Atomic norm denoising with applications to line spectral estimation

Motivated by recent work on atomic norms in inverse problems, we propose a new approach to line spectral estimation that provides theoretical guarantees for the mean-squared-error (MSE) performance in the presence of noise and without knowledge of the model order. We propose an abstract theory of denoising with atomic norms and specialize this theory to provide a convex optimization problem for estimating the frequencies and phases of a mixture of complex exponentials. We show that the associated convex optimization problem can be solved in polynomial time via semidefinite programming (SDP). We also show that the SDP can be approximated by an l1-regularized least-squares problem that achieves nearly the same error rate as the SDP but can scale to much larger problems. We compare both SDP and l1-based approaches with classical line spectral analysis methods and demonstrate that the SDP outperforms the l1 optimization which outperforms MUSIC, Cadzow's, and Matrix Pencil approaches in terms of MSE over a wide range of signal-to-noise ratios.Comment: 27 pages, 10 figures. A preliminary version of this work appeared in the Proceedings of the 49th Annual Allerton Conference in September 2011. Numerous numerical experiments added to this version in accordance with suggestions by anonymous reviewer

Iterative Solvers for Large, Dense Matrices

Stochastic Interpolation (SI) uses a continuous, centrally symmetric probability distribution function to interpolate a given set of data points, and splits the interpolation operator into a discrete deconvolution followed by a discrete convolution of the data. The method is particularly effective for large data sets, as it does not suffer from the problem of oversampling, where too many data points cause the interpolating function to oscillate wildly. Rather, the interpolation improves every time more data points are added. The method relies on the inversion of relatively large, dense matrices to solve Annx = b for x. Based on the probability distribution function chosen, the matrix Ann may have specific properties that make the problem of solving for x tractable. The iterative Shulz Jones Mayer (SJM) method relies on an initial guess, which is iterated to approximate A�1 nn . We present initial guesses that are guaranteed to converge quadratically for several classes of matrices, including diagonally and tri-diagonally dominant matrices and the structured matrices we encounter in the implementation of SI. We improve the method, creating the Polynomial Shulz Jones Mayer method, and take advantage of the more efficient matrix operations possible for Toeplitz matrices. We calculate error bounds and use those to improve the method’s accuracy, resulting in a method requiring O(nlogn) operations that returns x with double precision. The use of SI and PSJM is illustrated in interpolating functions and images in grey scale and color

Some characteristics and applications of a general purpose optimum multichannel stacking filter

Imperial Users onl

Second harmonic generation in novel optical waveguides

Second Harmonic Generation has traditionally been restricted to crystals; however, due to the development of highly sophisticated fabrication technology, poling and phase matching techniques, it has now become feasible in glass fibres and waveguides which are widely used in nonlinear optics. For instance, silica glass based Photonic Crystal Fibres (PCF) exhibit long coherence lengths and controllable optical properties such as chromatic dispersion despite low second order nonlinearity. Given such benefits, the numerical modelling and analysis of optical waveguides accurately and efficiently has become vital for the advancement of nonlinear optics. Hence, this thesis focuses on enhancing the Second Harmonic Generation in optical waveguides through the use of different structures and different materials, and demonstrating the same by numerical simulations. In this thesis, the accurate and numerically efficient Finite Element based Beam Propagation Method has been employed to investigate the evolution of Second Harmonic Generation in highly nonlinear SF57 soft glass Equiangular Spiral PCFs. Further, the H-field based Finite Element Method has been employed for the stationary analysis of Photonic Crystal Fibres. It is shown here that the second harmonic output power in highly nonlinea

V-cycle optimal convergence for certain (multilevel) structured linear systems

In this paper we are interested in the solution by multigrid strategies of multilevel linear systems whose coefficient matrices belong to the circulant, Hartley, or \u3c4 algebras or to the Toeplitz class and are generated by (the Fourier expansion of) a nonnegative multivariate polynomial f. It is well known that these matrices are banded and have eigenvalues equally distributed as f, so they are ill-conditioned whenever f takes the zero value; they can even be singular and need a low-rank correction. We prove the V-cycle multigrid iteration to have a convergence rate independent of the dimension even in presence of ill-conditioning. If the (multilevel) coefficient matrix has partial dimension nr at level r, r = 1, . . . ,d, then the size of the algebraic system is N(n) = \u3a0r=1 d nr, O(N(n)) operations are required by our technique, and therefore the corresponding method is optimal. Some numerical experiments concerning linear systems arising in applications, such as elliptic PDEs with mixed boundary conditions and image restoration problems, are considered and discussed.cussed

Wannier90 as a community code: new features and applications

Wannier90 is an open-source computer program for calculating maximally-localised Wannier functions (MLWFs) from a set of Bloch states. It is interfaced to many widely used electronic-structure codes thanks to its independence from the basis sets representing these Bloch states. In the past few years the development of Wannier90 has transitioned to a community-driven model; this has resulted in a number of new developments that have been recently released in Wannier90 v3.0. In this article we describe these new functionalities, that include the implementation of new features for wannierisation and disentanglement (symmetry-adapted Wannier functions, selectively-localised Wannier functions, selected columns of the density matrix) and the ability to calculate new properties (shift currents and Berry-curvature dipole, and a new interface to many-body perturbation theory); performance improvements, including parallelisation of the core code; enhancements in functionality (support for spinor-valued Wannier functions, more accurate methods to interpolate quantities in the Brillouin zone); improved usability (improved plotting routines, integration with high-throughput automation frameworks), as well as the implementation of modern software engineering practices (unit testing, continuous integration, and automatic source-code documentation). These new features, capabilities, and code development model aim to further sustain and expand the community uptake and range of applicability, that nowadays spans complex and accurate dielectric, electronic, magnetic, optical, topological and transport properties of materials.The WDG acknowledges financial support from the NCCR MARVEL of the Swiss National Science Foundation, the European Union’s Centre of Excellence E-CAM (Grant No. 676531), and the Thomas Young Centre for Theory and Simulation of Materials (Grant No. TYC-101).Peer reviewe

スパース時間周波数表現に関する研究

早大学位記番号:新9160博士（工学）早稲田大

Problems in Scattering and Imaging

Technology advances are always driven by the discovery of new materials, better understanding of their properties and improvements in processing power. This trend is reflected in this work, where I will demonstrate how new science and applications of both scattering and imaging are enabled by these frontiers. This thesis explores a broad spectrum of topics associated with the problems of scattering and imaging. The first topic concerns the fundamental study of the symmetry breaking and the nonlinear light scattering in the system of gold nanorod. In the most recent experiments, the intrinsic electrostatic asymmetry of gold nanorods was investigated by Ji-Young et al. using a variety of microscopy techniques, and the associated optical asymmetry was immediately demonstrated through the nonlinear optical experiments. The understanding of the symmetry breaking of gold nanorods, motivated the development of a model where the second order longitudinal plasmon resonance mode scatters with the electron gas and accounting for the plasmon damping effect. The new microscopic description self-consistently explains all the main features of the nonlinear optical components, and provides a fresh look that beautifully aligns with the recent observations of the nonlinear optical properties of nanorods. Next, we demonstrate an optical system that enables the control of monochromatic light transmission through highly scattering media, with Complex Semi-Definite Programming (SDP) introduced as a novel approach to solve the associated phase retrieval problem. In contrast to the conventional approach that employed an interferometric design which is vulnerable to system vibration, a simple optical setup without the need for a reference beam is proposed by Moussa et al. The SDP algorithm allows computation of the complex transmission matrix of the system from a sequence of intensity speckle patterns generated with phase-modulated wavefronts. We showed that once the transmission matrix is determined, optimal wavefronts can be computed that focuses the incident beam to any position on the far side of the scattering medium, without the need for subsequent measurements or wavefront shaping iterations. Finally, the optical properties and applications of graphene were explored. As a true 2D material, graphene has a unique electronic band structure and has been demonstrated by various research groups to be an interesting photonic building block. At first, we focused on the absorption saturation in optically excited graphene. The microscopic theory that includes Coulomb-scattering as the dominant relaxation mechanism at high carrier densities was developed and then verified by the optical transmission experiment. Then, we showed a novel scheme of a light field camera using a focal stack proposed by a team at the University of Michigan. The key enabling technology is the highly transparent graphene photodetector fabricated by Che-Hung et al., where graphene is used both as the photoconductive gain material and the circuit interconnects. Physically, we built the prototype single-pixel light field camera and demonstrated its operation through optical experiment. Computationally, a synthetic camera system was designed based on the Fourier slice analysis and the framework for the model-based light field reconstruction was provided.PHDElectrical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/138550/1/mblien_1.pd