Forward and Inverse Analysis of Chemical Transport Models

Assessing the discrepancy between modeled and observed distributions of aerosols is a persistent problem on many scales. Tools for analyzing the evolution of aerosol size distributions using the adjoint method are presented in idealized box model calculations. The ability to recover information about aerosol growth rates and initial size distributions is assessed given a range of simulated observations of evolving systems. While such tools alone could facilitate analysis of chamber measurements, improving estimates of aerosol sources on regional and global scales requires explicit consideration of many additional chemical and physical processes that govern secondary formation of atmospheric aerosols from emissions of gas-phase precursors. The adjoint of the global chemical transport model GEOS-Chem is derived, affording detailed analysis of the relationship between gas-phase aerosol precursor emissions (SOx, NOx, and NH3) and the subsequent distributions of sulfate - ammonium - nitrate aerosol. Assimilation of surface measurements of sulfate and nitrate aerosol is shown to provide valuable constraints on emissions of ammonia. Adjoint sensitivities are used to propose strategies for air quality control, suggesting, for example, that reduction of SOx emissions in the summer and NH3 emissions in the winter would most effectively reduce non-attainment of aerosol air quality standards. The ability of this model to estimate global distributions of carbonaceous aerosol is also addressed. Based on new yield data from environmental chamber studies, mechanisms for incorporating the dependence of secondary organic aerosol (SOA) formation on NOx concentrations are developed for use in global models. When NOx levels are appropriately accounted for, it is demonstrated that sources such as isoprene and aromatics, previously neglected as sources of aerosol in global models, significantly contribute to predicted SOA burdens downwind of polluted areas (owing to benzene and toluene) and in the free troposphere (owing to isoprene)

Discrete Adjoints: Theoretical Analysis, Efficient Computation, and Applications

The technique of automatic differentiation provides directional derivatives and discrete adjoints with working accuracy. A complete complexity analysis of the basic modes of automatic differentiation is available. Therefore, the research activities are focused now on different aspects of the derivative calculation, as for example the efficient implementation by exploitation of structural information, studies of the theoretical properties of the provided derivatives in the context of optimization problems, and the development and analysis of new mathematical algorithms based on discrete adjoint information. According to this motivation, this habilitation presents an analysis of different checkpointing strategies to reduce the memory requirement of the discrete adjoint computation. Additionally, a new algorithm for computing sparse Hessian matrices is presented including a complexity analysis and a report on practical experiments. Hence, the first two contributions of this thesis are dedicated to an efficient computation of discrete adjoints. The analysis of discrete adjoints with respect to their theoretical properties is another important research topic. The third and fourth contribution of this thesis focus on the relation of discrete adjoint information and continuous adjoint information for optimal control problems. Here, differences resulting from different discretization strategies as well as convergence properties of the discrete adjoints are analyzed comprehensively. In the fifth contribution, checkpointing approaches that are successfully applied for the computation of discrete adjoints, are adapted such that they can be used also for the computation of continuous adjoints. Additionally, the fifth contributions presents a new proof of optimality for the binomial checkpointing that is based on new theoretical results. Discrete adjoint information can be applied for example for the approximation of dense Jacobian matrices. The development and analysis of new mathematical algorithms based on these approximate Jacobians is the topic of the sixth contribution. Is was possible to show global convergence to first-order critical points for a whole class of trust-region methods. Here, the usage of inexact Jacobian matrices allows a considerable reduction of the computational complexity

ANISORROPIA: the adjoint of the aerosol thermodynamic model ISORROPIA

We present the development of ANISORROPIA, the discrete adjoint of the ISORROPIA thermodynamic equilibrium model that treats the Na<sup>+</sup>-SO<sub>4</sub><sup>2−</sup>- HSO<sub>4</sub><sup>−</sup>-NH<sub>4</sub><sup>+</sup> -NO<sub>3</sub><sup>−</sup>-Cl<sup>−</sup>-H<sub>2</sub>O aerosol system, and we demonstrate its sensitivity analysis capabilities. ANISORROPIA calculates sensitivities of an inorganic species in aerosol or gas phase with respect to the total concentrations of each species present with less than a two-fold increase in computational time over the concentration calculations. Due to the highly nonlinear and discontinuous solution surface of ISORROPIA, evaluation of the adjoint required a new, complex-variable version of the model, which determines first-order sensitivities with machine precision and avoids cancellation errors arising from finite difference calculations. The adjoint is verified over an atmospherically relevant range of concentrations, temperature, and relative humidity. We apply ANISORROPIA to recent field campaign results from Atlanta, GA, USA, and Mexico City, Mexico, to characterize the inorganic aerosol sensitivities of these distinct urban air masses. The variability in the relationship between fine mode inorganic aerosol mass and precursor concentrations shown has important implications for air quality and climate

Efficient Output-Based Adaptation Mechanics for High-Order Computational Fluid Dynamics Methods

As numerical simulations are applied to more complex and large-scale problems, solution verification becomes increasingly important in ensuring accuracy of the computed results. In addition, although improvements in computer hardware have brought expensive simulations within reach, efficiency is still paramount, especially in the context of design optimization and uncertainty quantification. This thesis addresses both of these needs through contributions to solution-based adaptive algorithms, in which the discretization is modified through a feedback of solution error estimates so as to improve the accuracy. In particular, new methods are developed for two discretizations relevant to Computational Fluid Dynamics: the Active Flux method and the discontinuous Galerkin method. For the Active Flux method, which is fully-discrete third-order discretization, both the discrete and continuous adjoint methods are derived and used to drive mesh (h) refinement and dynamic node movement, also known as $r$ adaptation. For the discontinuous Galerkin method, which is an arbitrary-order finite-element discretization, efficiency improvements are presented for computing and using error estimates derived from the discrete adjoint, and a new $r$-adaptation strategy is presented for unsteady problems. For both discretizations, error estimate efficacy and adaptive efficiency improvements are shown relative to other strategies.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/144065/1/dkaihua_1.pd

Reexamination of Quantum Bit Commitment: the Possible and the Impossible

Bit commitment protocols whose security is based on the laws of quantum mechanics alone are generally held to be impossible. In this paper we give a strengthened and explicit proof of this result. We extend its scope to a much larger variety of protocols, which may have an arbitrary number of rounds, in which both classical and quantum information is exchanged, and which may include aborts and resets. Moreover, we do not consider the receiver to be bound to a fixed "honest" strategy, so that "anonymous state protocols", which were recently suggested as a possible way to beat the known no-go results are also covered. We show that any concealing protocol allows the sender to find a cheating strategy, which is universal in the sense that it works against any strategy of the receiver. Moreover, if the concealing property holds only approximately, the cheat goes undetected with a high probability, which we explicitly estimate. The proof uses an explicit formalization of general two party protocols, which is applicable to more general situations, and a new estimate about the continuity of the Stinespring dilation of a general quantum channel. The result also provides a natural characterization of protocols that fall outside the standard setting of unlimited available technology, and thus may allow secure bit commitment. We present a new such protocol whose security, perhaps surprisingly, relies on decoherence in the receiver's lab.Comment: v1: 26 pages, 4 eps figures. v2: 31 pages, 5 eps figures; replaced with published version; title changed to comply with puzzling Phys. Rev. regulations; impossibility proof extended to protocols with infinitely many rounds or a continuous communication tree; security proof of decoherence monster protocol expanded; presentation clarifie

Sensitivity analysis of thermoacoustic instability with adjoint Helmholtz solvers

Gas turbines and rocket engines sometimes suffer from violent oscillations caused by feedback between acoustic waves and flames in the combustion chamber. These are known as thermoacoustic oscillations and they often occur late in the design process. Their elimination usually requires expensive tests and re-design. Full scale tests and laboratory scale experiments show that these oscillations can usually be stabilized by making small changes to the system. The complication is that, while there is often just one unstable natural oscillation (eigenmode), there are very many possible changes to the system. The challenge is to identify the optimal change systematically, cheaply, and accurately. This paper shows how to evaluate the sensitivities of a thermoacoustic eigenmode to all possible system changes with a single calculation by applying adjoint methods to a thermoacoustic Helmholtz solver. These sensitivities are calculated here with finite difference and finite element methods, in the weak form and the strong form, with the discrete adjoint and the continuous adjoint, and with a Newton method applied to a nonlinear eigenvalue problem and an iterative method applied to a linear eigenvalue problem. This is the first detailed comparison of adjoint methods applied to thermoacoustic Helmholtz solvers. Matlab codes are provided for all methods and all figures so that the techniques can be easily applied and tested. This paper explains why the finite difference of the strong form equations with replacement boundary conditions should be avoided and why all of the other methods work well. Of the other methods, the discrete adjoint of the weak form equations is the easiest method to implement; it can use any discretization and the boundary conditions are straightforward. The continuous adjoint is relatively easy to implement but requires careful attention to boundary conditions. The Summation by Parts finite difference of the strong form equations with a Simultaneous Approximation Term for the boundary conditions (SBP--SAT) is more challenging to implement, particularly at high order or on non-uniform grids. Physical interpretation of these results shows that the well-known Rayleigh criterion should be revised for a linear analysis. This criterion states that thermoacoustic oscillations will grow if heat release rate oscillations are sufficiently in phase with pressure oscillations. In fact, the criterion should contain the adjoint pressure rather than the pressure. In self-adjoint systems the two are equivalent. In non-self-adjoint systems, such as all but a special case of thermoacoustic systems, the two are different. Finally, the sensitivities of the growth rate of oscillations to placement of a hot or cold mesh are calculated, simply by multiplying the feedback sensitivities by a number. These sensitivities are compared successfully with experimental results. With the same technique, the influence of the viscous and thermal acoustic boundary layers is found to be negligible, while the influence of a Helmholtz resonator is found, as expected, to be considerable

Optimal anticipatory control as a theory of motor preparation

Supported by a decade of primate electrophysiological experiments, the prevailing theory of neural motor control holds that movement generation is accomplished by a preparatory process that progressively steers the state of the motor cortex into a movement-specific optimal subspace prior to movement onset. The state of the cortex then evolves from these optimal subspaces, producing patterns of neural activity that serve as control inputs to the musculature. This theory, however, does not address the following questions: what characterizes the optimal subspace and what are the neural mechanisms that underlie the preparatory process? We address these questions with a circuit model of movement preparation and control. Specifically, we propose that preparation can be achieved by optimal feedback control (OFC) of the cortical state via a thalamo-cortical loop. Under OFC, the state of the cortex is selectively controlled along state-space directions that have future motor consequences, and not in other inconsequential ones. We show that OFC enables fast movement preparation and explains the observed orthogonality between preparatory and movement-related monkey motor cortex activity. This illustrates the importance of constraining new theories of neural function with experimental data. However, as recording technologies continue to improve, a key challenge is to extract meaningful insights from increasingly large-scale neural recordings. Latent variable models (LVMs) are powerful tools for addressing this challenge due to their ability to identify the low-dimensional latent variables that best explain these large data sets. One shortcoming of most LVMs, however, is that they assume a Euclidean latent space, while many kinematic variables, such as head rotations and the configuration of an arm, are naturally described by variables that live on non-Euclidean latent spaces (e.g., SO(3) and tori). To address this shortcoming, we propose the Manifold Gaussian Process Latent Variable Model, a method for simultaneously inferring nonparametric tuning curves and latent variables on non-Euclidean latent spaces. We show that our method is able to correctly infer the latent ring topology of the fly and mouse head direction circuits.This work was supported by a Trinity-Henry Barlow scholarship and a scholarship from the Ministry of Education, ROC Taiwan