Catterina Dagnino

Catterina Dagnino, Angelo Dallefrate, Sara Remogna

In this paper we propose projection methods based on spline quasi-interpolating projectors of degree $d$ and class $C^{d-1}$ on a bounded interval for the numerical solution of nonlinear integral equations. We prove that they have high order of convergence $2d+2$ if $d$ is odd and $2d+3$ if $d$ is even. We also present the implementation details of the above methods. Finally, we provide numerical tests, that confirm the theoretical results. Moreover, we compare the theoretical and numerical results with those obtained by using a collocation method based on the same spline quasi-interpolating projectors.

Catterina Dagnino, Paola Lamberti, Sara Remogna

In this paper we investigate the problem of interpolating a B-spline curve network, in order to create a surface satisfying such a constraint and defined by blending functions spanning the space of bivariate $C^1$ quadratic splines on criss-cross triangulations. We prove the existence and uniqueness of the surface, providing a constructive algorithm for its generation. We also present numerical and graphical results and comparisons with other methods.

Catterina Dagnino, Paola Lamberti, Sara Remogna

In this paper, we present new quasi-interpolating spline schemes defined on 3D bounded domains, based on trivariate $C^2$ quartic box splines on type-6 tetrahedral partitions and with approximation order four. Such methods can be used for the reconstruction of gridded volume data. More precisely, we propose near-best quasi-interpolants, i.e. with coefficient functionals obtained by imposing the exactness of the quasi-interpolants on the space of polynomials of total degree three and minimizing an upper bound for their infinity norm. In case of bounded domains the main problem consists in the construction of the coefficient functionals associated with boundary generators (i.e. generators with supports not completely inside the domain), so that the functionals involve data points inside or on the boundary of the domain. We give norm and error estimates and we present some numerical tests, illustrating the approximation properties of the proposed quasi-interpolants, and comparisons with other known spline methods. Some applications with real world volume data are also provided.

Catterina Dagnino, Paul Sablonnière

Given a non-uniform criss-cross partition of a rectangular domain $Ω$, we analyse the error between a function $f$ defined on $Ω$ and two types of $C^1$-quadratic spline quasi-interpolants (QIs) obtained as linear combinations of B-splines with discrete functionals as coefficients. The main novelties are the facts that supports of B-splines are contained in $Ω$ and that data sites also lie inside or on the boundary of $Ω$. Moreover, the infinity norms of these QIs are small and do not depend on the triangulation: as the two QIs are exact on quadratic polynomials, they give the optimal approximation order for smooth functions. Our analysis is done for $f$ and its partial derivatives of the first and second orders and a particular effort has been made in order to give the best possible error bounds in terms of the smoothness of $f$ and of the mesh ratios of the triangulation.