The dominant problem in applied mathematics is the application of linear operators and solving linear equations. Dense linear systems arise in numerous applications: Boundary integral equations for elliptic partial differential equations, covariance matrices in statistics, Bayesian inversion, Kalman filtering, inverse problems, radial basis function interpolation, density functional theory, multi-frontal solvers for sparse linear systems, etc. As the problem size increases, large memory requirements scaling as $O(N^2)$ and extensive computational time to perform matrix algebra scaling as $O(N^2)$ or $O(N^3)$ make computations impractical, where $N$ is the underlying number of degrees of freedom. This talk will discuss new fast algorithms that scale as $O(N)$ for a class of problems, given a prescribed tolerance. Applications in the context of Gaussian processes, Integral equations for electromagnetic scattering, Symmetric factorization for Brownian dynamics, Bayesian inversion, Kalman filtering, multi-frontal solvers will also be presented.

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- Seminars for 2014