Data-driven quasi-interpolant spline surfaces for point cloud approximation

In this paper we investigate a local surface approximation, the Weighted Quasi Interpolant Spline Approximation (wQISA), specifically designed for large and noisy point clouds. We briefly describe the properties of the wQISA representation and introduce a novel data-driven implementation, which combines prediction capability and complexity efficiency. We provide an extended comparative analysis with other continuous approximations on real data, including different types of surfaces and levels of noise, such as 3D models, terrain data and digital environmental data

On the role of particles and radial basis functions in a finite element level set method for bubble dynamics

The aim of this presentation is to highlight the role that Particle-based simulations and Radial Basis Functions (RBFs) have played in the development of a computationally efficient, level-set, Finite Element method for the simulation of Newtonian and non-Newtonian interface flows. First, we introduce the mathemat- ical formulation and the interface-capturing technique used in the simulation of multiphase flows, underscoring the influence of marker particles on the enhanced definition of the interface. Then, we explore the effect of adding polymer parti- cles to the domain to perform Brownian Dynamics Simulations of polymer flows. Finally, we leverage RBFs to reconstruct, in an almost free-independent way the polymer stress tensor retrieved from the polymer particles. Numerical simulations of pure advection flows and bubble dynamics simulations of complex flows on two-dimensional configurations emphasize the improvements offered by this hybrid, Finite Element/RBF/Particle-based method

Radial Basis Functions Interpolation and Applications: An Incremental Approach

Radial Basis Functions (RBF) interpolation is primarily used for interpolation of scattered data in higher dimensions. The RBF interpolation is a non-separable interpolation which offers a smooth interpolation, generally in n-dimensional space. We present a new method for RBF computation using an incremental approach. The proposed method is especially convenient in cases when larger data sets are randomly updated as the proposed method is of O(N2) computational complexity instead of O(N3) for insert / remove operations only and therefore it is much faster than the standard approach. If t-varying data or vector data are to be interpolated, the proposed method offers a significant speed-up as well

Preconditioning for radial basis function partition of unity methods

Meshfree radial basis function (RBF) methods are of interest for solving partial differential equations due to attractive convergence properties, flexibility with respect to geometry, and ease of implementation. For global RBF methods, the computational cost grows rapidly with dimension and problem size, so localised approaches, such as partition of unity or stencil based RBF methods, are currently being developed. An RBF partition of unity method (RBF--PUM) approximates functions through a combination of local RBF approximations. The linear systems that arise are locally unstructured, but with a global structure due to the partitioning of the domain. Due to the sparsity of the matrices, for large scale problems, iterative solution methods are needed both for computational reasons and to reduce memory requirements. In this paper we implement and test different algebraic preconditioning strategies based on the structure of the matrix in combination with incomplete factorisations. We compare their performance for different orderings and problem settings and find that a no-fill incomplete factorisation of the central band of the original discretisation matrix provides a robust and efficient preconditioner

A Partition of Unity Method for Divergence-Free or Curl-Free Radial Basis Function Approximation

Divergence-free (div-free) and curl-free vector fields are pervasive in many areas of science and engineering, from fluid dynamics to electromagnetism. A common problem that arises in applications is that of constructing smooth approximants to these vector fields and/or their potentials based only on discrete samples. Additionally, it is often necessary that the vector approximants preserve the div-free or curl-free properties of the field to maintain certain physical constraints. Div/curl-free radial basis functions (RBFs) are a particularly good choice for this application as they are meshfree and analytically satisfy the div-free or curl-free property. However, this method can be computationally expensive due to its global nature. In this paper, we develop a technique for bypassing this issue that combines div/curl-free RBFs in a partition of unity framework, where one solves for local approximants over subsets of the global samples and then blends them together to form a div-free or curl-free global approximant. The method is applicable to div/curl-free vector fields in ℝ2 and tangential fields on two-dimensional surfaces, such as the sphere, and the curl-free method can be generalized to vector fields in ℝd. The method also produces an approximant for the scalar potential of the underlying sampled field. We present error estimates and demonstrate the effectiveness of the method on several test problems