Rigorous numerical approaches in electronic structure theory

Electronic structure theory concerns the description of molecular properties according to the postulates of quantum mechanics. For practical purposes, this is realized entirely through numerical computation, the scope of which is constrained by computational costs that increases rapidly with the size of the system. The significant progress made in this field over the past decades have been facilitated in part by the willingness of chemists to forego some mathematical rigour in exchange for greater efficiency. While such compromises allow large systems to be computed feasibly, there are lingering concerns over the impact that these compromises have on the quality of the results that are produced. This research is motivated by two key issues that contribute to this loss of quality, namely i) the numerical errors accumulated due to the use of finite precision arithmetic and the application of numerical approximations, and ii) the reliance on iterative methods that are not guaranteed to converge to the correct solution. Taking the above issues in consideration, the aim of this thesis is to explore ways to perform electronic structure calculations with greater mathematical rigour, through the application of rigorous numerical methods. Of which, we focus in particular on methods based on interval analysis and deterministic global optimization. The Hartree-Fock electronic structure method will be used as the subject of this study due to its ubiquity within this domain. We outline an approach for placing rigorous bounds on numerical error in Hartree-Fock computations. This is achieved through the application of interval analysis techniques, which are able to rigorously bound and propagate quantities affected by numerical errors. Using this approach, we implement a program called Interval Hartree-Fock. Given a closed-shell system and the current electronic state, this program is able to compute rigorous error bounds on quantities including i) the total energy, ii) molecular orbital energies, iii) molecular orbital coefficients, and iv) derived electronic properties. Interval Hartree-Fock is adapted as an error analysis tool for studying the impact of numerical error in Hartree-Fock computations. It is used to investigate the effect of input related factors such as system size and basis set types on the numerical accuracy of the Hartree-Fock total energy. Consideration is also given to the impact of various algorithm design decisions. Examples include the application of different integral screening thresholds, the variation between single and double precision arithmetic in two-electron integral evaluation, and the adjustment of interpolation table granularity. These factors are relevant to both the usage of conventional Hartree-Fock code, and the development of Hartree-Fock code optimized for novel computing devices such as graphics processing units. We then present an approach for solving the Hartree-Fock equations to within a guaranteed margin of error. This is achieved by treating the Hartree-Fock equations as a non-convex global optimization problem, which is then solved using deterministic global optimization. The main contribution of this work is the development of algorithms for handling quantum chemistry specific expressions such as the one and two-electron integrals within the deterministic global optimization framework. This approach was implemented as an extension to an existing open source solver. Proof of concept calculations are performed for a variety of problems within Hartree-Fock theory, including those in i) point energy calculation, ii) geometry optimization, iii) basis set optimization, and iv) excited state calculation. Performance analyses of these calculations are also presented and discussed

Fast, Optimal, and Safe Motion Planning for Bipedal Robots

Bipedal robots have the potential to traverse a wide range of unstructured environments, which are otherwise inaccessible to wheeled vehicles. Though roboticists have successfully constructed controllers for bipedal robots to walk over uneven terrain such as snow, sand, or even stairs, it has remained challenging to synthesize such controllers in an online fashion while guaranteeing their satisfactory performance. This is primarily due to the lack of numerical method that can accommodate the non-smooth dynamics, high degrees of freedom, and underactuation that characterize bipedal robots. This dissertation proposes and implements a family of numerical methods that begin to address these three challenges along three dimensions: optimality, safety, and computational speed. First, this dissertation develops a convex relaxation-based approach to solve optimal control for hybrid systems without a priori knowledge of the optimal sequence of transition. This is accomplished by formulating the problem in the space of relaxed controls, which gives rise to a linear program whose solution is proven to compute the globally optimal controller. This conceptual program is solved using a sequence of semidefinite programs whose solutions are proven to converge from below to the true optimal solution of the original optimal control problem. Moreover, a method to synthesize the optimal controller is developed. Using an array of examples, the performance of this method is validated on problems with known solutions and also compared to a commercial solver. Second, this dissertation constructs a method to generate safety-preserving controllers for a planar bipedal robot walking on flat ground by performing reachability analysis on simplified models under the assumption that the difference between the two models can be bounded. Subsequently, this dissertation describes how this reachable set can be incorporated into a Model Predictive Control framework to select controllers that result in safe walking on the biped in an online fashion. This method is validated on a 5-link planar model. Third, this dissertation proposes a novel parallel algorithm capable of finding guaranteed optimal solutions to polynomial optimization problems up to pre-specified tolerances. Formal proofs of bounds on the time and memory usage of such method are also given. Such algorithm is implemented in parallel on GPUs and compared against state-of-the-art solvers on a group of benchmark examples. An application of such method on a real-time trajectory-planning task of a mobile robot is also demonstrated. Fourth, this dissertation constructs an online Model Predictive Control framework that guarantees safety of a 3D bipedal robot walking in a forest of randomly-placed obstacles. Using numerical integration and interval arithmetic techniques, approximations to trajectories of the robot are constructed along with guaranteed bounds on the approximation error. Safety constraints are derived using these error bounds and incorporated in a Model Predictive Control framework whose feasible solutions keep the robot from falling over and from running into obstacles. To ensure that the bipedal robot is able to avoid falling for all time, a finite-time terminal constraint is added to the Model Predictive Control algorithm. The performance of this method is implemented and compared against a naive Model Predictive Control method on a biped model with 20 degrees of freedom. In summary, this dissertation presents four methods for control synthesis of bipedal robots with improvements in either optimality, safety guarantee, or computational speed. Furthermore, the performance of all proposed methods are compared with existing methods in the field.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/162880/1/pczhao_1.pd

Proceedings of the XIII Global Optimization Workshop: GOW'16

[Excerpt] Preface: Past Global Optimization Workshop shave been held in Sopron (1985 and 1990), Szeged (WGO, 1995), Florence (GO’99, 1999), Hanmer Springs (Let’s GO, 2001), Santorini (Frontiers in GO, 2003), San José (Go’05, 2005), Mykonos (AGO’07, 2007), Skukuza (SAGO’08, 2008), Toulouse (TOGO’10, 2010), Natal (NAGO’12, 2012) and Málaga (MAGO’14, 2014) with the aim of stimulating discussion between senior and junior researchers on the topic of Global Optimization. In 2016, the XIII Global Optimization Workshop (GOW’16) takes place in Braga and is organized by three researchers from the University of Minho. Two of them belong to the Systems Engineering and Operational Research Group from the Algoritmi Research Centre and the other to the Statistics, Applied Probability and Operational Research Group from the Centre of Mathematics. The event received more than 50 submissions from 15 countries from Europe, South America and North America. We want to express our gratitude to the invited speaker Panos Pardalos for accepting the invitation and sharing his expertise, helping us to meet the workshop objectives. GOW’16 would not have been possible without the valuable contribution from the authors and the International Scientific Committee members. We thank you all. This proceedings book intends to present an overview of the topics that will be addressed in the workshop with the goal of contributing to interesting and fruitful discussions between the authors and participants. After the event, high quality papers can be submitted to a special issue of the Journal of Global Optimization dedicated to the workshop. [...

The Numerical Simulation of Fluid Flow

This book collects the accepted contributions to the Special Issue "The Numerical Simulation of Fluid Flow" in the Energies journal of MDPI. It is focused more on practical applications of numerical codes than in its development. It covers a wide variety of topics, from aeroacoustics to aerodynamics and flow-particles interaction

Assessment of a hybrid numerical approach to estimate sound wave propagation in an enclosure and application of auralizations to evaluate acoustical conditions of a classroom to establish the impact of acoustic variables on cognitive processes

In this research, the concept of auralization is explored taking into account a hybrid numerical approach to establish good options for calculating sound wave propagation and the application of virtual sound environments to evaluate acoustical conditions of a classroom, in order to determine the impact of acoustic variables on cognitive processes. The hybrid approach considers the combination of well-established Geometrical Acoustic (GA) techniques and the Finite Element Method (FEM), contemplating for the latter the definition of a real valued impedance boundary condition related to absorption coefficients available in GA databases. The realised virtual sound environments are verified against real environment measurements by means of objective and subjective methods. The former is based on acoustic measurements according to international standards, in order to evaluate the numerical approaches used with established acoustic indicators to assess sound propagation in rooms. The latter comprises a subjective test comparing the virtual auralizations to the reference ones, which are obtained by means of binaural impulse response measurements. The first application of the auralizations contemplates an intelligibility and listening difficulty subjective test, considering different acoustic conditions of reverberation time and background noise levels. The second application studies the impact of acoustic variables on the cognitive processes of attention, memory and executive function, by means of psychological tests

Acceleration of MCMC-based algorithms using reconfigurable logic

Monte Carlo (MC) methods such as Markov chain Monte Carlo (MCMC) and sequential Monte Carlo (SMC) have emerged as popular tools to sample from high dimensional probability distributions. Because these algorithms can draw samples effectively from arbitrary distributions in Bayesian inference problems, they have been widely used in a range of statistical applications. However, they are often too time consuming due to the prohibitive costly likelihood evaluations, thus they cannot be practically applied to complex models with large-scale datasets. Currently, the lack of sufficiently fast MCMC methods limits their applicability in many modern applications such as genetics and machine learning, and this situation is bound to get worse given the increasing adoption of big data in many fields. The objective of this dissertation is to develop, design and build efficient hardware architectures for MCMC-based algorithms on Field Programmable Gate Arrays (FPGAs), and thereby bring them closer to practical applications. The contributions of this work include: 1) Novel parallel FPGA architectures of the state-of-the-art resampling algorithms for SMC methods. The proposed architectures allow for parallel implementations and thus improve the processing speed. 2) A novel mixed precision MCMC algorithm, along with a tailored FPGA architecture. The proposed design allows for more parallelism and achieves low latency for a given set of hardware resources, while still guaranteeing unbiased estimates. 3) A new variant of subsampling MCMC method based on unequal probability sampling, along with a highly optimized FPGA architecture. The proposed method significantly reduces off-chip memory access and achieves high accuracy in estimates for a given time budget. This work has resulted in the development of hardware accelerators of MCMC and SMC for very large-scale Bayesian tasks by applying the above techniques. Notable speed improvements compared to the respective state-of-the-art CPU and GPU implementations have been achieved in this work.Open Acces

Artificial Intelligence in Image-Based Screening, Diagnostics, and Clinical Care of Cardiopulmonary Diseases

Cardiothoracic and pulmonary diseases are a significant cause of mortality and morbidity worldwide. The COVID-19 pandemic has highlighted the lack of access to clinical care, the overburdened medical system, and the potential of artificial intelligence (AI) in improving medicine. There are a variety of diseases affecting the cardiopulmonary system including lung cancers, heart disease, tuberculosis (TB), etc., in addition to COVID-19-related diseases. Screening, diagnosis, and management of cardiopulmonary diseases has become difficult owing to the limited availability of diagnostic tools and experts, particularly in resource-limited regions. Early screening, accurate diagnosis and staging of these diseases could play a crucial role in treatment and care, and potentially aid in reducing mortality. Radiographic imaging methods such as computed tomography (CT), chest X-rays (CXRs), and echo ultrasound (US) are widely used in screening and diagnosis. Research on using image-based AI and machine learning (ML) methods can help in rapid assessment, serve as surrogates for expert assessment, and reduce variability in human performance. In this Special Issue, “Artificial Intelligence in Image-Based Screening, Diagnostics, and Clinical Care of Cardiopulmonary Diseases”, we have highlighted exemplary primary research studies and literature reviews focusing on novel AI/ML methods and their application in image-based screening, diagnosis, and clinical management of cardiopulmonary diseases. We hope that these articles will help establish the advancements in AI