Turbulent flows occur in various fields and are a central, yet an extremely complex, topic in fluid dynamics. Understanding the behaviour of fluids is vital for multiple research areas including, but not limited to: biological models, weather prediction, and engineering design models for automobiles and aircraft. In this thesis, we study a number of fundamental problems that arise in 2D turbulent flows, using the 2D Navier-Stokes system. Introducing optimization techniques for systems described by partial differential equations (PDE), we frame these problems such that they can be solved using computational methods. We utilize adjoint calculus to build the computational framework to be implemented in an iterative gradient flow procedure, using the "optimize-then-discretize" approach. Pseudospectral methods are employed for solving PDEs in a numerically efficient manner. The use of optimization methods together with computational mathematics in this work provides an illuminating perspective on fluid mechanics. We first apply these techniques to better understand enstrophy dissipation in 2D Navier-Stokes flows, in the limit of vanishing viscosity. By defining an optimization problem to determine optimal initial conditions, multiple branches of local maximizers were obtained each corresponding to a different mechanism producing maximum enstrophy dissipation. Viewing this quantity as a function of viscosity revealed quantitative agreement with an analytic bound, demonstrating the sharpness of this bound. We also introduce an extension of this problem, where enstrophy dissipation is maximized in the context of kinetic theory using the Boltzmann equation. Secondly, these PDE-constrained optimization techniques were used to probe the fundamental limitations on the performance of the Leith eddy-viscosity closure model for 2D Large-Eddy Simulations of the Navier-Stokes system. Obtained by solving an optimization problem with a non-standard structure, the results demonstrate the optimal eddy viscosities do not converge to a well-defined limit as regularization and discretization parameters are refined, hence the problem of determining an optimal eddy viscosity is ill-posed. Further extending the problem of finding optimal eddy-viscosity closures, we consider imposing an additional nonlinear constraint on the control variable in the problem, in the form of requiring the time-averaged enstrophy be preserved. To address this problem in a novel way, we employ adjoint calculus to characterize a subspace tangent to the constraint manifold, which allows one to approximately enforce the constraint. Not only do we demonstrate that this produces better results when compared to the case without constraints, but this also provides a flexible computational framework for approximate enforcement of general nonlinear constraints. Lastly in this thesis, we introduce an optimization problem to study the Kolmogorov-Richardson energy cascade, where a pathway towards solutions is outlined. / Thesis / Doctor of Philosophy (PhD)

Using derivative based numerical optimization routines to solve optimization problems governed by partial differential equations (PDEs) with uncertain coefficients is computationally expensive due to the large number of PDE solves required at each iteration. In this thesis, I present an adaptive stochastic collocation framework for the discretization and numerical solution of these PDE constrained optimization problems. This adaptive approach is based on dimension adaptive sparse grid interpolation and employs trust regions to manage the adapted stochastic collocation models. Furthermore, I prove the convergence of sparse grid collocation methods applied to these optimization problems as well as the global convergence of the retrospective trust region algorithm under weakened assumptions on gradient inexactness. In fact, if one can bound the error between actual and modeled gradients using reliable and efficient a posteriori error estimators, then the global convergence of the proposed algorithm follows. Moreover, I describe a high performance implementation of my adaptive collocation and trust region framework using the C++ programming language with the Message Passing interface (MPI). Many PDE solves are required to accurately quantify the uncertainty in such optimization problems, therefore it is essential to appropriately choose inexpensive approximate models and large-scale nonlinear programming techniques throughout the optimization routine. Numerical results for the adaptive solution of these optimization problems are presented.

In this dissertation we investigate methods of solving various optimization problems with PDE constraints, i.e. optimization problems that have a system of partial differential equations in the set of constraints, and develop frameworks for a number of practically inspired problems that were not considered in the literature before. Such problems arise in areas like fluid mechanics, chemical engineering, finance, and other areas where a physical system needs to be optimized. In most of the literature sources on PDE-constrained optimization only relatively simple systems of PDEs are considered, they are either linear, or the size of the system is small. On the contrary, in our case, we search for solution methods to problems constrained by large (8 to 10 equations) and non-linear systems of PDEs. More specifically, in the first part of the dissertation we consider a multiphysics phenomenon where electromagnetic and mechanical fields interact within an electrically conductive body, and develop the optimization framework to find an efficient way to control one field through another. We also apply the developed PDE-constrained optimization framework to a financial options portfolio optimization problem, and more specifically consider the case that to the best of our knowledge is not covered in the literature.

This project concerns the computational solution of inverse problems formulated as partial differential equation (PDE)-constrained optimization problems with interior data. The areas addressed are twofold. First, we present a novel software architecture designed to solve inverse problems constrained by an elliptic system of PDEs. These generally require the solution of forward and adjoint problems, evaluation of the objective function, and computation of its gradient, all of which are approximated numerically using finite elements. The creation of specialized "layered"' elements to perform these tasks leads to a modular software structure that improves code maintainability and promotes functional interoperability between different software components. Second, we address issues related to forward model definition in the presence of boundary condition (BC) uncertainty. We propose two variational formulations to accommodate that uncertainty: (a) a Bayesian formulation that assumes Gaussian measurement noise and a minimum strain energy prior, and (b) a Lagrangian formulation that is completely free of displacement and traction BCs. This work is motivated by applications in the field of biomechanical imaging, where the mechanical properties within soft tissues are inferred from observations of tissue motion. In this context, the constraint PDE is well accepted, but considerable uncertainty exists in the BCs. The approaches developed here are demonstrated on a variety of applications, including simulated and experimental data. We present modulus reconstructions of individual cells, tissue-mimicking phantoms, and breast tumors.

No description available.

Currently, researchers at Sandia National Laboratories are creating software that is designed to determine the source of a toxic release given sensor readings of the toxin concentration at fixed locations in the building. One of the most important concerns in solving such problems is computation time since even a crude approximation to the source, if found in a timely manner, will give emergency personnel the chance to take appropriate actions to contain the substance. The manner in which the toxin spreads depends on the air flow within the building. Due to the turbulence in the air flow, it is necessary to calculate the flow field on a fine mesh. Unfortunately, using a fine mesh for every calculation in this problem may result in prohibitively long computation times when other features are incorporated into the model. The goal of this thesis is to reduce the computation time required by the software mentioned above by applying two different mesh coarsening strategies after the flow field is computed. The first of these strategies is to use a uniformly coarse mesh and the second is to use our knowledge of the air flow in the building to construct an adaptive mesh. The objective of the latter strategy is to use a fine mesh only in areas where it is absolutely necessary, i.e., in areas where there is a great change in the flow field.

We develop novel numerical methods for optimization problems subject to constraints given by nonlinear hyperbolic systems of conservation and balance laws in one space dimension. These types of control problems arise in a variety of applications, in which inverse problems for the corresponding initial value problems are to be solved. The optimization method can be seen as a block Gauss-Seidel iteration. The optimization requires one to numerically solve the hyperbolic system forward in time and the corresponding linear adjoint system backward in time. We test the optimization method on a number of control problems constrained by nonlinear hyperbolic systems of PDEs with both smooth and discontinuous prescribed terminal states. The theoretical foundation of the introduced scheme is provided in the case of scalar hyperbolic conservation laws on an unbounded domain with a strictly convex flux. In addition, we empirically demonstrate that using a higher-order temporal discretization helps to substantially improve both the efficiency and accuracy of the overall numerical method. / acase@tulane.edu

Projecting the ice sheets' contribution to sea-level rise is difficult because of the complexity of accurately modeling ice sheet dynamics for the full polar ice sheets, because of the uncertainty in key, unobservable parameters governing those dynamics, and because quantifying the uncertainty in projections is necessary when determining the confidence to place in them. This work presents the formulation and solution of the Bayesian inverse problem of inferring, from observations, a probability distribution for the basal sliding parameter field beneath the Antarctic ice sheet. The basal sliding parameter is used within a high-fidelity nonlinear Stokes model of ice sheet dynamics. This model maps the parameters "forward" onto a velocity field that is compared against observations. Due to the continental-scale of the model, both the parameter field and the state variables of the forward problem have a large number of degrees of freedom: we consider discretizations in which the parameter has more than 1 million degrees of freedom. The Bayesian inverse problem is thus to characterize an implicitly defined distribution in a high-dimensional space. This is a computationally demanding problem that requires scalable and efficient numerical methods be used throughout: in discretizing the forward model; in solving the resulting nonlinear equations; in solving the Bayesian inverse problem; and in propagating the uncertainty encoded in the posterior distribution of the inverse problem forward onto important quantities of interest. To address discretization, a hybrid parallel adaptive mesh refinement format is designed and implemented for ice sheets that is suited to the large width-to-height aspect ratios of the polar ice sheets. An efficient solver for the nonlinear Stokes equations is designed for high-order, stable, mixed finite-element discretizations on these adaptively refined meshes. A Gaussian approximation of the posterior distribution of parameters is defined, whose mean and covariance can be efficiently and scalably computed using adjoint-based methods from PDE-constrained optimization. Using a low-rank approximation of the covariance of this distribution, the covariance of the parameter is pushed forward onto quantities of interest.

In this thesis, we develop preconditioned iterative methods for the solution of matrix systems arising from PDE-constrained optimization problems. In order to do this, we exploit saddle point theory, as this is the form of the matrix systems we wish to solve. We utilize well-known results on saddle point systems to motivate preconditioners based on effective approximations of the (1,1)-block and Schur complement of the matrices involved. These preconditioners are used in conjunction with suitable iterative solvers, which include MINRES, non-standard Conjugate Gradients, GMRES and BiCG. The solvers we use are selected based on the particular problem and preconditioning strategy employed. We consider the numerical solution of a range of PDE-constrained optimization problems, namely the distributed control, Neumann boundary control and subdomain control of Poisson's equation, convection-diffusion control, Stokes and Navier-Stokes control, the optimal control of the heat equation, and the optimal control of reaction-diffusion problems arising in chemical processes. Each of these problems has a special structure which we make use of when developing our preconditioners, and specific techniques and approximations are required for each problem. In each case, we motivate and derive our preconditioners, obtain eigenvalue bounds for the preconditioners where relevant, and demonstrate the effectiveness of our strategies through numerical experiments. The goal throughout this work is for our iterative solvers to be feasible and reliable, but also robust with respect to the parameters involved in the problems we consider.

We are concerned with the high-fidelity subsurface imaging of the soil, which commonly arises in geotechnical site characterization and geophysical explorations. Specifically, we attempt to image the spatial distribution of the Lame parameters in semi-infinite, three-dimensional, arbitrarily heterogeneous formations, using surficial measurements of the soil's response to probing elastic waves. We use the complete waveforms of the medium's response to drive the inverse problem. Specifically, we use a partial-differential-equation (PDE)-constrained optimization approach, directly in the time-domain, to minimize the misfit between the observed response of the medium at select measurement locations, and a computed response corresponding to a trial distribution of the Lame parameters. We discuss strategies that lend algorithmic robustness to the proposed inversion schemes. To limit the computational domain to the size of interest, we employ perfectly-matched-layers (PMLs). The PML is a buffer zone that surrounds the domain of interest, and enforces the decay of outgoing waves. In order to resolve the forward problem, we present a hybrid finite element approach, where a displacement-stress formulation for the PML is coupled to a standard displacement-only formulation for the interior domain, thus leading to a computationally cost-efficient scheme. We discuss several time-integration schemes, including an explicit Runge-Kutta scheme, which is well-suited for large-scale problems on parallel computers. We report numerical results demonstrating stability and efficacy of the forward wave solver, and also provide examples attesting to the successful reconstruction of the two Lame parameters for both smooth and sharp profiles, using synthetic records. We also report the details of two field experiments, whose records we subsequently used to drive the developed inversion algorithms in order to characterize the sites where the field experiments took place. We contrast the full-waveform-based inverted site profile against a profile obtained using the Spectral-Analysis-of-Surface-Waves (SASW) method, in an attempt to compare our methodology against a widely used concurrent inversion approach. We also compare the inverted profiles, at select locations, with the results of independently performed, invasive, Cone Penetrometer Tests (CPTs). Overall, whether exercised by synthetic or by physical data, the full-waveform inversion method we discuss herein appears quite promising for the robust subsurface imaging of near-surface deposits in support of geotechnical site characterization investigations.