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Hodge Laplacians on Simplicial Meshes and Graphs © 2015 Kaushik Kalyanaraman HODGE LAPLACIANS ON SIMPLICIAL MESHES AND GRAPHS BY KAUSHIK KALYANARAMAN DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science in the Graduate College of the University of Illinois at Urbana-Champaign, 2015 Urbana, Illinois Doctoral Committee: Associate Professor Anil N. Hirani, Chair, Director of Research Associate Professor Alan Demlow, Texas A&M University Professor Jeff Erickson Professor Michael Heath Abstract We present in this dissertation some developments in the discretizations of exterior calculus for problems posed on simplicial discretization (meshes) of geometric manifolds and analogous prob- lems on abstract simplicial complexes. We are primarily interested in discretizations of elliptic type partial differential equations, and our model problem is the Hodge Laplacian Poisson problem on differential 푘-forms on 푛-dimensional manifolds. One of our major contributions in this work is the computational quantification of the solution using the weak mixed formulation of this problem on simplicial meshes using discrete exterior calculus (DEC), and its comparisons with the solution due to a different discretization framework, namely, finite element exterior calculus (FEEC). Conse- quently, our important computational result is that the solution of the Poisson problem on different manifolds in two- and three-dimensions due to DEC recovers convergence properties on many se- quences of refined meshes similar to that of FEEC. We also discuss some potential attempts for showing this convergence theoretically. In particular, we demonstrate that a certain formulation of a variational crimes approach that can be used for showing convergence for a generalized FEEC may not be directly applicable to DEC convergence in its current formulation. In order to perform computations using DEC, a key development that we present is exhibiting sign rules that allow for the computation of the discrete Hodge star operators in DEC on Delaunay meshes in a piecewise manner. Another aspect of computationally solving the Poisson problem using the mixed formulation with either DEC or FEEC requires knowing the solution to the cor- responding Laplace’s problem, namely, the harmonics. We present a least squares method for com- puting a basis for the space of such discrete harmonics via their isomorphism to cohomology. We also provide some numerics to quantify the efficiency of this solution in comparison with previously known methods. Finally, we demonstrate an application to obtain the ranking of pairwise compar- ison data. We model this data as edge weights on graphs with 3-cliques included and perform its Hodge decomposition by solving two least squares problems. An outcome of this exploration is also providing some computational evidence that algebraic multigrid linear solvers for the resulting linear systems on Erdős-Rényi random graphs and on Barabási-Albert graphs do not perform very well in comparison with iterative Krylov solvers. ii To no one in particular, and everyone in general. To humanity! iii Acknowledgments A Ph.D. is not a degree but an experience! This experience would not have come to fruition without a large number of people having played a significant role in my life and having supported me in very many ways all through the years. Such support has also not been limited to those from my graduate school but, in some instances, all my life! Even professionally,this work is not an individual achievement, and I could not have written it without amply crediting, and sharing credit with, several individuals, and it bearing my name is no testimony to all the collaborations it entailed. This is my little tribute to all those wonderful people who have a made a difference to my couple of years on the third rock in the star system of Sol, a fairly ordinary middle aged average star located in one of the outer arms of a reasonable size, nevertheless unremarkable, standard elliptic galaxy tucked away in some uninteresting part of the local Lanakea supercluster in a Universe otherwise seemingly utterly devoid of life. (This is a narcissistic tribute to my failed childhood dream of becoming an astrophysicist!) Also, if not anything else, I hope that at least this section of my thesis is read by others because, when younger, often times, it was the Acknowledgments section of a thesis that I would first turn to. On a meta level, I wish to also acknowledge those several theses that I have come across which alluded to this fact! To begin with, I would like to thank my advisor Anil N. Hirani for having me as his student, and also for having been the best advisor a student could have! Advising is hard work, and I have the highest appreciation for his sense of patience, interest in my professional and personal well being, and for having carefully imbibed in me all the technical know-how that eventually led to this thesis. I have to specifically acknowledge that this work is joint with him in every sense. I could not have accomplished it without the unequivocal attention and care provided by him throughout these years. Having to write a tribute to him alone would entail far many pages, and in the interest of professionalism, I would like to leave it at that during even desperate times, Anil never let me feel like I made a mistake in my choice or manner of pursuing a Ph.D.! I look forward to maintaining more than simply a professional advisor-advisee relationship for all future! I wish to next thank Alan Demlow,Jeff Erickson and Michael Heath for having agreed to serve on my thesis committee. Alan has been a fantastic collaborator, and I wish to especially thank him for providing valuable suggestions at several stages of my Ph.D., and for funding my travel and hosting me twice at Kentucky in 2013. Jeff has been inspiring in my chats with him. In particular, Jeff was assigned as the initial faculty member with whom I would have to interact about the program, and provided me with the first pep talk about research in Illinois. Mike’s inputs have been constant and consistent throughout the scientific computing seminar series, which in addition has helped me in slowly and thoroughly improving my overall presentation skills. Mark Schubel has been my best friend in doing professional and later personal things at UIUC! I believe that after Mark joined Anil’s group, my relationship with him was essentially shaped by iv our spending six months or so carefully reading two of V. I. Arnold’s books, Lectures on Partial Differential Equations and Ordinary Differential Equations, every week for a few hours or so. (We got through the first chapter in each of the two books!) Mark has been my go to person for any technical discussion on geometry, physics or computation with a fellow student, and I consider myself quite lucky in having had him to toss ideas around throughout my research. Alas, at the time of this writing, it is a bit disappointing that we do not have a joint paper hiding, therefore, the depth of our collaboration. I wish to also thank Mark for grabbing a drink or two on innumerable occasions, and for helping me appreciate the art and style of responsible drinking! Also, I suspect I cannot thank Mark without thanking Mary Christoph at the same time for being a friend whose views I cherish. Thanks to you both folks for making my stay at Illinois so much better! Han Wang was my office mate for a number of years at the beginning of graduate school, and I wish to thank him for several scholarly discussions over the years. Sean Shahkarami was a fairly recent academic companion whose help in the end years is something I greatly benefitted from. I wish to also thank Seth Watts for many valuable discussions especially during the group meetings. I wish to thank my academic sibling Evan VanderZee for discussions in my first semester of grad- uate school when we overlapped, and later during our collaboration. Evan had already set the aspi- ration bar high for us future students! Andrew Gillette has been instrumental in shaping some of my understanding of finite element exterior calculus over the years, and for having provided me with confidence and encouragement all through my graduate years. I wish to thank Doug Arnold and Ragnar Winther for having been kind enough to indulge me in discussions whenever I have met with them, and for helping me understand possibly trite issues in finite element exterior calculus. Numerous fellow graduate students in scientific computing have been incredible to me over the years, and I would like to thank Nana Arizumi, Eleana Caraba, Jehanzeb Chaudhry, Steven Dalton, Mark Gates, James Lai, and Dmitry Yershov for providing avenues of discussion in CS. I wish to particularly acknowledge Natalie Beams for having been an office mate par extraordinaire during my last year, and for generally being a wonderful person with whom I could discuss everything under the sun. Thanks Natalie for inspiring me in your own way and my apologies if I was too bothersome! I wish to also thank the crop of younger fellow students in scientific computing as well as fellow TAs over the years for their own roles in the cog that has been my graduate school experience. These include Amanda Bienz, Fatih Boyaci, Nathan Bowman, Erin Carrier, Rajiv Giri, Shelley Goel, Erin Molloy, and Saurabh Nangia. I wish to thank Harish Chintakunta for his professional collaboration with me as well as for being a friend, and for leading several challenging and interesting discussions during our many formal and informal meetings. I have been a recipient of many travel grants for attending conferences and workshops from mul- tiple institutes throughout my graduate years.
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