Collaborative Research: Minimum Sobolov Norm Methods

合作研究：最小索波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解设计快速而准确的数值算法。特别是，主要关注的是复数域上积分-微分方程的求解以及信号和图像处理问题。这种方法的基础是根据给定的数据，将方程在某一点的解的估计表示为该点的最光滑解(平均)的值。由此得到的离散方程可以显示为具有特殊结构的矩阵，可以利用这些矩阵来创建这些方程的快速解算器。这些方法有两个主要的计算优势。首先，它们可以设计成避免复杂区域的网格化或三角剖分。其次，这些方法表现出局部收敛；即，在某一点上，近似值收敛到解的程度仅取决于解的局部光滑性。这些优点使得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表示一个元素具有S分数导数的Sobolv-Hilbert空间。设HS中的未知函数f满足方程L(F)=g，其中L是线性算子，g是已知函数。设Ln表示Hr上的n个线性泛函，Q表示Hs上的线性泛函。然后，从满足约束Ln(L(P))=Ln(G)的HS中的最小Soblev范数函数p可以求出Q(F)的最佳极大极小估计。这个p可以非常快速地计算，因为最优p是由一组很好的方程给出的，当写成适当的表示时，这些方程具有快速多极子方法(FMM)结构。此外，还可以处理p=1的Lp Sobolev空间。在这些情况下，优化问题更加复杂，可以归结为线性规划问题，为此正在开发利用约束矩阵的基本FMM结构的快速求解器。理论工作包括研究当n变大时解的收敛，以及改进所得到的离散方程的FMM结构。算法的工作包括设计构造FMM表示的快速算法，然后设计这些方程的直接(非迭代)解的快速算法。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，对于二维复数区域上的积分方程组和椭圆型偏微分方程组的求解，也发展了无网格、局部收敛的格式。

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.