Collaborative Research: Minimum Sobolev Norm Methods

合作研究：最小索博列夫范数方法本研究项目的目的是为工程和科学中出现的大类数学方程的求解设计快速准确的数值算法。特别是，主要关注的是复杂域上的积分微分方程的解以及信号和图像处理问题。该方法基于将方程在某一点的解的估计表述为基于给定数据的该点最平滑解（平均）的值。由此产生的离散方程可以显示为具有特殊结构的矩阵，可以利用它来创建这些方程的快速求解器。由此产生的方法有两个主要的计算优势。首先，它们可以设计成避免网格化或三角化的复杂域。其次，这些方法具有局部收敛性；也就是说，近似收敛于某一点解的速率仅取决于解的局部平滑性。这些优点使该方法能够相对容易地处理具有复杂奇异结构的方程。设Hs表示一个索博列夫希尔伯特空间它的元素有s阶导数。假设h中的一个未知函数f满足方程L(f) = g，其中L是一个线性算子，g是一个已知函数。设Ln表示Hr上的n个线性泛函。设q表示Hs上的线性泛函。然后由Hs中满足约束Ln(L(p)) = Ln(g)的最小Sobolev范数函数p计算出q(f)的最佳最小估计。这个p可以非常快速地计算出来，因为最优p是由一组具有快速多极方法（FMM）结构的方程给出的。同样，p = 1的Lp Sobolev空间也可以工作。在这些情况下，优化问题更加复杂，可以简化为线性规划问题，因此正在开发利用约束矩阵的底层fmm结构的快速求解器。理论工作包括研究n变大时解的收敛性，以及改进得到的离散方程的FMM结构。算法工作包括设计快速算法来构造FMM表示，然后为这些方程的直接（非迭代）解设计快速算法。应用工作包括将这些思想应用于图像分割和多速率信号处理。此外，对于二维复域上的积分方程和椭圆型偏微分方程的解，也开发了无网格的局部收敛格式。

Collaborative Research: Minimum Sobolev Norm MethodsThe aim of this research project is to design fast and accuratenumerical algorithms for the solution of large classes of mathematicalequations that arise in engineering and science. In particular, themain concerns are the solution of integro-differential equations oncomplex domains and of signal and image processing problems. Theapproach is based on formulating the estimate of the solution of theequation at a point as the value of the smoothest solution (onaverage) at that point based on the given data. The resulting discreteequations can be shown to have specially structured matrices, whichcan be exploited to create fast solvers for these equations. Theresulting methods have two main computational advantages. First, theycan be designed to avoid gridding or triangulation of the complexdomain. Second, these methods exhibit local convergence; that is, therate at which the approximant converges to the solution at a pointdepends only on the local smoothness of the solution. These advantagesenable the method to tackle equations with complicated singularitystructures with relative ease.Let Hs denote a Sobolev Hilbert space whose elements have s 1fractional derivatives. Suppose an unknown function f in Hs satisfiesthe equation L(F) = g, where L is a linear operator and g is a knownfunction. Let Ln denote n linear functionals on Hr. Let q denote alinear functional on Hs. Then the best minmax estimate for q(f) can becomputed from the minimum Sobolev norm function p in Hs that satisfiesthe constraints Ln(L(p)) = Ln(g). This p can be computed very rapidlysince the optimal p is given by a nice set of equations that has FastMultipole Method (FMM) structure when written in the properrepresentation. Also, it is possible to work with Lp Sobolev spaceswith p = 1. In these cases the optimization problem is morecomplicated and can be reduced to linear programming problems, forwhich fast solvers are being developed that exploit the underlying FMMstructure of the constraint matrix. The theoretical work consists ofstudying the convergence of the solution as n gets bigger, and also inproving the FMM structure of the resulting discrete equations. Thealgorithmic work consists of designing fast algorithms forconstructing the FMM representation and then designing fast algorithmsfor the direct (non-iterative) solution of these equations. Theapplication work consists of applying these ideas to imagesegmentation and multi-rate signal processing. Also, mesh free,locally convergent schemes are being developed for the solution ofintegral equations and elliptic partial differential equations oncomplex domains in two dimensions.