Advances in barycentric rational interpolation of a function and its derivatives

Linear barycentric rational interpolants are a particular kind of rational interpolants, defined by weights that are independent of the function f. Such interpolants have recently proved to be a viable alternative to more classical interpolation methods, such as global polynomial interpolants and splines, especially in the equispaced setting. Other kinds of interpolants might indeed suffer from the use of floating point arithmetic, while the particular form of barycentric rational interpolants guarantees that the interpolation of data is achieved even if rounding errors affect the computation of the weights, as long as they are non zero. This dissertation is mainly concerned with the analysis of the convergence of a particular family of barycentric rational interpolants, the so-called Floater-Hormann family. Such functions are based on the blend of local polynomial interpolants of fixed degree d with rational blending functions, and we investigate their behavior in the interpolation of the derivatives of a function f. In the first part we focus on the approximation of the k-th derivative of the function f with classical Floater-Hormann interpolants. We first introduce the Floater-Hormann interpolation scheme and present the main advantages and disadvantages of these functions compared to polynomial and classical rational interpolants. We then proceed by recalling some previous result regarding the convergence rate of the k-th derivatives of these interpolants and extend these results. In particular, we prove that the k-th derivative of the Floater-Hormann interpolant converges to f^(k) at the rate of O(h_j^(d+1-k), for any k >= 0 and any set of well-spaced nodes, where h_j is the local mesh size. In the second part we instead focus on the interpolation of the derivatives of a function up to some order m. We first present several theorems regarding this kind of interpolation, both for polynomials and barycentric rational functions, and then we introduce a new iterative approach that allows us to generalise the Floater-Hormann family to this new setting. The resulting rational Hermite interpolants have numerator and denominator of degree at most (m+1)(n+1)-1 and (m+1) (n-d), respectively, and converge to the function at the rate of O(h^((m+1)(d+1))) as the mesh size h converges to zero. Next, we focus on the conditioning of the interpolants, presenting some classical results regarding polynomials and showing the reasons that make these functions unsuited to fit any kind of equispaced data. We then compare these results with the ones regarding Floater-Hormann interpolants at equispaced nodes, showing again the advantages of this interpolation scheme in this setting. Finally, we extend these conclusions to the Hermite setting, first introducing the generalisation of the results presented for polynomial Lagrange interpolants and then bounding the condition number of our Hermite interpolant at equispaced nodes by a constant independent of n. The comparison between this result and the equivalent for polynomials shows that our barycentric rational interpolants should be in many cases preferred to polynomials

On the Lebesgue constant of barycentric rational Hermite interpolants at equidistant nodes

Barycentric rational Floater–Hormann interpolants compare favourably to classical polynomial interpolants in the case of equidistant nodes, because the Lebesgue constant associated with these interpolants grows logarithmically in this setting, in contrast to the exponential growth experienced by polynomials. In the Hermite setting, in which also the first derivatives of the interpolant are prescribed at the nodes, the same exponential growth has been proven for polynomial interpolants, and the main goal of this paper is to show that much better results can be obtained with a recent generalization of Floater–Hormann interpolants. After summarizing the construction of these barycentric rational Hermite interpolants, we study the behaviour of the corresponding Lebesgue constant and prove that it is bounded from above by a constant. Several numerical examples confirm this result

Synthesis of the trisaccharide outer core fragment of Burkholderia cepacia pv. vietnamiensis lipooligosaccharide

he synthesis of beta-Gal-(1 -> 3)-alpha-GalNAc-(1 -> 3)-beta-GalNAc allyl trisaccharide as the outer core fragment of Burkholderia cepacia pv. vietnamiensis lipooligosaccharide was accomplished through a concise, optimized, multi-step synthesis, having as key steps three glycosylations, that were in-depth studied performing them under several conditions. The target trisaccharide was designed with an allyl aglycone in order to open a future access to the conjugation with an immunogenic protein en route to the development of a synthetic neoglycoconjugate vaccine against this Burkholderia pathogen

Heterogeneous parametric trivariate fillets

Blending and filleting are well established operations in solid modeling and computer-aided geometric design. The creation of a transition surface which smoothly connects the boundary surfaces of two (or more) objects has been extensively investigated. In this work, we introduce several algorithms for the construction of, possibly heterogeneous, trivariate fillets, that support smooth filleting operations between pairs of, possibly heterogeneous, input trivariates. Several construction methods are introduced that employ functional composition algorithms as well as introduce a half Volumetric Boolean sum operation. A volumetric fillet, consisting of one or more tensor product trivariate(s), is fitted to the boundary surfaces of the input. The result smoothly blends between the two inputs, both geometrically and material-wise (properties of arbitrary dimension). The application of encoding heterogeneous material information into the constructed fillet is discussed and examples of all proposed algorithms are presented. (C) 2021 Elsevier B.V. All rights reserved

Acetolysis of 6-deoxysugar disaccharide building-block: exo versus endo activation

Two different protocols for the mild and selective acetolysis of 6-deoxysugar methyl disaccharides under thermodynamic or kinetic control have been developed. The structures of the disaccharides obtained depend on the protocol used and, in the kinetically controlled cases, on the 6-deoxysugar configuration and protecting group pattern too. The behavior of 6-deoxyhexose oligosaccharides of different series (rhamno, quinovo, and fuco) under these two reaction conditions has been studied and rationalized based on the competition between exo versus endo oxygen activation in the acetolysis mechanism

Versatile and self-assembling urea-linked neosaccharides from sugar aminoalcohols

The increasing interest in urea compounds as self-assembling molecules, ion transporters and organocatalysts prompted several efforts towards synthetic urea-linked glycomimetics. In this frame we studied in details a novel two steps dimerization reaction of sugar vicinal aminoalcohol building blocks, opening a synthetic path to a series of urea-linked neosaccharides. Glucosamine neodisaccharide possessing an oxazolidinoneeureaeoxazolidinone system could be transformed into both cyclic and higher linear neosaccharides. Furthermore, a series of six urea-linked glucosamine and galactosamine neodisaccharides was tested for self-assembling properties by measuring NMR spectra at different temperatures and concentrations as well as by gelation of several organic solvent