A parallel numerical solver using hierarchically tiled arrays

Solving linear systems is an important problem for scientific computing. Exploiting parallelism is essential for solving complex systems, and this traditionally involves writing parallel algorithms on top of a library such as MPI. The SPIKE family of algorithms is one well-known example of a parallel solver for linear systems. The Hierarchically Tiled Array data type extends traditional data-parallel array operations with explicit tiling and allows programmers to directly manipulate tiles. The tiles of the HTA data type map naturally to the block nature of many numeric computations, including the SPIKE family of algorithms. The higher level of abstraction of the HTA enables the same program to be portable across different platforms. Current implementations target both shared-memory and distributed-memory models. In this thesis we present a proof-of-concept for portable linear solvers. We implement two algorithms from the SPIKE family using the HTA library. We show that our implementations of SPIKE exploit the abstractions provided by the HTA to produce a compact, clean code that can run on both shared-memory and distributed-memory models without modification. We discuss how we map the algorithms to HTA programs as well as examine their performance. We compare the performance of our HTA codes to comparable codes written in MPI as well as current state-of-the-art linear algebra routines.

Numerical methods for solar tomography in the STEREO era

In this thesis, we propose several advances in the numerical and computational algorithms that are used to determine tomographic estimates of physical parameters in the solar corona. We focus on methods for both global dynamic estimation of the coronal electron density and estimation of local transient phenomena, such as coronal mass ejections, from empirical observations acquired by instruments onboard the STEREO spacecraft. We present a first look at tomographic reconstructions of the solar corona from multiple points-of-view, which motivates the developments in this thesis. In particular, we propose a method for linear equality constrained state estimation that leads toward more physical global dynamic solar tomography estimates. We also present a formulation of the local static estimation problem, i.e., the tomographic estimation of local events and structures like coronal mass ejections, that couples the tomographic imaging problem to a phase field based level set method. This formulation will render feasible the 3D tomography of coronal mass ejections from limited observations. Finally, we develop a scalable algorithm for ray tracing dense meshes, which allows efficient computation of many of the tomographic projection matrices needed for the applications in this thesis.