Two-dimensional Setting Research Articles

Extended Kalman Filter-Based Active Alignment Control for LED Optical Communication

Light-emitting diode (LED)-based optical communication is emerging as a low-power, low-cost, and high-data rate alternative to acoustic communication for mobile applications underwater. However, it requires a close-to- line-of-sight (LOS) link between the transmitter and the receiver. Alignment for maintaining LOS is challenging due to the constant movement of underlying mobile platforms caused by propulsion and unwanted disturbances. In this paper, we present a novel, compact LED-based communication system with active alignment control, in a two-dimensional setting, that maintains the LOS despite the underlying platform movement. An extended Kalman filter-based algorithm is proposed to estimate the angle between the receiver orientation and the receiver–transmitter line, which is used subsequently to adjust the receiver orientation. The algorithm uses only the measured light intensity from a single photodiode, where successive measurements are obtained via a scanning technique. A simple proportional controller is designed for alignment that also ensures the observability of the system. The effectiveness of the proposed active alignment algorithm is verified in simulation and experiments. In particular, its robustness in the presence of measurement noise is demonstrated via comparison with two alternative algorithms that are based on hill-climbing and three-point-averaging.

Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves

This paper is devoted to the analysis of convergence of Schwarz Waveform Relaxation (SWR) domain decomposition methods (DDM) for solving the stationary linear and nonlinear Schrödinger equations by the imaginary-time method. Although SWR are extensively used for numerically solving high-dimensional quantum and classical wave equations, the analysis of convergence and of the rate of convergence is still largely open for linear equations with variable coefficients and nonlinear equations. The aim of this paper is to tackle this problem for both the linear and nonlinear Schrödinger equations in the two-dimensional setting. By extending ideas and concepts presented earlier [X. Antoine and E. Lorin,Numer. Math.137(2017) 923–958] and by using pseudodifferential calculus, we prove the convergence and determine some approximate rates of convergence of the two-dimensional Classical SWR method for two subdomains with smooth boundary. Some numerical experiments are also proposed to validate the analysis.

Linear and nonlinear coherent perfect absorbers on simple layers

We consider linear and nonlinear coherent perfect absorbers (CPAs) in multidimensional geometries and construct explicitly the respective perfectly absorbed solutions. The multidimensional CPAs have a structure of the so-called simple layers which represent the generalization of the point $\delta$ function potential to higher dimensions. The considered examples include broadband CPAs confined to a straight line (in a two-dimensional setting) and to a plane (in the three-dimensional space); CPAs for topological vortices on an absorbing circle; as well as axially-symmetric CPAs on a sphere. Additionally, it is shown that a paraxial beam propagating along a surface nonlinear CPA embedded in the three-dimensional space can be stable against perturbations. The results are interpreted in applications to optical and acoustic systems.

Characterising particulate random media from near-surface backscattering: A machine learning approach to predict particle size and concentration

To what extent can particulate random media be characterised using direct wave backscattering from a single receiver/source? Here, in a two-dimensional setting, we show using a machine learning approach that both the particle radius and concentration can be accurately measured when the boundary condition on the particles is of Dirichlet type. Although the methods we introduce could be applied to any particle type. In general backscattering is challenging to interpret for a wide range of particle concentrations, because multiple scattering cannot be ignored, except in the very dilute range. Across the concentration range from 1% to 20% we find that the mean backscattered wave field is sufficient to accurately determine the concentration of particles. However, to accurately determine the particle radius, the second moment, or average intensity, of the backscattering is necessary. We are also able to determine what is the ideal frequency range to measure a broad range of particles sizes. To get rigorous results with supervised machine learning requires a large, highly precise, dataset of backscattered waves from an infinite half-space filled with particles. We are able to create this dataset by introducing a numerical approach which accurately approximates the backscattering from an infinite half-space.

Shape optimized inclined single and double wall wave barriers for ground vibration mitigation

Stiff wave barriers are capable of reducing the transmission of ground vibrations. Most designs consist of a single vertical wall, although double walls are also being considered. This paper investigates the shape optimization (position, inclination, length and thickness) of these topologies in a two-dimensional setting, for a point source and a point receiver placed symmetrically with respect to the design domain. Three types of sources are studied: a single-frequency source, a broadband source and a harmonic source within a given frequency range. An economical constraint on the maximum material use is considered. A multi-region BEM methodology is used for evaluating the objective function and its gradient. Analytical expressions are presented for the sensitivities, providing a very effective simulation tool for this type of problem. It is found that significant improvement can be achieved by repositioning and inclining the walls when compared to the reference cases. It is also found that optimized double wall barriers outperform single wall barriers. The improvement is insignificant for sources which generate Rayleigh wavelengths similar to the design domain depth, but it greatly increases as frequency increases and the penetration depth decreases.

On the use of reduced integration in combination with discontinuous Galerkin discretization: application to volumetric and shear locking problems

In the present work, the discontinuous Galerkin (DG) method is applied to linear elasticity for two-dimensional and three-dimensional settings. A locking-free element formulation based on reduced integration and physically-based hourglass stabilization (Q1SP) is coupled for the first time with the DG framework. The incomplete interior penalty Galerkin method is chosen, being one example of different variations of DG methods. Several 2D and 3D typical benchmark problems of linear elasticity are investigated. A selection of numerical integration schemes for the boundary terms is presented, namely reduced and mixed integration schemes. The treatment of the surface terms by means of different rules of integration shows a significant influence on the performance of the resulting DG method in combination with the standard Q1 element. This intelligent treatment of the surface part leads to a DG variant with very good convergence properties.

Spontaneous motion of localized structures induced by parity symmetry breaking transition.

We consider a paradigmatic nonvariational scalar Swift-Hohenberg equation that describes short wavenumber or large wavelength pattern forming systems. This work unveils evidence of the transition from stable stationary to moving localized structures in one spatial dimension as a result of a parity breaking instability. This behavior is attributed to the nonvariational character of the model. We show that the nature of this transition is supercritical. We characterize analytically and numerically this bifurcation scenario from which emerges asymmetric moving localized structures. A generalization for two-dimensional settings is discussed.

The method of transformed angular basis function for solving the Laplace equation

In this paper, we propose a new approach to improve the method of angular basis function (MABF) proposed by Young et al. (2015) for the Laplace equation in two-dimensional settings. Instead of the fundamental solution ln r used in the traditional Method of Fundamental Solution (MFS), MABF employs a different basis function θ and produces good approximate solutions on the domains with acute, narrow regions and exterior problems (Young et al., 2015). However, the definition of θ inevitably incurs a singularity situation for many different types of domains. Therefore, the selection of source points of MABF is not as convenient as the traditional MFS. To avoid the singularity situation in implementing, we introduce a transformation so that the transformed angular basis function does not exhibit this type of singularity for commonly used distributions of source points. As a result, source points for the method of transformed angular basis function (MTABF) can then be chosen in a similar way to traditional MFS. Numerical experiments demonstrate that the proposed approach significantly simplifies the selection of source points in MABF for different types of domains, which makes MABF more applicable. Numerical results of MTABF and MFS are presented for comparison purposes.

Topology in colored tensor models via crystallization theory

The aim of this paper is twofold. On the one hand, it provides a review of the links between random tensor models, seen as quantum gravity theories, and the PL-manifolds representation by means of edge-colored graphs (crystallization theory). On the other hand, the core of the paper is to establish results about the topological and geometrical properties of the Gurau-degree (or G-degree) of the represented manifolds, in relation with the motivations coming from physics. In fact, the G-degree appears naturally in higher dimensional tensor models as the quantity driving their 1/N expansion, exactly as it happens for the genus of surfaces in the two-dimensional matrix model setting. In particular, the G-degree of PL-manifolds is proved to be finite-to-one in any dimension, while in dimension 3 and 4 a series of classification theorems are obtained for PL-manifolds represented by graphs with a fixed G-degree. All these properties have specific relevance in the tensor models framework, showing a direct fruitful interaction between tensor models and discrete geometry, via crystallization theory.

Reconstruction of cloud geometry from high-resolution multi-angle images

We consider the reconstruction of the interface of compact, connected clouds from satellite or airborne light intensity measurements. In a two-dimensional setting, the cloud is modeled by an interface, locally represented as a graph, and an outgoing radiation intensity that is consistent with a diffusion model for light propagation in the cloud. Light scattering inside the cloud and the internal optical parameters of the cloud are not modeled explicitly. The main objective is to understand what can or cannot be reconstructed in such a setting from intensity measurements in a finite (on the order of 10) number of directions along the path of a satellite or an aircraft. Numerical simulations illustrate the theoretical predictions. Finally, we explore a kinematic extension of the algorithm for retrieving cloud motion (wind) along with its geometry.

Analytical validation of a $2+1$ dimensional continuum model for epitaxial growth with elastic substrate

An analiyical validation is obtained for the evolution equation ht = ∆[F−1(−aEF(h))− r/h2 −∆h], introduced in [18] by W.T. Tekalign and B.J. Spencer to describe the heteroepitaxial growth of a two-dimensional thin film on an elastic substrate. In the expression above, h denotes the surface height of the film, F is the Fourier transform, and a, E, r are positive material constants. Existence, uniqueness, and Lipschitz regularity in time for weak solutions are proved, under suitable assumptions on the initial datum. Introduction The epitaxial deposition of a thin film on a relatively thick substrate has gained much interest in the recent years due to its applications to semiconductor electronics and quantum dots. Roughly speaking, the morphology of the film is known to be the result of a competition between the elastic energy associated to the mismatch between film and substrate, and the surface mass transport due to the film deposition. An extensive mathematical analysis of the mechanism associated to epitaxial film growth has been carried out in [2, 3, 10, 11, 15, 16] in the context of plane linear elasticity, and regularity results have been established for volume-constrained minimizers. Short time existence for a surface diffusion type geometric evolution equation keeping into account elasticity has first been analyzed in [12] in a two-dimensional setting (see also [17]). The previous result has been recently extended in [13] to the three-dimensional case. The central aim of this work is to study existence and Lipschitz regularity in time of weak solutions to the 2+1 dimensional evolution equation ht = ∆[F−1(−aEF(h))− r/h2 −∆h] (0.1) derived by W.T. Tekalign and B.J. Spencer in [18], where h denotes the surface height of the film and F is the Fourier transform. The quantities a, E, r are positive material constants. To be precise, a (resp. r) is the wavenumber (resp. wetting coefficient) associated to the equation, and E is defined as E := 2μ F (1 + ν )(1− ν) (1− νF )μS , where μ and ν (resp. μ and ν) are the elastic shear modulus and the Poisson’s 2010 Mathematics Subject Classification. 35K55, 35K67, 44A15, 74K35.

Histological and Morphological Characterization of Developing Dermal Lymphatic Vessels.

The capacity to visualize the lymphatic vasculature in three-dimensions has revolutionized our understanding of the morphogenetic mechanisms important for constructing the lymphatic vascular network during development. Two complementary approaches are commonly employed to assess the function of genes and signaling pathways important for development of the dermal lymphatic vasculature in the mouse embryo. The first of these is whole-mount immunostaining of embryonic skin to analyze dermal lymphatic vessel network patterning and morphology in two and three dimensions. The second is immunostaining of thin tissue sections to examine lymphatic vessel identity, lumen formation and protein localization within discrete lymphatic endothelial cells in a two-dimensional setting. Here we present detailed protocols for multicolor immunofluorescent immunostaining of embryonic dorsal skin and thin tissue cryosections. Each of these methods generates high-resolution images of the dermal lymphatic vasculature, yielding information integral to in-depth characterization of lymphatic vessel phenotypes in the developing mouse embryo.

Two-Dimensional Surface Wave Propagation over Arbitrary Ridge-Like Topographies

We extend the conformal mapping technique for water waves over topography, from its natural two-dimensional (2D) setting to three dimensions (3D) where the Laplacian, from potential theory, is no l...

A Numerical Framework for Diffusive Transport in Rotational Symmetric Systems with Discontinuous Interlayer Conditions

A numerical method based on a finite element approach for modelling interface conditions in the context of diffusive transport is presented. Such interface conditions may result in concentration jumps due to drug partitioning and/or mass transfer effects. The paper extends previous work in a one-dimensional multilayer setting to the two-dimensional one of cylindrical coordinates with rotational symmetry, applicable e.g. to drug transport in and out of optical lenses with possible surface barriers. Simulations of such a case are presented. The method derived is also applicable to a two-dimensional Cartesian setting.

On the Optimal Management of Public Debt: a Singular Stochastic Control Problem

Consider the problem of a government that wants to reduce the debt-to-GDP (gross domestic product) ratio of a country. The government aims at choosing a debt reduction policy which minimizes the total expected cost of having debt, plus the total expected cost of interventions on the debt ratio. We model this problem as a singular stochastic control problem over an infinite time horizon. In a general not necessarily Markovian framework, we first show by probabilistic arguments that the optimal debt reduction policy can be expressed in terms of the optimal stopping rule of an auxiliary optimal stopping problem. We then exploit such a link to characterize the optimal control in a two-dimensional Markovian setting in which the state variables are the level of the debt-to-GDP ratio and the current inflation rate of the country. The latter follows uncontrolled Ornstein--Uhlenbeck dynamics and affects the growth rate of the debt ratio. We show that it is optimal for the government to adopt a policy that keeps th...

Geometric frustration and compatibility conditions for two-dimensional director fields.

Bent core (or banana shaped) liquid-crystal-forming-molecules locally favor an ordered state of zero splay and constant bend. Such a state, however, cannot be realized in the plane and the resulting liquid-crystalline phase is frustrated and must exhibit some compromise of these two mutually contradicting local intrinsic tendencies. This constitutes one of the most well-studied examples in which the intrinsic geometry of the constituents of a material gives rise to a geometrically frustrated assembly. Such geometric frustration is not only natural and ubiquitous but also leads to a striking variety of morphologies of ground states and exotic response properties. In this work we establish the necessary and sufficient conditions for two scalar functions, s and b to describe the splay and bend of a director field in the plane. We generalize these compatibility conditions for geometries with non-vanishing constant Gaussian curvature, and provide a reconstruction formula for the director field depending only on the splay and bend fields and their derivatives. Finally, we discuss optimal compromises for simple incompatible cases where the locally preferred values of the splay and bend cannot be simultaneously achieved.

Parallel space–time hp adaptive discretization scheme for parabolic problems

We introduce an integration scheme for parabolic problems. Our parallelizable method uses adaptive h p -finite elements in space, and finite differences in time. The strategy can also be combined with a classical finite element method parallelization technique based on domain decomposition. We verified the performance of our method against two different benchmarks, in both two-dimensional (model problem on an L-shaped domain) and three-dimensional (Pennes bioheat equation) settings. Results show a significant speedup in computational time when compared with the sequential version of the algorithm. Moreover, we develop a mathematical framework to analyze similar schemes which include h p spatial adaptivity. Our framework describes error propagation rigorously, and as such allows to analyze convergence properties of these mixed methods.

Disclosure and Choice

An agent chooses among projects with random outcomes. His payoff is increasing in the outcome and in an observer's expectation of the outcome. With some probability, the agent will be able to disclose some information about the true outcome to the observer. We show that choice is inefficient in general. We illustrate this point with a characterization of the inefficiencies that result when the agent can perfectly disclose the outcome with some probability and can disclose nothing otherwise as in Dye (1985a). In this case, the agent favours riskier projects even with lower expected returns. On the other hand, if information can also be disclosed by a challenger who prefers lower beliefs of the observer, the chosen project is excessively risky when the agent has better access to information, excessively risk-averse when the challenger has better access, and efficient otherwise. We also characterize the agent's worst-case equilibrium payoff. We give examples of alternative disclosure technologies illustrating other forms the inefficiencies can take. For example, in a two-dimensional setting, we demonstrate a “hitting for the fences” effect where the agent systematically focuses on the “harder” dimension at the expense of success on the easier.

Neutrophil-inspired propulsion in a combined acoustic and magnetic field

Systems capable of precise motion in the vasculature can offer exciting possibilities for applications in targeted therapeutics and non-invasive surgery. So far, the majority of the work analysed propulsion in a two-dimensional setting with limited controllability near boundaries. Here we show bio-inspired rolling motion by introducing superparamagnetic particles in magnetic and acoustic fields, inspired by a neutrophil rolling on a wall. The particles self-assemble due to dipole–dipole interaction in the presence of a rotating magnetic field. The aggregate migrates towards the wall of the channel due to the radiation force of an acoustic field. By combining both fields, we achieved a rolling-type motion along the boundaries. The use of both acoustic and magnetic fields has matured in clinical settings. The combination of both fields is capable of overcoming the limitations encountered by single actuation techniques. We believe our method will have far-reaching implications in targeted therapeutics.

Large-Scale Structured Sparsity via Parallel Fused Lasso on Multiple GPUs

ABSTRACTWe present a massively parallel algorithm for the fused lasso, powered by a multiple number of graphics processing units (GPUs). Our method is suitable for a class of large-scale sparse regression problems on which a two-dimensional lattice structure among the coefficients is imposed. This structure is important in many statistical applications, including image-based regression in which a set of images are used to locate image regions predictive of a response variable such as human behavior. Such large datasets are increasingly common. In our study, we employ the split Bregman method and the fast Fourier transform, which jointly have a high data-level parallelism that is distinct in a two-dimensional setting. Our multi-GPU parallelization achieves remarkably improved speed. Specifically, we obtained as much as 433 times improved speed over that of the reference CPU implementation. We demonstrate the speed and scalability of the algorithm using several datasets, including 8100 samples of 512 × 512 images. Compared to the single GPU counterpart, our method also showed improved computing speed as well as high scalability. We describe the various elements of our study as well as our experience with the subtleties in selecting an existing algorithm for parallelization. It is critical that memory bandwidth be carefully considered for multi-GPU algorithms. Supplementary material for this article is available online.