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Full-Text Articles in Physical Sciences and Mathematics
Numerical Methods For Optimal Transport And Optimal Information Transport On The Sphere, Axel G. R. Turnquist
Numerical Methods For Optimal Transport And Optimal Information Transport On The Sphere, Axel G. R. Turnquist
Dissertations
The primary contribution of this dissertation is in developing and analyzing efficient, provably convergent numerical schemes for solving fully nonlinear elliptic partial differential equation arising from Optimal Transport on the sphere, and then applying and adapting the methods to two specific engineering applications: the reflector antenna problem and the moving mesh methods problem. For these types of nonlinear partial differential equations, many numerical studies have been done in recent years, the vast majority in subsets of Euclidean space. In this dissertation, the first major goal is to develop convergent schemes for the sphere. However, another goal of this dissertation is …
Periodic Fast Multipole Method, Ruqi Pei
Periodic Fast Multipole Method, Ruqi Pei
Dissertations
Applications in electrostatics, magnetostatics, fluid mechanics, and elasticity often involve sources contained in a unit cell C, centered at the origin, on which periodic boundary condition are imposed. The free-space Green’s functions for many classical partial differential equations (PDE), such as the modified Helmholtz equation, are well-known. Among the existing schemes for imposing the periodicity, three common approaches are: direct discretization of the governing PDE including boundary conditions to yield a large sparse linear system of equations, spectral methods which solve the governing PDE using Fourier analysis, and the method of images based on tiling the plane with copies of …
Nystrom Methods For High-Order Cq Solutions Of The Wave Equation In Two Dimensions, Erli Wind-Andersen
Nystrom Methods For High-Order Cq Solutions Of The Wave Equation In Two Dimensions, Erli Wind-Andersen
Dissertations
An investigation of high order Convolution Quadratures (CQ) methods for the solution of the wave equation in unbounded domains in two dimensions is presented. These rely on Nystrom discretizations for the solution of the ensemble of associated Laplace domain modified Helmholtz problems. Two classes of CQ discretizations are considered: one based on linear multistep methods and the other based on Runge-Kutta methods. Both are used in conjunction with Nystrom discretizations based on Alpert and QBX quadratures of Boundary Integral Equation (BIE) formulations of the Laplace domain Helmholtz problems with complex wavenumbers. CQ in conjunction with BIE is an excellent candidate …
Eigenvalue Problems For Fully Nonlinear Elliptic Partial Differential Equations With Transport Boundary Conditions, Jacob Lesniewski
Eigenvalue Problems For Fully Nonlinear Elliptic Partial Differential Equations With Transport Boundary Conditions, Jacob Lesniewski
Dissertations
Fully nonlinear elliptic partial differential equations (PDEs) arise in a number of applications. From mathematical finance to astrophysics, there is a great deal of interest in solving them. Eigenvalue problems for fully nonlinear PDEs with transport boundary conditions are of particular interest as alternative formulations of PDEs that require data to satisfy a solvability condition, which may not be known explicitly or may be polluted by noisy data. Nevertheless, these have not yet been well-explored in the literature. In this dissertation, a convergence framework for numerically solving eigenvalue problems for fully nonlinear PDEs is introduced. In addition, existing two-dimensional methods …