Multi-Moment Methods for PDEs and GPUs for Large-Scale Scientific Computations

Sarah Amalie King
North Carolina State University
North Carolina State University, 2012


   title={Multi-Moment Methods for PDEs and GPUs for Large-Scale Scientific Computations},

   author={King, S.A.},



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The scope of this thesis is broad but focuses on developing effective numerical methods and efficient implementations. We investigate numerical solution methods for hyperbolic partial differential equations, numerical optimization methods, and implementation of fast numerical algorithms on graphics processor units (GPUs). For partial differential equations we develop numerical methods for the transport and advection equations. Our method is based on the method of characteristics and multi-moment approximation of functions. At uniform grid points update formulas for solutions and their derivatives are derived for variable wave speed. For numerical optimization we develop a nonsmooth optimization method for solving the elastic contact problem. The Signorini contact problem is a variational problem that minimizes the elastic deformation energy subject to the contact inequality. The Coulomb friction problem is a minimization of the deformable energy at the boundary. We develop a numerical optimization method of the form of Primal-Dual active set methods for Lagrange multiplier methods and semi-smooth newton method for these variational problems. To solve these problems numerically we approximate the variational problems with a multi-moment scheme based on Adini’s elements which involves the use of the function values as well as the gradient values at nodes. These solution methods are then combined to solve the full contact problem. Lastly we develop GPU implementations that combine algorithmic efficiency and computing power. We look at constrained and nonsmooth optimization algorithms as well as an inverse Hamiltonian based Riccati solver. Our efforts enhance numerical optimization and control problems for large-scale scientific systems.
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