Signal Reconstruction from Frame and Sampling Erasures

This dissertation is concerned with the efficient reconstruction of signals from frame coefficient erasures at known locations. Three methods of perfect reconstruction from frame coefficient erasures will be discussed. These reconstructions are more efficient than older methods in the literature because they only require an L × L matrix inversion, where L denotes the cardinality of the erased set of indices. This is a significant improvement over older methods which require an n × n matrix inversion, where n denotes the dimension of the underlying Hilbert space. The first of these methods, called Nilpotent Bridging, uses a small subset of the non-erased coefficients to reconstruct the erased coefficients. This subset is called the bridge set. To perform the reconstruction an equation, known as the bridge equation, must be solved. A proof is given that under a very mild assumption there exists a bridge set of size L for which the bridge equation has a solution. A stronger result is also proven that shows that for a very large class of frames, no bridge set search is required. We call this set of frames the set of full skew-spark frames. Using the Baire Category Theorem and tools from Matrix Theory, the set of full skew-spark frames is shown to be an open, dense subset of the set of all frames in finite dimensions. The second method of reconstruction is called Reduced Direct Inversion because it provides a basis-free, closed-form formula for inverting a particular n × n matrix, which only requires the inversion of an L × L matrix. By inverting this matrix we obtain another efficient reconstruction formula. The final method considered is a continuation of work by Han and Sun. The method utilizes an Erasure Recovery Matrix, which is a matrix that annihilates the range of the analysis operator for a frame. Because of this, the erased coefficients can be reconstructed using a simple pseudo-inverse technique. For each method, a discussion of the stability of our algorithms is presented. In particular, we present numerical experiments to investigate the effects of normally distributed additive channel noise on our reconstruction. For Reduced Direct Inversion and Erasure Recovery Matrices, using the Restricted Isometry Property, we construct classes of frames which are numerically robust to sparse channel noise. For these frames, we provide error bounds for sparse channel noise

Image Appraisal for 2D and 3D Electromagnetic Inversion

Linearized methods are presented for appraising image resolution and parameter accuracy in images generated with two and three dimensional non-linear electromagnetic inversion schemes. When direct matrix inversion is employed, the model resolution and posterior model covariance matrices can be directly calculated. A method to examine how the horizontal and vertical resolution varies spatially within the electromagnetic property image is developed by examining the columns of the model resolution matrix. Plotting the square root of the diagonal of the model covariance matrix yields an estimate of how errors in the inversion process such as data noise and incorrect a priori assumptions about the imaged model map into parameter error. This type of image is shown to be useful in analyzing spatial variations in the image sensitivity to the data. A method is analyzed for statistically estimating the model covariance matrix when the conjugate gradient method is employed rather than a direct inversion technique (for example in 3D inversion). A method for calculating individual columns of the model resolution matrix using the conjugate gradient method is also developed. Examples of the image analysis techniques are provided on 2D and 3D synthetic cross well EM data sets, as well as a field data set collected at the Lost Hills Oil Field in Central California

Quantum Process Tomography: Resource Analysis of Different Strategies

Characterization of quantum dynamics is a fundamental problem in quantum physics and quantum information science. Several methods are known which achieve this goal, namely Standard Quantum Process Tomography (SQPT), Ancilla-Assisted Process Tomography (AAPT), and the recently proposed scheme of Direct Characterization of Quantum Dynamics (DCQD). Here, we review these schemes and analyze them with respect to some of the physical resources they require. Although a reliable figure-of-merit for process characterization is not yet available, our analysis can provide a benchmark which is necessary for choosing the scheme that is the most appropriate in a given situation, with given resources. As a result, we conclude that for quantum systems where two-body interactions are not naturally available, SQPT is the most efficient scheme. However, for quantum systems with controllable two-body interactions, the DCQD scheme is more efficient than other known QPT schemes in terms of the total number of required elementary quantum operations.Comment: 15 pages, 5 figures, published versio

Efficient approximation of functions of some large matrices by partial fraction expansions

Some important applicative problems require the evaluation of functions $\Psi$ of large and sparse and/or \emph{localized} matrices $A$. Popular and interesting techniques for computing $\Psi(A)$ and $\Psi(A)\mathbf{v}$, where $\mathbf{v}$ is a vector, are based on partial fraction expansions. However, some of these techniques require solving several linear systems whose matrices differ from $A$ by a complex multiple of the identity matrix $I$ for computing $\Psi(A)\mathbf{v}$ or require inverting sequences of matrices with the same characteristics for computing $\Psi(A)$. Here we study the use and the convergence of a recent technique for generating sequences of incomplete factorizations of matrices in order to face with both these issues. The solution of the sequences of linear systems and approximate matrix inversions above can be computed efficiently provided that $A^{-1}$ shows certain decay properties. These strategies have good parallel potentialities. Our claims are confirmed by numerical tests

Numerical methods for multiscale inverse problems

We consider the inverse problem of determining the highly oscillatory coefficient $a^\epsilon$ in partial differential equations of the form $-\nabla\cdot (a^\epsilon\nabla u^\epsilon)+bu^\epsilon = f$ from given measurements of the solutions. Here, $\epsilon$ indicates the smallest characteristic wavelength in the problem ($0<\epsilon\ll1$). In addition to the general difficulty of finding an inverse, the oscillatory nature of the forward problem creates an additional challenge of multiscale modeling, which is hard even for forward computations. The inverse problem in its full generality is typically ill-posed and one common approach is to replace the original problem with an effective parameter estimation problem. We will here include microscale features directly in the inverse problem and avoid ill-posedness by assuming that the microscale can be accurately represented by a low-dimensional parametrization. The basis for our inversion will be a coupling of the parametrization to analytic homogenization or a coupling to efficient multiscale numerical methods when analytic homogenization is not available. We will analyze the reduced problem, $b = 0$, by proving uniqueness of the inverse in certain problem classes and by numerical examples and also include numerical model examples for medical imaging, $b > 0$, and exploration seismology, $b < 0$

Robustness of nuclear core activity reconstruction by data assimilation

We apply a data assimilation techniques, inspired from meteorological applications, to perform an optimal reconstruction of the neutronic activity field in a nuclear core. Both measurements, and information coming from a numerical model, are used. We first study the robustness of the method when the amount of measured information decreases. We then study the influence of the nature of the instruments and their spatial repartition on the efficiency of the field reconstruction