tabu search methods, noted by GATS is proposed to find an optimal or an approximate solution for RCBIH, V problem. GATS starts from a set of solutions called 'population' initialised by using an extension of the network flow model, incorporating a cost function. Two operators, namely crossover and mutation are used to explore the search space, then the quality of each individual in the population is improved by using another local search method named tabu search operator. In this paper we describe the proposed algorithm, then we evaluate and compare its performance with other optimisation techniques. The analysis of the experimental results shows the advantages of our GATS approach in terms of reconstruction quality and computational time.

Discrete orthogonal projections on multiple knot periodic splines

This paper establishes properties of discrete orthogonal projections on periodic spline spaces of order r, with knots that are equally spaced and of arbitrary multiplicity M ⩽ r . The discrete orthogonal projection is expressed in terms of a quadrature rule formed by mapping a fixed J-point rule to each sub-interval. The results include stability with respect to discrete and continuous norms, convergence, commutator and superapproximation properties. A key role is played by a novel basis for the spline space of multiplicity M, which reduces to a familiar basis when M = 1 .

Fully discrete spectral boundary integral methods for Helmholtz problems on smooth closed surfaces in ${\mathbb R}^3$

In this paper we describe and analyse a class of spectral methods, based on spherical polynomial approximation, for second-kind weakly singular boundary integral equations arising from the Helmholtz equation on smooth closed 3D surfaces diffeomorphic to the sphere. Our methods are fully discrete Galerkin methods, based on the application of special quadrature rules for computing the outer and inner integrals arising in the Galerkin matrix entries. For the outer integrals we use, for example, product-Gauss rules. For the inner integrals, a variant of the classical product integration procedure is employed to remove the singularity arising in the kernel. The key to the analysis is a recent result of Sloan and Womersley on the norm of discrete orthogonal projection operators on the sphere. We prove that our methods are stable for continuous data and superalgebraically convergent for smooth data. Our theory includes as a special case a method closely related to one of those proposed by Wienert (1990) for the fast computation of direct and inverse acoustic scattering in 3D.

Reconstructing (h,v)-convex 2-dimensional patterns of objects from approximate horizontal and vertical projections

The problem of reconstructing a pattern of an object from its approximate discrete orthogonal projections in a 2-dimensional grid, may have no solution because the inaccuracy in the measurements of the projections may generate an inconsistent problem. To attempt to overcome this difficulty, one seeks to reconstruct a pattern with projection values having possibly some bounded differences with the given projection values and minimizing the sum of the absolute differences.This paper addresses the problem of reconstructing a pattern with a difference at most equal to +1 or −1 between each of its projection values and the corresponding given projection value. We deal with the case of patterns which have to be horizontally and vertically convex and the case of patterns which have to be moreover connected, the so-called convex polyominoes. We show that in both cases, the problem of reconstructing a pattern can be transformed into a Satisfiability (SAT) Problem. This is done in order to take advantage of the recent advances in the design of solvers for the SAT Problem. We show, experimentally, that by adding two important features to CSAT (an efficient SAT solver), optimal patterns can be found if there exist feasible ones. These two features are: first, a method that extracts in linear time an optimal pattern from a set of feasible patterns grouped in a generic pattern (obtaining a generic pattern may be exponential in the worst case) and second, a method that computes actively a lower bound of the sum of absolute differences that can be obtained from a partially defined pattern. This allows to prune the search tree if this lower bound exceeds the best sum of absolute differences found so far.

The use of discrete orthogonal projections in boundary element methods

In recent papers by Sloan and Wendland (1999), Grigorieff and Sloan (1998) and Grigorieff, Sloan and Brandts (2000), a formalism was developed that serves many important and interesting applications in boundary element methods: the ‘commutator property’ for splines. Based on superapproximation results, this property is, for example, a tool of central importance in stability and convergence proofs for qualocation methods for boundary integral equations with variable coefficients. Another application is the transfer of superconvergence properties from constant-coefficient boundary integral equations to the variable coefficient case. The heart of the theory is formed by the concept ‘discrete orthogonal projection’, that arises when the L 2-orthogonal inner product is discretized by possibly non-standard quadrature rules. In this paper, we present an overview of the theory of discrete orthogonal projections, and a new set of numerical experiments that confirm the theory. The main conclusion is that the presence of variable coefficients of a certain smoothness does not influence superconvergence in a negative way, and that henceforth the use of superconvergence-based a posteriori error estimators in this particular case is theoretically justified.

Superapproximation and Commutator Properties of Discrete Orthogonal Projections for Continuous Splines

This paper builds upon the Lp-stability results for discrete orthogonal projections on the spaces Sh of continuous splines of order r obtained by R. D. Grigorieff and I. H. Sloan in (1998, Bull. Austral. Math. Soc.58, 307–332). Properties of such projections were proved with a minimum of assumptions on the mesh and on the quadrature rule defining the discrete inner product. The present results, which include superapproximation and commutator properties, are similar to those derived by I. H. Sloan and W. Wendland (1999, J. Approx. Theory97, 254–281) for smoothest splines on uniform meshes. They are expected to have applications (as in I. H. Sloan and W. Wendland, Numer. Math. (1999, 83, 497–533)) to qualocation methods for non-constant-coefficient boundary integral equations, as well as to the wide range of other numerical methods in which quadrature is used to evaluate L2-inner products. As a first application, we consider the most basic variable-coefficient boundary integral equation, in which the constant-coefficient operator is the identity. The results are also extended to the case of periodic boundary conditions, in order to allow appplication to boundary integral equations on closed curves.

Stability of discrete orthogonal projections for continuous splines

In this paper Lp stability and convergence properties of discrete orthogonal projections on the finite element space Sh of continuous polynomial splines of order r are proved. The discrete inner products are defined by composite quadrature rules with positive weights on a sequence of nonuniform grids. It is assumed that the basic quadrature rule Q has at least r quadrature points in order to resolve Sh, but no accuracy is required. The main results are derived under minimal further assumptions, for example the rule Q is allowed to be non-symmetric, and no quasi-uniformity of the mesh is required. The corresponding stability of the orthogonal L2-projections has been studied by de Boor [1] and by Crouzeix and Thomee [2]. Stability of the first derivative of the projection is also proved, under an assumption (unless p = 1) of local quasi-uniformity of the mesh.

Discrete Bernstein bases and Hahn polynomials

In this paper, we study the representation and evaluation of Hahn polynomials in discrete Bernstein bases of various degrees defined on different intervals of integers. Basic tools are degree elevation and subdivision algorithms. They only involve linear convex combinations with simple rational coefficients and are numerically stable. Their computational complexity is also evaluated. Applications to the computation of discrete orthogonal projections and to a property of the max-norm of discrete Chebyshev polynomials are given.