径向基函数多尺度配点算法研究

径向基函数配点方法在实际数值计算中具有精度高、不受问题维数限制等优势，但往往带来高度病态的离散代数系统。为了处理该离散方法中逼近精度与数值稳定性之间的矛盾(即trade-off principle)，本研究将提出一类多尺度配点算法。类似于有限元空间的多水平分裂，我们将分裂再生核Hilbert空间（即径向基函数的本性空间），从而得到一族嵌套的等级径向基函数空间。将其用于散乱数据插值时，全局试探空间上的插值误差将被给出，而且研究将给出等级径向基函数离散矩阵的条件数；将其用于求解适定的线性与非线性椭圆偏微分方程时，研究将给出强配点方法的稳定性条件，以及Sobolev模下的误差估计。该研究将加速径向基函数无网格离散系统的代数求解，从而提升用径向基函数数值求解偏微分方程的计算效率。该研究也将为后续研究无网格局部Petrov-Galerkin方法的收敛性以及发展稠密矩阵迭代技术等提供重要理论基础。

Radial basis functions collocation methods allow the easy construction of approximation spaces in arbitrary dimensions with highly accurate solutions. But unfortunately, we are facing severely ill-conditioned algebraic systems when using radial basis functions approximation. To deal with the contradiction between approximation accuracy and numerical stability (or briefly, trade-off principle), the research will derive a class of multiscale collocation algorithms. Similar to multilevel splitting of finite element spaces, we will obtain a nested hierarchical radial basis functions spaces by splitting the reproducing kernel Hilbert spaces (native spaces of radial basis functions). The interpolation errors in the global trial spaces will be given and the condition numbers of the discrete matrices produced by hierarchical radial basis functions discretization will be estimated, when solving scattered data interpolation problems by the newly-developed functions spaces. The research will provide the stability conditions and the error estimates under Sobolev norms of the strongly collocation methods , when solving linear and nonlinear elliptic partial differential equations by the same trial spaces. The research will speed up the solution of discrete system produced by the radial basis functions meshfree methods, thereby enhance the computational efficiency for solving partial differential equations by radial basis functions. The research also will provide the future works including the convergence analysis of meshfree local Petrov-Galerkin method and the iterative techniques of dense matrices with the important theoretical foundation.