Multidimensional Case Research Articles

Large time step wave adding scheme for systems of hyperbolic conservation laws

The current paper presents a large time step wave adding scheme (hereafter LTS-WA) originated in LeVeque's large time Godunov scheme, i.e. constructing numerical schemes via adding strong waves of discontinuity decomposition. Compared with the original scheme, we use a different strategy for wave adding and extend it to multi-dimensional cases. For rarefaction waves, we propose a grid cell decomposition method which can automatically satisfy the entropy condition and avoid nonphysical solutions. We give detailed formulae of the scheme, and make numerical experiments using scalar equations and Euler equations in one and multi-dimensional cases. Numerical results show that, besides the advantage of large time step, the new scheme has a lower numerical dissipation and a higher resolution of shocks and contact discontinuities with the increase of CFL number in certain range.

Decoupled local/global energy-preserving schemes for the N-coupled nonlinear Schrödinger equations

We develop two local energy-preserving integrators and a global energy-preserving integrator for the general multisymplectic Hamiltonian system. When applied to the 1D and multi-dimensional N-coupled nonlinear Schrödinger equations, the given schemes have the exact preservation of the local/global conservation law and are decoupled in the components ψn, n=1,2,…,N, i.e., each of the components can be solved independently. The decoupled feature is significant and helpful for overcoming the computational difficulty of the N-coupled (N≥3) nonlinear Schrödinger equations, especially of the multi-dimensional case. The composition method is employed to improve the accuracy of the schemes in time and the discrete fast Fourier transform is used to reduce the computational complexity. Several numerical experiments are carried out to exhibit the behaviors of the wave solutions. Numerical results confirm the theoretical results.

Initial–boundary value problem of Euler equations with damping in [formula omitted

The asymptotic wave behavior of solution for the Euler equations with damping in R+n is investigated around a given constant equilibrium in this paper. Three classical boundary condition cases are considered here. New approaches and techniques are introduced to deal with the multi-dimensional case for the system. For the linearized problem, by comparing symbols in the transformed tangential–spatial and time space, we show that its Green’s function can be described in terms of the fundamental solution for the Cauchy problem and reflected fundamental solution coupled with a boundary operator. Our approaches help us to simplify the Green’s function to the essential part and benefit the follow-up nonlinear analysis. For the nonlinear problem, we prove the pointwise decaying rate with the help of the Duhamel’s principle.

Highly efficient family of iterative methods for solving nonlinear models

In this study, we present a new highly efficient sixth-order family of iterative methods for solving nonlinear equations along with convergence properties. Further, we extend this family to the multidimensional case preserving the same order of convergence. In order to illustrate the applicability of our methods, we choose some real world problems namely, kinematic syntheses, boundary value, Bratu’s 2D, Fisher’s and Hammerstein integral problems in the case of nonlinear systems (see Farhane et al., 2017 [ 1 ]; Kaur and Arora, 2017 [ 2 ] for more applications) . In addition, numerical comparisons are made to show the performance of our iterative techniques with the existing ones. Moreover, we find that our techniques perform better as compared to the existing ones of same order in terms of residual error and the errors between two consecutive iterations. Finally, the stability analysis of our methods also support this to great extent.

An EMG Pattern Classification Method Based on a Mixture of Variance Distribution Models.

This paper proposes an electromyogram (EMG) pattern classification method based on a mixture of variance distribution models. A variance distribution model is a stochastic model of raw surface EMG signals in which the EMG variance is taken as a random variable, allowing the representation of uncertainty in the variance. In this paper, we extend the variance distribution model to the multidimensional case and enhance its flexibility for multichannel and processed EMG signals. The enhanced model enables the accurate classification of EMG patterns while considering the uncertainty in the EMG variance. The robustness and applicability of the proposed method are demonstrated through a simulation experiment using artificially generated data and EMG classification experiments using two real datasets.

Existence, uniqueness and regularity of multi-dimensional nonlocal balance laws with damping

In this article, we generalize some existence and uniqueness results in [22] for scalar nonlocal balance laws to multi-dimensional nonlocal balance laws. Also in the multi-dimensional case it is possible to get rid of the Entropy condition which is usually postulated to guarantee a unique weak solution of (local) balance laws but which we prove to be redundant in the nonlocal case. We start with a well known existence theorem for linear balance laws and the explicit formula for a solution to obtain a fixed-point problem in the nonlocal term. Using Banach's fixed-point theorem we prove existence and uniqueness of a fixed-point. This existence and uniqueness result is used for constructing and proving a unique solution of the multi-dimensional nonlocal balance law on a sufficiently small time horizon. While in many publications the nonlocal term will be considered over the entire space domain, we present an existence and uniqueness result which allows rather general integration areae for the nonlocal term. Eventually, we obtain a unique weak solution on every finite time horizon. Existence and uniqueness are complemented by stability results and an exhaustive study on more regular weak solutions in case the data is more regular.

Correction to: On the ambiguous treatment of the Schrödinger equation for the infinite potential well and an alternative via singular potentials: the multi-dimensional case

We show that the existence of flat solutions for the Schrodinger equation with super-singular potentials

Cross-impact and no-dynamic-arbitrage

We extend the ‘No-dynamic-arbitrage and market impact’-framework of Gatheral [Quant. Finance, 2010, 10(7), 749–759] to the multi-dimensional case where trading in one asset has a cross-impact on the price of other assets. From the condition of absence of dynamical arbitrage we derive theoretical limits for the size and form of cross-impact that can be directly verified on data. For bounded decay kernels we find that cross-impact must be an odd and linear function of trading intensity and cross-impact from asset i to asset j must be equal to the one from j to i. To test these constraints we estimate cross-impact among sovereign bonds traded on the electronic platform MOT. While we find significant violations of the above symmetry condition of cross-impact, we show that these are not arbitrageable with simple strategies because of the presence of the bid-ask spread.

A New Revised Scheme for SPH

Based on the smoothed kernel approximation of the Smoothed Particle Hydrodynamics (SPH) method and Taylor series expansion, a new revised scheme for SPH method is proposed which significantly improves its accuracy, especially near the boundaries where the particle points do not entirely cover the compact support domain of kernel function and particles are irregularly distributed. The revised scheme is derived up to the first-and second-order derivatives both in one-dimensional and multi-dimensional cases. In order to demonstrate the ability of the proposed revised scheme, the scheme is applied to the interpolation of functions and the numerical solution of the convection–diffusion equations both in one- and two-dimensional cases.

VirtualDesk: A Comfortable and Efficient Immersive Information Visualization Approach

Abstract3D representations are potentially useful under many circumstances, but suffer from long known perception and interaction challenges. Current immersive technologies, which combine stereoscopic displays and natural interaction, are being progressively seen as an opportunity to tackle this issue, but new guidelines and studies are still needed, especially regarding information visualization. Many proposed approaches are impractical for actual usage, resulting in user discomfort or requiring too much time or space. In this work, we implement and evaluate an alternative data exploration metaphor where the user remains seated and viewpoint change is only realisable through physical movements. All manipulation is done directly by natural mid‐air gestures, with the data being rendered at arm's reach. The virtual reproduction of the analyst's desk aims to increase immersion and enable tangible interaction with controls and two dimensional associated information. A comparative user study was carried out against a desktop‐based equivalent, exploring a set of 9 perception and interaction tasks based on previous literature and a multidimensional projection use case. We demonstrate that our prototype setup, named VirtualDesk, presents excellent results regarding user comfort and immersion, and performs equally or better in all analytical tasks, while adding minimal or no time overhead and amplifying user subjective perceptions of efficiency and engagement. Results are also contrasted to a previous experiment employing artificial flying navigation, with significant observed improvements.

Multidimensional Motion Parameters’ Estimation for Multi-SAR System Using the Cubic Phase Function

In this paper, the motion parameters’ estimation problem in synthetic aperture radar (SAR) imaging is extended to the multidimensional case, and the azimuth signal of a moving target with velocities and accelerations in three directions is modeled as a polynomial-phase signal (PPS). To estimate the multidimensional motion parameters of the moving target from the azimuth PPS, a multi-SAR system is required. On this basis, a new method based on the cubic phase function (CPF) is proposed to estimate the Doppler phase coefficients (DPCs) of the PPS. And, all the first- and second-order motion terms can be derived from the DPC. Experimental results have shown that, using the proposed scheme, the motion parameters can be accurately estimated in high signal-to-noise ratio scenarios.

Some higher-order iteration functions for solving nonlinear models

In this paper, we present a new efficient sixth-order family of Jarratt type methods for solving scalar equations. Then, we extend this family to the multidimensional case preserving the same order of convergence. We also discuss the theoretical convergence properties of the proposed scheme in the case of scalar as well as multidimensional case. The derivation of these schemes are based on weight function approach and free disposable parameters. We also demonstrate the applicability of them on total six number of problems: first five are real life problems namely, continuous stirred tank reactor (CSTR), chemical engineering, the trajectory of an electron in the air gap between two parallel plates, Hammerstein integration and boundary value problems; last one is the standard academic test problem. In addition, numerical comparisons are made to show the performance of the proposed iterative techniques with the existing techniques of the same order in the scalar as well as multi-dimensional case. Finally on the basis of numerical results, we conclude that our techniques perform better in terms of residual error, error between the two consecutive iterations, asymptotic error constant term and approximated root as compared to the existing ones of same order in scalar as well as multidimensional case.

Lie Symmetries of Nonlinear Parabolic-Elliptic Systems and Their Application to a Tumour Growth Model

A generalisation of the Lie symmetry method is applied to classify a coupled system of reaction-diffusion equations wherein the nonlinearities involve arbitrary functions in the limit case in which one equation of the pair is quasi-steady but the other is not. A complete Lie symmetry classification, including a number of the cases characterised as being unlikely to be identified purely by intuition, is obtained. Notably, in addition to the symmetry analysis of the PDEs themselves, the approach is extended to allow the derivation of exact solutions to specific moving-boundary problems motivated by biological applications (tumour growth). Graphical representations of the solutions are provided and a biological interpretation is briefly addressed. The results are generalised on multi-dimensional case under the assumption of the radially symmetrical shape of the tumour.

Stability of One-Phase States in Variational Problem of the Theory of Elasticity in Two-Phase Media. The Multi-Dimensional Case

We study the phenomenon of loss of stability of one-phase critical points of the energy functional of a two-phase elastic medium under perturbations of the temperature.

Psychological distress increases the risk of falling into poverty amongst older Australians: the overlooked costs-of-illness

BackgroundThis paper aimed to identify whether high psychological distress is associated with an increased risk of income and multidimensional poverty amongst older adults in Australia.MethodsWe undertook longitudinal analysis of the nationally representative Household Income and Labour Dynamics in Australian (HILDA) survey using modified Poisson regression models to estimate the relative risk of falling into income poverty and multidimensional poverty between 2010 and 2012 for males and females, adjusting for age, employment status, place of residence, marital status and housing tenure; and Population Attributable Risk methodology to estimate the proportion of poverty directly attributable to psychological distress, measured by the Kessler 10 scale.ResultsFor males, having high psychological distress increased the risk of falling into income poverty by 1.68 (95% CI: 1.02 to 2.75) and the risk of falling into multidimensional poverty by 3.40 (95% CI: 1.91 to 6.04). For females, there was no significant difference in the risk of falling into income poverty between those with high and low psychological distress (p = 0.1008), however having high psychological distress increased the risk of falling into multidimensional poverty by 2.15 (95% CI: 1.30 to 3.55). Between 2009 and 2012, 8.0% of income poverty cases for people aged 65 and over (95% CI: 7.8% to 8.4%), and 19.5% of multidimensional poverty cases for people aged 65 and over (95% CI: 19.2% to 19.9%) can be attributed to high psychological distress.ConclusionsThe elevated risk of falling into income and multidimensional poverty has been an overlooked cost of poor mental health.

A multi-physics methodology for the simulation of reactive flow and elastoplastic structural response

We propose a numerical methodology for the numerical simulation of distinct, interacting physical processes described by a combination of compressible, inert and reactive forms of the Euler equations, multiphase equations and elastoplastic equations. These systems of equations are usually solved by coupling finite element and CFD models. Here we solve them simultaneously, by recasting all the equations in the same, hyperbolic form and solving them on the same grid with the same finite-volume numerical schemes. The proposed compressible, multiphase, hydrodynamic formulation can employ a hierarchy of five reactive and non-reactive flow models, which allows simple to more involved applications to be directly described by the appropriate selection. The communication between the hydrodynamic and elastoplastic systems is facilitated by means of mixed-material Riemann solvers at the boundaries of the systems, which represent physical material boundaries. To this end we derive approximate mixed Riemann solvers for each pair of the above models based on characteristic equations. The components for reactive flow and elastoplastic solid modelling are validated separately before presenting validation for the full, coupled systems. Multi-dimensional use cases demonstrate the suitability of the reactive flow-solid interaction methodology in the context of impact-driven initiation of reactive flow and structural response due to violent reaction in automotive (e.g. car crash) or defence (e.g. explosive reactive armour) applications. Several types of explosives (C4, Detasheet, nitromethane, gaseous fuel) in gaseous, liquid and solid state are considered.

Finding well approximating lattices for a finite set of points

In this paper we address the problem of finding well approximating lattices for a given finite set $A$ of points in ${\mathbb R}^n$. More precisely, we search for $ǒ,\v{d_1}, \dots,\v{d_n}\in \mathbb{R}^n$ such that $ǎ-ǒ$ is close to $\Lambda=\v{d_1}\mathbb{Z}+\dots+\v{d_n}\mathbb{Z}$ for every $ǎ\in A$. First we deal with the one-dimensional case, where we show that in a sense the results are almost the best possible. These results easily extend to the multi-dimensional case where the directions of the axes are given, too. Thereafter we treat the general multi-dimensional case. Our method relies on the LLL algorithm. Finally we apply the least squares algorithm to optimize the results. We give several examples to illustrate our approach.

Teager–Kaiser energy methods for signal and image analysis: A review

This paper provides a review of the Teager–Kaiser (TK) energy operator and its extensions for signals and images processing. This class of operators possesses simplicity and good time-resolution and is very efficient in instantaneously estimating AM–FM signals and images. We point out the importance of the concept of energy from the point of view of the generation of the signal. More precisely, we emphasize the importance of analyzing signals from the point of view of the energy of the system needed to produce them. We show how this class of TK energy operators can be used to estimate useful features for signals and images analysis in time, space and frequency domains such as instantaneous frequency, second-order moment frequency, coherence function or spatial envelope and phase. We also show the importance of the higher derivative order of TK energy operator in terms of demodulation for both mono and multi-dimensional signals. Most of the developed tools around TK energy operators deal with real and complex-valued signals and some of them extended to multi-dimensional case. Due to their low complexity and their instantaneous-adapting nature, the class of TK energy operators offers valuable processing tools for time (space), frequency and time–frequency analysis.

Adaptive clustering based breast cancer detection with ANFIS classifier using mammographic images

Breast cancer is the most invasive cancer in women and second leading cause of mortality in women. Advances in detection and treatment have improved survival rates dramatically in patients. Now-a-days mammographic image based analysis is widely used to predict the breast cancer. The diagnosis is simple and survivals of the patient are high if the breast cancer is predicted at the early stage accurately. This paper heavily focuses on methodologies which provides accurate detection of cancerous tissues and thereby reduces the death rates. The algorithms are non-subsampled Shearlet transform (NSST) for image preprocessing, Adaptive clustering for image segmentation and finally adaptive neuro-fuzzy inference system (ANFIS) for image classification. NSST is an extension of wavelet transform in multidimensional and multidirectional case wherein the components are processed with the help of thresholding scheme. Adaptive clustering algorithm separates the pixels in the image into clusters based on both their intensity and sensitivity and thereby makes the segmentation much simpler and easier. ANFIS algorithm is the classifier which is a combination of both neural network adaptive capabilities and the fuzzy logic qualitative approach which is used to classify the normal and abnormal image accurately. The robustness of the proposed methodology is examined using classification accuracy, sensitivity and specificity.

Stein's method for negatively associated random variables with applications to second-order stationary random fields

Abstract Let ξ = (ξ1, . . ., ξm) be a negatively associated mean-zero random vector with components that obey the bound |ξi| ≤ B, i = 1, . . ., m, and whose sum W = ∑i=1mξi has variance 1. The bound d1(ℒ(W), ℒ(Z)) ≤ 5B - 5.2∑i≠ jσij is obtained, where Z has the standard normal distribution and d1(∙, ∙) is the L1 metric. The result is extended to the multidimensional case with the L1 metric replaced by a smooth functions metric. Applications to second-order stationary random fields with exponential decreasing covariance are also presented.