A High-Order Radial Basis Function (RBF) Leray Projection Method for the Solution of the Incompressible Unsteady Stokes Equations

A new projection method based on radial basis functions (RBFs) is presented for discretizing the incompressible unsteady Stokes equations in irregular geometries. The novelty of the method comes from the application of a new technique for computing the Leray-Helmholtz projection of a vector field using generalized interpolation with divergence-free and curl-free RBFs. Unlike traditional projection methods, this new method enables matching both tangential and normal components of divergence-free vector fields on the domain boundary. This allows incompressibility of the velocity field to be enforced without any time-splitting or pressure boundary conditions. Spatial derivatives are approximated using collocation with global RBFs so that the method only requires samples of the field at (possibly scattered) nodes over the domain. Numerical results are presented demonstrating high-order convergence in both space (between 5th and 6th order) and time (up to 4th order) for some model problems in two dimensional irregular geometries.Comment: 34 pages, 8 figure

RBF multiscale collocation for second order elliptic boundary value problems

In this paper, we discuss multiscale radial basis function collocation methods for solving elliptic partial differential equations on bounded domains. The approximate solution is constructed in a multi-level fashion, each level using compactly supported radial basis functions of smaller scale on an increasingly fine mesh. On each level, standard symmetric collocation is employed. A convergence theory is given, which builds on recent theoretical advances for multiscale approximation using compactly supported radial basis functions. We are able to show that the convergence is linear in the number of levels. We also discuss the condition numbers of the arising systems and the effect of simple, diagonal preconditioners, now proving rigorously previous numerical observations

Multilevel interpolation of divergence-free vector fields

We introduce a multilevel technique for interpolating scattered data of divergence-free vector fields with the help of matrix-valued compactly supported kernels. The support radius at a given level is linked to the mesh norm of the data set at that level. There are at least three advantages of this method: no grid structure is necessary for the implementation, the multilevel approach is computationally cheaper than solving a large one-shot system and the interpolant is guaranteed to be analytically divergence-free. Furthermore, though we will not pursue this here, our multiscale approach is able to represent multiple scales in the data if present. We will prove convergence of the scheme, stability estimates and give a numerical example

Kernel-based discretisation for solving matrix-valued PDEs

In this paper, we discuss the numerical solution of certain matrix-valued partial differential equations. Such PDEs arise, for example, when constructing a Riemannian contraction metric for a dynamical system given by an autonomous ODE. We develop and analyse a new meshfree discretisation scheme using kernel-based approximation spaces. However, since these pproximation spaces have now to be matrix-valued, the kernels we need to use are fourth order tensors. We will review and extend recent results on even more general reproducing kernel Hilbert spaces. We will then apply this general theory to solve a matrix-valued PDE and derive error estimates for the approximate solution. The paper ends with applications to typical examples from dynamical system

A Radial Basis Function Method for Computing Helmholtz-Hodge Decompositions

A radial basis function (RBF) method based on matrix-valued kernels is presented and analyzed for computing two types of vector decompositions on bounded domains: one where the normal component of the divergence-free part of the field is specified on the boundary, and one where the tangential component of the curl-free part of the field specified. These two decompositions can then be combined to obtain a full Helmholtz-Hodge decomposition of the field, i.e. the sum of divergence-free, curl-free, and harmonic fields. All decompositions are computed from samples of the field at (possibly scattered) nodes over the domain, and all boundary conditions are imposed on the vector fields, not their potentials, distinguishing this technique from many current methods. Sobolev-type error estimates for the various decompositions are provided and demonstrated with numerical examples

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

Refinement of Operator-valued Reproducing Kernels

This paper studies the construction of a refinement kernel for a given operator-valued reproducing kernel such that the vector-valued reproducing kernel Hilbert space of the refinement kernel contains that of the given one as a subspace. The study is motivated from the need of updating the current operator-valued reproducing kernel in multi-task learning when underfitting or overfitting occurs. Numerical simulations confirm that the established refinement kernel method is able to meet this need. Various characterizations are provided based on feature maps and vector-valued integral representations of operator-valued reproducing kernels. Concrete examples of refining translation invariant and finite Hilbert-Schmidt operator-valued reproducing kernels are provided. Other examples include refinement of Hessian of scalar-valued translation-invariant kernels and transformation kernels. Existence and properties of operator-valued reproducing kernels preserved during the refinement process are also investigated