Accurate prediction of tsunami waves requires complete boundary and initial condition data, coupled with the appropriate mathematical model. However, necessary data is often missing or inaccurate, and may not have sufficient resolution to capture the dynamics of such nonlinear waves accurately. In this thesis we demonstrate that variational data assimilation for the continuous shallow water equations (SWE) is a feasible approach for recovering both initial conditions and bathymetry data from sparse observations. Using a Sadourny finite-difference finite volume discretisation for our numerical implementation, we show that convergence to true initial conditions can be achieved for sparse observations arranged in multiple configurations, for both isotropic and anisotropic initial conditions, and with realistic bathymetry data in two dimensions. We demonstrate that for the 1-D SWE, convergence to exact bathymetry is improved by including a low-pass filter in the data assimilation algorithm designed to remove scale-scale noise, and with a larger number of observations. A necessary condition for a relative L2 error less than 10% in bathymetry reconstruction is that the amplitude of the initial conditions be less than 1% of the bathymetry height. We perform Second Order Adjoint Sensitivity Analysis and Global Sensitivity Analysis to comprehensively assess the sensitivity of the surface wave to errors in the bathymetry and perturbations in the observations. By demonstrating low sensitivity of the surface wave to the reconstruction error, we found that reconstructing the bathymetry with a relative error of about 10% is sufficiently accurate for surface wave modelling in most cases. These idealised results with simplified 2-D and 1-D geometry are intended to be a first step towards more physically realistic settings, and can be used in tsunami modelling to (i) maximise accuracy of tsunami prediction through sufficiently accurate reconstruction of the necessary data, (ii) attain a priori knowledge of how different bathymetry and initial conditions can affect the surface wave error, and (iii) provide insight on how these can be mitigated through optimal configuration of the observations. / Thesis / Candidate in Philosophy

Many important engineering phenomena such as turbulent flow, fluid-structure interactions, and climate diagnostics are chaotic and sensitivity analysis of such systems is a challenging problem. Computational methods have been proposed to accurately and efficiently estimate the sensitivity analysis of these systems which is of great scientific and engineering interest. In this thesis, a new approach is applied to compute the direct and adjoint sensitivities of time-averaged quantities defined from the chaotic response of the Lorenz system and the double pendulum system. A stabilized time-integrator with adaptive time-step control is used to maintain stability of the sensitivity calculations. A study of convergence of a quantity of interest and its square is presented. Results show that the approach computes accurate sensitivity values with a computational cost that is multiple orders-of-magnitude lower than competing approaches based on least-squares-shadowing approach.

<p> This thesis addresses recent trends and developments of the adjoint-variable method (AVM) for microwave structures with the time-domain transmission-line modeling (TD-TLM) method. </p> <p> Design sensitivity analysis of high-frequency (HF) structures is concerned with estimating the sensitivity of the response with respect to the design parameters. This information is essential at different stages of the design cycle such as the optimization, tolerance analysis, and yield analysis. </p> <p> Traditional approaches of sensitivity calculations involve estimating the sensitivities thought fmite-difference approximations. They suffer from formidable simulation time, as the full-wave analysis of practical HF structure requires extensive computational time. For a structure with N design parameters, at least N+l system analyses are required to extract the design response and its sensitivities. The adjoint variable method, on the other hand, supplies the sensitivity information in a very efficient way. Using at most two system analysis, the algorithm provides the design responses and its sensitivities, regardless of the number of the design parameters. </p> <p> In this thesis two contributions have been achieved which aims at enhancing the efficiency of the TLM-A VM framework. The first contribution is a reformulation of the AVM. This reformulation results in casting both the original and the adjoint systems in mathematically identical forms. It is shown that both systems can thus be modeled using a single TLM simulator with the only difference in the excitation. The second contribution focuses on generalizing the A VM algorithm by employing it for more advanced TLM nodes. The compatibility of the symmetrical condensed node (SCN) with the AVM algorithm has been verified in previous work for a general 3-D problem. Here, this is extended to include the hybrid symmetrical condensed node (HSCN), which is more efficient in terms of memory saving and simulation time. The new approaches are all illustrated through sensitivity estimation of different waveguide structures. </p> / Thesis / Master of Applied Science (MASc)

The Community Multiscale Air Quality (CMAQ) system is the Environmental Protection Agency's main modeling tool for atmospheric pollution studies. CMAQ-ADJ, the adjoint model of CMAQ, offers new analysis capabilities such as receptor-oriented sensitivity analysis and chemical data assimilation. This thesis presents the construction, validation, and properties of new adjoint modules in CMAQ, and illustrates their use in sensitivity analyses and data assimilation experiments. The new module of discrete adjoint of advection is implemented with the aid of automatic differentiation tool (TAMC) and is fully validated by comparing the adjoint sensitivities with finite difference values. In addition, adjoint sensitivity with respect to boundary conditions and boundary condition scaling factors are developed and validated in CMAQ. To investigate numerically the impact of the continuous and discrete advection adjoints on data assimilation, various four dimensional variational (4D-Var) data assimilation experiments are carried out with the 1D advection PDE, and with CMAQ advection using synthetic and real observation data. The results show that optimization procedure gives better estimates of the reference initial condition and converges faster when using gradients computed by the continuous adjoint approach. This counter-intuitive result is explained using the nonlinearity properties of the piecewise parabolic method (the numerical discretization of advection in CMAQ). Data assimilation experiments are carried out using real observation data. The simulation domain encompasses Texas and the simulation period is August 30 to September 1, 2006. Data assimilation is used to improve both initial and boundary conditions. These experiments further validate the tools developed in this thesis. / Master of Science

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Nos dias atuais a Ãgua se constitui em um bem natural que limita o desenvolvimento socioeconÃmico e, atà mesmo, a subsistÃncia da populaÃÃo. Como tentativa de minimizar o problema da escassez de Ãgua tem-se utilizado a explotaÃÃo da Ãgua subterrÃnea. Entretanto, esse crescimento da utilizaÃÃo de Ãguas subterrÃneas foi feito de forma desordenada e com a construÃÃo inadequada de poÃos. Essa prÃtica acabou por colocar em risco a qualidade das Ãguas subterrÃneas. Assim, a gestÃo dos recursos hÃdricos subterrÃneos tem se tornado um grande desafio. Essa tese propÃe o desenvolvimento um modelo para a simulaÃÃo de fluxo hÃdrico e de transporte de massa para problemas transientes em aqÃÃferos costeiros sujeitos à intrusÃo salina, por meio do desenvolvimento de um modelo numÃrico. Em seguida à desenvolvida uma anÃlise de sensitividade com o objetivo de possibilitar, atravÃs do melhor conhecimento dos parÃmetros locais e suas influÃncias, uma melhor adequaÃÃo do modelo à realidade. / Today the water is a natural well which limits the socioeconomic development and even the subsistence of the population. An attempt to minimize the problem of water scarcity has used the farming of groundwater. However, this growth of the use of groundwater was done inappropriately and with inadequate wells construction. This practice was eventually put at risk the quality of groundwater. Thus, the management of groundwater resources has become a major challenge. This thesis proposes developing a model for the simulation of water flow and mass transport for transient problems in coastal aquifers subject to saline intrusion, through the development of a numerical model. Then we developed a sensitivity analysis with the goal of enabling through better knowledge of local parameters and their influences, a best fit of model to reality.

Optical fibers play a vital role in modern telecommunication systems and networks. An optical fiber link imposes some linear and nonlinear distortions on the propagating light-wave signal due to the inherent dispersive nature and nonlinear behavior of the fiber. These distortions impede the increasing demand for higher data rate transmission over longer distances. Developing efficient and computationally non-expensive digital signal processing (DSP) techniques to effectively compensate for the fiber impairments is therefore essential and of preeminent importance. This thesis proposes two DSP-based approaches for mitigating the induced distortions in short-reach and long-haul fiber-optic communication systems. The first approach introduces a powerful digital nonlinear feed-forward equalizer (NFFE), exploiting multilayer artificial neural network (ANN). The proposed ANN-NFFE mitigates nonlinear impairments of short-haul optical fiber communication systems, arising due to the nonlinearity introduced by direct photo-detection. In a direct detection system, the detection process is nonlinear due to the fact that the photo-current is proportional to the absolute square of the electric field intensity. The proposed equalizer provides the most efficient computational cost with high equalization performance. Its performance is comparable to the benchmark compensation performance achieved by maximum-likelihood sequence estimator. The equalizer trains an ANN to act as a nonlinear filter whose impulse response removes the intersymbol interference (ISI) distortions of the optical channel. Owing to the proposed extensive training of the equalizer, it achieves the ultimate performance limit of any feed-forward equalizer. The performance and efficiency of the equalizer are investigated by applying it to various practical short-reach fiber-optic transmission system scenarios. These scenarios are extracted from practical metro/media access networks and data center applications. The obtained results show that the ANN-NFFE compensates for the received BER degradation and significantly increases the tolerance to the chromatic dispersion distortion. The second approach is devoted for blindly combating impairments of long-haul fiber-optic systems and networks. A novel adjoint sensitivity analysis (ASA) approach for the nonlinear Schrödinger equation (NLSE) is proposed. The NLSE describes the light-wave propagation in optical fiber communication systems. The proposed ASA approach significantly accelerates the sensitivity calculations in any fiber-optic design problem. Using only one extra adjoint system simulation, all the sensitivities of a general objective function with respect to all fiber design parameters are estimated. We provide a full description of the solution to the derived adjoint problem. The accuracy and efficiency of our proposed algorithm are investigated through a comparison with the accurate but computationally expensive central finite-differences (CFD) approach. Numerical simulation results show that the proposed ASA algorithm has the same accuracy as the CFD approach but with a much lower computational cost. Moreover, we propose an efficient, robust, and accelerated adaptive digital back propagation (A-DBP) method based on adjoint optimization technique. Provided that the total transmission distance is known, the proposed A-DBP algorithm blindly compensates for the linear and nonlinear distortions of point-to-point long-reach optical fiber transmission systems or multi-point optical fiber transmission networks, without knowing the launch power and channel parameters. The NLSE-based ASA approach is extended for the sensitivity analysis of general multi-span DBP model. A modified split-step Fourier scheme method is introduced to solve the adjoint problem, and a complete analysis of its computational complexity is studied. An adjoint-based optimization (ABO) technique is introduced to significantly accelerate the parameters extraction of the A-DBP. The ABO algorithm utilizes a sequential quadratic programming (SQP) technique coupled with the extended ASA algorithm to rapidly solve the A-DBP training problem and optimize the design parameters using minimum overhead of extra system simulations. Regardless of the number of A-DBP design parameters, the derivatives of the training objective function with respect to all parameters are estimated using only one extra adjoint system simulation per optimization iterate. This is contrasted with the traditional finite-difference (FD)-based optimization methods whose sensitivity analysis calculations cost per iterate scales linearly with the number of parameters. The robustness, performance, and efficiency of the proposed A-DBP algorithm are demonstrated through applying it to mitigate the distortions of a 4-span optical fiber communication system scenario. Our results show that the proposed A-DBP achieves the optimal compensation performance obtained using an ideal fine-mesh DBP scheme utilizing the correct channel parameters. Compared to A-DBPs trained using SQP algorithms based on forward, backward, and central FD approaches, the proposed ABO algorithm trains the A-DBP with 2.02 times faster than the backward/forward FD-based optimizers, and with 3.63 times faster than the more accurate CFD-based optimizer. The achieved gain further increases as the number of design parameters increases. A coarse-mesh A-DBP with less number of spans is also adopted to significantly reduce the computational complexity, achieving compensation performance higher than that obtained using the coarse-mesh DBP with full number of spans. / Thesis / Doctor of Philosophy (PhD) / This thesis proposes two powerful and computationally efficient digital signal processing (DSP)-based techniques, namely, artificial neural network nonlinear feed forward equalizer (ANN-NFFE) and adaptive digital back propagation (A-DBP) equalizer, for mitigating the induced distortions in short-reach and long-haul fiber-optic communication systems, respectively. The ANN-NFFE combats nonlinear impairments of direct-detected short-haul optical fiber communication systems, achieving compensation performance comparable to the benchmark performance obtained using maximum-likelihood sequence estimator with much lower computational cost. A novel adjoint sensitivity analysis (ASA) approach is proposed to significantly accelerate sensitivity analyses of fiber-optic design problems. The A-DBP exploits a gradient-based optimization method coupled with the ASA algorithm to blindly compensate for the distortions of coherent-detected fiber-optic communication systems and networks, utilizing the minimum possible overhead of performed system simulations. The robustness and efficiency of the proposed equalizers are demonstrated using numerical simulations of varied examples extracted from practical optical fiber communication systems scenarios.

<p>In this thesis, inverse problems related to microwave imaging and microwave component design are investigated. Our contribution in microwave imaging for breast tumor detection can be divided into two parts. In the first part, a vectorial 3D near-field microwave holography is proposed which is an improvement over the existing holography algorithms. In the second part, a simple and fast post-processing algorithm based on the principle of blind de-convolution is proposed for removing the integration effect of the antenna aperture. This allows for the data collected by the antennas to be used in 3D holography reconstruction. The blind deconvolution algorithm is a well-known algorithm in signal processing and our contribution here is its adaptation to microwave data processing.</p> <p>Second, a procedure for accelerating the space-mapping optimization process is presented. By exploiting both fine- and surrogate-model sensitivity information, a good mapping between the two model spaces is efficiently obtained. This results in a significant speed-up over direct gradient-based optimization of the original fine model and enhanced performance compared with other space-mapping approaches. Our approach utilizes commercially available software with adjoint-sensitivity analysis capabilities.</p> / Thesis / Master of Applied Science (MASc)

The overall goals of this dissertation are to advance the field of chemical data assimilation, and to develop efficient computational tools that allow the atmospheric science community benefit from state of the art assimilation methodologies. Data assimilation is the procedure to combine data from observations with model predictions to obtain a more accurate representation of the state of the atmosphere. As models become more complex, determining the relationships between pollutants and their sources and sinks becomes computationally more challenging. The construction of an adjoint model ( capable of efficiently computing sensitivities of a few model outputs with respect to many input parameters ) is a difficult, labor intensive, and error prone task. This work develops adjoint systems for two of the most widely used chemical transport models: Harvard's GEOS-Chem global model and for Environmental Protection Agency's regional CMAQ regional air quality model. Both GEOS-Chem and CMAQ adjoint models are now used by the atmospheric science community to perform sensitivity analysis and data assimilation studies. Despite the continuous increase in capabilities, models remain imperfect and models alone cannot provide accurate long term forecasts. Observations of the atmospheric composition are now routinely taken from sondes, ground stations, aircraft, and satellites, etc. This work develops three and four dimensional variational data assimilation capabilities for GEOS-Chem and CMAQ which allow to estimate chemical states that best fit the observed reality. Most data assimilation systems to date use diagonal approximations of the background covariance matrix which ignore error correlations and may lead to inaccurate estimates. This dissertation develops computationally efficient representations of covariance matrices that allow to capture spatial error correlations in data assimilation. Not all observations used in data assimilation are of equal importance. Erroneous and redundant observations not only affect the quality of an estimate but also add unnecessary computational expense to the assimilation system. This work proposes techniques to quantify the information content of observations used in assimilation; information-theoretic metrics are used. The four dimensional variational approach to data assimilation provides accurate estimates but requires an adjoint construction, and uses considerable computational resources. This work studies versions of the four dimensional variational methods (Quasi 4D-Var) that use approximate gradients and are less expensive to develop and run. Variational and Kalman filter approaches are both used in data assimilation, but their relative merits and disadvantages in the context of chemical data assimilation have not been assessed. This work provides a careful comparison on a chemical assimilation problem with real data sets. The assimilation experiments performed here demonstrate for the first time the benefit of using satellite data to improve estimates of tropospheric ozone. / Ph. D.

<div>The rapid development of topology optimization in photonics has greatly expanded the number of photonic structures with extraordinary performance. The optimization is usually solved by using a gradient-based optimization algorithm, where gradients are evaluated by the adjoint sensitivity analysis. While the adjoint sensitivity analysis has been demonstrated to provide reliable gradients for designs of dielectrics, there has not been too much success in plasmonics. The difficulty of obtaining accurate field solutions near sharp edges and corners in plasmonic structures, and the strong field enhancement jointly increase the numerical error of gradients, leading to failure of convergence for any gradient-based algorithm. </div><div> </div><div>We present a new method of calculating accurate sensitivity with the FDTD method by direct differentiation of the time-marching system in frequency domain. This new method supports general frequency-domain objective functions, does not relay on implementation details of the FDTD method, works with general isotropic materials and can be easily incorporated into both level-set-based and density-based topology optimizations. The method is demonstrated to have superior accuracy compared to the traditional continuous sensitivity. Next, we present a framework to carry out density-based topology optimization using our new sensitivity formula. We use the non-linear material interpolation to counter the rough landscape of plasmonics, adopt the filteringand-projection regularization to ensure manufacturability and perform the optimization with a continuation scheme to improve convergence. </div><div> </div><div>We give two examples involving reconstruction of near fields of plasmonic structures to illustrate the robustness of the sensitivity formula and the optimization framework. In the end, we apply our method to generate a rectangular temperature profile in the recording medium of the HAMR system. </div>