Seeking Space Aliens and the Strong Approximation Property: a (Disjoint) Study in Dust Plumes on Planetary Satellites and Nonsymmetric Algebraic Multigrid

PART I: One of the most fascinating questions to humans has long been whether life exists outside of our planet. To our knowledge, water is a fundamental building block of life, which makes liquid water on other bodies in the universe a topic of great interest. In fact, there are large bodies of water right here in our solar system, underneath the icy crust of moons around Saturn and Jupiter. The NASA-ESA Cassini Mission spent two decades studying the Saturnian system. One of the many exciting discoveries was a “plume” on the south pole of Enceladus, emitting hundreds of kg/s of water vapor and frozen water-ice particles from Enceladus’ subsurface ocean. It has since been determined that Enceladus likely has a global liquid water ocean separating its rocky core from icy surface, with conditions that are relatively favorable to support life. The plume is of particular interest because it gives direct access to ocean particles from space, by flying through the plume. Recently, evidence has been found for similar geological activity occurring on Jupiter’s moon Europa, long considered one of the most likely candidate bodies to support life in our solar system. Here, a model for plume-particle dynamics is developed based on studies of the Enceladus plume and data from the Cassini Cosmic Dust Analyzer. A C++, OpenMP/MPI parallel software package is then built to run large scale simulations of dust plumes on planetary satellites. In the case of Enceladus, data from simulations and the Cassini mission provide insight into the structure of emissions on the surface, the total mass production of the plume, and the distribution of particles being emitted. Each of these are fundamental to understanding the plume and, for Europa and Enceladus, simulation data provide important results for the planning of future missions to these icy moons. In particular, this work has contributed to the Europa Clipper mission and proposed Enceladus Life Finder.PART II: Solving large, sparse linear systems arises often in the modeling of biological and physical phenomenon, data analysis through graphs and networks, and other scientific applications. This work focusesprimarily on linear systems resulting from the discretization of partial differential equations (PDEs). Because solving linear systems is the bottleneck of many large simulation codes, there is a rich field of research in developing “fast” solvers, with the ultimate goal being a method that solves an n × n linear system in O(n) operations. One of the most effective classes of solvers is algebraic multigrid (AMG), which is a multilevel iterative method based on projecting the problem into progressively smaller spaces, and scales like O(n) or O(nlogn) for certain classes of problems. The field of AMG is well-developed for symmetric positive definite matrices, and is typically most effective on linear systems resulting from the discretization of scalar elliptic PDEs, such as the heat equation. Systems of PDEs can add additional difficulties, but the underlying linear algebraic theory is consistent and, in many cases, an elliptic system of PDEs can be handled well by AMG with appropriate modifications of the solver. Solving general, nonsymmetric linear systems remains the wild west of AMG (and other fast solvers), lacking significant results in convergence theory as well as robust methods. Here, we develop new theoretical motivation and practical variations of AMG to solve nonsymmetric linear systems, often resulting from the discretization of hyperbolic PDEs. In particular, multilevel convergence of AMG for nonsymmetric systems is proven for the first time. A new nonsymmetric AMG solver is also developed based on an approximate ideal restriction, referred to as AIR, which is able to solve advection-dominated, hyperbolic-type problems that are outside the scope of existing AMG solvers and other fast iterative methods. AIR demonstrate