C2 piecewise cubic quasi-interpolants on a 6-direction mesh

We study two kinds of quasi-interpolants (abbr. QI) in the space of C2 piecewise cubics in the plane, or in a rectangular domain, endowed with the highly symmetric triangulation generated by a uniform 6-direction mesh. It has been proved recently that this space is generated by the integer translates of two multi-box splines. One kind of QIs is of differential type and the other of discrete type. As those QIs are exact on the space of cubic polynomials, their approximation order is 4 for sufficiently smooth functions. In addition, they exhibit nice superconvergent properties at some specific points. Moreover, the infinite norms of the discrete QIs being small, they give excellent approximations of a smooth function and of its first order partial derivatives. The approximation properties of the QIs are illustrated by numerical examples

Near-best C2C^2 quartic spline quasi-interpolants on type-6 tetrahedral partitions of bounded domains

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.Comment: In the new version of the paper, we have done some minor revisions with respect to the previous version, CALCOLO, Published online: 10 October 201

Recent Results on Near-Best Spline Quasi-Interpolants

Roughly speaking, a near-best (abbr. NB) quasi-interpolant (abbr. QI) is an approximation operator of the form $Q_af=\sum_{\alpha\in A} \Lambda_\alpha (f) B_\alpha$ where the $B_\alpha$'s are B-splines and the $\Lambda_\alpha (f)$'s are linear discrete or integral forms acting on the given function $f$. These forms depend on a finite number of coefficients which are the components of vectors $a_\alpha$ for $\alpha\in A$. The index $a$ refers to this sequence of vectors. In order that $Q_a p=p$ for all polynomials $p$ belonging to some subspace included in the space of splines generated by the $B_\alpha$'s, each vector $a_\alpha$ must lie in an affine subspace $V_\alpha$, i.e. satisfy some linear constraints. However there remain some degrees of freedom which are used to minimize $\Vert a_\alpha \Vert_1$ for each $\alpha\in A$. It is easy to prove that $\max \{\Vert a_\alpha \Vert_1 ; \alpha\in A\}$ is an upper bound of $\Vert Q_a \Vert_{\infty}$: thus, instead of minimizing the infinite norm of $Q_a$, which is a difficult problem, we minimize an upper bound of this norm, which is much easier to do. Moreover, the latter problem has always at least one solution, which is associated with a NB QI. In the first part of the paper, we give a survey on NB univariate or bivariate spline QIs defined on uniform or non-uniform partitions and already studied by the author and coworkers. In the second part, we give some new results, mainly on univariate and bivariate integral QIs on {\sl non-uniform} partitions: in that case, NB QIs are more difficult to characterize and the optimal properties strongly depend on the geometry of the partition. Therefore we have restricted our study to QIs having interesting shape properties and/or infinite norms uniformly bounded independently of the partition

Multilevel Sparse Grid Methods for Elliptic Partial Differential Equations with Random Coefficients

Stochastic sampling methods are arguably the most direct and least intrusive means of incorporating parametric uncertainty into numerical simulations of partial differential equations with random inputs. However, to achieve an overall error that is within a desired tolerance, a large number of sample simulations may be required (to control the sampling error), each of which may need to be run at high levels of spatial fidelity (to control the spatial error). Multilevel sampling methods aim to achieve the same accuracy as traditional sampling methods, but at a reduced computational cost, through the use of a hierarchy of spatial discretization models. Multilevel algorithms coordinate the number of samples needed at each discretization level by minimizing the computational cost, subject to a given error tolerance. They can be applied to a variety of sampling schemes, exploit nesting when available, can be implemented in parallel and can be used to inform adaptive spatial refinement strategies. We extend the multilevel sampling algorithm to sparse grid stochastic collocation methods, discuss its numerical implementation and demonstrate its efficiency both theoretically and by means of numerical examples

Mesh generation using algebraic techniques

The solution of partial differential equations which describe physical phenomena by the use of coordinate transformations is described. The constraints of the problems are stated in geometric terms and include boundary constraints, uniformity constraints, and internal constraints. Algebraic mesh generations are very satisfactory for these constraints

B-spline-like bases for C2C^2 cubics on the Powell-Sabin 12-split

For spaces of constant, linear, and quadratic splines of maximal smoothness on the Powell-Sabin 12-split of a triangle, the so-called S-bases were recently introduced. These are simplex spline bases with B-spline-like properties on the 12-split of a single triangle, which are tied together across triangles in a B\'ezier-like manner. In this paper we give a formal definition of an S-basis in terms of certain basic properties. We proceed to investigate the existence of S-bases for the aforementioned spaces and additionally the cubic case, resulting in an exhaustive list. From their nature as simplex splines, we derive simple differentiation and recurrence formulas to other S-bases. We establish a Marsden identity that gives rise to various quasi-interpolants and domain points forming an intuitive control net, in terms of which conditions for $C^0$-, $C^1$-, and $C^2$-smoothness are derived