Seeking Space Aliens and the Strong Approximation Property

221 Pages·2017·12.41 MB·English

University of Colorado, Boulder
CU Scholar
Applied Mathematics Graduate Theses &
Applied Mathematics
Dissertations
Spring 1-1-2017
Seeking Space Aliens and the Strong
Approximation Property: a (Disjoint) Study in
Dust Plumes on Planetary Satellites and
Nonsymmetric Algebraic Multigrid
Benjamin Scott Southworth
University of Colorado at Boulder, [email protected]
Follow this and additional works at:https://scholar.colorado.edu/appm_gradetds
Part of theApplied Mathematics Commons, and thePhysics Commons
Recommended Citation
Southworth, Benjamin Scott, "Seeking Space Aliens and the Strong Approximation Property: a (Disjoint) Study in Dust Plumes on
Planetary Satellites and Nonsymmetric Algebraic Multigrid" (2017).Applied Mathematics Graduate Theses & Dissertations. 99.
https://scholar.colorado.edu/appm_gradetds/99
This Dissertation is brought to you for free and open access by Applied Mathematics at CU Scholar. It has been accepted for inclusion in Applied
Mathematics Graduate Theses & Dissertations by an authorized administrator of CU Scholar. For more information, please contact
[email protected].
Seeking Space Aliens and the Strong Approximation
Property: A (disjoint) Study in Dust Plumes on Planetary
Satellites and Nonsymmetric Algebraic Multigrid
by
Benjamin Scott Southworth
B.A. Mathematics, Dartmouth College, 2013
M.S. Applied Mathematics, University of Colorado, 2015
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Department of Applied Mathematics
2017
This thesis entitled:
Seeking Space Aliens and the Strong Approximation Property: A (disjoint) Study in Dust Plumes on
Planetary Satellites and Nonsymmetric Algebraic Multigrid
written by Benjamin Scott Southworth
has been approved for the Department of Applied Mathematics
Professor Thomas Manteuffel
Professor Sascha Kempf
Professor Steve McCormick
Dr. Jacob Schroder
Dr. John Ruge
Professor Stephen Becker
Date
The final copy of this thesis has been examined by the signatories, and we find that both the content and
the form meet acceptable presentation standards of scholarly work in the above mentioned discipline.
Southworth, Benjamin Scott (Ph.D., Applied Mathematics)
Seeking Space Aliens and the Strong Approximation Property: A (disjoint) Study in Dust Plumes on Plan-
etary Satellites and Nonsymmetric Algebraic Multigrid
Thesis directed by Professor Thomas Manteuffel and Professor Sascha Kempf
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
bodiesintheuniverseatopicofgreatinterest. Infact, therearelargebodiesofwaterrighthereinoursolar
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
massproductionoftheplume,andthedistributionofparticlesbeingemitted. Eachofthesearefundamental
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 phe-
nomenon, data analysis through graphs and networks, and other scientific applications. This work focuses
iv
primarily on linear systems resulting from the discretization of partial differential equations (PDEs). Be-
cause 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 demonstrates scalable convergence on unstructured
meshes, in multiple dimensions, and with high-order finite elements, expanding the applicability of AMG to
a new class of problems.
Dedication
To all of my teachers and mentors along the way.
“If a mistake is found, the reader should substitute what was originally intended.”
vi
Acknowledgements
I want to extend my gratitude to all of the teachers and professors I have had that encouraged and
supported my interest in mathematics, through grade school, college, and graduate school. At Dartmouth,
I discovered how stimulating research can be while working with Professors Dan Rockmore, Alex Barnett,
Dorothy Wallace, and Brenden Epps, and learned that there can be a future in just studying mathematics
(as opposed to moving to the dark side of, for example, engineering). While working towards my PhD, I
have been fortunate enough to continue working with great scientists and great people, which has made
the entire experience very enjoyable. My advisors, Tom and Sascha, are both excellent academic mentors,
as well as people who I admire and consider role models. Through working with them, I have also also
been able to meet and collaborate with many of their colleagues. In no particular order, I have appreciated
working with Jacob Schroder, Rob Falgout, and friends, at Lawrence Livermore National Lab; officially
retired and Professor Emeritus, Steve McCormick, permanent post-doc and AMG-whisperer, John Ruge;
our German friend, Steffen Mu¨nzenmaier; Joe Spitale, of the Planetary Sciences Institute; Ju¨rgen Schmidt,
at the University of Oulu; and many others, indirectly. I would also like to thank Stephen Becker for being
on my committee even though we didn’t get the chance to work together, and Mih´aly Hor´anyi for general
guidance as I started my graduate career. Finally, I would like to thank my wife, Laura Spector, for her
continuedsupportwhilesheworkstowardsaPhDingenetics, andmyfamilyfortheirenthusiasmaboutmy
research, even if they don’t understand it.
Contents
Chapter
PART I: Modeling dust plumes on planetary satellites 1
1 Model and software development 2
1.1 Dust plumes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 What are plumes and why do we care? . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Models of plume dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Software package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Coordinate systems and transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Integrating and tracking trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3 Parallel simulation of jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.4 Notes on usability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Surface deposition and the angle of emissions 20
2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Particle flux and surface deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Surface maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Jets vs. curtains and tilting of emissions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.1 Competing theories of emission structure . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.2 Surface deposition and highly-tilted emissions . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Moving forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
viii
3 Plume density and spacecraft impact rates 34
3.1 CDA and simulation data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.1 Integrating impact rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.2 Size distribution slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 A comparison of model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.1 Electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.2 Speed distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 CDA encounters the Enceladus plume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.1 Active areas of emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.2 Particle distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.3 Mass production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Plumes on Europa? 49
4.1 Evidence for plume activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Plume Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3 Implications for Mission Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3.1 Large particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
PART II: Broadening the applicability of algebraic multigrid 64
5 Algebraic multigrid 65
5.1 Iterative methods and algebraic multigrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2 AMG: The SPD case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3 AMG: The nonsymmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.4 Computational complexity and PyAMG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6 Energy-minimization in AMG 77
6.1 The role of energy minimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
ix
6.1.1 The nonsymmetric setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.2 Weighted energy-minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.2.1 Minimization theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.2.2 Comparison with constrained energy minimization . . . . . . . . . . . . . . . . . . . . 89
6.3 Constrained energy minimization and root-node AMG . . . . . . . . . . . . . . . . . . . . . . 96
6.3.1 Sparsity pattern filtering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.3.2 Totally anisotropic diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.3.3 Recirculating flow (non-symmetric) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7 Convergence of nonsymmetric algebraic multigrid 107
7.0.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.1 Two-grid convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.1.1 Approximation properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.1.2 General conditions for stability and two-grid convergence . . . . . . . . . . . . . . . . 113
7.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.2 Multigrid convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8 Ideal restriction part I: reduction and upwind discretizations 136
8.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8.1.1 Ideal restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.1.2 Relation to existing AMG methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.2 Algebraic multigrid based on ideal restriction (AMGir) . . . . . . . . . . . . . . . . . . . . . . 143
8.2.1 Triangular structure and approximate ideal restriction . . . . . . . . . . . . . . . . . . 143
8.2.2 Nilpotency of error propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.2.3 CF-splitting and L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
ff
8.2.4 Schur-complement preconditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152