In this thesis, we study three classes of problems within the general area of sequential decision making with limited information, namely (i) sequential model-based optimization, (ii) active learning and (iii) active resource allocation. For all the problems considered, we propose and analyze new algorithms and also characterize the fundamental performance limits by obtaining algorithm independent impossibility results.

For the problem of sequential model-based optimization, we propose a general algorithmic strategy which proceeds by combining global and local models over an adaptively constructed non-uniform partition of the input space. For the special cases of Gaussian Process~(GP) bandits and kernelized bandits, this approach leads to improved regret bounds compared to the state-of-the-art. Next, we quantify the significance of incorporating gradient information in GP bandits by first deriving an algorithm independent lower bound on the regret, and then obtaining an upper bound on a new first-order algorithm. Finally, we end this part by obtaining the first instance-dependent regret lower bounds for kernelized bandits, and then proposing an algorithm whose performance matches this lower bound under some parameter regimes.

In the next part, we show that the general algorithmic strategy we developed for sequential optimization can also be useful for active learning problems. In particular, we first propose an algorithm for the GP level set estimation problem, and obtain upper bounds on the uniform estimation error which improves upon the prior results. Next, we propose an active learning strategy for the problem of classification with abstention and demonstrate the our proposed strategy is minimax near-optimal under certain smoothness and margin assumptions.

Finally, in the last part we consider the problem of active resource allocation for ensuring uniformly good performance of certain statistical tasks. In particular, we first design and analyze a sample allocation strategy to estimate several discrete distributions uniformly well in terms of common distance measures such as $\ell_2^2$, $\ell_1$, $f-$divergence and separation distance. Next, we propose a strategy of actively constructing a training data-set consisting of members from several sub-groups to ensure that the classifier trained on the resulting dataset is fair in a minimax sense.

In the wide spectrum of excited state quantum chemistry methods, there is no direct analogue to Hartree-Fock theory. This dissertation presents the theory and initial applications for a self consistent framework for Excited State Mean Field (ESMF) theory. This method presents a self consistent equation analogous to the Roothaan-Hall equation, that is constructed with mean-field one-electron operators. The convergence criteria is described by a commutator condition between Fock-type operators and density operators, just like in Hartree-Fock theory. Finally, this method is accelerated via direct inversion of the iterative subspace (DIIS), akin to acceleration in the ground state theory. Futhermore, this work discusses applications of ESMF to larger solvated systems, afforded by ESMF's scaling -- the method costs roughly twice the cost of a Hartree-Fock calculation. Getting an accurate physical picture of excited states in solvated systems is challenging, and the second half of this dissertation focuses on a comparative analysis of various methods, their degree of correlation, and their ability to qualitatively describe donor and acceptor regions for a charge transfer excitation. This comparison shows that ESMF can accurately describe the degree of participation of solvent water molecules in the excitation, unlike Density Functional Theory (DFT) based methods. However, when there is minimal participation from the solvent, Restricted Open-shell Kohn Sham methods fare better, indicating that the lack of correlation in ESMF prevents the method from providing a more quantitatively accurate picture. Using ESMF as a stepping rung for developing a hierarchy of excited state specific methods is a promising platform to achieve affordable excited state specific calculations.

Vision begins in the neural retina, a three layered film in the back of the eye composed of five major classes of neurons: photoreceptors, horizontal cells, bipolar cells, amacrine cells, and retinal ganglion cells (RGCs). Each major neuronal class can be further divided into multiple different neuronal cell types based off morphological, physiological, and molecular properties. While the basic plan of the retina is conserved across vertebrates, species differ profoundly in their visual needs. Retinal cell types may have evolved to accommodate these varied needs, but this has not been systematically studied. We collected and analyzed single-cell transcriptomic atlases of the retina from 17 species: humans, two non-human primates, four rodents, three ungulates, opossum, ferret, tree shrew, a bird, a reptile, a teleost fish and a lamprey. In this thesis, I present insights into the evolution of retinal cell types gained from these studies.First, we systematically classified retinal ganglion cells in adult and larval zebrafish using single-cell transcriptomics, identifying marker genes for >30 mature types and several developmental intermediates. These analyses guided collaborators to develop transgenic driver lines, enabling specific experimental access to a subset of RGC types. They found that expression of one or few transcription factors predicted dendrite morphologies and axonal projections to specific tectal layers and extratectal targets, and in vivo calcium imaging revealed that molecularly defined RGCs exhibit specific functional tuning. Second, we studied a specialized downstream circuit used for vision in low-light conditions called the primary rod pathway. We demonstrate that the mammalian primary rod pathway is conserved in zebrafish, which diverged from extant mammals ∼400 million years ago. Using single-cell RNA sequencing, we identified two bipolar cell types in zebrafish that are related to mammalian rod bipolar cell (RBC), the only bipolar type that directly carries rod signals from the outer to the inner retina in the primary rod pathway. Guided by this prediction, coauthors combined electrophysiology, histology and ultrastructural reconstruction experiments of zebrafish RBCs to show that one of the two RBC types is homologous to the mammalian RBC. This suggests that the cell types and circuit design of the primary rod pathway emerged before the divergence of teleost fish and mammals. Third, we compared the visual systems of two closely related murid species that occupy different visual environments: Rhabdomys pumilio are diurnal and have substantially thicker inner retina than nocturnal Mus musculus. Comparative analysis of single cell transcriptomic atlases between the two species revealed an increase in the relative abundance of bipolar and ganglion cell types supporting OFF and ON-OFF responses in Rhabdomys. Collaborators found that this shift towards OFF and transient responses extends into the dorsal lateral geniculate nucleus (dLGN) using high-density electrophysiological recordings. These measurements further revealed that Rhabdomys attains higher spatiotemporal acuity both by denser coverage of the visual scene and a selective expansion of elements of the visual code characterised by non-linear spatiotemporal summation, illustrating how sensory systems can adapt to different ecologies by adjusting the proportion of cell types comprising a neural circuit. Finally, we performed an integrative analysis of all 17 species. Collaborators were able to collect the sequencing datasets and found that the structure of the retina as well as all major retinal classes are conserved across all species using histology. We identify conserved molecular signatures of these cell classes, and find that transcriptomic variation across species is related to evolutionary distance. We built retinal atlases for each of these species and find that the degree of variation among cell types increased from the outer retina (photoreceptors) to the inner retina (RGCs), suggesting that evolution acts preferentially to shape the retinal output. To identify conserved cell types, we performed an integrative analysis to identify groups of related cell types that we name orthotypes, which we believe represent a cell type blueprint that derived from an early ancestral vertebrate. We identify 14 bipolar orthotypes, including an orthotype corresponding to the rod bipolar cell and a non-canonical OFF bipolar cell described in mice that has yet to be identified in other species, revealing that numerous cell types are shared across species based on conserved gene expression programs. RGC types across species were clustered into 21 orthotypes, including an orthotype containing orthologs of an evolutionarily ancient set of RGC types called intrinsically photosensitive RGCs. Importantly, we identified rodent orthologs of midget RGCs, which comprise more than 80% of RGCs in the human retina, subserve high-acuity vision, and were previously believed to be restricted to primates. We further argue this correspondence using molecular, functional, and morphological evidence and propose a correlation between the expansion of this cell type in primates and an increased role of the cortex in visual processing.

Modeling groundwater is challenging: it is not readily visible and is difficult to measure, with limited sets of observations available. Even though groundwater models can reproduce water table and head variations, considerable drift in modeled land surface states can nonetheless result from partially known geologic structure, errors in the input forcing fields, and imperfect Land Surface Model (LSM) parameterizations. These models frequently have biased results that are very different from observations. While many hydrologic groups are grappling with developing better models to resolve these issues, it is also possible to make models more robust through data assimilation of observation groundwater data. The goal of this project is to develop a methodology for high-resolution land surface model runs over large spatial region and improve hydrologic modeling through observation data assimilation, and then to apply this methodology to improve groundwater monitoring and banking.

The high-resolution LSM modeling in this dissertation shows that model physics performs well at these resolutions and actually leads to better modeling of water/energy budget terms. The overarching goal of assimilation methodology is to resolve the critical issue of how to improve groundwater modeling in LSMs that lack sub-surface parameterizations and also run them on global scales. To achieve this, the research in this dissertation has been divided into three parts. The first goal was to run a commonly used land surface model at hyper resolution (1 km or finer) and show that this improves the modeling results without breaking the model. The second goal was to develop an observation data assimilation methodology to improve the high-resolution model. The third was to show real-world applications of this methodology.

The need for improved accuracy is currently driving the development of hyper-resolution land surface models that can be implemented at a continental scale with resolutions of 1 km or finer. In Chapter 2, I describe our research incorporating fine-scale grid resolutions and surface data into the National Center for Atmospheric Research (NCAR) Community Land Model (CLM v4.0) for simulations at 1 km, 25 km, and 100 km resolution using 1 km soil and topographic information. Multi-year model runs were performed over the southwestern United States, including the entire state of California and the Colorado River basin. Results show changes in the total amount of CLM-modeled water storage and in the spatial and temporal distributions of water in snow and soil reservoirs, as well as in surface fluxes and energy balance. We also demonstrate the critical scales at which important hydrological processes--such as snow water equivalent, soil moisture content, and runoff--begin to more accurately capture the magnitude of the land water balance for the entire domain. This proves that grid resolution itself is also a critical component of accurate model simulations, and of hydrologic budget closure.

To inform future model progress, we compare simulation outputs to station and gridded observations of model fields. Although the higher grid resolution model is not driven by high-resolution forcing, grid resolution changes alone yield significant reductions in the Root Mean Square Error (RMSE) between model outputs and actual observations: the RMSE decreases by more than 35% for soil moisture, 36% for terrestrial water storage anomaly, 34% for sensible heat, and 12% for snow water equivalent. The results of a 100 m resolution simulation over a spatial sub-domain indicate that parameters such as drainage, runoff, and infiltration are significantly impacted when hillslope scales of ~100 meters or finer are considered. We further show how limitations in the current model physics, including no lateral flow between grid cells, can affect model simulation accuracy.

The results presented in Chapter 2 are encouraging, but also highlight the limitations in improving an LSM by only increasing spatial resolution of the model and the surface datasets. As was shown with the water table depth analysis, increasing model resolution cannot compensate for parameterization errors and lack of sub-surface information in CLM. However, this problem can be solved by providing additional information to the model in the form of water table depth via data assimilation.

In Chapter 3, I discuss the development and verification of a methodology for assimilating observed groundwater depth measurements from multiple wells into the high spatial resolution LSM. A kriging-based interpolation technique is employed to overcome the problem of spatially and temporally sparse observations, and the interpolated data is assimilated into the CLM4.0 at 1 km resolution in a test region in northern California. Direct insertion and Ensemble Adjusted Kalman Filter (EAKF) based techniques are used for assimilation with direct insertion, producing better results and demonstrating major improvement in the simulation of water table depth. The Linear Pearson correlation coefficient between the observed well data and the assimilated model is 0.810, as opposed to only 0.107 for the non-assimilated model. This improvement is most significant where the water table depth is greater than 5 m. Assimilation also improves soil moisture content, especially in the dry season when the water table is at its lowest. Other variables, including sensible heat flux, ground evaporation, infiltration, and runoff are also analyzed in order to quantify the effect of this assimilation methodology. Though the changes in these variables seem small, they can be very important in coupled models, and the improved simulation of groundwater in the assimilated model can explain the changes in these results.

This assimilation technique has been designed for use in regions with sparse and varied observation data, and it can be easily adapted to work in LSMs with a functional groundwater component. This gives us the capability to better model groundwater for the recent past and present, and also the potential to apply climate projections to probabilistically predict groundwater for future climate-change scenarios.

We have collaborated with Wellintel Inc. to implement our methodology on the ground using their observation data. We are in the process of setting up our model over a large region along the central California coast, where for the past few months Wellintel has implemented a pilot with measurements based on its water table depth measuring devices. The aim of this collaboration is to provide users with actionable water table depth data in and around their properties for the past, present, and near future. We are combining this methodology with Wellintel data to create a groundwater-management and groundwater-banking monitoring tool.

This is the first time that groundwater assimilation has been simulated in a high-resolution LSM, and as such this project provides proof-of-concept and application of a unique methodology that can be run at hyper resolution with data assimilation. The assimilation method is a very powerful tool that researchers can now apply to model land surface parameters much better than previously.

The explosion of protein diversity through domain rearrangements and inter-domain coupling supported the evolution of multicellular organisms. To perform the advanced signaling necessary for multicellularity, sometimes unrelated protein domains combined to form novel domain architectures that over time evolved tight mechanisms of allosteric coupling. One such protein, the voltage sensing phosphatase (VSP), developed a sophisticated mechanism of inter-domain coupling, which enabled cells to integrate changes in membrane potential into chemical changes in a class of secondary signaling lipids called phosphatidylinositol-phosphates (PIPs).

Phosphoinositol phosphate signaling lipids (PIPs) are important second messengers that regulate ion channels, transporters, cell motility and endo/exocytosis. PIP concentrations are controlled by enzymes, including VSP, which has broad specificity for a diverse class of PIPs. VSP is a novel lipid phosphatase, which contains a voltage sensing domain (VSD) homologous to voltage-gated ion channels, and a lipid phosphatase domain (PD). Until now it was not known what properties of the cytosolic PD were allosterically regulated by the membrane-associated VSD. Using a pair of new PIP sensors to monitor enzyme activity and voltage clamp fluorometry to monitor conformational changes in the VSD, it becomes clear the Ciona intestinalis VSP (Ci-VSP) has two distinct voltage regulated enzyme active states: a faster low-voltage state with substrate preference for PIP3 and a slower high-voltage state with preference for PIP2. This novel 2-step allosteric switch for enzyme specificity enables membrane potential to function as an allosteric effector that dynamically regulates PIP concentration.

In this work, it is show that two unrelated domains, a VSD from voltage dependent ion channels and a lipid PD homologous to protein tyrosine phosphatases evolved a tight mechanism of allosteric regulation that transduces fluctuations in membrane potential into changes in the enzyme selectivity of a novel lipid phosphatase. This regulation of active site specificity in the PD by an allosteric effector domain represents a significant advancement in our understanding of allosteric regulation, which has previously been restricted to control of activity on only one type of substrate.