Gradient Based Iterative Algorithm to Solve General Coupled Discrete-Time Periodic Matrix Equations over Generalized Reflexive Matrices

The discrete-time periodic matrix equations are encountered in periodic state feedback problems and model reduction of periodic descriptor systems. The aim of this paper is to compute the generalized reflexive solutions of the general coupled discrete-time periodic matrix equations. We introduce a gradient-based iterative (GI) algorithm for finding the generalized reflexive solutions of the general coupled discretetime periodic matrix equations. It is shown that the introduced GI algorithm always converges to the generalized reflexive solutions for any initial generalized reflexive matrices. Finally, two numerical examples are investigated to confirm the efficiency of GI algorithm

A Preconditioned Iteration Method for Solving Sylvester Equations

A preconditioned gradient-based iterative method is derived by judicious selection of two auxil- iary matrices. The strategy is based on the Newton’s iteration method and can be regarded as a generalization of the splitting iterative method for system of linear equations. We analyze the convergence of the method and illustrate that the approach is able to considerably accelerate the convergence of the gradient-based iterative method

A gradient based iterative solutions for Sylvester tensor equations

This paper is concerned with the numerical solution of the Sylvester tensor equation, which includes the Sylvester matrix equation as special case. By applying hierarchical identification principle proposed by Ding and Chen, 2005, and by using tensor arithmetic concepts, an iterative algorithm and its modification are established to solve the Sylvester tensor equation. Convergence analysis indicates that the iterative solutions always converge to the exact solution for arbitrary initial value. Finally, some examples are provided to show that the proposed algorithms are effective

A Gradient Based Iterative Solutions for Sylvester Tensor Equations

This paper is concerned with the numerical solution of the Sylvester tensor equation, which includes the Sylvester matrix equation as special case. By applying hierarchical identification principle proposed by Ding and Chen, 2005, and by using tensor arithmetic concepts, an iterative algorithm and its modification are established to solve the Sylvester tensor equation. Convergence analysis indicates that the iterative solutions always converge to the exact solution for arbitrary initial value. Finally, some examples are provided to show that the proposed algorithms are effective

Calibration of spatial relationships between multiple robots and sensors

Classic hand-eye calibration methods have been limited to single robots and sensors. Recently a new calibration formulation for multiple robots has been proposed that solves for the extrinsic calibration parameters for each robot simultaneously instead of sequentially. The existing solutions for this new problem required data to have correspondence, but Ma, Goh and Chirikjian (MGC) proposed a probabilistic method to solve this problem which eliminated the need for correspondence. In this thesis, the literature of the various robot-sensor calibration problems and solutions are surveyed, and the MGC method is reviewed in detail. Lastly comparison with other methods using numerical simulations were carried out to draw some conclusions

Seeking Space Aliens and the Strong Approximation Property: a (Disjoint) Study in Dust Plumes on Planetary Satellites and Nonsymmetric Algebraic Multigrid

PART I: One of the most fascinating questions to humans has long been whether life exists outside of our planet. To our knowledge, water is a fundamental building block of life, which makes liquid water on other bodies in the universe a topic of great interest. In fact, there are large bodies of water right here in our solar system, underneath the icy crust of moons around Saturn and Jupiter. The NASA-ESA Cassini Mission spent two decades studying the Saturnian system. One of the many exciting discoveries was a “plume” on the south pole of Enceladus, emitting hundreds of kg/s of water vapor and frozen water-ice particles from Enceladus’ subsurface ocean. It has since been determined that Enceladus likely has a global liquid water ocean separating its rocky core from icy surface, with conditions that are relatively favorable to support life. The plume is of particular interest because it gives direct access to ocean particles from space, by flying through the plume. Recently, evidence has been found for similar geological activity occurring on Jupiter’s moon Europa, long considered one of the most likely candidate bodies to support life in our solar system. Here, a model for plume-particle dynamics is developed based on studies of the Enceladus plume and data from the Cassini Cosmic Dust Analyzer. A C++, OpenMP/MPI parallel software package is then built to run large scale simulations of dust plumes on planetary satellites. In the case of Enceladus, data from simulations and the Cassini mission provide insight into the structure of emissions on the surface, the total mass production of the plume, and the distribution of particles being emitted. Each of these are fundamental to understanding the plume and, for Europa and Enceladus, simulation data provide important results for the planning of future missions to these icy moons. In particular, this work has contributed to the Europa Clipper mission and proposed Enceladus Life Finder.PART II: Solving large, sparse linear systems arises often in the modeling of biological and physical phenomenon, data analysis through graphs and networks, and other scientific applications. This work focusesprimarily on linear systems resulting from the discretization of partial differential equations (PDEs). Because solving linear systems is the bottleneck of many large simulation codes, there is a rich field of research in developing “fast” solvers, with the ultimate goal being a method that solves an n × n linear system in O(n) operations. One of the most effective classes of solvers is algebraic multigrid (AMG), which is a multilevel iterative method based on projecting the problem into progressively smaller spaces, and scales like O(n) or O(nlogn) for certain classes of problems. The field of AMG is well-developed for symmetric positive definite matrices, and is typically most effective on linear systems resulting from the discretization of scalar elliptic PDEs, such as the heat equation. Systems of PDEs can add additional difficulties, but the underlying linear algebraic theory is consistent and, in many cases, an elliptic system of PDEs can be handled well by AMG with appropriate modifications of the solver. Solving general, nonsymmetric linear systems remains the wild west of AMG (and other fast solvers), lacking significant results in convergence theory as well as robust methods. Here, we develop new theoretical motivation and practical variations of AMG to solve nonsymmetric linear systems, often resulting from the discretization of hyperbolic PDEs. In particular, multilevel convergence of AMG for nonsymmetric systems is proven for the first time. A new nonsymmetric AMG solver is also developed based on an approximate ideal restriction, referred to as AIR, which is able to solve advection-dominated, hyperbolic-type problems that are outside the scope of existing AMG solvers and other fast iterative methods. AIR demonstrate

Resource Allocation in Wireless Networks: Theory and Applications

Limited wireless resources, such as spectrum and maximum power, give rise to various resource allocation problems that are interesting both from theoretical and application viewpoints. While the problems in some of the wireless networking applications are amenable to general resource allocation methods, others require a more specialized approach suited to their unique structural characteristics. We study both types of the problems in this thesis. We start with a general problem of alpha-fair packing, namely, the problem of maximizing sum_j {w_j f_α(x_j)}, where w_j > 0, ∀j, and (i) f_α(x_j)=ln(x_j), if α = 1, (ii) f_α(x_j)= {x_j^(1-α)}/{1-α}, if α ≠ 1,α > 0, subject to positive linear constraints of the form Ax ≤ b, x ≥ 0, where A and b are non-negative. This problem has broad applications within and outside wireless networking. We present a distributed algorithm for general alpha that converges to an epsilon-approximate solution in time (number of distributed iterations) that has an inverse polynomial dependence on the approximation parameter epsilon and poly-logarithmic dependence on the problem size. This is the first distributed algorithm for weighted alpha-fair packing with poly-logarithmic convergence in the input size. We also obtain structural results that characterize alpha-fair allocations as the value of alpha is varied. These results deepen our understanding of fairness guarantees in alpha-fair packing allocations, and also provide insights into the behavior of alpha-fair allocations in the asymptotic cases when alpha tends to zero, one, and infinity. With these general tools on hand, we consider an application in wireless networks where fairness is of paramount importance: rate allocation and routing in energy-harvesting networks. We discuss the importance of fairness in such networks and cases where our results on alpha-fair packing apply. We then turn our focus to rate allocation in energy harvesting networks with highly variable energy sources and that are used for applications such as monitoring and tracking. In such networks, it is essential to guarantee fairness over both the network nodes and the time slots and to be as fair as possible -- in particular, to require max-min fairness. We first develop an algorithm that obtains a max-min fair rate assignment for any routing that is specified at the input. Then, we consider the problem of determining a "good'' routing. We consider various routing types and either provide polynomial-time algorithms for finding such routings or prove that the problems are NP-hard. Our results reveal an interesting trade-off between the complexities of computation and implementation. The results can also be applied to other related fairness problems. The second part of the thesis is devoted to the study of resource allocation problems that require a specialized approach. The problems we focus on arise in wireless networks employing full-duplex communication -- the simultaneous transmission and reception on the same frequency channel. Our primary goal is to understand the benefits and complexities tied to using this novel wireless technology through the study of resource (power, time, and channel) allocation problems. Towards that goal, we introduce a new realistic model of a compact (e.g., smartphone) full-duplex receiver and demonstrate its accuracy via measurements. First, we focus on the resource allocation problems with the objective of maximizing the sum of uplink and downlink rates, possibly over multiple orthogonal channels. For the single-channel case, we quantify the rate improvement as a function of the remaining self-interference and signal-to-noise ratios and provide structural results that characterize the sum of uplink and downlink rates on a full-duplex channel. Building on these results, we consider the multi-channel case and develop a polynomial time algorithm which is nearly optimal in practice under very mild restrictions. To reduce the running time, we develop an efficient nearly-optimal algorithm under the high SINR approximation. Then, we study the achievable capacity regions of full-duplex links in the single- and multi-channel cases. We present analytical results that characterize the uplink and downlink capacity region and efficient algorithms for computing rate pairs at the region's boundary. We also provide near-optimal and heuristic algorithms that "convexify'' the capacity region when it is not convex. The convexified region corresponds to a combination of a few full-duplex rates (i.e., to time sharing between different operation modes). The analytical results provide insights into the properties of the full-duplex capacity region and are essential for future development of fair resource allocation and scheduling algorithms in Wi-Fi and cellular networks incorporating full-duplex

The Large Hadron-Electron Collider at the HL-LHC

The Large Hadron-Electron Collider (LHeC) is designed to move the field of deep inelastic scattering (DIS) to the energy and intensity frontier of particle physics. Exploiting energy-recovery technology, it collides a novel, intense electron beam with a proton or ion beam from the High-Luminosity Large Hadron Collider (HL-LHC). The accelerator and interaction region are designed for concurrent electron-proton and proton-proton operations. This report represents an update to the LHeC's conceptual design report (CDR), published in 2012. It comprises new results on the parton structure of the proton and heavier nuclei, QCD dynamics, and electroweak and top-quark physics. It is shown how the LHeC will open a new chapter of nuclear particle physics by extending the accessible kinematic range of lepton-nucleus scattering by several orders of magnitude. Due to its enhanced luminosity and large energy and the cleanliness of the final hadronic states, the LHeC has a strong Higgs physics programme and its own discovery potential for new physics. Building on the 2012 CDR, this report contains a detailed updated design for the energy-recovery electron linac (ERL), including a new lattice, magnet and superconducting radio-frequency technology, and further components. Challenges of energy recovery are described, and the lower-energy, high-current, three-turn ERL facility, PERLE at Orsay, is presented, which uses the LHeC characteristics serving as a development facility for the design and operation of the LHeC. An updated detector design is presented corresponding to the acceptance, resolution, and calibration goals that arise from the Higgs and parton-density-function physics programmes. This paper also presents novel results for the Future Circular Collider in electron-hadron (FCC-eh) mode, which utilises the same ERL technology to further extend the reach of DIS to even higher centre-of-mass energies.Peer reviewe

The Large Hadron–Electron Collider at the HL-LHC

Se incluye contenido parcial de los autores, (contiene más de 300 autores)The Large Hadron–Electron Collider (LHeC) is designed to move the field of deep inelastic scattering (DIS) to the energy and intensity frontier of particle physics. Exploiting energy-recovery technology, it collides a novel, intense electron beam with a proton or ion beam from the High-Luminosity Large Hadron Collider (HL-LHC). The accelerator and interaction region are designed for concurrent electron–proton and proton–proton operations. This report represents an update to the LHeC’s conceptual design report (CDR), published in 2012. It comprises new results on the parton structure of the proton and heavier nuclei, QCD dynamics, and electroweak and top-quark physics. It is shown how the LHeC will open a new chapter of nuclear particle physics by extending the accessible kinematic range of lepton–nucleus scattering by several orders of magnitude. Due to its enhanced luminosity and large energy and the cleanliness of the final hadronic states, the LHeC has a strong Higgs physics programme and its own discovery potential for new physics. Building on the 2012 CDR, this report contains a detailed updated design for the energy-recovery electron linac (ERL), including a new lattice, magnet and superconducting radio-frequency technology, and further components. Challenges of energy recovery are described, and the lower-energy, high-current, three-turn ERL facility, PERLE at Orsay, is presented, which uses the LHeC characteristics serving as a development facility for the design and operation of the LHeC. An updated detector design is presented corresponding to the acceptance, resolution, and calibration goals that arise from the Higgs and parton-density-function physics programmes. This paper also presents novel results for the Future Circular Collider in electron–hadron (FCC-eh) mode, which utilises the same ERL technology to further extend the reach of DIS to even higher centre-of-mass energies