A study on spline quasi-interpolation based quadrature rules for the isogeometric Galerkin BEM

Two recently introduced quadrature schemes for weakly singular integrals [Calabr\`o et al. J. Comput. Appl. Math. 2018] are investigated in the context of boundary integral equations arising in the isogeometric formulation of Galerkin Boundary Element Method (BEM). In the first scheme, the regular part of the integrand is approximated by a suitable quasi--interpolation spline. In the second scheme the regular part is approximated by a product of two spline functions. The two schemes are tested and compared against other standard and novel methods available in literature to evaluate different types of integrals arising in the Galerkin formulation. Numerical tests reveal that under reasonable assumptions the second scheme convergences with the optimal order in the Galerkin method, when performing $h$-refinement, even with a small amount of quadrature nodes. The quadrature schemes are validated also in numerical examples to solve 2D Laplace problems with Dirichlet boundary conditions

The Sensitivity of a Spline Function to Perturbations of the Knots

In this paper we study the sensitivity of a spline function,r ep r sented in ter s of Bsplines, to per tur( tions of the knots. We do this by bounding the di#er ence between a pr imar spline, and a secondar spline with the same B-spline coe#cients, but di#erR t knots. We give a number of bounds for this di#er ence, both local bounds and global bounds in gener l L p -spaces. All the bounds ar e based on a simple identity for divided di#er nces. AMS subject classificat on: 41A15. Key words: B-splines, divided di#er2 ces, per u r ation of knots. 1I ntroduction. All floatingp oint comp#fiE tions are infested with round-o# error, and comp #F) tions with sp line functions are noexcep#+F n. To limit the round-o# error insp#3Ffi comp#3C tions it isimp ortant to choose a basis with the p# op erty that smallp erturbations of the coe#cients lead to a smallp erturbation of the sp line rep#C)fiI ted by the coe#cients. TheB-sp#3)+ basis is widely recognized as satisfying this criterion, in short it is said to be well conditioned. In thisp ap er we investigate the conditioning of theB-sp line basis from a di#erentp oint of view, namely its sensitivity top erturbations of the knots. Simp#E stated, the conclusion is that the B-sp#)(E basis is also well conditioned with resp ect to the knots, under mild conditions on thep erturbations. The motivation for this investigation isp ractical. In some constructions of sp lines the knots result from floatingp oint comp#Efi tions, and because of roundo # error the se p# ration between some of the knots can be very small. Since some sp line comp#fiO tions can be troublesome when the knot sp acing is highly nonuniform it is thentemp#fifi( to let knots that are very close coalesce, or move # Received Augus t 1998. + The authors would lik..

Spline Wavelets 1 Theory and Algorithms for Non-Uniform Spline Wavelets

Abstract. We investigate mutually orthogonal spline wavelet spaces on non-uniform partitions of a bounded interval, addressing the existence, uniqueness and construction of bases of minimally supported spline wavelets. The relevant algorithms for decomposition and reconstruction are considered as well as some stability-related questions. In addition, we briefly review the bivariate case for tensor products and arbitrary triangulations. We conclude the paper with a discussion of some special cases. Splines have become the standard mathematical tool for representing smooth shapes in computer graphics and geometric modelling. Wavelets have been introduced more recently, but are by now well established both in mathematics and in applied sciences like signal processing and numerical analysis. The two concepts are closely related as spline

Basic image features (BIFs) arising from approximate symmetry type

We consider detection of local image symmetry using linear filters. We prove a simple criterion for determining if a filter is sensitive to a group of symmetries. We show that derivative-of-Gaussian (DtG) filters are excellent at detecting local image symmetry. Building on this, we propose a very simple algorithm that, based on the responses of a bank of six DtG filters, classifies each location of an image into one of seven Basic Image Features (BIFs). This effectively and efficiently realizes Marr's proposal for an image primal sketch. We summarize results on the use of BIFs for texture classification, object category detection, and pixel classification