Multiple Spatial Dimensions Research Articles

YAMATO: The First SDN Control Plane for Independent, Joint, and Fractional-Joint Switched SDM Optical Networks

The deployment of space division multiplexed (SDM) optical networks hinges on this technology being able to offer a beneficial ratio between the expected capacity gains and the resulting cost increase. In order to keep the cost of SDM low, several integrated devices have been proposed to perform the functions of transmission, reception, amplification, and switching; however, little or no work has been carried out in creating a network model to represent their behavior, and in actually managing and controlling them, especially with respect to those devices that are able to jointly switch multiple spatial dimensions. This paper describes YAMATO, the first control plane designed to handle any type of SDM fiber and switching paradigm. To do so, we present a network model capable of encoding any SDM link and node, and describe the design and functions of our network controller, which is based on the OpenDaylight framework. Furthermore, we investigate the additional complexity faced by optical restoration schemes for SDM compared to standard optical networks. We experimentally evaluate the proposed system via an emulated testbed, and report some performance measurements of our implementation.

The Piecewise Cubic Method (PCM) for computational fluid dynamics

We present a new high-order finite volume reconstruction method for hyperbolic conservation laws. The method is based on a piecewise cubic polynomial which provides its solutions a fifth-order accuracy in space. The spatially reconstructed solutions are evolved in time with a fourth-order accuracy by tracing the characteristics of the cubic polynomials. As a result, our temporal update scheme provides a significantly simpler and computationally more efficient approach in achieving fourth order accuracy in time, relative to the comparable fourth-order Runge–Kutta method. We demonstrate that the solutions of PCM converges at fifth-order in solving 1D smooth flows described by hyperbolic conservation laws. We test the new scheme on a range of numerical experiments, including both gas dynamics and magnetohydrodynamics applications in multiple spatial dimensions.

Improving Performance of Spatially Joint-Switched Space Division Multiplexing Optical Networks via Spatial Group Sharing

Space division multiplexing (SDM) is a promising approach to overcome the looming fiber capacity crunch. Joint and fractional joint switching (JoS and FJoS) architectures, which switch multiple spatial dimensions together as atomic units, have been proposed to reduce the cost of implementing SDM and cater to strongly coupled transmission media. However, they have been shown to impose significant penalties to overall network performance, at least when many relatively small connections are used in the context of uncoupled spatial dimensions, while performing nearly as well as the independent switching (InS) architecture if the average connection size is sufficiently large to efficiently exploit their increased granularity. In this work we tackle the performance penalty that JoS and FJoS architectures impose on SDM networks with uncoupled spatial dimensions compared to InS, by proposing two novel routing, space, and spectrum allocation (RSSA) heuristics that effectively share jointly switched resources, adapted to use different types of SDM reconfigurable optical add/drop multiplexer architectures requiring only a small additional cost. We investigate the performance of these new RSSA algorithms compared to an existing one under several traffic conditions and models using simulations. We find that our proposals can achieve significant improvements in the performance of SDM networks carrying large numbers of relatively small connections, up to parity of performance with InS designs, at a lower implementation cost, under all traffic conditions under study, including different average connection sizes and size diversity.

Moving forward: insights and applications of moving‐habitat models for climate change ecology

Summary Predicting and managing species’ responses to climate change is one of the most significant challenges of our time. Tools are needed to address problems associated with novel climatic conditions, biotic interactions and greater climate velocities. We present a spatially explicit moving‐habitat model (MHM) and demonstrate its versatility in tackling critical questions in climate change research, including dispersal in multiple spatial dimensions, population stage structure, interspecific interactions, asymmetric range shifts, Allee effects and the presence of infectious diseases. The model utilizes integrodifference equations to track changes in population density over time in a habitat that is moving. The model is quite flexible and can accommodate variation in demography, dispersal patterns, biotic interaction and stochasticity in the velocity of climate change. The methods provide a general mechanistic understanding of the underlying ecological processes that drive a system. Field data can be readily incorporated into the model to give insight into specific populations of interest and inform management decisions. Synthesis. Moving‐habitat models unite ecological theory, data‐centred modelling and conservation decision support under a single framework. Their ability to generate testable hypotheses, incorporate data and inform best management practices proves that these models provide a valuable framework for climate change biologists.

Structured regression models for high-dimensional spatial spectroscopy data

Modeling and analysis of spectroscopy data is an active area of research with applications to chemistry and biology. This paper focuses on modelling high-dimensional spectra for the purpose of noise reduction and prediction in problems where the spectra can be used as covariates. We propose a functional representation of the spectra as well as functional regression model that accommodates multiple spatial dimensions. Both steps emphasize sparsity to reduce the number of parameters and mitigate over-fitting. The motivating application for these models, discussed in some detail, is predicting bone-mineral-density (BMD), an important indicator of fracture healing, from Raman spectra, in both the in vivo and ex vivo settings of a bone fracture healing experiment. To illustrate the general applicability of the method, we also use it to predict lipoprotein concentrations from spectra obtained by nuclear magnetic resonance (NMR) spectroscopy.

Evaluating multiple spatial dimensions of economic growth in Brazil using spatial panel data models

The goal of this paper is to evaluate the results of regional economic growth model estimations at multiple spatial scales using spatial panel data models. The spatial scales examined are minimum comparable areas, microregions, mesoregions and states between 1970 and 2000. Alternative spatial panel data models with fixed effects were systematically estimated across those spatial scales to demonstrate that the estimated coefficients change with the scale level. The results show that the conclusions obtained from growth regressions depend on the choice of spatial scale. First, the values of spatial spillover coefficients vary according to the spatial scale under analysis. In general, such coefficients are statistically significant at the MCA, microregional and mesoregional levels, however, at state level those coefficients are no longer statistically significant, suggesting that spatial spillovers are bounded in space. Moreover, the positive average-years-of-schooling direct effect coefficient increases as more aggregate spatial scales are used. Population density coefficients show that higher populated areas are harmful to economic growth, indicating that congestion effects are operating in all spatial scales, but their magnitudes vary across geographic scales. Finally, the club convergence hypothesis cannot be rejected suggesting that there are differences in the convergence processes between the north and south in Brazil. Furthermore, the paper discusses the potential theoretical reasons for different results found across estimations at different spatial scales.

Distribution Functions of Saturation for Stochastic Nonlinear Two-Phase Flow

We derive analytical expressions for the one-point cumulative distribution function and probability density function of water saturation in the one-dimensional stochastic immiscible two-phase (Buckley–Leverett) problem. The sources of uncertainty are the spatial distributions of porosity and total velocity. The derived distribution functions involve integrals of the input random parameters. Comparisons with standard Monte Carlo simulation demonstrate that the method is applicable for input parameters with large variance and arbitrary correlation lengths. We also show that the proposed method is superior to the low-order statistical moment equations approach. We also outline a streamline-based strategy to extend our distribution-based method to multiple spatial dimensions.

Commentary: Developmental connectomics to advance our understanding of typical and atypical brain development--a commentary on Vértes and Bullmore (2015).

Vértes and Bullmore's article lays a framework for applying connectomics, the study of brain function from the perspective of underlying network organization, to advance understanding of healthy and maladaptive brain development. They elucidate the power of connectomics for bridging both different levels of analysis (e.g. from synapses to brain regions) and multiple academic fields. In this commentary, we highlight important themes and remaining questions stemming from Vértes and Bullmore's work, including: (a) the application of connectomics in the context of integrating analyses across multiple spatial and temporal dimensions, (b) the extent to which connectomics might be applied in translational and clinical studies of development, (c) growth connectomics and the Developmental Origins of Health and Disease (DOHaD) hypothesis, and (d) the importance and complexity of sound methodological practices in applying connectomics to developmental and clinical science. Ongoing work in these areas will be important for fulfilling the promise of connectomics as a bridge between neuroscience, developmental science, and translational and clinical research.

A note on an accelerated exponential Euler method for parabolic SPDEs with additive noise

This note aims to present further results on the accelerated exponential Euler method proposed in Jentzen & Kloeden (2009). In contrast to very restrictive assumptions made there, we reformulate appropriate conditions on the drift coefficient of SPDEs to include a large class of nonlinear Nemytskii operators. In our setting, the method achieves the convergence order in time of 1 2 − ϵ for arbitrarily small ϵ > 0 in the case of space–time white noise. For the trace-class noise case, multiple spatial dimensions are allowed and an optimal convergence rate is attained based on optimal regularity results of the mild solution, which improves the corresponding convergence results in Jentzen et al. (2011).

Visual Overlay on OpenStreetMap Data to Support Spatial Exploration of Urban Environments

Increasing volumes of spatial data about urban areas are captured and made available via volunteered geographic information (VGI) sources, such as OpenStreetMap (OSM). Hence, new opportunities arise for regional exploration that can lead to improvements in the lives of citizens through spatial decision support. We believe that the VGI data of the urban environment could be used to present a constructive overview of the regional infrastructure with the advent of web technologies. Current location-based services provide general map-based information for the end users with conventional local search functionality, and hence, the presentation of the rich urban information is limited. In this work, we analyze the OSM data to classify the geo entities into consequential categories with facilities, landscape and land use distribution. We employ a visual overlay of heat map and interactive visualizations to present the regional characterization on OSM data classification. In the proposed interface, users are allowed to express a variety of spatial queries to exemplify their geographic interests. They can compare the characterization of urban areas with respect to multiple spatial dimensions of interest and can search for the most suitable region. The search experience is further enhanced via efficient optimization and interaction methods to support the decision making of end users. We report the end user acceptability and efficiency of the proposed system via usability studies and performance analysis comparison.

Distributed-memory parallelization of the Wigner Monte Carlo method using spatial domain decomposition

The Wigner Monte Carlo method, based on the generation and annihilation of particles, has emerged as a promising approach to treat transient problems of quantum electron transport in nanostructures. Tackling these simulations in multiple spatial dimensions demands a parallelized approach to facilitate a practical application of the method in order to investigate realistic problems, due to the otherwise exorbitant execution-times and memory requirements. Because of the annihilation step, a straight-forward parallelization of the Wigner Monte Carlo code is not possible, since sub-ensembles of particles can not be treated independently. Moreover, the large memory requirements of the annihilation procedure presents challenges when working in a distributed-memory setting. A solution to this problem is presented here with a parallelization approach using a spatial domain decomposition, implemented using the message passing interface. The presented benchmark results, based on standard one-dimensional examples, exhibit a good efficiency in the scalability of not only speed, but also memory consumption, which is paramount for the simulation of realistic devices.

Open boundary conditions for the Diffuse Interface Model in 1-D

New techniques are developed for solving multi-phase flows in unbounded domains using the Diffuse Interface Model in 1-D. They extend two open boundary conditions originally designed for the Navier–Stokes equations. The non-dimensional formulation of the DIM generalizes the approach to any fluid. The equations support a steady state whose analytical approximation close to the critical point depends only on temperature. This feature enables the use of detectors at the boundaries switching between conventional boundary conditions in bulk phases and a multi-phase strategy in interfacial regions. Moreover, the latter takes advantage of the steady state approximation to minimize the interface–boundary interactions. The techniques are applied to fluids experiencing a phase transition and where the interface between the phases travels through one of the boundaries. When the interface crossing the boundary is fully developed, the technique greatly improves results relative to cases where conventional boundary conditions can be used. Limitations appear when the interface crossing the boundary is not a stable equilibrium between the two phases: the terms responsible for creating the true balance between the phases perturb the interior solution. Both boundary conditions present good numerical stability properties: the error remains bounded when the initial conditions or the far field values are perturbed. For the PML, the influence of its main parameters on the global error is investigated to make a compromise between computational costs and maximum error. The approach can be extended to multiple spatial dimensions.

The spectral theory of tensors and its applications

Over the last few years, there has been a realization that there is a reasonably complete and consistent theory of eigenvalues and singular values for tensors of higher order. This theory, which generalizes the theory of matrix eigenvalues and singular values in various manners and extent, was proposed by Lim 1 and Qi 2 independently. We will use the term ‘spectral theory of tensors’ to refer to this body of work. This special issue contains nine papers with new results on the spectral theory of tensors and its applications. The first paper 3 by Chang et al. is a survey on some recent developments in the spectral theory of nonnegative tensors: H-eigenvalue and Z-eigenvalue problems, Perron–Frobenius theory, applications to higher order Markov chains, spectral theory of hypergraphs, and quantum entanglement. The next two papers are about the spectral theory of nonnegative tensors. In 4, Zhou et al. proposed new spectral characterizations of symmetric nonnegative tensors, while in 5, Zhang et al. extended the concept of essential nonnegativity from matrices to higher order tensors and established the convexity and log convexity of dominant eigenvalues for such a class of tensors. The next three papers concern the numerical computations of tensor eigenvalues or singular values. In 6, Friedland et al. proposed an algorithm called the alternating singular value decomposition for computing a best rank-one approximation of tensors based on computations of maximal singular values and singular vectors of matrices. In 7, Dupont and Scott studied the angular orientations of convergent iterates generated by Newton's method in multiple spatial dimensions for solving a tensor eigenproblem. In 8, Hu et al. studied tensor conic linear programming and employed it to solve the extreme Z-eigenvalue problem. In 9, Li et al. studied the perturbation bound for the Perron vector of a transition probability tensor P, and derived a new perturbation bound for the Perron vector of a transition probability matrix. Finally, the last papers are related to the applications of the spectral theory of tensors to hypergraphs. In 10, Li et al. examined some fundamental analytic properties of Z-eigenvalues of even order symmetric tensors; they showed that the maximum Z-eigenvalue function of the associated characteristic tensor provides a link between the combinatorial and analytic structures of the underlying hypergraph. In 11, Xie and Chang generalized the signless Laplacian matrices for graphs to an analogous notion for even uniform hypergraphs and deduced some of its fundamental properties. We hope that the reader will find these articles interesting and be spurred to either commence or continue working in this exciting new area.

Communication: Two-dimensional gas-phase coherent anti-Stokes Raman spectroscopy (2D-CARS): Simultaneous planar imaging and multiplex spectroscopy in a single laser shot

Coherent anti-Stokes Raman spectroscopy (CARS) has been widely used as a powerful tool for chemical sensing, molecular dynamics measurements, and rovibrational spectroscopy since its development over 30 years ago, finding use in fields of study as diverse as combustion diagnostics, cell biology, plasma physics, and the standoff detection of explosives. The capability for acquiring resolved CARS spectra in multiple spatial dimensions within a single laser shot has been a long-standing goal for the study of dynamical processes, but has proven elusive because of both phase-matching and detection considerations. Here, by combining new phase matching and detection schemes with the high efficiency of femtosecond excitation of Raman coherences, we introduce a technique for single-shot two-dimensional (2D) spatial measurements of gas phase CARS spectra. We demonstrate a spectrometer enabling both 2D plane imaging and spectroscopy simultaneously, and present the instantaneous measurement of 15,000 spatially correlated rotational CARS spectra in N2 and air over a 2D field of 40 mm(2).

Subcell resolution in simplex stochastic collocation for spatial discontinuities

Subcell resolution has been used in the Finite Volume Method (FVM) to obtain accurate approximations of discontinuities in the physical space. Stochastic methods are usually based on local adaptivity for resolving discontinuities in the stochastic dimensions. However, the adaptive refinement in the probability space is ineffective in the non-intrusive uncertainty quantification framework, if the stochastic discontinuity is caused by a discontinuity in the physical space with a random location. The dependence of the discontinuity location in the probability space on the spatial coordinates then results in a staircase approximation of the statistics, which leads to first-order error convergence and an underprediction of the maximum standard deviation. To avoid these problems, we introduce subcell resolution into the Simplex Stochastic Collocation (SSC) method for obtaining a truly discontinuous representation of random spatial discontinuities in the interior of the cells discretizing the probability space. The presented SSC–SR method is based on resolving the discontinuity location in the probability space explicitly as function of the spatial coordinates and extending the stochastic response surface approximations up to the predicted discontinuity location. The applications to a linear advection problem, the inviscid Burgers’ equation, a shock tube problem, and the transonic flow over the RAE 2822 airfoil show that SSC–SR resolves random spatial discontinuities with multiple stochastic and spatial dimensions accurately using a minimal number of samples.

Using Generalized Dimensional Analysis to Obtain Reduced Effective Model Equations for Condensation in Slender Tubes With Rotational Symmetry

In this paper, we study the continuum physics model equations for condensation (two phase flow problems) in vertical tubes with small diameter and obtain reduced model equations. We found that generalization of dimensional analysis to multiple spatial dimensions is an excellent tool for that purpose, so that a review of this method is also part of the paper. We obtain the nondimensional numbers of the problem and derive reduced bulk and interface equations. The problem is characterized by three length scales, tube radius R, tube length L, and initial film thickness H. For small ratio ɛ=H/L, we derive a single ordinary differential equation for the condensate film thickness as function of axial position with tube radius as parameter, which agrees well with commonly used (parametric) models from literature. Our model is based on the physical dimensions of the problem which gives a greater geometrical flexibility and a wider range of applicability. We also discuss the effect of surface tension and the limit of the model.

Optimal control of one‐dimensional cellular uptake in tissue engineering

A control problem motivated by tissue engineering is formulated and solved in which control of the uptake of growth factors (signaling molecules) is necessary to spatially and temporally regulate cellular processes for the desired growth or regeneration of a tissue. Four approaches are compared for determining 1D optimal boundary control trajectories for a distributed parameter model with reaction, diffusion, and convection: (i) basis function expansion, (ii) method of moments, (iii) internal model control (IMC), and (iv) model predictive control (MPC). The proposed method-of-moments approach is computationally efficient while enforcing a non-negativity constraint on the control input. While more computationally expensive than methods (i)-(iii), the MPC formulation significantly reduced the computational cost compared to simultaneous optimization of the entire control trajectory. A comparison of the pros and cons of each of the four approaches suggests that an algorithm that combines multiple approaches is most promising for solving the optimal control problem for multiple spatial dimensions.

A NEW MONTE CARLO METHOD FOR TIME-DEPENDENT NEUTRINO RADIATION TRANSPORT

Monte Carlo approaches to radiation transport have several attractive properties such as simplicity of implementation, high accuracy, and good parallel scaling. Moreover, Monte Carlo methods can handle complicated geometries and are relatively easy to extend to multiple spatial dimensions, which makes them potentially interesting in modeling complex multi-dimensional astrophysical phenomena such as core-collapse supernovae. The aim of this paper is to explore Monte Carlo methods for modeling neutrino transport in core-collapse supernovae. We generalize the Implicit Monte Carlo photon transport scheme of Fleck & Cummings and gray discrete-diffusion scheme of Densmore et al. to energy-, time-, and velocity-dependent neutrino transport. Using our 1D spherically-symmetric implementation, we show that, similar to the photon transport case, the implicit scheme enables significantly larger timesteps compared with explicit time discretization, without sacrificing accuracy, while the discrete-diffusion method leads to significant speed-ups at high optical depth. Our results suggest that a combination of spectral, velocity-dependent, Implicit Monte Carlo and discrete-diffusion Monte Carlo methods represents a robust approach for use in neutrino transport calculations in core-collapse supernovae. Our velocity-dependent scheme can easily be adapted to photon transport.

Neighbourhoods and potential access to health care: The role of spatial and aspatial factors

The availability of, and access to, primary health care is one neighbourhood characteristic that has the potential to impact health thus representing an important area of focus for neighbourhood-health research. This research examines neighbourhood access to primary health care in the city of Mississauga, Ontario, Canada. A modification of the Two Step Floating Catchment Area method is used to measure multiple spatial and aspatial (social) dimensions of potential access to primary health care in natural neighbourhoods of Mississauga. The analysis reveals that neighbourhood-level potential access to primary care is dependant on spatial and aspatial dimensions of access selected for examination. The results also show that potential accessibility is reduced for linguistic minorities as well as for recent immigrant populations who appear, on the surface, to have better access to walk-in clinics than dedicated physicians. The research results reinforce the importance of focusing on intra-urban variations in access to care and demonstrate the utility of a new approach for studying neighbourhood impacts that better represents spatial variations in health care access and demand.

Vlasov simulation in multiple spatial dimensions

A long-standing challenge encountered in modeling plasma dynamics is achieving practical Vlasov equation simulation in multiple spatial dimensions over large length and time scales. While direct multi-dimension Vlasov simulation methods using adaptive mesh methods [M. Gutnic et al., Comput. Phys. Commun. 164, 214 (2004)] have recently shown promising results in two dimensions (2D) [J. W. Banks et al., Phys. Plasmas 18, 052102 (2011); B. I. Cohen et al., November 10, 2010, http://meetings.aps.org/link/BAPS.2010.DPP.NP9.142], in this paper, we present an alternative, the Vlasov multi dimensional (VMD) model, that is specifically designed to take advantage of solution properties in regimes when plasma waves are confined to a narrow cone, as may be the case for stimulated Raman scatter in large optic f# laser beams. Perpendicular grid spacing large compared to a Debye length is then possible without instability or loss of accuracy, enabling an order 10 decrease in required computational resources compared to standard particle in cell (PIC) methods in 2D, with another reduction of that order in 3D. Further advantage compared to PIC methods accrues in regimes where particle noise is an issue. VMD and PIC results in a 2D model of localized Langmuir waves are in qualitative agreement.