THE UNIVERSITY OF CHICAGO FAST NUMERICAL METHODS AND BIOLOGICAL PROBLEMS A DISSERTATION SUBMITTED TO THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF COMPUTER SCIENCE BY PETER BRUNE CHICAGO, ILLINOIS AUGUST 2011 ABSTRACT This thesis encompasses a number of efforts towards the development of fast numer- ical methods and their applications, particularly in light of the simulation of bio- chemical systems. Scientific computing as a discipline requires considerations from computer science; namely those of algorithmic efficiency and software automation. Also required is knowledge of applied mathematics in order to bridge the gap be- tween computer science and specific application. This thesis spans these fields, with the study and implementation of optimal numerical techniques encompassing two chapters, and the development and application of numerical techniques to biochemi- cal problems encompassing another two. The first of these efforts is the construction of robust, optimal geometric unstructured multigrid methods in the face of difficult problem and mesh conditions. The second was the construction of optimal discrete function spaces for problems arising in quantum mechanics and electronic structure calculations. The third and fourth were the development of fast and flexible meth- ods for nanoscale implicit solvent electrostatics. The development of fast and parallel methods for an important quantity of interest in classical density functional theory calculations is discussed. Also, the derivation and implementation of a finite element method for improved solvent models using automated scientific computing tools is described. ii TABLE OF CONTENTS ABSTRACT . ii LIST OF TABLES . vii LIST OF FIGURES . viii ACKNOWLEDGEMENTS . x Chapter 1 ROADMAP . 1 1.1 Geometric Unstructured Multigrid . .1 1.2 Exponential Meshes . .2 1.3 Classical Density Functional Theory . .2 1.4 Nonlocal Bioelectrostatics . .3 1.5 Disciplines . .4 1.5.1 Computer Science . .4 1.5.2 Computational and Applied Mathematics . .5 1.5.3 Finite Element Methods . .6 2 MULTILEVEL METHODS . 7 2.1 Multigrid . .8 2.2 FEM & Multilevel Methods . 11 2.2.1 Efficiency and High Performance Computing . 12 2.2.2 Feasibility and Low Performance Computing . 12 2.2.3 Coarsening and Multigrid Components . 13 2.3 Vertex Selection . 14 2.4 Multigrid Mesh Conditions . 16 2.4.1 Function-Based Coarsening . 18 2.4.2 Graph Coarsening Algorithm . 20 2.4.3 Mesh Boundaries . 24 2.5 Remeshing . 26 2.5.1 Remeshing Options . 27 2.5.2 Simple Vertex Removal . 28 iii 2.5.3 Quality Measure Possibilities . 30 2.5.4 Anisotropic Quality Measures . 32 2.5.5 Remeshing the Boundary . 34 2.5.6 Mesh Cleanup . 35 2.6 Adaptive Refinement . 37 2.7 Interpolation Operators . 40 2.7.1 Construction by Traversal . 41 2.8 Experiments . 46 2.8.1 Experimental Setup . 47 2.8.2 Test Problems . 48 2.8.3 Mesh Quality Experiments . 49 2.8.4 Mesh Grading Experiments . 52 2.8.5 Multigrid Performance . 55 2.8.6 Anisotropic Multigrid Performance . 56 2.9 Outlook . 58 3 EXPONENTIAL MESHES AND QUANTUM MECHANICS . 61 3.1 Approximation of Exponential Decay . 62 3.1.1 Optimal Spacing . 63 3.1.2 The Higher Order Case . 65 3.2 Mesh Representation . 67 3.2.1 Simple Radial Meshes . 70 3.3 Quantum Mechanics . 71 3.3.1 The Spectrum . 72 3.3.2 Solving the Schr¨odingerEquation . 72 3.4 General-Dimensional Finite Elements . 76 3.4.1 Tensor Product Geometries and Elements . 76 3.4.2 Barycentric Coordinates . 77 3.4.3 Tensor Product Simplices . 77 3.4.4 The Finite Element Basis . 78 3.4.5 Lagrange Elements . 78 3.4.6 Quadrature on Tensor Product Cells . 80 3.5 Experiments . 82 3.5.1 The Hydrogen Atom . 83 3.5.2 The Hydrogen Spectrum . 84 3.6 Outlook . 84 iv 4 EFFICIENT CLASSICAL DENSITY FUNCTIONAL THEORY . 87 4.0.1 Contribution . 90 4.1 Problem Setup and Parameters . 91 4.1.1 The Hard Wall Case . 92 4.1.2 Reduced Model for the Ion Channel . 93 4.1.3 Numerical Approaches . 94 4.2 Parallel Algorithm . 96 4.2.1 GPU Computation . 98 4.2.2 The Algorithm . 101 4.2.3 Optimization . 103 4.3 Initial Results . 105 4.4 Fast Algorithm for the Reference Density . 107 4.4.1 Parameter gridding . 107 4.4.2 Error Estimation . 108 4.4.3 Complexity of Approximation . 111 4.4.4 Results . 112 4.5 Outlook . 114 5 FINITE ELEMENT METHODS FOR NONLOCAL ELECTROSTATICS 116 5.1 Biochemical Continuum Electrostatics . 117 5.1.1 Domain of Interest . 118 5.2 Models and Formulations . 121 5.2.1 The Fourier-Lorentzian Model . 122 5.2.2 Parameters of the FL Model . 125 5.2.3 Discretization of the FL Model . 126 5.3 FEM Approach . 128 5.3.1 Formulation Details . 129 5.3.2 Code Generation and Automation . 131 5.3.3 Fast Solvers . 133 5.4 Atomic Meshing . 133 5.5 Experiments . 135 5.6 Outlook . 137 5.6.1 Model Improvements . 138 Appendix v A MULTIGRID CODE OVERVIEW . 139 A.1 Multigrid Infrastructure . 139 A.1.1 MultigridProblem . 139 A.1.2 Interpolation . 141 A.2 Coarsening Infrastructure . 142 A.2.1 CoarsenedHierarchy . 142 B OPTIMAL GRIDS FOR OTHER FUNCTIONS . 144 B.1 Geometric Grids . 144 B.2 Grids for Algebraic Functions . 145 C MOLECULAR MESHING . 147 C.0.1 Surface Meshing . 148 C.0.2 Inputs . 149 C.0.3 Algorithm . 149 REFERENCES . 150 vi LIST OF TABLES 2.1 Hierarchy quality metrics for the Pacman mesh. 50 2.2 Hierarchy quality metrics for the Fichera mesh. 51 2.3 Calculated Cl and Ch for the Pacman mesh hierarchy. 53 2.4 Calculated Cl and Ch for the Fichera mesh hierarchy. 54 2.5 Multigrid performance on the Pacman and Fichera problems. 55 2.6 Anisotropic problem multigrid performance. 57 2.7 Galerkin vs. Rediscretization Multigrid for the Sinker Problem. 60 3.1 Relative L2 errors. 82 3.2 Number of unknowns. 82 4.1 Table of species parameters. 91 4.2 Error and refinement in R on the hard wall. 114 5.1 Constants used in various studies of the nonlocal model. 126 5.2 Free energies of solvation for example ions . 137 A.1 Solver parameters. 140 A.2 Multigrid cycle type options. ..