Cell Average Data Research Articles

High order recovery of geometric interfaces from cell-average data

We consider the problem of recovering multivariate characteristic functions u := χΩ from cell-average data on a coarse grid, motivated in particular by the accurate treatment of interfaces in finite volume schemes. While linear recovery methods are known to perform poorly, nonlinear strategies based on local reconstructions of the jump interface Γ := ∂Ω by geometrically simpler interfaces may offer significant improvements. We study two main families of local reconstruction schemes the first one based on nonlinear least-squares fitting, the second one based on the explicit computation of a polynomial-shaped curve fitting the data, which yields simpler numerical computations and high order geometric fitting. For each of them, we derive a general theoretical framework which allows us to control the recovery error by the error of best approximation up to a fixed multiplicative constant. Numerical tests in 2d illustrate the expected approximation order of these strategies. Several extensions are discussed, in particular the treatment of piecewise smooth interfaces with corners.

Nonlinear approximation spaces for inverse problems

This paper is concerned with the ubiquitous inverse problem of recovering an unknown function u from finitely many measurements, possibly affected by noise. In recent years, inversion methods based on linear approximation spaces were introduced in [1, 2] with certified recovery bounds. It is however known that linear spaces become ineffective for approximating simple and relevant families of functions, such as piecewise smooth functions, that typically occur in hyperbolic PDEs (shocks) or images (edges). For such families, nonlinear spaces [3] are known to significantly improve the approximation performance. The first contribution of this paper is to provide with certified recovery bounds for inversion procedures based on nonlinear approximation spaces. The second contribution is the application of this framework to the recovery of general bidimensional shapes from cell-average data. We also discuss how the application of our results to n-term approximation relates to classical results in compressed sensing.

Explicit multivariate approximations from cell-average data

Given gridded cell-average data of a smooth multivariate function, we present a constructive explicit procedure for generating a high-order global approximation of the function. One contribution is the derivation of high-order approximations to point-values of the function directly from the cell-average data. The second contribution is the development of univariate B-spline-based high-order quasi-interpolation operators using cell-average data. Multivariate spline quasi-interpolation approximation operators are obtained by tensor product of the univariate operators.

Global and explicit approximation of piecewise-smooth two-dimensional functions from cell-average data

Abstract Given cell-average data values of a piecewise-smooth bivariate function $f$ within a domain $\varOmega $, we look for a piecewise adaptive approximation to $f$. We are interested in an explicit and global (smooth) approach. Bivariate approximation techniques, as trigonometric or splines approximations, achieve reduced approximation orders near the boundary of the domain and near curves of jump singularities of the function or its derivatives. Whereas the boundary of $\varOmega $ is assumed to be known, the subdivision of $\varOmega $ to subdomains on which $f$ is smooth is unknown. The first challenge of the proposed approximation algorithm would be to find a good approximation to the curves separating the smooth subdomains of $f$. In the second stage, we simultaneously look for approximations to the different smooth segments of $f$, where on each segment we approximate the function by a linear combination of basis functions $\{p_i\}_{i=1}^M$, considering the corresponding cell averages. A discrete Laplacian operator applied to the given cell-average data intensifies the structure of the singularity of the data across the curves separating the smooth subdomains of $f$. We refer to these derived values as the signature of the data, and we use it for both approximating the singularity curves separating the different smooth regions of $f$. The main contributions here are improved convergence rates to the approximation of the singularity curves and the approximation of $f$, an explicit and global formula, and, in particular, the derivation of a piecewise-smooth high-order approximation to the function.

Development of a WENO scheme based on radial basis function with an improved convergence order

In this article, we present a novel RBF-WENO scheme improving the fifth-order WENO techniques for solving hyperbolic conservation laws. The numerical flux is implemented by incorporating radial basis function (RBF) interpolation to cell average data. To do this, the classical RBF interpolation is amended to be suitable for cell average data setting. With the aid of a locally fitting parameter in the RBF, the RBF-WENO reconstruction attains an additional one order of improvement, resulting in the sixth-order of accuracy. In addition, on the purpose of detecting small scale structures and steep gradients more accurately, we present new smoothness indicators by devising a method of generalized undivided differences with exponential vanishing moments. Several experimental results are performed to confirm the effectiveness of the proposed WENO method.

Thermal Energy Production and Heat Exchange between an Electrochemical Cell and Its Surroundings

The theoretical formulation presented in this paper was developed to predict the cell average temperature as a function of time for a given cell discharge current and its initial temperature under adiabatic and nonadiabatic conditions. The cell average temperature versus time data calculated from the derived formulation is presented in the form of plots for an ideal lithium-based button cell (for example, lithium(s)/electrolyte/carbon monofluoride(s)) during its discharge period. The presented data are briefly discussed in light of cell component stability and safe discharge operation.

Gaussian Mixture Model for Millimeter-Wave Cellular Communication Networks

As compared to the microwave communication networks, the theoretical analyses of system performances such as the cell coverage and the cell average data rate are more difficult due to the unique propagation path loss model for mmWave cellular communication networks. In this paper, based on the unique transmission states of the distance-dependent piecewise propagation probability functions, the analytical six-state analytical mixture distribution of the propagation loss including the distance-dependent path loss, shadowing and small-scale fading is obtained. For each state, a novel Kullback–Leibler divergence based Gaussian approximation method is proposed to model the distribution of the propagation loss. The distribution of the propagation loss including the large-scale fading and small-scale fading is approximated via a six-state Gaussian mixture model, then the closed-form expression for the signal-to-noise ratio can be easily obtained. The cell coverage is expressed as the weighted sum of the error functions, and the cell average data rate is expressed as the weighted sum of the moments of the propagation loss for the six different states. Numerical results show that the cell coverage mainly depends on the none line-of-sight transmission and the cell average data rate mainly depends on the line-of-sight transmission.

On Weighted Average Interpolation with Cardinal Splines

Given a sequence of data $\{ y_{n} \} _{n \in \mathbb{Z}}$ with polynomial growth and an odd number $d$ , Schoenberg proved that there exists a unique cardinal spline $f$ of degree $d$ with polynomial growth such that $f ( n ) =y_{n}$ for all $n\in \mathbb{Z}$ . In this work, we show that this result also holds if we consider weighted average data $f\ast h ( n ) =y_{n}$ , whenever the average function $h$ satisfies some light conditions. In particular, the interpolation result is valid if we consider cell-average data $\int_{n-a}^{n+a}f ( x ) dx=y_{n}$ with $0< a\leq 1/2$ . The case of even degree $d$ is also studied.

User Scheduling for Cooperative Base Station Transmission Exploiting Channel Asymmetry

We study low-signalling overhead scheduling for downlink coordinated multi-point (CoMP) transmission with multi-antenna base stations (BSs) and single-antenna users. By exploiting the asymmetric channel feature, i.e., the path-loss differences towards different BSs, we derive a metric to judge orthogonality among users only using their average channel gains, based on which we propose a semi-orthogonal scheduler that can be applied in a two-stage transmission strategy. Simulation results demonstrate that the proposed scheduler performs close to the semi-orthogonal scheduler with full channel information, especially when each BS is with more antennas and the cell-edge region is large. Compared with other overhead reduction strategies, the proposed scheduler requires much less training overhead to achieve the same cell-average data rate.

Resource allocation for clustered network MIMO OFDMA systems

In this article, we address the resource allocation problem for the downlink of a large network multiple input multiple output orthogonal frequency division multiplexing system with 3-sector base stations. The system is statically divided into a number of disjoint clusters of sectors. A two-step resource allocation scheme is proposed involving the inter-cluster and the intra-cluster levels. As a first step or inter-cluster level, two cooperative frequency reuse approaches are designed to mitigate the inter-cluster interference. A user partition method is proposed to divide the users of each cluster into cluster-edge and cluster-center users. To balance the cell-edge and the cell-average performance, a fairness jug function is introduced to determine the frequency partition of the cooperative frequency reuse approaches. Then, as a second step or intra-cluster level, a utility-based joint scheduling and power allocation algorithm is proposed for each cluster, to maximize the sum utility of all users in the cluster under per-sector power constraints. Zero-forcing joint transmission is used across multiple sectors within the same cluster. Simulation results show that the proposed scheme can efficiently reduce the inter-cluster interference and provide considerable performance improvement in terms of both the cell-edge and cell-average user data rate. The proposed two-step resource allocation scheme can be implemented independently in each cluster without inter-cluster information exchange, which is an attractive property for practical systems, since it reduces both the network signaling overhead and the computational complexity.

A New fourth order central WENO method for 3D hyperbolic conservation laws

In this paper, a new fourth order central weighted non oscillatory (CWENO) scheme is presented to solve non-linear hyperbolic conservation laws in three spatial dimensions. The main idea of this method is a genuine three dimensional reconstruction procedure, in which a three dimensional interpolant is reconstructed from the cell average data. The procedure is performed on a staggered grid configuration and therefore, this method can be viewed as an extension of the previous work proposed by Levy et al. [5,6] to the three dimensional problems. The performance of the method is demonstrated through some standard numerical test cases.

Evaluation and bias correction of satellite rainfall data for drought monitoring in Indonesia

Abstract. The accuracy of three satellite rainfall products (TMPA 3B42RT, CMORPH and PERSIANN) was investigated through comparison with grid cell average ground station rainfall data in Indonesia, with a focus on their ability to detect patterns of low rainfall that may lead to drought conditions. Each of the three products underestimated rainfall in dry season months. The CMORPH and PERSIANN data differed most from ground station data and were also very different from the TMPA 3B42RT data. It proved possible to improve TMPA 3B42RT estimates by applying a single empirical bias correction equation that was uniform in space and time. For the six regions investigated, this reduced the root mean square error for estimates of dry season rainfall totals by a mean 9% (from 44 to 40 mm) and for annual totals by 14% (from 77 to 66 mm). The resulting errors represent 10% and 3% of mean dry season and annual rainfall, respectively. The accuracy of these bias corrected TMPA 3B42RT data is considered adequate for use in real-time drought monitoring in Indonesia. Compared to drought monitoring with only ground stations, this use of satellite-based rainfall estimates offers important advantages in terms of accuracy, spatial coverage, timeliness and cost efficiency.

A Modified Third-Order Semi-Discrete Central-Upwind Scheme for MHD Simulation

The Kurganov scheme is a third-order semi-discrete central numerical algorithm. The high solution of the scheme is ensured by a piecewise quadratic non-oscillatory reconstruction which consists of the cell-average data. We employ a modification of the smooth limiter of reconstruction in a simple way. The modified limiter possesses rigorous positivity and the reformulation does not change the non-oscillatory property of reconstruction. In order to explore the potential capability of application of the modified Kurganov scheme to magnetohydrodynamics (MHD) and resistive magnetohydrodynamics (RMHD) equations, two numerical problems are simulated in two dimensions (2D). These numerical simulations demonstrate that the modified Kurganov scheme keeps high precision and has stable reliable results for MHD and RMHD applications.

Vector cell-average multiresolution based on Hermite interpolation

Harten’s interpolatory multiresolution representation of data has been extended in the case of point-value discretization to include Hermite interpolation by Warming and Beam in [17]. In this work we extend Harten’s framework for multiresolution analysis to the vector case for cell-averaged data, focusing on Hermite interpolatory techniques.

Spectral (Finite) Volume Method for Conservation Laws on Unstructured Grids. Basic Formulation: Basic Formulation

A high-order, conservative, yet efficient method named the spectral volume (SV) method is developed for conservation laws on unstructured grids. The concept of a “spectral volume” is introduced to achieve high-order accuracy in an efficient manner similar to spectral element and multidomain spectral methods. Each spectral volume is further subdivided into control volumes, and cell-averaged data from these control volumes are used to reconstruct a high-order approximation in the spectral volume. Then Riemann solvers are used to compute the fluxes at spectral volume boundaries. Cell-averaged state variables in the control volumes are updated independently. Furthermore, total variation diminishing and total variation bounded limiters are introduced in the SV method to remove/reduce spurious oscillations near discontinuities. Unlike spectral element and multidomain spectral methods, the SV method can be applied to fully unstructured grids. A very desirable feature of the SV method is that the reconstruction is carried out analytically, and the reconstruction stencil is always nonsingular, in contrast to the memory and CPU-intensive reconstruction in a high-order k-exact finite volume method. Fundamental properties of the SV method are studied and high-order accuracy is demonstrated for several model problems with and without discontinuities.

On the use of Mühlbach expansions in the recovery step of ENO methods

The recovery step is the most expensive algorithmic ingredient in modern essentially non-oscillatory (ENO) shock capturing methods on triangular meshes for the numerical simulation of compressible fluid flow. While recovery polynomials in Newton form are used in one-dimensional ENO schemes it is a priori not clear whether such useful as well as numerically stable form of polynomials exists in multiple dimensions. As was observed in [1] a very general answer to this question was provided by Muhlbach in two subsequent papers [15] and [16]. We generalise his interpolation theory further to the general recovery problem and outline the use of Muhlbach's expansion in ENO schemes. Numerical examples show the usefulness of this approach in the problem of recovery from cell average data.

The flux integral method for multidimensional convection and diffusion

The flux integral method is a procedure for constructing an explicit single-step forward-in-time conservative control-volume update of the unsteady multidimensional convection-diffusion equation. The convective-plus-diffusive flux at each face of a control-volume cell is estimated by integrating the transported variable and its face-normal derivative over the volume swept out by the convecting velocity field. This yields a unique description of the fluxes, whereas other conservative methods rely on nonunique, arbitrary pseudoflux-difference splitting procedures. The accuracy of the resulting scheme depends on the form of the subcell interpolation assumed, given cell-average data. Cell-wise constant behavior results in a (very artificially diffusive) first-order convection scheme. Second-order convection-diffusion schemes correspond to cellwise linear (or bilinear) subcell interpolation. Cellwise quadratic subcell interpolants generate a highly accurate convection-diffusion scheme with excellent phase accuracy. Under constant coefficient conditions, this is a uniformly third-order polynomial interpolation algorithm (UTOPIA).

Spatial large‐scale correlations for optimal interpolation of temperature, salinity and nutrient concentrations in the Gulf of Finland

AbstractSpatial correlations of temperature, salinity and nutrient concentrations in the Gulf of Finland are estimated for winter, spring, summer and fall. The correlation estimates are based on the results of observations in the Gulf of Finland for the period 1972–1991. The time and space resolution of the correlation model are determined by averaging the original data in time over each month and in space over the cells with horizontal dimensions equal to 10′ in the east–west and 20′ in the south–north direction and to the standard sampling depth intervals in the vertical direction. The fluctuations of the averaged data are defined as deviations of the month and cell average data values from the seasonal 20‐year overall mean values. The model of spatial correlation is based on the assumption that the fluctuations of the month‐and‐cell average data form a random sample out of a spatially homogeneous, horizontally isotropic and temporally stationary field. Most of the estimated correlations are positive, except for the vertical correlation component of salinity in summer, which shows considerable negative values at the distance lags exceeding 30 m. As an example of assimilation of the estimated correlations, the statistical (optimal) linear interpolation approach was applied for estimation of the optimal location of a measurement site, additional to the three currently used international monitoring stations, commonly known as LL12, LL7 and LL3A, in the Gulf of Finland. The optimal location of the additional fourth measurement site, which leads to the least average interpolation error, was found to be within the vertical column of cells with co‐ordinates 60° 00′ N 28° 10′ E for salinity and 60° 00′ N 28° 20′ E for total phosphorus concentration measurements. The same average error cannot be achieved by increasing the sampling frequency at the three present international monitoring stations.

Triangle based adaptive stencils for the solution of hyperbolic conservation laws

A triangle based adaptive difference stencil for the numerical approximation of hyperbolic conservation laws in two space dimensions is constructed. The novelty of the resulting scheme lies in the nature of the preprocessing of the cell averaged data, which is accomplished via a nearest neighbor linear interpolation followed by a slope limiting procedure. Two such limiting procedures are suggested. The resulting method is considerably more simple than other triangle based non-oscillatory approximations which, like this scheme, approximate the flux up to second-order accuracy. Numerical results for constant and variable coefficient linear advection, as well as for nonlinear flux functions (Burgers' equation and the Buckley-Leverett equation), are presented. The observed order of convergence, after local averaging, is from 1.7 to 2.0 in L 1.