Multiscale model reduction methods for deterministic and stochastic partial differential equations

Partial differential equations (PDEs) with multiscale coefficients are very difficult to solve due to the wide range of scales in the solutions. In the thesis, we propose some efficient numerical methods for both deterministic and stochastic PDEs based on the model reduction technique.

For the deterministic PDEs, the main purpose of our method is to derive an effective equation for the multiscale problem. An essential ingredient is to decompose the harmonic coordinate into a smooth part and a highly oscillatory part of which the magnitude is small. Such a decomposition plays a key role in our construction of the effective equation. We show that the solution to the effective equation is smooth, and could be resolved on a regular coarse mesh grid. Furthermore, we provide error analysis and show that the solution to the effective equation plus a correction term is close to the original multiscale solution.

For the stochastic PDEs, we propose the model reduction based data-driven stochastic method and multilevel Monte Carlo method. In the multiquery, setting and on the assumption that the ratio of the smallest scale and largest scale is not too small, we propose the multiscale data-driven stochastic method. We construct a data-driven stochastic basis and solve the coupled deterministic PDEs to obtain the solutions. For the tougher problems, we propose the multiscale multilevel Monte Carlo method. We apply the multilevel scheme to the effective equations and assemble the stiffness matrices efficiently on each coarse mesh grid. In both methods, the $\KL$ expansion plays an important role in extracting the main parts of some stochastic quantities.

For both the deterministic and stochastic PDEs, numerical results are presented to demonstrate the accuracy and robustness of the methods. We also show the computational time cost reduction in the numerical examples.

Optimal procurement and contracting with research and development

Government procurement of a new good or service is a process that usually includes basic research, development, and production. Empirical evidences indicate that investments in research and development (R and D) before production are significant in many defense procurements. Thus, optimal procurement policy should not be only to select the most efficient producer, but also to induce the contractors to design the best product and to develop the best technology. It is difficult to apply the current economic theory of optimal procurement and contracting, which has emphasized production, but ignored R and D, to many cases of procurement.

In this thesis, I provide basic models of both R and D and production in the procurement process where a number of firms invest in private R and D and compete for a government contract. R and D is modeled as a stochastic cost-reduction process. The government is considered both as a profit-maximizer and a procurement cost minimizer. In comparison to the literature, the following results derived from my models are significant. First, R and D matters in procurement contracting. When offering the optimal contract the government will be better off if it correctly takes into account costly private R and D investment. Second, competition matters. The optimal contract and the total equilibrium R and D expenditures vary with the number of firms. The government usually does not prefer infinite competition among firms. Instead, it prefers free entry of firms. Third, under a R and D technology with the constant marginal returns-to-scale, it is socially optimal to have only one firm to conduct all of the R and D and production. Fourth, in an independent private values environment with risk-neutral firms, an informed government should select one of four standard auction procedures with an appropriate announced reserve price, acting as if it does not have any private information.

Paths Towards High Efficiency Silicon Photovoltaics

While photovoltaics hold much promise as a sustainable electricity source, continued cost reduction is necessary to continue the current growth in deployment. A promising path to continuing to reduce total system cost is by increasing device efficiency. This thesis explores several silicon-based photovoltaic technologies with the potential to reach high power conversion efficiencies. Silicon microwire arrays, formed by joining millions of micron diameter wires together, were developed as a low cost, low efficiency solar technology. The feasibility of transitioning this to a high efficiency technology was explored. In order to achieve high efficiency, high quality silicon material must be used. Lifetimes and diffusion lengths in these wires were measured and the action of various surface passivation treatments studied. While long lifetimes were not achieved, strong inversion at the silicon / hydrofluoric acid interface was measured, which is important for understanding a common measurement used in solar materials characterization.

Cryogenic deep reactive ion etching was then explored as a method for fabricating high quality wires and improved lifetimes were measured. As another way to reach high efficiency, growth of silicon-germanium alloy wires was explored as a substrate for a III-V on Si tandem device. Patterned arrays of wires with up to 12% germanium incorporation were grown. This alloy is more closely lattice matched to GaP than silicon and allows for improvements in III-V integration on silicon.

Heterojunctions of silicon are another promising path towards achieving high efficiency devices. The GaP/Si heterointerface and properties of GaP grown on silicon were studied. Additionally, a substrate removal process was developed which allows the formation of high quality free standing GaP films and has wide applications in the field of optics.

Finally, the effect of defects at the interface of the amorphous silicon heterojuction cell was studied. Excellent voltages, and thus efficiencies, are achievable with this system, but the voltage is very sensitive to growth conditions. We directly measured lateral transport lengths at the heterointerface on the order of tens to hundreds of microns, which allows carriers to travel towards any defects that are present and recombine. This measurement adds to the understanding of these types of high efficiency devices and may aid in future device design.

Development of Metalloenzyme Dioxygen Reduction Cathodes

The prime thrust of this dissertation is to advance the development of fuel cell dioxygen reduction cathodes that employ some variant of multicopper oxidase enzymes as the catalyst. The low earth-abundance of platinum metal and its correspondingly high market cost has prompted a general search amongst chemists and materials scientists for reasonable alternatives to this metal for facilitating catalytic dioxygen reduction chemistry. The multicopper oxidases (MCOs), which constitute a class of enzyme that naturally catalyze the reaction O₂ + 4H⁺ + 4e^- → 2H₂O, provide a promising set of biochemical contenders for fuel cell cathode catalysts. In MCOs, a substrate reduces a copper atom at the type 1 site, where charge is then transferred to a trinuclear copper cluster consisting of a mononuclear type 2 or “normal copper” site and a binuclear type 3 copper site. Following the reduction of all four copper atoms in the enzyme, dioxygen is then reduced to water in two two-electron steps, upon binding to the trinuclear copper cluster. We identified an MCO, a laccase from the hyperthermophilic bacterium Thermus thermophilus strain HB27, as a promising candidate for cathodic fuel cell catalysis. This protein demonstrates resilience at high temperatures, exhibiting no denaturing transition at temperatures high as 95°C, conditions relevant to typical polymer electrolyte fuel cell operation.

In Chapter I of this thesis, we discuss initial efforts to physically characterize the enzyme when operating as a heterogeneous cathode catalyst. Following this, in Chapter II we then outline the development of a model capable of describing the observed electrochemical behavior of this enzyme when operating on porous carbon electrodes. Developing a rigorous mathematical framework with which to describe this system had the potential to improve our understanding of MCO electrokinetics, while also providing a level of predictive power that might guide any future efforts to fabricate MCO cathodes with optimized electrochemical performance. In Chapter III we detail efforts to reduce electrode overpotentials through site-directed mutagenesis of the inner and outer-sphere ligands of the Cu sites in laccase, using electrochemical methods and electronic spectroscopy to try and understand the resultant behavior of our mutant constructs. Finally, in Chapter IV, we examine future work concerning the fabrication of enhanced MCO cathodes, exploring the possibility of new cathode materials and advanced enzyme deposition techniques.