Extending Data to Improve Stability and Error Estimates Using Asymmetric Kansa-like Methods to Solve PDEs

ArXiv.org · 2025-07-20

In this paper, a theoretical framework is presented for the use of a Kansa-like method to numerically solve elliptic partial differential equations on spheres and other manifolds. The theory addresses both the stability of the method and provides error estimates for two different approximation methods. A Kansa-like matrix is obtained by replacing the test point set $X$, used in the traditional Kansa method, by a larger set $Y$, which is a norming set for the underlying trial space. This gives rise to a rectangular matrix. In addition, if a basis of Lagrange (or local Lagrange) functions is used for the trial space, then it is shown that the stability of the matrix is comparable to the stability of the elliptic operator acting on the trial space. Finally, two different types of error estimates are given. Discrete least squares estimates of very high accuracy are obtained for solutions that are sufficiently smooth. The second method, giving similar error estimates, uses a rank revealing factorization to create a ``thinning algorithm'' that reduces $\#Y$ to $\#X$. In practice, this algorithm doesn't need $Y$ to be a norming set.

Highly localized RBF Lagrange functions for finite difference methods on spheres

BIT Numerical Mathematics · 2024-03-15 · 2 citations

Norming Sets and Spherical Cubature Formulas

Advances in Computational Mathematics · 2023 · 8 citations

• Mathematics
• Medicine
• Orthodontics

We investigate the construction of cubature formulas for the unit sphere in https://www.w3.org/1998/Math/MathML" display="inline"> R n https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419839/4baaeab4-d6e3-45ed-9dec-1deb90c341dd/content/inline-math373.tif"/> that have almost equal weights. The corresponding knots are taken from equidistributed point sets on the sphere. The notion of norming sets in connection with the Markov inequality of spherical harmonics is used in order to provide a general result on uniformly stable cubature formulas. We also present some numerical evidence that there exist stable and almost equally-weighted cubature formulas, if the number of knots is slightly larger than required by the exactness conditions for spherical harmonics of a certain degree.

Highly Localized RBF Lagrange Functions for Finite Difference Methods on Spheres

arXiv (Cornell University) · 2023-02-16

The aim of this paper is to show how rapidly decaying RBF Lagrange functions on the spheres can be used to create effective, stable finite difference methods based on radial basis functions (RBF-FD). For certain classes of PDEs this approach leads to precise convergence estimates for stencils which grow moderately with increasing discretization fineness.

Meshfree Extrapolation with Application to Enhanced Near-Boundary Approximation with Local Lagrange Kernels

Foundations of Computational Mathematics · 2021 · 2 citations

• Computer Science
• Mathematics
• Applied mathematics

Locally supported, quasi-interpolatory bases on graphs.

arXiv (Cornell University) · 2021-01-06

Lagrange functions are localized bases that have many applications in signal processing and data approximation. Their structure and fast decay make them excellent tools for constructing approximations. Here, we propose perturbations of Lagrange functions on graphs that maintain the nice properties of Lagrange functions while also having the added benefit of being locally supported. Moreover, their local construction means that they can be computed in parallel, and they are easily implemented via quasi-interpolation.

Interpolating splines on graphs for data science applications

Applied and Computational Harmonic Analysis · 2020-06-04

A high-order meshless Galerkin method for semilinear parabolic equations on spheres

Numerische Mathematik · 2019-01-23 · 13 citations

Direct and Inverse Results on Bounded Domains for Meshless Methods via Localized Bases on Manifolds

2018-01-01 · 16 citations

A meshless Galerkin method for non-local diffusion using localized kernel bases

Mathematics of Computation · 2017-11-22

We introduce a meshless method for solving both continuous and discrete variational formulations of a volume constrained, non-local diffusion problem. We use the discrete solution to approximate the continuous solution. Our method is non-conforming and uses a localized Lagrange basis that is constructed out of radial basis functions. By verifying that certain inf-sup conditions hold, we demonstrate that both the continuous and discrete problems are well-posed, and also present numerical and theoretical results for the convergence behavior of the method. The stiffness matrix is assembled by a special quadrature routine unique to the localized basis. Combining the quadrature method with the localized basis produces a well-conditioned, symmetric matrix. This then is used to find the discretized solution.