Collaborative Research: Minimum Sobolov Norm Methods

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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.