Carla Manni

On a class of weak Tchebycheff systems

AbstractIn this paper we study the approximation power, the existence of a normalized B-basis and the structure of a degree-raising process for spaces of the form requiring suitable assumptions on the functions u and v. The results about degree raising are detailed for special spaces of this form which have been recently introduced in the area of CAGD.

Quadratic spline quasi-interpolants on Powell-Sabin partitions

In this paper we address the problem of constructing quasi-interpolants in the space of quadratic Powell-Sabin splines on nonuniform triangulations. Quasi-interpolants of optimal approximation order are proposed and numerical tests are presented.

Efficient Solution of the Complex Quadratic Tridiagonal System for C2 PH Quintic Splines

The construction of C2 Pythagorean-hodograph (PH) quintic spline curves that interpolate a sequence of points p0,...,pN and satisfy prescribed end conditions incurs a “tridiagonal” system of N quadratic equations in N complex unknowns. Albrecht and Farouki [1] invoke the homotopy method to compute all 2N+k solutions to this system, among which there is a unique “good” PH spline that is free of undesired loops and extreme curvature variations (k∈{−1,0,+1} depends on the adopted end conditions). However, the homotopy method becomes prohibitively expensive when N≳10, and efficient methods to construct the “good” spline only are desirable. The use of iterative solution methods is described herein, with starting approximations derived from “ordinary” C2 cubic splines. The system Jacobian satisfies a global Lipschitz condition in CN, yielding a simple closed-form expression of the Kantorovich condition for convergence of Newton–Raphson iterations, that can be evaluated with O(N2) cost. These methods are also generalized to the case of non-uniform knots.

Robust and optimal multi-iterative techniques for IgA Galerkin linear systems

We consider fast solvers for large linear systems arising from the Galerkin approximation based on B-splines of classical d-dimensional elliptic problems, d ≥ 1, in the context of isogeometric analysis. Our ultimate goal is to design iterative algorithms with the following two properties. First, their computational cost is optimal, that is linear with respect to the number of degrees of freedom, i.e. the resulting matrix size. Second, they are totally robust, i.e., their convergence speed is substantially independent of all the relevant parameters: in our case, these are the matrix size (related to the fineness parameter), the spline degree (associated to the approximation order), and the dimensionality d of the problem. We review several methods like PCG, multigrid, multiiterative algorithms, and we show how their numerical behavior (in terms of convergence speed) can be understood through the notion of spectral distribution, i.e., through a compact symbol which describes the global eigenvalue behavior of the considered stiffness matrices. As a final step, we show how we can design an optimal and totally robust multi-iterative method, by taking into account the analytic features of the symbol. A wide variety of numerical experiments, few open problems and perspectives are presented and critically discussed.

Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics,

Abstract It is shown that, depending upon the orientation of the end tangents t0,t1 relative to the end point displacement vector Δp=p1−p0, the problem of G1 Hermite interpolation by PH cubic segments may admit zero, one, or two distinct solutions. For cases where two interpolants exist, the bending energy may be used to select among them. In cases where no solution exists, we determine the minimal adjustment of one end tangent that permits a spatial PH cubic Hermite interpolant. The problem of assigning tangents to a sequence of points p0,. . .,pn in R3, compatible with a G1 piecewise-PH-cubic spline interpolating those points, is also briefly addressed. The performance of these methods, in terms of overall smoothness and shape-preservation properties of the resulting curves, is illustrated by a selection of computed examples.

Effortless quasi-interpolation in hierarchical spaces

We present a general and simple procedure to construct quasi-interpolants in hierarchical spaces. Such spaces are composed of a hierarchy of nested spaces and provide a flexible framework for local refinement. The proposed hierarchical quasi-interpolants are described in terms of the so-called truncated hierarchical basis. Assuming a quasi-interpolant is selected for each space associated with a particular level in the hierarchy, the hierarchical quasi-interpolants are obtained without any additional manipulation. The main properties (like polynomial reproduction) of the quasi-interpolants selected at each level are locally preserved in the hierarchical construction. We show how to construct hierarchical local projectors, and the local approximation order of the underling hierarchical space is also investigated. The presentation is detailed for the truncated hierarchical B-spline basis, and we discuss its extension to a more general framework.

Quasi-interpolation in isogeometric analysis based on generalized B-splines

Isogeometric analysis is a new method for the numerical simulation of problems governed by partial differential equations. It possesses many features in common with finite element methods (FEM) but takes some inspiration from Computer Aided Design tools. We illustrate how quasi-interpolation methods can be suitably used to set Dirichlet boundary conditions in isogeometric analysis. In particular, we focus on quasi-interpolant projectors for generalized B-splines, which have been recently proposed as a possible alternative to NURBS in isogeometric analysis.

On the spectrum of stiffness matrices arising from isogeometric analysis

We study the spectral properties of stiffness matrices that arise in the context of isogeometric analysis for the numerical solution of classical second order elliptic problems. Motivated by the applicative interest in the fast solution of the related linear systems, we are looking for a spectral characterization of the involved matrices. In particular, we investigate non-singularity, conditioning (extremal behavior), spectral distribution in the Weyl sense, as well as clustering of the eigenvalues to a certain (compact) subset of

Shape-preserving C 2 functional interpolation via parametric cubics

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