ASKIT: Approximate Skeletonization Kernel-Independent Treecode in High Dimensions

We present a fast algorithm for kernel summation problems in high-dimensions. These problems appear in computational physics, numerical approximation, non-parametric statistics, and machine learning. In our context, the sums depend on a kernel function that is a pair potential defined on a dataset of points in a high-dimensional Euclidean space. A direct evaluation of the sum scales quadratically with the number of points. Fast kernel summation methods can reduce this cost to linear complexity, but the constants involved do not scale well with the dimensionality of the dataset. The main algorithmic components of fast kernel summation algorithms are the separation of the kernel sum between near and far field (which is the basis for pruning) and the efficient and accurate approximation of the far field. We introduce novel methods for pruning and approximating the far field. Our far field approximation requires only kernel evaluations and does not use analytic expansions. Pruning is not done using bounding boxes but rather combinatorially using a sparsified nearest-neighbor graph of the input. The time complexity of our algorithm depends linearly on the ambient dimension. The error in the algorithm depends on the low-rank approximability of the far field, which in turn depends on the kernel function and on the intrinsic dimensionality of the distribution of the points. The error of the far field approximation does not depend on the ambient dimension. We present the new algorithm along with experimental results that demonstrate its performance. We report results for Gaussian kernel sums for 100 million points in 64 dimensions, for one million points in 1000 dimensions, and for problems in which the Gaussian kernel has a variable bandwidth. To the best of our knowledge, all of these experiments are impossible or prohibitively expensive with existing fast kernel summation methods.Comment: 22 pages, 6 figure

Locality of forces in molecular systems

Locality is a basic requirement for most modern methods for modelling systems of thousands of atoms or more. Despite its widespread use, the assumption of locality remains largely unfounded in the underlying quantum mechanical description. This work presents an algorithm for quantifying the locality of forces in a general molecular system. The algorithm was tested on linear hydrocarbon systems; it confirmed the locality of the tight-binding model DFTB and quantified the extent to which chemical changes, such as bond conjugation and oxygen addition, introduce long-range effects within the more accurate DFT model. The results motivated the development of an intramolecular hydrocarbon potential using the GAP machine learning method; the new potential is a promising alternative to existing models used in hydrocarbon simulation

Modeling and simulation of disordered light management structures in optoelectronic devices

Um die Lichtausbreitung innerhalb optoelektronischer Bauelemente gezielt zu manipulieren greift Lichtmanagement zunehmend auf ungeordnete Strukturen und Materialien zurück. Die quantitative Beschreibung dieser ungeordneten Teilchensysteme wird jedoch maßgeblich durch das Fehlen von Symmetrien erschwert. Hierdurch verlangt insbesondere die Diskrepanz der einzelnen Größenordnungen innerhalb eines Systems Modellierungswerkzeuge mit einem breiten Anwendungsbereich. Um die Streuprobleme in den typischen Dünnschichtsystemen optoelektronischer Bauelemente abzubilden, wird in dieser Arbeit eine Simulationsmethode genutzt, welche die gestreuten elektromagnetischen Felder in Kugelwellen abbildet und mit einem Formalismus für ebene Wellen kombiniert. Im Vergleich zu den etablierten differentiellen Methoden und Integralansätzen profitiert der gewählte Reihenansatz maßgeblich von einer stark reduzierten Anzahl an Unbekannten, erweist sich allerdings im Falle komplexer Streugeometrien bisher als nicht ausreichend flexibel. Bei Streuanordnungen aus nichtkugelförmigen Partikeln erfordert die T-Matrix-Methode beispielsweise einen Mindestabstand zwischen benachbarten Partikeln, um die Mehrfachstreuung richtig auflösen zu können und erweist sich daher ungeeignet für das Modellieren von dichten Partikelanhäufungen. In der Praxis kann die Methode zur optischen Modellierung somit nicht immer ihrem Ziel der Optimierung und Unterstützung der Bauelementeherstellung gerecht werden. In dieser Arbeit wird ein alternatives Verfahren zur Berücksichtigung direkter Wechselwirkungen zwischen nichtkugelförmigen Teilchen vorgestellt. Der Formalismus basiert auf einer zwischenzeitlichen Umwandlung des Translationsoperators für Kugelwellen in ein System ebener Wellen. Hierdurch können die sich ausbreitenden Felder vom evaneszenten Feld getrennt und die direkten Wechselwirkungen zwischen nichtsphärischen, konvexen Partikeln für beliebige Abstände ermittelt werden. Um den Rechenaufwand weiter zu reduzieren, werden periodische Randbedingungen für die T-matrix-Methode auf Basis von Ewald-Summen in das bestehende Modell integriert. Neben der Modellierung streng periodische Systeme kann der Reihenansatz somit ebenfalls auf die Untersuchung großer, periodischer Einheitszellen erweitert werden. Es wird untersucht, inwieweit sich eine weitreichende Periodizität auf die lokale Unordnung innerhalb der Einheitszellen auswirkt und unter welchen Bedingungen solch eine Periodizität geeignet ist um ungeordnete Partikelsysteme zu beschreiben. Die numerischen Herausforderungen der vorgestellten Techniken zur optischen Modellierung ungeordneter Partikelsysteme werden erörtert und anschließend anhand zweier praxisrelevanter Fallbeispiele illustriert. Zunächst wird ein Vergleich zwischen planarisierten Extraktionsschichten mit niedrigem und hohen Brechungsindex zur Auskopplung von Licht aus einer organischen Leuchtdiode für unterschiedliche Dichten der Streutextur gezogen. Anschließend werden poröse Polymere in eine Perowskit-Solarzelle integriert um eine diffuse und breitbandige Reflexion zu ermöglichen, wie sie für die Gebäudeintegration von Photovoltaikanlagen wünschenswert sein kann

Fast algorithms for biophysically-constrained inverse problems in medical imaging

We present algorithms and software for parameter estimation for forward and inverse tumor growth problems and diffeomorphic image registration. Our methods target the following scenarios: automatic image registration of healthy images to tumor bearing medical images and parameter estimation/calibration of tumor models. This thesis focuses on robust and scalable algorithms for these problems. Although the proposed framework applies to many problems in oncology, we focus on primary brain tumors and in particular low and high-grade gliomas. For the tumor model, the main quantity of interest is the extent of tumor infiltration into the brain, beyond what is visible in imaging. The inverse tumor problem assumes that we have patient images at two (or more) well-separated times so that we can observe the tumor growth. Also, the inverse problem requires that the two images are segmented. But in a clinical setting such information is usually not available. In a typical case, we just have multimodal magnetic resonance images with no segmentation. We address this lack of information by solving a coupled inverse registration and tumor problem. The role of image registration is to find a plausible mapping between the patient's tumor-bearing image and a normal brain (atlas), with known segmentation. Solving this coupled inverse problem has a prohibitive computational cost, especially in 3D. To address this challenge we have developed novel schemes, scaled up to 200K cores. Our main contributions is the design and implementation of fast solvers for these problems. We also study the performance for the tumor parameter estimation and registration solvers and their algorithmic scalability. In particular, we introduce the following novel algorithms: An adjoint formulation for tumor-growth problems with/without mass-effect; The first parallel 3D Newton-Krylov method for large diffeomorphic image registration; A novel parallel semi-Lagrangian algorithm for solving advection equations in image registration and its parallel implementation on shared and distributed memory architectures; and Accelerated FFT (AccFFT), an open-source parallel FFT library for CPU and GPUs scaled up to 131,000 cores with optimized kernels for computing spectral operators. The scientific outcomes of this thesis, has appeared in the proceedings of three ACM/IEEE SCxy conferences (two best student paper finalist, and one ACM SRC gold medal), two journal papers, two papers in review, four papers in preparation (coupling, mass effect, segmentation, and multi-species tumor model), and seven conference presentations.Computational Science, Engineering, and Mathematic

Minimal Basis Iterative Stockholder: Atoms in Molecules for Force-Field Development

Atomic partial charges appear in the Coulomb term of many force-field models and can be derived from electronic structure calculations with a myriad of atoms-in-molecules (AIM) methods. More advanced models have also been proposed, using the distributed nature of the electron cloud and atomic multipoles. In this work, an electrostatic force field is defined through a concise approximation of the electron density, for which the Coulomb interaction is trivially evaluated. This approximate "pro-density" is expanded in a minimal basis of atom-centered s-type Slater density functions, whose parameters are optimized by minimizing the Kullback-Leibler divergence of the pro-density from a reference electron density, e.g. obtained from an electronic structure calculation. The proposed method, Minimal Basis Iterative Stockholder (MBIS), is a variant of the Hirshfeld AIM method but it can also be used as a density-fitting technique. An iterative algorithm to refine the pro-density is easily implemented with a linear-scaling computational cost, enabling applications to supramolecular systems. The benefits of the MBIS method are demonstrated with systematic applications to molecular databases and extended models of condensed phases. A comparison to 14 other AIM methods shows its effectiveness when modeling electrostatic interactions. MBIS is also suitable for rescaling atomic polarizabilities in the Tkatchenko-Sheffler scheme for dispersion interactions.Comment: 61 pages, 12 figures, 2 table

Non-empirical Force-Field Development for Weakly-Bound Organic Molecules

This thesis pioneers the development of non-empirical anisotropic atom-atom force-fields for organic molecules, and their use as state-of-the-art intermolecular potentials for modelling the solid-state. The long-range electrostatic, polarization and dispersion terms have been derived directly from the molecular charge density, while the short-range terms are obtained through fitting to the symmetry-adapted perturbation theory (SAPT(DFT)) intermolecular interaction energies of a large number of different dimer configurations. This study aims to establish how far this approach, previously used for small molecules, could be applied to specialty molecules, and whether these potentials improve on the current empirical force-fields FIT and WILLIAMS01. The scaling of the underlying electronic structure calculations with system size means many adaptions have been made. This project aims to generate force-fields suitable for use in Crystal Structure Prediction (CSP) and for modelling possible polymorphs, particularly high-pressure polymorphs. By accurately modelling the repulsive wall of the potential energy surface, the high pressure/temperature conditions typically sampled by explosive materials could be studied reliably, as shown in a CSP study of pyridine using a non-empirical potential. This thesis also investigates the transferability of these potentials from the gas to condensed-phase, as well as the transferability and importance of the intermolecular interactions of flexible functional groups, in particular NO2 groups. The charge distribution was found to be strongly influenced by variations in the observed NO2 torsion angle and the conformation of the rest of the molecule. This conformation dependence coupled with the novelty of the methods and size of the molecules has made developing non-empirical models for flexible nitro-energetic materials very challenging. The thesis culminates in the development of a bespoke non-empirical force-field for rigid trinitrobenzene and its use in a CSP study

Tools for Biomolecular Modeling and Simulation

Electrostatic interactions play a pivotal role in understanding biomolecular systems, influencing their structural stability and functional dynamics. The Poisson-Boltzmann (PB) equation, a prevalent implicit solvent model that treats the solvent as a continuum while describes the mobile ions using the Boltzmann distribution, has become a standard tool for detailed investigations into biomolecular electrostatics. There are two primary methodologies: grid-based finite difference or finite element methods and body-fitted boundary element methods. This dissertation focuses on developing fast and accurate PB solvers, leveraging both methodologies, to meet diverse scientific needs and overcome various obstacles in the field

Developments in multiscale ONIOM and fragment methods for complex chemical systems

Multiskalenprobleme werden in der Computerchemie immer allgegenwärtiger und bestimmte Klassen solcher Probleme entziehen sich einer effizienten Beschreibung mit den verfügbaren Berechnungsansätzen. In dieser Arbeit wurden effiziente Erweiterungen der Multilayer-Methode ONIOM und von Fragmentmethoden als Lösungsansätze für derartige Probleme entwickelt. Dabei wurde die Kombination von ONIOM und Fragmentmethoden im Rahmen der Multi-Centre Generalised ONIOM entwickelt sowie die eine Multilayer-Variante der Fragment Combinatio Ranges. Außerdem wurden Schemata für elektronische Einbettung derartiger Multilayer-Systeme entwickelt. Der zweite Teil der Arbeit beschreibt die Implementierung im Haskell-Programm "Spicy" und demonstriert Anwendungen derartiger Multiskalen-Methoden

Annual Research Briefs: 1995

This report contains the 1995 annual progress reports of the Research Fellows and students of the Center for Turbulence Research (CTR). In 1995 CTR continued its concentration on the development and application of large-eddy simulation to complex flows, development of novel modeling concepts for engineering computations in the Reynolds averaged framework, and turbulent combustion. In large-eddy simulation, a number of numerical and experimental issues have surfaced which are being addressed. The first group of reports in this volume are on large-eddy simulation. A key finding in this area was the revelation of possibly significant numerical errors that may overwhelm the effects of the subgrid-scale model. We also commissioned a new experiment to support the LES validation studies. The remaining articles in this report are concerned with Reynolds averaged modeling, studies of turbulence physics and flow generated sound, combustion, and simulation techniques. Fundamental studies of turbulent combustion using direct numerical simulations which started at CTR will continue to be emphasized. These studies and their counterparts carried out during the summer programs have had a noticeable impact on combustion research world wide

The 1995 NASA-ODU American Society for Engineering Education (ASEE) Summer Faculty Fellowship Program

Since 1964, the National Aeronautics and Space Administration (NASA) has supported a program of summer faculty fellowships for engineering and science educators. In a series of collaborations between NASA research and development centers and nearby universities, engineering faculty members spend 10 weeks working with professional peers on research. The Summer Faculty Program Committee of the American Society for Engineering Education supervises the programs. The objectives of this program are: (1) To further the professional knowledge of qualified engineering and science faculty members; (2) To stimulate and exchange ideas between participants and NASA; (3) To enrich and refresh the research and teaching activities of participants' institutions; and (4) To contribute to the research objectives of the NASA center. College or university faculty members will be appointed as Research Fellows to spend 10 weeks in cooperative research and study at the NASA Langley Research Center. The Fellow will devote approximately 90 percent of the time to a research problem and the remaining time to a study program. The study program will consist of lectures and seminars on topics of interest or that are directly relevant to the Fellows' research topics. The lectures and seminar leaders will be distinguished scientists and engineers from NASA, education, or industry