Page 131 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 131
APPROXIMATION THEORY AND APPLICATIONS (MS-78)
sets nearly as good as Fekete points of the domain. Optimal admissible meshes have been con-
structed on many polynomially determining compact sets, e.g., sections of discs, ball, convex
bodies, sets with regular boundary, by different analytical and geometrical techniques. Regard-
ing subsets of algebraic varieties admissible meshes are known only for a few compacts like
sections of a sphere, a torus, a circle and curves in C with analytic parametrization. We con-
struct polynomial weakly admissible meshes on compact subsets of algebraic hypersurfaces in
CN+1. These meshes are optimal in some cases. We present also partial results for algebraic
sets of codimension greater than one.
An iso-geometric radial basis function partition of unity method for PDEs
in thin structures
Elisabeth Larsson, elisabeth.larsson@it.uu.se
Uppsala University, Sweden
Coauthors: Igor Tominec, Ulrika Sundin, Nicola Cacciani, Pierre-Frédéric Villard
The application that motivates this work is numerical simulation of the biomechanics of the
respiratory system. The main respiratory muscle is the diaphragm, which is a thin structure.
There are several challenges associated with the geometry, including its representation. Here,
we first use a radial basis function partition of unity method (RBF-PUM) to make a smooth
reconstruction of the geometry from noisy medical image data. Then we use RBF-PUM to
approximate the solution of a PDE problem posed in this geometry. In a PUM, the global
approximation is expressed as a weighted combination of local approximations over patches
that form a cover of the domain. A particular benefit of RBF-PUM is that we can adapt each
local approximation to the local properties of the problem. For this thin, curved, non-trivial
geometry, we can scale the local problems to ensure sufficient local resolution of the thickness
dimension. We show results for a simple Poisson test problem and show that we can achieve
high-order convergence with an appropriate choice of method parameters.
Bernstein-Chlodovsky operators preserving exponentials
Vita Leonessa, vita.leonessa@unibas.it
University of Basilicata, Italy
The aim of this talk is to illustrate a generalization of Bernstein-Chlodovsky operators, intro-
duced and studied in [6, 4], that preserves the exponential function e−2x (x ≥ 0).
In particular, in [6] we studied its approximation properties in several function spaces, also
evaluating the rate of convergence by means of suitable moduli of continuity. As a consequence,
we proved better error estimation than the original operators on certain intervals.
In [4] we continued the study of such operators by proving some Voronovskaya type the-
orems and deducing saturation results. A comparison of this new class of operators with the
classical Bernstein-Chlodovsky ones is also made, proving that the new operators provide bet-
ter approximation results for certain functions on [0, +∞).
References
[1] T. Acar, M. Cappelletti Montano, P. Garrancho, V. L., On Bernstein-Chlodovsky operators
preserving e−2x, Bull. Belg. Math. Soc. Simon Stevin 26 (2019), no. 5, 681–698.
129
sets nearly as good as Fekete points of the domain. Optimal admissible meshes have been con-
structed on many polynomially determining compact sets, e.g., sections of discs, ball, convex
bodies, sets with regular boundary, by different analytical and geometrical techniques. Regard-
ing subsets of algebraic varieties admissible meshes are known only for a few compacts like
sections of a sphere, a torus, a circle and curves in C with analytic parametrization. We con-
struct polynomial weakly admissible meshes on compact subsets of algebraic hypersurfaces in
CN+1. These meshes are optimal in some cases. We present also partial results for algebraic
sets of codimension greater than one.
An iso-geometric radial basis function partition of unity method for PDEs
in thin structures
Elisabeth Larsson, elisabeth.larsson@it.uu.se
Uppsala University, Sweden
Coauthors: Igor Tominec, Ulrika Sundin, Nicola Cacciani, Pierre-Frédéric Villard
The application that motivates this work is numerical simulation of the biomechanics of the
respiratory system. The main respiratory muscle is the diaphragm, which is a thin structure.
There are several challenges associated with the geometry, including its representation. Here,
we first use a radial basis function partition of unity method (RBF-PUM) to make a smooth
reconstruction of the geometry from noisy medical image data. Then we use RBF-PUM to
approximate the solution of a PDE problem posed in this geometry. In a PUM, the global
approximation is expressed as a weighted combination of local approximations over patches
that form a cover of the domain. A particular benefit of RBF-PUM is that we can adapt each
local approximation to the local properties of the problem. For this thin, curved, non-trivial
geometry, we can scale the local problems to ensure sufficient local resolution of the thickness
dimension. We show results for a simple Poisson test problem and show that we can achieve
high-order convergence with an appropriate choice of method parameters.
Bernstein-Chlodovsky operators preserving exponentials
Vita Leonessa, vita.leonessa@unibas.it
University of Basilicata, Italy
The aim of this talk is to illustrate a generalization of Bernstein-Chlodovsky operators, intro-
duced and studied in [6, 4], that preserves the exponential function e−2x (x ≥ 0).
In particular, in [6] we studied its approximation properties in several function spaces, also
evaluating the rate of convergence by means of suitable moduli of continuity. As a consequence,
we proved better error estimation than the original operators on certain intervals.
In [4] we continued the study of such operators by proving some Voronovskaya type the-
orems and deducing saturation results. A comparison of this new class of operators with the
classical Bernstein-Chlodovsky ones is also made, proving that the new operators provide bet-
ter approximation results for certain functions on [0, +∞).
References
[1] T. Acar, M. Cappelletti Montano, P. Garrancho, V. L., On Bernstein-Chlodovsky operators
preserving e−2x, Bull. Belg. Math. Soc. Simon Stevin 26 (2019), no. 5, 681–698.
129
