On numerical solution of Fredholm and Hammerstein integral equations via Nyström method and Gaussian quadrature rules for splines

The first author acknowledges partial financial support by the IMAG-Maria de Maeztu grant CEX2020-001105-M/AEI/10.13039/501100011033. The second author has been partially supported by Spanish State Research Agency (Spanish Min-istry of Science, Innovation and Universities) : BCAM Severo Ochoa excellence accreditation SEV-2017-0718 and by Ramon y Cajal with reference RYC-2017-22649. The fourth author is member of GNCS-INdAM.Nyström method is a standard numerical technique to solve Fredholm integral equations of the second kind where the integration of the kernel is approximated using a quadrature formula. Traditionally, the quadrature rule used is the classical polynomial Gauss quadrature. Motivated by the observation that a given function can be better approximated by a spline function of a lower degree than a single polynomial piece of a higher degree, in this work, we investigate the use of Gaussian rules for splines in the Nyström method. We show that, for continuous kernels, the approximate solution of linear Fredholm integral equations computed using spline Gaussian quadrature rules converges to the exact solution for m →∞, m being the number of quadrature points. Our numerical results also show that, when fixing the same number of quadrature points, the approximation is more accurate using spline Gaussian rules than using the classical polynomial Gauss rules. We also investigate the non-linear case, considering Hammerstein integral equations, and present some numerical tests.IMAG-Maria de Maeztu grant CEX2020-001105-M/AEI/10.13039/501100011033Spanish State Research Agency (Spanish Min-istry of Science, Innovation and Universities) SEV-2017-0718Spanish Government RYC-2017-2264

Superconvergent Nyström and Degenerate Kernel Methods for Integro-Differential Equations

This research received no external funding and APC was funded by University of Granada.The aim of this paper is to carry out an improved analysis of the convergence of the Nystrom and degenerate kernel methods and their superconvergent versions for the numerical solution of a class of linear Fredholm integro-differential equations of the second kind. By using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomial functions of degree <= r - 1, we obtain convergence order 2r for degenerate kernel and Nystrom methods, while, for the superconvergent and the iterated versions of theses methods, the obtained convergence orders are 3r + 1 and 4r, respectively. Moreover, we show that the optimal convergence order 4r is restored at the partition knots for the approximate solutions. The obtained theoretical results are illustrated by some numerical examples.University of Granad

On C2 cubic quasi-interpolating splines and their computation by subdivision via blossoming

We discuss the construction of C2 cubic spline quasi-interpolation schemes defined on a refined partition. These schemes are reduced in terms of degrees of freedom compared to those existing in the literature. Namely, we provide a rule for reducing them by imposing super-smoothing conditions while preserving full smoothness and cubic precision. In addition, we provide subdivision rules by means of blossoming. The derived rules are designed to express the B-spline coefficients associated with a finer partition from those associated with the former one."Maria de Maeztu" Excellence Unit IMAG (University of Granada, Spain) CEX2020-001105-MICIN/AEI/10.13039/501100011033University of Granada University of Granada/CBU

C-1-Cubic Quasi-Interpolation Splines over a CT Refinement of a Type-1 Triangulation

C1 continuous quasi-interpolating splines are constructed over Clough–Tocher refinement of a type-1 triangulation. Their Bernstein–Bézier coefficients are directly defined from the known values of the function to be approximated, so that a set of appropriate basis functions is not required. The resulting quasi-interpolation operators reproduce cubic polynomials. Some numerical tests are given in order to show the performance of the approximation scheme

A new approach to deal with C2 cubic splines and its application to super-convergent quasi-interpolation

The authors wish to thank the anonymous referees for their very pertinent and useful comments which helped them to improve the original manuscript. The first and third authors are members of the research group FQM 191 Matematica Aplicada funded by the PAIDI programme of the Junta de Andalucia, Spain. The second author would like to thank the University of Granada, Spain for the financial support for the research stay during which this work was carried out. Funding for open access charge: Universidad de Granada/CBUAIn this paper, we construct a novel normalized B-spline-like representation for C2-continuous cubic spline space defined on an initial partition refined by inserting two new points inside each sub-interval. The basis functions are compactly supported non-negative functions that are geometrically constructed and form a convex partition of unity. With the help of the control polynomial theory introduced herein, a Marsden identity is derived, from which several families of super-convergent quasi-interpolation operators are defined.Junta de AndaluciaUniversity of Granada, SpainUniversidad de Granada/CBU

Non-uniform UE-spline quasi-interpolants and their application to the numerical solution of integral equations

A construction of Marsden’s identity for UE-splines is developed and a complete proof is given. With the help of this identity, a new non-uniform quasi-interpolant that repro-duces the spaces of polynomial, trigonometric and hyperbolic functions are defined. Effi-cient quadrature rules based on integrating these quasi-interpolation schemes are derived and analyzed. Then, a quadrature formula associated with non-uniform quasi-interpolation along with Nyström’s method is used to numericallysolve Hammerstein and Fredholm integral equations. Numerical results that illustrate the effectiveness of these rules are pre-sented.Universidad de Granada / CBU

Near-best univariate spline discrete quasi-interpolants on non-uniform partitions

International audienceUnivariate spline discrete quasi-interpolants (abbr. dQIs) are approximation operators using B-spline expansions with coefficients which are linear combinations of discrete values of the function to be approximated. When working with nonuniform partitions, the main challenge is to find dQIs which have both good approximation orders and bounded uniform norms independent of the given partition. Near-best dQIs are obtained by minimizing an upper bound of the infinite norm of dQIs depending on a certain number of free parameters, thus reducing this norm. This paper is devoted to the study of some families of near-best dQIs of approximation order 2

Near-best bivariate spline quasi-interpolants on a four-directional mesh of the plane

Spline quasi-interpolants (QIs) are practical and effective approximation operators. In this paper, we construct QIs with optimal approximation orders and small infinity norms called near-best discrete and integral quasi-interpolants which are based on $\Omega$-~splines, i.e. B-splines with regular lozenge supports on the uniform four directional mesh of the plane. These quasi-interpolants are obtained so as to be exact on some space of polynomials and to minimize an upper bound of their infinity norms which depend on a finite number of free parameters. We show that this problem has always a solution, which is not unique in general. Concrete examples of these types of quasi-interpolants are given in the last section

Near minimally normed spline quasi-interpolants on uniform partitions

International audienceSpline quasi-interpolants are local approximating operators for functions or discrete data. We consider the construction of discrete and integral spline quasi-interpolants on uniform partitions of the real line having small infinite norms. We call them near minimally normed quasi-interpolants: they are exact on polynomial spaces and minimize a simple upper bound of their infinite norms. We give precise results for cubic and quintic quasi-interpolants. Also the quasi-interpolation error is considered, as well as the advantage that these quasi-interpolants present when approximating functions with isolated discontinuities

Near-best quasi-interpolants associated with H-splines on a three-directional mesh.

International audienceSpline quasi-interpolants with best approximation orders and small norms are useful in several applications. In this paper, we construct the so-called near-best discrete and integral quasi-interpolants based on H-splines, i.e., B-splines with regular hexagonal supports on the uniform three-directional mesh of the plane. These quasi-interpolants are obtained so as to be exact on some space of polynomials, and minimize an upper bound of their infinite norms depending on a finite number of free parameters. We show that this problem has always a solution, but it is not unique in general. Concrete examples of these types of quasi-interpolants are given in the two last sections