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Register a new userA Tchebycheffian extension of multi-degree B-splines: Algorithmic computation and properties
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Added by
Deepesh Toshniwal
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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In this paper we present an efficient and robust approach to compute a normalized B-spline-like basis for spline spaces with pieces drawn from extended Tchebycheff spaces. The extended Tchebycheff spaces and their dimensions are allowed to change from interval to interval. The approach works by constructing a matrix that maps a generalized Bernstein-like basis to the B-spline-like basis of interest. The B-spline-like basis shares many characterizing properties with classical univariate B-splines and may easily be incorporated in existing spline codes. This may contribute to the full exploitation of Tchebycheffian splines in applications, freeing them from the restricted role of an elegant theoretical extension of polynomial splines. Numerical examples are provided that illustrate the procedure described.
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In mathematics there is a wide class of knot invariants that may be expressed in the form of multiple line integrals computed along the trajectory C describing the spatial conformation of the knot. In this work it is addressed the problem of evaluating invariants of this kind in the case in which the knot is discrete, i.e. its trajectory is constructed by joining together a set of segments of constant length. Such discrete knots appear almost everywhere in numerical simulations of systems containing one dimensional ring-shaped objects. Examples are polymers, the vortex lines in fluids and superfluids like helium and other quantum liquids. Formally, the trajectory of a discrete knot is a piecewise smooth curve characterized by sharp corners at the joints between contiguous segments. The presence of these corners spoils the topological invariance of the knot invariants considered here and prevents the correct evaluation of their values. To solve this problem, a smoothing procedure is presented, which eliminates the sharp corners and transforms the original path C into a curve that is everywhere differentiable. The procedure is quite general and can be applied to any discrete knot defined off or on lattice. This smoothing algorithm is applied to the computation of the Vassiliev knot invariant of degree 2 denoted here with the symbol r(C). This is the simplest knot invariant that admits a definition in terms of multiple line integrals. For a fast derivation of r(C), it is used a Monte Carlo integration technique. It is shown that, after the smoothing, the values of r(C) may be evaluated with an arbitrary precision. Several algorithms for the fast computation of the Vassiliev knot invariant of degree 2 are provided.
We introduce intrinsic interpolatory bases for data structured on graphs and derive properties of those bases. Polyharmonic Lagrange functions are shown to satisfy exponential decay away from their centers. The decay depends on the density of the zeros of the Lagrange function, showing that they scale with the density of the data. These results indicate that Lagrange-type bases are ideal building blocks for analyzing data on graphs, and we illustrate their use in kernel-based machine learning applications.
In this paper, we demonstrate the construction of generalized Rough Polyhamronic Splines (GRPS) within the Bayesian framework, in particular, for multiscale PDEs with rough coefficients. The optimal coarse basis can be derived automatically by the randomization of the original PDEs with a proper prior distribution and the conditional expectation given partial information on edge or derivative measurements. We prove the (quasi)-optimal localization and approximation properties of the obtained bases, and justify the theoretical results with numerical experiments.
In this paper we derive stability estimates in $L^{2}$- and $L^{infty}$- based Sobolev spaces for the $L^{2}$ projection and a family of quasiinterolants in the space of smooth, 1-periodic, polynomial splines defined on a uniform mesh in $[0,1]$. As a result of the assumed periodicity and the uniform mesh, cyclic matrix techniques and suitable decay estimates of the elements of the inverse of a Gram matrix associated with the standard basis of the space of splines, are used to establish the stability results.
We consider the uniqueness of solution (i.e., nonsingularity) of systems of $r$ generalized Sylvester and $star$-Sylvester equations with $ntimes n$ coefficients. After several reductions, we show that it is sufficient to analyze periodic systems having, at most, one generalized $star$-Sylvester equation. We provide characterizations for the nonsingularity in terms of spectral properties of either matrix pencils or formal matrix products, both constructed from the coefficients of the system. The proposed approach uses the periodic Schur decomposition, and leads to a backward stable $O(n^3r)$ algorithm for computing the (unique) solution.
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