# Quasi-Interpolation

Quasi-interpolation is one of the most useful and often applied methods for the approximation of functions and data in mathematics and applications. Its advantages are manifold: quasi-interpolants are able to approximate in any number of dimensions, they are efficient and relatively easy to formulate for scattered and meshed nodes and for any number of data. In our numerical analysis group, research into quasi-interpolation with multivariate approximation algorithms provides a field for work of graduate students and contributions from researchers, using all the mathematical background and methods of implementation. The mathematical analysis of quasi-interpolation is studied in three directions, namely on the basis (spline spaces, radial basis functions) from which the approximation is taken, on the form and computation of the quasi-interpolants (point evaluations, averages, least squares), and on the mathematical properties (existence, locality, convergence questions, precision). We study which type of quasi-interpolation to use in different contexts and how to optimise its features to suit applications in medicine, physics and engineering.

# Radial basis functions

It is necessary to estimate parameters by approximation and interpolation in many areas-from computer graphics to inverse methods to signal processing. Radial basis functions are modern, powerful tools which are being used more widely as the limitations of other methods become apparent. In the numerical analysis group we are interested in analyses of radial basic functions from the theoretical and practical implementation viewpoints. This includes in particular the ubiquitous multiquadric function and its generalisations. The particular advantages of radial basis functions lie in their availability in any space dimensions and in their versatility in applications such as solving partial differential equations, neural networks and deep learning. The need for approximations in high dimensions appear in particular in the current interest in big data analysis where multivariate approximation theory tools such as radial basis functions are state-of-the-art algorithms. For applications in PDEs, both collocation and Galerkin approaches are available methods.

# Meshless Finite Difference Methods

Discretization of partial differential equations by point clouds without creating any partitions or networks offers major advantages over mesh-based methods in terms of the computational efficiency and geometric flexibility. We develop meshless methods of this type based on finite difference formulas for irregular points, and apply them to various types of boundary value problems.

# Piecewise polynomial methods

Many successful methods of approximation and numerical simulation are based on piecewise polynomials, both in one and many variables. We develop such algorithms for data fitting with wavelets, numerical solution of partial differential equations with finite elements and splines, as well as explore the theoretical aspects of nonlinear approximation on anisotropic partitions.

# Drittmittel

Unsere Forschung wird durch folgende Drittmittelprojekte gefördert:
• 2023-2025: Auftragsforschung Fraunhofer ITWM „Netzfreie Methoden für partielle Differentialgleichungen“ -- 161.367,- €
• 2023-2025: MSCA4Ukraine „A new approach to optimal numerical differentiation and its application for solving equations of mathematical physics“ -- 173.847,- €
• 2023-2025: MSCA4Ukraine „Modules over group rings of linear groups of finite rank and their applications in Modern Harmonic Analysis“
-- 173.847,- €

# Ausgewählte Puplikationen

• Davydov O. (2023) Error bounds for a least squares meshless finite difference method on closed manifolds. Adv Comput Math 49: 48.
• Davydov O, Oanh DT, Tuong NM. (2023) Improved stencil selection for meshless finite difference methods in 3D. J Comput Appl Math 425: 115031.
• Suchde P, and Jacquemin T, Davydov O. (2022) Point cloud generation for meshfree methods: An overview. Arch Comput Method E
• Davydov O, Kozynenko O, Skorokhodov D. (2020) Optimal approximation order of piecewise constants on convex partitions. J Complexity, 58: 101444.
• Sokolov A, Davydov O, Kuzmin D, Westermann A, Turek S. (2019) A flux-corrected RBF-FD method for convection dominated problems in domains and on manifolds. J Numer Math 27: 253-269.
• Oanh DT, Davydov O, Phu HX. (2017) Adaptive RBF-FD method for elliptic problems with point singularities in 2D. Appl Math Comput 313: 474-497.
• Davydov O, Schaback R. (2016) Error bounds for kernel-based numerical differentiation. Numer Math 132: 243-269.
• Ainsworth M, Davydov O, Schumaker LL. (2016) Bernstein-Bézier finite elements on tetrahedral–hexahedral–pyramidal partitions. Comput Method Appl M 304: 140-170.
• Ainsworth M, Andriamaro G, Davydov O. (2015) A Bernstein–Bézier basis for arbitrary order Raviart–Thomas finite elements. Constr Approx 41: 1-22.