Effective and Efficient Reconstruction Schemes for the Inverse Medium Problem in Scattering

This thesis challenges with the development of a computational framework facilitating the solution for the inverse medium problem in time-independent scattering in two- and three-dimensional setting. This includes three main application cases: the simulation of the scattered field for a given transmitter-receiver geometry; the generation of simulated data as well as the handling of real-world data; the reconstruction of the refractive index of a penetrable medium from several measured, scattered fields. We focus on an effective and efficient reconstruction algorithm. Therefore we set up a variational reconstruction scheme. The underlying paradigm is to minimize the discrepancy between the predicted data based on the reconstructed refractive index and the given data while taking into account various structural a priori information via suitable penalty terms, which are designed to promote information expected in real-world environments. Finally, the scheme relies on a primal-dual algorithm. In addition, information about the obstacle's shape and position obtained by the factorization method can be used as a priori information to increase the overall effectiveness of the scheme. An implementation is provided as MATLAB toolbox IPscatt. It is tailored to the needs of practitioners, e.g. a heuristic algorithm for an automatic, data-driven choice of the regularization parameters is available. The effectiveness and efficiency of the proposed approach are demonstrated for simulated as well as real-world data by comparisons with existing software packages

Calderon's reproducing formulas for the Spherical mean L^2-multiplier operators

First we study the spherical mean L^2-multiplier operators on [0,+∞[xR^n. Next, we give for these operators Calderon's reproducing formulas and best approximation formulas

Fast, High-Order Accurate Integral Equation Methods and Application to PDE-Constrained Optimization

Over the last several decades, the development of fast, high-order accurate, and robust integral equation methods for computational physics has gained increasing attention. Using integral equation formulation as a global statement in contrast to a local partial differential equation (PDE) formulation offers several unique advantages. For homogeneous PDEs, the boundary integral equation (BIE) formulation allows accurate handling of complex and moving geometries, and it only requires a mesh on the boundary, which is much easier to generate as a result of the dimension reduction. With the acceleration of fast algorithms like the Fast Multipole Method (FMM), the computational complexity can be reduced to O(N), where N is the number of degrees of freedom on the boundary. Using standard potential theory decomposition, inhomogeneous PDEs can be solved by evaluating a volume potential over the inhomogeneous source domain, followed by a solution of the homogeneous part. Despite the advantages of BIE methods in easy meshing, near-optimal efficiency, and well conditioning, the accurate evaluation of nearly singular integrals is a classical problem that needs to be addressed to enable simulations for practical applications. In the first half of this thesis, we develop a series of product integration schemes to solve this close evaluation problem. The use of differential forms provides a dimensional-agnostic way of integrating the nearly singular kernels against polynomial basis functions analytically. So the problem of singular integration gets reduced to a matter of source function approximation. In 2D, this procedure has been traditionally portrayed by building a connection to complex Cauchy integral, then supplemented by a complex monomial approximation. In $3$D, the closed differential form requirement leads to the design of a new function approximation scheme based on harmonic polynomials and quaternion algebra. Under a similar framework, we develop a high-order accurate product integration scheme for evaluating singular and nearly singular volume integral equations (VIE) in complex domains using regular Cartesian grids discretization. A high-order accurate source term approximation scheme matching smooth volume integrals on irregular cut cells is developed, which requires no function extension. BIE methods have been widely used for studying Stokes flows, incompressible flows at low Reynolds' number, in both biological systems and microfluidics. In the second half of this thesis, we employ the BIE methods to simulate and optimize Stokes fluid-structure interactions. In 2D, a hybrid computational method is presented for simulating cilia-generated fluid mixing as well as the cilia-particle hydrodynamics. The method is based on a BIE formulation for confining geometries and rigid particles, and the method of regularized Stokeslets for the cilia. In 3D, we use the time-independent envelop model for arbitrary axisymmetric microswimmers to minimize the power loss while maintaining a target swimming speed. This is a quadratic optimization problem in terms of the slip velocity due to the linearity of Stokes flow. Under specified reduced volume constraint, we find prolate spheroids to be the most efficient micro-swimmer among various families of shapes we considered. We then derive an adjoint-based formulation for computing power loss sensitivities in terms of a time-dependent slip profile by introducing an auxiliary time-periodic function, and find that the optimal swimmer displays one or multiple traveling waves, reminiscent of the typical metachronal waves observed in ciliated microswimmers.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169695/1/hszhu_1.pd

Fractional thoughts

In this note we present some of the most basic aspects of the fractional Laplacean with a self-contained and purely didactic intent, and with a somewhat different slant from the several excellent existing references. Given the interest that nonlocal operators have generated since the 2007 extension paper of Caffarelli and Silvestre, we feel it is appropriate offering to young researchers a quick additional guide to the subject which, we hope, will nicely complement the existing ones.Comment: several typos fixed, some references corrected, some references adde

On generalized Hardy spaces associated with singular partial differential operators

We define and study the Hardy spaces associated with singular partial differential operators. Also, a characterization by mean of atomic decomposition is investigated

Development of Computed Tomography Head and Body Phantom for Organ Dosimetry

Introduction: Quality assurance in Computed tomography (CT) centers in developing countries are largely hindered by the unavailability of CT phantoms. The development of a local CT phantom for the measurement of organ radiation absorbed dose is therefore requisite. Material and Methods: Local CT phantoms were designed to meet the standard criteria of 32 cm diameter for body, 16 cm diameter for head, and 14 cm in length respectively. The outer plastic shell was made using poly (methyl methacrylate [PMMA]) sheet. The developed CT phantoms were validated against a standard phantom. Radiation absorbed dose was determined by scanning the setup with the same protocol used for the standard phantom. The local phantoms were then verified for organ radiation absorbed dose measurement using bovine tissues. The set up was CT-scanned, and Hounsfield units (HU) for bovine tissues were obtained. Results: There was no significant difference between the local and standard head phantoms (P=0.060). Similarly, no difference was noted between the local and standard body phantoms (P=0.795). The percentage difference in volume CT dose index (CTDIvol) between the body (local and standard) phantoms was higher than that for the head phantoms. There were no significant differences in HU between bovine and human brain, liver, kidney and lung tissues (P=0.938). Conclusion: The local phantoms showed good agreement with the standard ones. The developed phantoms can be used for CT organ radiation absorbed dose measurement in radiology departments in Nigeria

Representations of semisimple Lie algebras in prime characteristic and the noncommutative Springer resolution

We prove most of Lusztig’s conjectures on the canonical basis in homology of a Springer fiber. The conjectures predict that this basis controls numerics of representations of the Lie algebra of a semisimple algebraic group over an algebraically closed field of positive characteristic. We check this for almost all characteristics. To this end we construct a noncommutative resolution of the nilpotent cone which is derived equivalent to the Springer resolution. On the one hand, this noncommutative resolution is closely related to the positive characteristic derived localization equivalences obtained earlier by the present authors and Rumynin. On the other hand, it is compatible with the t-structure arising from an equivalence with the derived category of perverse sheaves on the affine flag variety of the Langlands dual group. This equivalence established by Arkhipov and the first author fits the framework of local geometric Langlands duality. The latter compatibility allows one to apply Frobenius purity theorem to deduce the desired properties of the basis. We expect the noncommutative counterpart of the Springer resolution to be of independent interest from the perspectives of algebraic geometry and geometric Langlands duality.United States. Air Force Office of Scientific Research (Grant FA9550-08-1-0315)National Science Foundation (U.S.) (Grant DMS-0854764)National Science Foundation (U.S.) (Grant DMS-1102434

New Developments in Geometric Function Theory

The book contains papers published in a Special Issue of Axioms, entitled "New Developments in Geometric Function Theory". An Editorial describes the 14 papers devoted to the study of complex-valued functions which present new outcomes related to special classes of univalent and bi-univalent functions, new operators and special functions associated with differential subordination and superordination theories, fractional calculus, and certain applications in geometric function theory

Appearance-based image splitting for HDR display systems

High dynamic range displays that incorporate two optically-coupled image planes have recently been developed. This dual image plane design requires that a given HDR input image be split into two complementary standard dynamic range components that drive the coupled systems, therefore there existing image splitting issue. In this research, two types of HDR display systems (hardcopy and softcopy HDR display) are constructed to facilitate the study of HDR image splitting algorithm for building HDR displays. A new HDR image splitting algorithm which incorporates iCAM06 image appearance model is proposed, seeking to create displayed HDR images that can provide better image quality. The new algorithm has potential to improve image details perception, colorfulness and better gamut utilization. Finally, the performance of the new iCAM06-based HDR image splitting algorithm is evaluated and compared with widely spread luminance square root algorithm through psychophysical studies