Applications of Scalable Bases in Kernel Approximation

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1716927Hangelbroek This project concerns research in the area of kernel-based meshless approximation methods, and applications to some large-scale scientific computing problems: numerical solution of partial differential equations (e.g., equations governing fluid flow), tomography (e.g., medical and seismic imaging), and scattered data approximation (modeling of irregularly sampled scientific data). The main focus is on generation and use of scalable bases for kernel spaces; this is a new mathematical tool meant to stabilize and accelerate kernel-based algorithms. Graduate students participate in the work of the project. The classical kernel approach is prized for its ability to provide accurate solutions to computational problems with complicated geometry; in this sense, it is a meshless method, which does not require sampling at uniformly placed sites or the careful construction of triangulations, meshes, or other apparatus. However, it can suffer from instability and heavy computational costs when the underlying problems grow in size. It has been shown that in some cases these drawbacks can be mitigated by construction of scalable bases (as developed by the investigator and collaborators), which can be efficiently generated and lead to stabilization of the underlying calculations. The primary applications considered are threefold. First is the development of an adaptive, meshless method for treating elliptic PDEs. The notion of adaptive refinement of meshes is well understood for classical finite elements; this aspect of the project seeks to use the scalable bases local construction (where basis functions decay at a rate determined by the local density of the scattered centers) to develop an adaptive algorithm where the centers are refined, but no remeshing is needed. Second, the investigator employs kernel-based quadrature, accelerated by using the scalable basis as a preconditioner, to treat tomographic problems. The aim here is to develop parameter selection and error analysis for approximate filtered backprojection of tomographic data acquired from a certain class of phantom images. Third, the investigator develops algorithms for scattered data-fitting that reflect local sampling density of the data. Graduate students participate in the work of the project.