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Graph Based Scalable Algorithms with Applications

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  • UNIVERSITY OF CALIFORNIA, MERCED

    Graph Based Scalable Algorithms with Applications

    A dissertation submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy

    in

    Applied Mathematics

    by

    Garnet Jason Vaz

    Committee in charge:

    Professor Harish S. Bhat, Chair

    Professor Mayya Tokman

    Professor Arnold D. Kim

    2014

  • All Chapters c 2014 Garnet Jason Vaz

  • The Dissertation of Garnet Jason Vaz is approved, and it is acceptable

    in quality and form for publication on microfilm and electronically:

    Mayya Tokman

    Arnold D. Kim

    Harish S. Bhat, Chair

    University of California, Merced

    2014

    iii

  • To my Aunt, Lynette

    iv

  • Contents

    0.1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

    1 Introduction 1

    2 Frequency Response and Gap Tuning for Nonlinear Electrical OscillatorNetworks 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

    2.1.1 Connections to Other Systems . . . . . . . . . . . . . . . . . . . . . . 72.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Algorithms for the forward problem . . . . . . . . . . . . . . . . . . . . . . . . 10

    2.3.1 Perturbative Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 Iterative Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    2.4 Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.1 Gap Tuning: Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 17

    2.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5.1 Comparison of Steady-State Algorithms . . . . . . . . . . . . . . . . . 202.5.2 Gap Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

    2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

    3 FVFD Method for Nonlinear Maxwells Equations 313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Finite Volume Discretization of Maxwells Equations . . . . . . . . . . . . . . 32

    3.2.1 Boundary conditions & forcing terms . . . . . . . . . . . . . . . . . . . 373.3 Assembly and Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

    3.4.1 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4.2 Convergence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

    3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

    4 Quantile Regression Tree 494.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

    4.2.1 Decision trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.2.2 Decision tree algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 51

    4.3 Qtree algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.4 Computational results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

    4.4.1 Scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

    iii

  • 4.4.2 Model accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

    A FVFD implementation 63A.0.1 Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63A.0.2 Loading the mesh into PETSc . . . . . . . . . . . . . . . . . . . . . . . 63A.0.3 Computing the dual of the mesh . . . . . . . . . . . . . . . . . . . . . 64A.0.4 Algorithm for computing . . . . . . . . . . . . . . . . . . . . . . . . 65A.0.5 Linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65A.0.6 Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

    iv

  • 0.1 Acknowledgements

    First and foremost I would like to thank my advisor Harish Bhat for his support and guidancethroughout my PhD career. He has been the advisor that I wanted and the one that I neededequally well and has helped me grow both professionally and personally in ways beyonddescription and I will forever be in his debt.

    I would like to thank the guidance provided by my other committee members MayyaTokman and Arnold Kim. Their patience in listening towards my concerns and providingguidance in my research has been invaluable.

    I would like to thank the helpful staff at the School of Natural Sciences office andespecially Carrie King for making all the paper work disappear. The staff at the InternationalOffice have been extremely helpful in making my stay here hassle free.

    My studies would not have been possible without the support and love from my Dadand my aunt. They have always believed in me and encouraged me to search my own path.

    Being away from my school friends was hard but my colleagues here including NiteshKumar, Jane Hyojin Lee and Derya ahin have taught me how to smile. They have beenaround to share in my laughter and more importantly supported me when I was down. Itwould be unfair to call them friends and so to me they will always be family.

    I would also like to thank the UC Merced Open Access Fund Pilot and U.S. Departmentof Energy (Contract No. DE-AC02-05CH11231, Subaward 7041635) for supporting myresearch.

    v

  • Graph Based Scalable Algorithms with Applications.

    by

    Garnet Jason Vaz

    University of California, Merced, 2014

    Prof. Harish S. Bhat, Chair

    ABSTRACT OF THE DISSERTATION

    In this thesis, we propose various algorithms for problems arising in nonlinear circuits,nonlinear electromagnetics and data mining. Through the design and implementation ofthese algorithms, we show that the algorithms developed are scalable.

    In the first part of the thesis we provide two solutions to the forward problem of findingthe steady-state solution of nonlinear RLC circuits subjected to harmonic forcing. The workgeneralizes and provides a mathematical theory bridging prior work on structured graphs andextending it to random graphs. Both algorithms are shown to be orders of magnitude fasterthan time stepping. We introduce an inverse problem of maximizing the energy/voltage atcertain nodes of the graph without altering the graph structure. By altering the eigenvaluesassociated with the weighted graph Laplacian of the underlying circuit using a Newton-typealgorithm, we solve the inverse problem. Extensive results verify that a majority of randomgraph circuits are capable of causing amplitude boosts.

    Next, we connect nonlinear Maxwells equations in 2D to the RLC circuit problem.This relationship is achieved by considering the finite volume decomposition of nonlinearMaxwells equations. When we consider a discretization of the domain, the dual graph of thisdiscretization provides us with a planar random graph structure very similar to our previouswork. Thus, algorithms developed in the previous work become applicable. Using distributedcomputing, we develop an implementation of one of the algorithms that scales to large-scaleproblems allowing us to obtain accurate and fast solutions. Simulations are conducted forstructured and unstructured meshes, and we verify that the method is first-order in space.

    Our final application is in the field of supervised learning for regression problems.Regression trees have been used extensively since their introduction and form the basis ofseveral state-of-the-art machine learning methods today. Regression trees minimize the losscriterion (objective function) using a greedy heuristic algorithm. The usual form of theloss criterion is the squared error. While it has been known that minimizing the absolutedeviation provides more robust trees in the presence of outliers trees based on absolute lossminimization have been ignored because they were believed to be computationally expensive.We provide the first implementation which has the same algorithmic complexity as comparedto trees built with the squared error loss function. Besides computing absolute deviationtrees, our algorithm generalizes and can be used as a non-parametric alternative to quantileregression.

    vi

  • Chapter 1

    Introduction

    The increase in computational power over the last two decades has led to massive advancesin our ability to solve a variety of mathematical problems. The growing computational powerin turn has resulted in a desire to solve even larger problems. The size of the problemswe solve routinely nowadays might have seemed impossible 23 decades ago. With suchimpressive advances it may seem that in order to solve problems of current interest, we mayjust have to wait for another decade. This line of reasoning is flawed. The ability of hardwareto speed up computations has stalled due to the inability to increase computational speedbeyond its current limit while providing energy efficient processors. For almost a decadenow processor speeds have not increased according to prior trends. This has impeded ourability to speed up computations. Instead of merely relying on hardware advances to speedour work, the scientific community has branched out towards alternate methods to feed ourcomputational hunger. Rather than rely on a single technique there now exist a varietyof methods depending on our needs. For example, if our applications lie in an area whichincludes high structured computing like BLAS based operations we can now use GPUs. A