arXiv subjects

Shixiao Willing Jiang

Publications and source records attributed to Shixiao Willing Jiang.

4 recordsLinked to original sources

arXiv2026

RBF-Generated Finite Difference Method Coupled with Quadratic Programming for Solving PDEs on Surfaces with Derivative Boundary Conditions

Derivative boundary conditions introduce challenges for mesh-free discretizations of PDEs on surfaces, especially when the domain is represented by randomly sampled point clouds. The recently developed two-step tangent-space RBF-generated finite difference (RBF-FD) method provides high accuracy on closed surfaces. However, it may lose stability when applied directly to surface PDEs with derivative boundary conditions. To enhance numerical stability, we develop a mesh-free method that couples the two-step tangent-space RBF-FD discretization with a quadratic programming (QP) procedure to stabilize the operator approximation for interior points near boundaries. For boundary points, we construct restricted nearest-neighbor stencils biased in the co-normal direction and employ a constrained quadratic program to approximate outward co-normal derivatives. The resulting method avoids using ghost points and does not require quasi-uniform node distributions. We validate the approach on elliptic problems, eigenvalue problems, time-dependent diffusion equations, and elliptic interface problems on surfaces with boundary. Numerical experiments demonstrate stable performance and high-order accuracy across a variety of surfaces.

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arXiv2025

Two-step Generalized RBF-Generated Finite Difference Method on Manifolds

Solving partial differential equations (PDEs) on manifolds defined by randomly sampled point clouds is a challenging problem in scientific computing and has broad applications in various fields. In this paper, we develop a two-step generalized radial basis function-generated finite difference (gRBF-FD) method for solving PDEs on manifolds without boundaries, identified by randomly sampled point cloud data. The gRBF-FD is based on polyharmonic spline kernels and multivariate polynomials (PHS+Poly) defined over the tangent space in a local Monge coordinate system. The first step is to regress the local target function using a generalized moving least squares (GMLS) while the second step is to compensate for the residual using a PHS interpolation. Our gRBF-FD method has the same interpolant form with the standard RBF-FD but differs in interpolation coefficients. Our approach utilizes a specific weight function in both the GMLS and PHS steps and implements an automatic tuning strategy for the stencil size K (i.e., the number of nearest neighbors) at each point. These strategies are designed to produce a Laplacian matrix with a specific coefficient structure, thereby enhancing stability and reducing the solution error. We establish an error bound for the operator approximation in terms of the so-called local stencil diameter as well as in terms of the number of data. We further demonstrate the high accuracy of gRBF-FD through numerical tests on various smooth manifolds.

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arXiv2023

Radial basis approximation of tensor fields on manifolds: From operator estimation to manifold learning

In this paper, we study the Radial Basis Function (RBF) approximation to differential operators on smooth tensor fields defined on closed Riemannian submanifolds of Euclidean space, identified by randomly sampled point cloud data. {The formulation in this paper leverages a fundamental fact that the covariant derivative on a submanifold is the projection of the directional derivative in the ambient Euclidean space onto the tangent space of the submanifold. To differentiate a test function (or vector field) on the submanifold with respect to the Euclidean metric, the RBF interpolation is applied to extend the function (or vector field) in the ambient Euclidean space. When the manifolds are unknown, we develop an improved second-order local SVD technique for estimating local tangent spaces on the manifold. When the classical pointwise non-symmetric RBF formulation is used to solve Laplacian eigenvalue problems, we found that while accurate estimation of the leading spectra can be obtained with large enough data, such an approximation often produces irrelevant complex-valued spectra (or pollution) as the true spectra are real-valued and positive. To avoid such an issue,} we introduce a symmetric RBF discrete approximation of the Laplacians induced by a weak formulation on appropriate Hilbert spaces. Unlike the non-symmetric approximation, this formulation guarantees non-negative real-valued spectra and the orthogonality of the eigenvectors. Theoretically, we establish the convergence of the eigenpairs of both the Laplace-Beltrami operator and Bochner Laplacian {for the symmetric formulation} in the limit of large data with convergence rates. Numerically, we provide supporting examples for approximations of the Laplace-Beltrami operator and various vector Laplacians, including the Bochner, Hodge, and Lichnerowicz Laplacians.

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arXiv2021

Kernel-based methods for Solving Time-Dependent Advection-Diffusion Equations on Manifolds

In this paper, we extend the class of kernel methods, the so-called diffusion maps (DM) and ghost point diffusion maps (GPDM), to solve the time-dependent advection-diffusion PDE on unknown smooth manifolds without and with boundaries. The core idea is to directly approximate the spatial components of the differential operator on the manifold with a local integral operator and combine it with the standard implicit time difference scheme. When the manifold has a boundary, a simplified version of the GPDM approach is used to overcome the bias of the integral approximation near the boundary. The Monte-Carlo discretization of the integral operator over the point cloud data gives rise to a mesh-free formulation that is natural for randomly distributed points, even when the manifold is embedded in high-dimensional ambient space. Here, we establish the convergence of the proposed solver on appropriate topologies, depending on the distribution of point cloud data and boundary type. We provide numerical results to validate the convergence results on various examples that involve simple geometry and an unknown manifold. Additionally, we also found positive results in solving the one-dimensional viscous Burger's equation where GPDM is adopted with a pseudo-spectral Galerkin framework to approximate nonlinear advection term.

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