Pseudospectral Methods and Radial Basis Functions

伪谱（PS）方法和径向基函数（RBFs）都是在20世纪70年代早期分别与湍流模拟和制图学有关而首次提出的。只有在过去的几年中，这两个主题开始合并，在偏微分方程（PDE）的高精度解决方案的领域。PS方法的上级性能在早期的应用中得到了证明，例如在非常简单的几何形状中的波型方程的精确长期积分。与通常的谱基函数不同，不同的径向基函数的区别并不在于它们振荡的速度，而是在于它们是一个单一函数的不同平移。虽然失去了某些正交性，但谱精度不仅保持不变，而且现在还可以在具有任意节点分布的复杂域上达到。一个新的（和违反直觉的）结果告诉，在RBFs变得越来越平坦的限制，经典的PS方法恢复。最近取得了很大的进展，在克服高的计算机成本和数值病态，以前被认为是严重限制这种方法。上面的观察指向几乎所有数值方法的扩展和推广，传统上，这些方法是基于多项式的。挑战已经从探索RBFs的潜力为任意几何谱方法的偏微分方程，开发它。在科学，工程，科学和社会的大多数现象可以近似的一些种类的数学模型。这些通常涉及一个被称为偏微分方程（PDE）的结构。这类方程只有在极少情况下才能不借助计算方法来求解。在上个世纪，发展了几类主要的求解方法。一个相当新的类别-伪谱（PS）方法-出现在大约30年前。大约在同一时间，径向基函数（RBFs）是在一个完全不同的背景下发明的。后者仍然只是探索性地用于PDE。然而，在过去的三年期间，PI表明，PS方法可以被视为RBFs的一个特例，后者的方法大大扩展了PS方法的范围和通用性。继续研究将扩大这一基本结果，与生产实用的算法，能够成为常规工具的高精度解决方案的偏微分方程在不规则的多维域的长期目标。

Both pseudospectral (PS) methods and radial basis functions (RBFs) were first proposed in the early 1970's, in connection with turbulence modeling and cartography respectively. Only in the last few years have the two topics begun to merge, in the area of high-accuracy solutions of partial differential equations (PDEs). Superior performance of PS methods was demonstrated early on in applications such as accurate long term integrations of wave- type equations in very simple geometries. In contrast to usual spectral basis functions, different RBFs are not distinguished by how rapidly they oscillate, but instead by being different translates of one single function. Although certain orthogonalities are lost, spectral accuracy not only remains, but can now also be reached on complex domains with arbitrary node distributions. A new (and counterintuitive) result tells that, in the limit of RBFs becoming increasingly flat, the classical PS methods are recovered. Recently great progress has been made in overcoming the high computer cost and numerical ill-conditioning that earlier were thought to severely limit this approach. The observations above point towards extensions and generalizations of almost all numerical methods which, by tradition, have been polynomial-based. The challenge has changed from exploring the potential of RBFs for arbitrary-geometry spectral methods for PDEs, into exploiting it.Most phenomena in science, engineering, sciences, and society can be approximated by some kinds of mathematical models. These often involve a construct known as partial differential equations (PDEs). Equations of this kind can only rarely be solved without resorting to computational methods. During the last century, a few main classes of such solution methods have evolved. A quite new class - pseudospectral (PS) methods - emerged some 30 years ago. Around the same time, radial basis functions (RBFs) were invented in a quite different context. The latter have still seen only exploratory use for PDEs. However, during the last three-year period, the PI demonstrated that PS methods can be seen as a special case of RBFs, with the latter approach greatly extending the PS methods' scope and generality. Continuing research will expand on this fundamental result, with the long-term goal of producing practical algorithms which are capable of becoming routine tools for high-precision solutions of PDEs in irregular multidimensional domains.