Collaborative Research: Minimum Sobolov Norm Methods

合作研究：最小Sobolev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代数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和信号和图像处理问题的解决方案。该方法是基于制定的估计的解决方案的方程在一个点的值的最平滑的解决方案（平均）在该点的基础上给定的数据。由此产生的离散方程可以被证明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，它们可以被设计为避免网格化或三角化的复杂域。其次，这些方法表现出局部收敛性，也就是说，逼近收敛到解的速度只取决于解的局部光滑性。设Hs表示Sobolev Hilbert空间，其元素具有s1阶分数阶导数.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.则q（f）的最佳minmax估计可由满足约束Ln（L（p））= Ln（g）的Hs中的最小Sobolev范数函数p给出。这个p可以非常快速地计算出来，因为最优的p是由一组很好的方程给出的，当用正确的表示法写出来时，这个方程具有快速多极方法（FMM）的结构。同样，也可以使用Lp Sobolev空间，p = 1。在这些情况下，优化问题是更复杂的，可以减少到线性规划问题，快速求解器正在开发，利用底层的FMM结构的约束矩阵。理论工作包括研究当n变大时解的收敛性，以及改进所得离散方程的FMM结构。算法工作包括设计快速算法构造FMM表示，然后设计快速算法直接（非迭代）解决这些方程。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，网格自由，局部收敛的计划正在开发的解决方案的积分方程和椭圆型偏微分方程在复杂的区域在二维。

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.