Finding correctable encoding that protects against a quantum process is in general a difficult task. Two main obstacles are that an exponential number of experiments are needed to gain complete information about the quantum process, and known algorithmic methods for finding correctable encodings involve operations on exponentially large matrices. In this thesis we discuss how useful partial information of a quantum channel can be systematically extracted by averaging the channel under the action of a set of unitaries in a process known as twirling. We show that in some cases it is possible to find correctable encodings for the channel using the partial information obtained via twirling. We investigate the particular case of twirling over the set of Pauli operators and qubit permutations, and show that the resulting quantum operation can be characterized experimentally in a scalable manner. A post-processing scheme for finding unitarily correctable codes for these twirled channels is presented which does not involve exponentially large matrices. A test for non-Markovian noise using such a twirling process is also discussed.

Recently there has been a lot of interest in the potential applications of performing computations on systems whose governing physical laws are quantum, rather than classical in nature. These quantum computers would have the ability to perform some calculations which would not be feasible for their classical counterparts. To date, however, a quantum computer large enough to perform useful calculations has yet to be built. Before this can be accomplished, a method must be developed to control the underlying quantum systems. This is a problem which can naturally be formulated in the language of control theory. This report outlines the basic control-theoretic approach to time-optimally controlling quantum systems evolving under the dynamics of the Schr\"{o}dinger operator equation. It is found that under the assumption of non-singularity, the controls which produce time-optimal trajectories are bang-bang. With this in mind, a switching time algorithm is implemented to find optimal bang-bang controls.

Processing and inpainting of sparse data as applied to atomic force microscopy imaging

The Spherical Mean Value Operators on Euclidean and Hyperbolic Spaces

This thesis introduces and applies model reduction techniques to problems associated with simulation of realistic single neurons. Neurons have complicated dendritic structures and spatially-distributed ionic kinetics that give rise to highly nonlinear dynamics. However, existing model reduction methods compromise the geometry, and thus sacrifice the original input-output relationship. I demonstrate that linear and nonlinear model reduction techniques yield systems that capture the salient dynamics of morphologically accurate neuronal models and preserve the input-output maps while using significantly fewer variables than the full systems. Two main dynamic regimes characterize the voltage response of a neuron, and I demonstrate that different model reduction techniques are well-suited to each regime. Small perturbations from the neuron's rest state fall into the subthreshold regime, which can be accurately described by a linear system. By applying Balanced Truncation (BT), a model reduction technique for general linear systems, I recover subthreshold voltage dynamics, and I provide an efficient Iterative Rational Krylov Algorithm (IRKA), which makes large problems of interest tractable. However, these approximations are not valid once the input to the neuron is sufficient to drive the voltage into the spiking regime, which is characterized by highly nonlinear behavior. To reproduce spiking dynamics, I use a proper orthogonal decomposition (POD) to reduce the number of state variables and a discrete empirical interpolation method (DEIM) to reduce the complexity of the nonlinear terms. The techniques described above are successful, but they inherently assume that the whole neuron is either passive (linear) or active (nonlinear). However, in realistic cells the voltage response at distal locations is nearly linear, while at proximal locations it is very nonlinear. With this observation, I fuse the aforementioned models together to create a reduced coupled model in which each reduction technique is used where it is most advantageous, thereby making it possible to more accurately simulate a larger class of cortical neurons.

This thesis describes an implementation of the discontinuous Galerkin finite element time domain (DGTD) method on unstructured meshes to solve acoustic wave equations in heterogeneous media. In oil industry people use finite difference time domain (FDTD) methods to simulate seismic surveys, the first step to explore oil and gas in the earth's subsurface, conducted either in land or sea. The results in this thesis indicate that the first order time shift effect resulting from misalignment between numerical meshes and material interfaces in the DGTD method occurs in the same way as interface errors in the finite difference simulation of wave propagation. This thesis describes two approaches: interface-fitting mesh and local mesh refinement, without modifying the DGTD scheme, to decrease this troublesome effect with verifications of 2D examples. The comparison of the DGTD method on the piecewise linear interface-fitting mesh and the staggered FDTD method both applied to a square-circle model and a 2D dome model in this thesis confirms the fact that the DGTD method can achieve a suboptimal second order convergence rate while the error in the staggered FDTD method is dominated by the first order interface error when the curved material interfaces are presented. I conclude that the DGTD method is more efficient than the staggered FDTD method for the two solutions to have roughly the same accuracy when the accuracy requirement becomes more and more strict and the model becomes more and more complex.

This thesis provides a rigorous framework for the solution of stochastic elliptic partial differential equation (SPDE) constrained optimization problems. In modeling physical processes with differential equations, much of the input data is uncertain (e.g. measurement errors in the diffusivity coefficients). When uncertainty is present, the governing equations become a family of equations indexed by a stochastic variable. Since solutions of these SPDEs enter the objective function, the objective function usually involves statistical moments. These optimization problems governed by SPDEs are posed as a particular class of optimization problems in Banach spaces. This thesis discusses Monte Carlo, stochastic Galerkin, and stochastic collocation methods for the numerical solution of SPDEs and identifies the stochastic collocation method as particularly useful for the optimization of SPDEs. This thesis extends the stochastic collocation method to the optimization context and explores the decoupling nature of this method for gradient and Hessian computations.

This thesis proposes a model for the interaction between ground flow and surface flow using a coupled system of the Navier-Stokes and Darcy equations. The coupling of surface flow with porous media flow has important applications in science and engineering. This work is motivated by applications to geo-sciences. This work couples the two flows using interface conditions that incorporate the continuity of the normal component, the balance of forces and the Beaver-Joseph-Saffman Law. The balance of forces condition can be written with or without inertial forces from the free fluid region. This thesis provides both theoretical and numerical analysis of the effect of the inertial forces on the model. Flow in porous media is often simulated over large domains in which the actual permeability is heterogeneous with discontinuities across the domain. The discontinuous Galerkin method is well suited to handle this problem. On the other hand, the continuous finite element is adequate for the free flow problems considered in this work. As a result this thesis proposes coupling the continuous finite element method in the free flow region with the discontinuous Galerkin method in the porous medium. Existence and uniqueness results of a weak solution and numerical scheme are proved. This work also provides derivations of optimal a priori error estimates for the numerical scheme. A two-grid approach to solving the coupled problem is analyzed. This method will decouple the problem naturally into two problems, one in the free flow domain and other in the porous medium. In applications for this model, it is often the case that the areas of interest (faults, kinks) in the porous medium are small compared to the rest of the domain. In view of this fact, the rest of the thesis is dedicated to a coupling of the Discontinuous Galerkin method in the problem areas with a cheaper method on the rest of the domain. The finite volume method will be coupled with the Discontinuous Galerkin method on parts of the domain on which the permeability field varies gradually to decrease the problem sizes and thus make the scheme more efficient.

We outline the use of Young tableaux to describe geometric and algebraic objects using combinatorial methods. In particular, we discuss applications to representations of the symmetric group and the general linear group, flag varieties, and Schubert varieties. We also describe some recent work, including proofs of the Saturation Conjecture and a theorem on the eigenvalues of sums of Hermitian matrices.

This thesis develops a new computation of the objective function and gradient for normal moveout-based differential semblance (DS). The DS principle underlies a class of algorithms for seismic velocity analysis. The simplest variant of DS is based on a drastic approximation to the scattering of waves, called "normal moveout" (NMO) in the seismic literature. This simple NMO-driven DS algorithm is very fast relative to other variants based on more faithful approximations to wave physics, but nonetheless accurate enough to be used to process field data. A recent implementation of NMO-based DS demonstrated these capabilities, but it also exhibited numerical irregularity which may have affected the stability of its velocity estimates. My alternative approach avoids interpolation noise that existed in previous work and so results in more stable numerical optimization.