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Dr. Credit King Credit Connection

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Israel Sukhanov
Israel Sukhanov

Numerical And Statistical Methods For Bioengine...



Abstract:Discontinuous Galerkin (DG) methods are specialized finite element methods that utilize discontinuous piecewise polynomial spaces to approximate the solutions of differential equations, with boundary conditions and inter-element continuity weakly imposed through bilinear forms. DG methods have recently gained popularity for many attractive properties. First of all, the methods are locally mass conservative and have small numerical diffusion. In addition, they support general nonconforming spaces including unstructured meshes, nonmatching grids and variable degrees of local approximations, thus allowing efficient h-, p-, and hp-adaptivities. Moreover, DG algorithms treat rough coefficient problems and effectively capture discontinuities in solutions. They have excellent parallel efficiency since data communications are relatively local. For time-dependent problems in particular, their mass matrices are block diagonal, providing a substantial computational advantage if explicit time integrations are used.




Numerical and Statistical Methods for Bioengine...



In this talk, we consider a family of discontinuous Galerkin (DG) applied to reactive transport problems. They are the four primal discontinuous Galerkin schemes for the space discretization: Symmetric Interior Penalty Galerkin (SIPG), Oden-Baumann-Babuska DG formulation (OBB-DG), Nonsymmetric Interior Penalty Galerkin (NIPG) and Incomplete Interior Penalty Galerkin (IIPG) methods. We address a priori error bounds in the L2(H1), L2(L2) and negative norms, and a posteriori error estimates in L2(H1) and L2(L2). Efficient implementation issues are discussed with emphasis on dynamic mesh adaptation strategies. A number of numerical examples are presented to illustrate various features of DG methods including their sharp a posteriori error indicators and effective adaptivities.


Abstract:We describe two regularization techniques based on optimal control for solving two types of ill-posed problems. The two ill-posed problems we consider are in signal processing and parameter identification. Our new mathematical formulations of these ill-posed problems lead to new efficient numerical methods. We compare our numerical results to the same examples using the well-known Tikhonov regularization.


It is the combinatorial properties of the Schur function that ultimately allowed us to develop the first efficient algorithm for computing