Efficient methods for stochastic optimal control

The Hamilton Jacobi Bellman (HJB) equation is central to stochastic optimal control (SOC) theory, yielding the optimal solution to general problems specified by known dynamics and a specified cost functional. Given the assumption of quadratic cost on the control input, it is well known that the HJB reduces to a particular partial differential equation (PDE). While powerful, this reduction is not commonly used as the PDE is of second order, is nonlinear, and examples exist where the problem may not have a solution in a classical sense. Furthermore, each state of the system appears as another dimension of the PDE, giving rise to the curse of dimensionality. Since the number of degrees of freedom required to solve the optimal control problem grows exponentially with dimension, the problem becomes intractable for systems with all but modest dimension.

In the last decade researchers have found that under certain, fairly non-restrictive structural assumptions, the HJB may be transformed into a linear PDE, with an interesting analogue in the discretized domain of Markov Decision Processes (MDP). The work presented in this thesis uses the linearity of this particular form of the HJB PDE to push the computational boundaries of stochastic optimal control.

This is done by crafting together previously disjoint lines of research in computation. The first of these is the use of Sum of Squares (SOS) techniques for synthesis of control policies. A candidate polynomial with variable coefficients is proposed as the solution to the stochastic optimal control problem. An SOS relaxation is then taken to the partial differential constraints, leading to a hierarchy of semidefinite relaxations with improving sub-optimality gap. The resulting approximate solutions are shown to be guaranteed over- and under-approximations for the optimal value function. It is shown that these results extend to arbitrary parabolic and elliptic PDEs, yielding a novel method for Uncertainty Quantification (UQ) of systems governed by partial differential constraints. Domain decomposition techniques are also made available, allowing for such problems to be solved via parallelization and low-order polynomials.

The optimization-based SOS technique is then contrasted with the Separated Representation (SR) approach from the applied mathematics community. The technique allows for systems of equations to be solved through a low-rank decomposition that results in algorithms that scale linearly with dimensionality. Its application in stochastic optimal control allows for previously uncomputable problems to be solved quickly, scaling to such complex systems as the Quadcopter and VTOL aircraft. This technique may be combined with the SOS approach, yielding not only a numerical technique, but also an analytical one that allows for entirely new classes of systems to be studied and for stability properties to be guaranteed.

The analysis of the linear HJB is completed by the study of its implications in application. It is shown that the HJB and a popular technique in robotics, the use of navigation functions, sit on opposite ends of a spectrum of optimization problems, upon which tradeoffs may be made in problem complexity. Analytical solutions to the HJB in these settings are available in simplified domains, yielding guidance towards optimality for approximation schemes. Finally, the use of HJB equations in temporal multi-task planning problems is investigated. It is demonstrated that such problems are reducible to a sequence of SOC problems linked via boundary conditions. The linearity of the PDE allows us to pre-compute control policy primitives and then compose them, at essentially zero cost, to satisfy a complex temporal logic specification.

Advanced density functional theory methods for materials science

In this work we chiefly deal with two broad classes of problems in computational materials science, determining the doping mechanism in a semiconductor and developing an extreme condition equation of state. While solving certain aspects of these questions is well-trodden ground, both require extending the reach of existing methods to fully answer them. Here we choose to build upon the framework of density functional theory (DFT) which provides an efficient means to investigate a system from a quantum mechanics description.

Zinc Phosphide (Zn₃P₂) could be the basis for cheap and highly efficient solar cells. Its use in this regard is limited by the difficulty in n-type doping the material. In an effort to understand the mechanism behind this, the energetics and electronic structure of intrinsic point defects in zinc phosphide are studied using generalized Kohn-Sham theory and utilizing the Heyd, Scuseria, and Ernzerhof (HSE) hybrid functional for exchange and correlation. Novel 'perturbation extrapolation' is utilized to extend the use of the computationally expensive HSE functional to this large-scale defect system. According to calculations, the formation energy of charged phosphorus interstitial defects are very low in n-type Zn₃P₂ and act as 'electron sinks', nullifying the desired doping and lowering the fermi-level back towards the p-type regime. Going forward, this insight provides clues to fabricating useful zinc phosphide based devices. In addition, the methodology developed for this work can be applied to further doping studies in other systems.

Accurate determination of high pressure and temperature equations of state is fundamental in a variety of fields. However, it is often very difficult to cover a wide range of temperatures and pressures in an laboratory setting. Here we develop methods to determine a multi-phase equation of state for Ta through computation. The typical means of investigating thermodynamic properties is via ’classical’ molecular dynamics where the atomic motion is calculated from Newtonian mechanics with the electronic effects abstracted away into an interatomic potential function. For our purposes, a ’first principles’ approach such as DFT is useful as a classical potential is typically valid for only a portion of the phase diagram (i.e. whatever part it has been fit to). Furthermore, for extremes of temperature and pressure quantum effects become critical to accurately capture an equation of state and are very hard to capture in even complex model potentials. This requires extending the inherently zero temperature DFT to predict the finite temperature response of the system. Statistical modelling and thermodynamic integration is used to extend our results over all phases, as well as phase-coexistence regions which are at the limits of typical DFT validity. We deliver the most comprehensive and accurate equation of state that has been done for Ta. This work also lends insights that can be applied to further equation of state work in many other materials.

Monitoring fish populations in Lower Lough Erne, Northern Ireland: applicability of current methods and implications for future monitoring under the EC Water Framework Directive

The proposed EC Water Framework Directive (WFD) will require member states to monitor both biotic and abiotic components of lake environments. With adoption of the WFD some measurement of fish populations will also be required. This paper describes work carried out since 1971, and particularly since 1991, on the status of fish populations in Lower Lough Erne, Northern Ireland, with an emphasis on defining change over time due to human impacts on the lake. This offers a reasonable starting point from which to develop a monitoring programme suitable for the needs of the WFD in this lake. The implications for as yet unmonitored fish populations in lakes are also determined.

Radar methods for the exploration of glaciers

The problem of finding the depths of glaciers and the current methods are discussed briefly. Radar methods are suggested as a possible improvement for, or adjunct to, seismic and gravity survey methods. The feasibility of propagating electromagnetic waves in ice and the maximum range to be expected are then investigated theoretically with the aid of experimental data on the dielectric properties of ice. It is found that the maximum expected range is great enough to measure the depth of many glaciers at the lower radar frequencies if there is not too much liquid water present. Greater ranges can be attained by going to lower frequencies.

The results are given of two expeditions in two different years to the Seward Glacier in the Yukon Territory. Experiments were conducted on a small valley glacier whose depth was determined by seismic sounding. Many echoes were received but their identification was uncertain. Using the best echoes, a profile was obtained each year, but they were not in exact agreement with each other. It could not be definitely established that echoes had been received from bedrock. Agreement with seismic methods for a considerable number of glaciers would have to be obtained before radar methods could be relied upon. The presence of liquid water in the ice is believed to be one of the greatest obstacles. Besides increasing the attenuation and possibly reflecting energy, it makes it impossible to predict the velocity of propagation. The equipment used was far from adequate for such purposes, so many of the difficulties could be attributed to this. Partly because of this, and the fact that there are glaciers with very little liquid water present, radar methods are believed to be worthy of further research for the exploration of glaciers.

Methods for melting temperature calculation

Melting temperature calculation has important applications in the theoretical study of phase diagrams and computational materials screenings. In this thesis, we present two new methods, i.e., the improved Widom's particle insertion method and the small-cell coexistence method, which we developed in order to capture melting temperatures both accurately and quickly.

We propose a scheme that drastically improves the efficiency of Widom's particle insertion method by efficiently sampling cavities while calculating the integrals providing the chemical potentials of a physical system. This idea enables us to calculate chemical potentials of liquids directly from first-principles without the help of any reference system, which is necessary in the commonly used thermodynamic integration method. As an example, we apply our scheme, combined with the density functional formalism, to the calculation of the chemical potential of liquid copper. The calculated chemical potential is further used to locate the melting temperature. The calculated results closely agree with experiments.

We propose the small-cell coexistence method based on the statistical analysis of small-size coexistence MD simulations. It eliminates the risk of a metastable superheated solid in the fast-heating method, while also significantly reducing the computer cost relative to the traditional large-scale coexistence method. Using empirical potentials, we validate the method and systematically study the finite-size effect on the calculated melting points. The method converges to the exact result in the limit of a large system size. An accuracy within 100 K in melting temperature is usually achieved when the simulation contains more than 100 atoms. DFT examples of Tantalum, high-pressure Sodium, and ionic material NaCl are shown to demonstrate the accuracy and flexibility of the method in its practical applications. The method serves as a promising approach for large-scale automated material screening in which the melting temperature is a design criterion.

We present in detail two examples of refractory materials. First, we demonstrate how key material properties that provide guidance in the design of refractory materials can be accurately determined via ab initio thermodynamic calculations in conjunction with experimental techniques based on synchrotron X-ray diffraction and thermal analysis under laser-heated aerodynamic levitation. The properties considered include melting point, heat of fusion, heat capacity, thermal expansion coefficients, thermal stability, and sublattice disordering, as illustrated in a motivating example of lanthanum zirconate (La₂Zr₂O₇). The close agreement with experiment in the known but structurally complex compound La₂Zr₂O₇ provides good indication that the computation methods described can be used within a computational screening framework to identify novel refractory materials. Second, we report an extensive investigation into the melting temperatures of the Hf-C and Hf-Ta-C systems using ab initio calculations. With melting points above 4000 K, hafnium carbide (HfC) and tantalum carbide (TaC) are among the most refractory binary compounds known to date. Their mixture, with a general formula Ta_xHf_1-xC_y, is known to have a melting point of 4215 K at the composition Ta₄HfC₅, which has long been considered as the highest melting temperature for any solid. Very few measurements of melting point in tantalum and hafnium carbides have been documented, because of the obvious experimental difficulties at extreme temperatures. The investigation lets us identify three major chemical factors that contribute to the high melting temperatures. Based on these three factors, we propose and explore a new class of materials, which, according to our ab initio calculations, may possess even higher melting temperatures than Ta-Hf-C. This example also demonstrates the feasibility of materials screening and discovery via ab initio calculations for the optimization of "higher-level" properties whose determination requires extensive sampling of atomic configuration space.

Relativistic mean field theory: methods and applications

We develop a method for performing one-loop calculations in finite systems that is based on using the WKB approximation for the high energy states. This approximation allows us to absorb all the counterterms analytically and thereby avoids the need for extreme numerical precision that was required by previous methods. In addition, the local approximation makes this method well suited for self-consistent calculations. We then discuss the application of relativistic mean field methods to the atomic nucleus. Self-consistent, one loop calculations in the Walecka model are performed and the role of the vacuum in this model is analyzed. This model predicts that vacuum polarization effects are responsible for up to five percent of the local nucleon density. Within this framework the possible role of strangeness degrees of freedom is studied. We find that strangeness polarization can increase the kaon-nucleus scattering cross section by ten percent. By introducing a cutoff into the model, the dependence of the model on short-distance physics, where its validity is doubtful, is calculated. The model is very sensitive to cutoffs around one GeV.

Development and Application of Embedding Methods for the Simulation of Large Chemical Systems

The high computational cost of correlated wavefunction theory (WFT) calculations has motivated the development of numerous methods to partition the description of large chemical systems into smaller subsystem calculations. For example, WFT-in-DFT embedding methods facilitate the partitioning of a system into two subsystems: a subsystem A that is treated using an accurate WFT method, and a subsystem B that is treated using a more efficient Kohn-Sham density functional theory (KS-DFT) method. Representation of the interactions between subsystems is non-trivial, and often requires the use of approximate kinetic energy functionals or computationally challenging optimized effective potential calculations; however, it has recently been shown that these challenges can be eliminated through the use of a projection operator. This dissertation describes the development and application of embedding methods that enable accurate and efficient calculation of the properties of large chemical systems.

Chapter 1 introduces a method for efficiently performing projection-based WFT-in-DFT embedding calculations on large systems. This is accomplished by using a truncated basis set representation of the subsystem A wavefunction. We show that naive truncation of the basis set associated with subsystem A can lead to large numerical artifacts, and present an approach for systematically controlling these artifacts.

Chapter 2 describes the application of the projection-based embedding method to investigate the oxidative stability of lithium-ion batteries. We study the oxidation potentials of mixtures of ethylene carbonate (EC) and dimethyl carbonate (DMC) by using the projection-based embedding method to calculate the vertical ionization energy (IE) of individual molecules at the CCSD(T) level of theory, while explicitly accounting for the solvent using DFT. Interestingly, we reveal that large contributions to the solvation properties of DMC originate from quadrupolar interactions, resulting in a much larger solvent reorganization energy than that predicted using simple dielectric continuum models. Demonstration that the solvation properties of EC and DMC are governed by fundamentally different intermolecular interactions provides insight into key aspects of lithium-ion batteries, with relevance to electrolyte decomposition processes, solid-electrolyte interphase formation, and the local solvation environment of lithium cations.