Supplementary data of my PhD thesis "Multilevel sparse grid Lagrange collocation method with positive definite radial basis functions for the solution of elliptic boundary value problems."

We introduce the Lagrange collocation method with radial basis functions (LRBF) as a novel approach to solving $n$-dimensional PDEs. Our method addresses the fundamental challenge -- the `trade-off principle' -- often associated with standard RBF collocation methods. We aim to maintain the accuracy and convergence of the numerical solution while improving stability and efficiency.
In 1D, we establish its existence and uniqueness for specific differential operators, including the Laplacian operator, and positive definite RBFs. In 2D, we provide some theoretical proofs and validate our claims through numerical experiments.
A pivotal innovation is to perturb the primary matrix, thus defining the perturbed LRBF method (PLRBF). This perturbation enables Cholesky decomposition, reducing the condition number to its square root, leading to the CPLRBF. This allows us to select larger shape parameters without compromising stability and accuracy. Consequently, we achieve highly accurate solutions early in the process, thus saving time. Surprisingly, the stability of CPLRBF in 2D is not solely influenced by the main matrix but also affected by a second matrix that does not play a key role in 1D. For sufficiently large shape parameters, this second matrix dominates the overall stability, posing limitations to be addressed in future work.
To address stagnation issues arising from using an adaptive shape parameter inversely proportional to the node spacing, we combine PLRBF/CPLRBF with multilevel techniques, resulting in the MuPLRBF/\allowbreak MuCPLRBF. Our findings reveal that MuCPLRBF significantly enhances accuracy early in the process.
In this context, the Multilevel Sparse Grid method yields improvement in terms of accuracy and efficiency for the Gaussian. However, no accuracy improvement was observed for IMQ when using a sparse grid.
Through a series of numerical experiments, we underscore the effectiveness of MuCPLRBF in achieving stability, accuracy, and efficiency in 1D and 2D.
Excel files containing results from testing of our method (MuCPLRBF) for numerically approximating solutions of partial differential equations against other methods using RBF , as discussed in my thesis, with data about their performance across a range of test cases and performance metrics.

本文提出一种带径向基函数的拉格朗日配点法（Lagrange collocation method with radial basis functions，LRBF），用于求解n维偏微分方程。该方法解决了标准径向基函数配点法中常见的核心难题——“权衡原则”，旨在保证数值解精度与收敛性的同时，提升求解的稳定性与计算效率。在一维场景下，针对拉普拉斯算子等特定微分算子以及正定径向基函数，我们证明了该方法解的存在性与唯一性；在二维场景下，我们给出了部分理论证明，并通过数值实验验证了所提方法的有效性。本方法的核心创新在于对主矩阵进行扰动，由此定义了扰动型LRBF方法（PLRBF）。该扰动使得主矩阵可进行乔列斯基分解，将矩阵条件数降至原有的平方根级别，进而得到CPLRBF方法。这一特性使得我们可以选用更大的形状参数，同时不会牺牲求解的稳定性与精度，由此可在求解前期快速获得高精度数值解，节省计算耗时。值得注意的是，二维场景下CPLRBF的稳定性并非仅由主矩阵决定，同时还受另一矩阵的影响——该矩阵在一维场景中并不会起到关键作用。当形状参数足够大时，该次要矩阵将主导整体稳定性，这一局限有待后续研究加以解决。针对因使用与节点间距成反比的自适应形状参数而引发的停滞问题，我们将PLRBF/CPLRBF与多层级技术相结合，由此得到MuPLRBF/MuCPLRBF方法。研究结果表明，MuCPLRBF可在求解前期显著提升数值解精度。在此框架下，多层级稀疏网格法对高斯径向基函数可实现精度与效率的双重提升；但对于逆多二次（Inverse Multiquadric，IMQ）径向基函数，采用稀疏网格并未带来精度提升。我们通过一系列数值实验，验证了MuCPLRBF在一维与二维场景下均可实现稳定性、精度与计算效率的协同优化。本数据集为Excel文件格式，收录了本文所提MuCPLRBF方法求解偏微分方程数值解的测试结果，并与其他基于径向基函数的方法进行了对比，相关内容已在我的学位论文中详细阐述；数据集涵盖了多种测试案例下各方法的性能指标与运行数据。