Space-time Grid Research Articles

RETRACTED: Zhang et al. Robot Path Planning Method Based on Indoor Spacetime Grid Model. Remote Sens. 2022, 14, 2357

Remote Sensing retracts the article “Robot Path Planning Method Based on Indoor Spacetime Grid Model” [...]

A Digital Grid Model for Complex Time-Varying Environments in Civil Engineering Buildings

The indoor environment is typically a complex time-varying environment. At present, the problem of indoor modeling is still a hot research topic for scholars at home and abroad. This paper primarily studies indoor time-varying space. On the basis of the Beidou grid framework and time coding model, in the first scenario, a local space subdivision framework based on Beidou is proposed. The necessity of local space subdivision framework is analyzed. In the second scenario, based on the time coding model needle, a local temporal subdivision model, more suitable for a short time domain, is proposed. Then, for the spatial modeling of an indoor time-varying environment, an indoor time-varying mesh frame based on global subdivision, local space subdivision, and local time subdivision is proposed. Using this framework, the indoor environment is represented by the space–time grid, and the basic storage data structure is designed. Finally, the experiment of local subdivision coding in the indoor space–time grid, indoor space–time grid modeling, and an organization experiment is carried out using real data and simulation data. The experimental results verify the feasibility and correctness of the encoding and decoding algorithm of local subdivision encoding in space–time encoding and the calculation algorithm of the space–time relationship. The experimental results also verify the multi-space organization and the management ability of the indoor space–time grid model.

ВЫБОР ОПТИМАЛЬНЫХ ПАРАМЕТРОВ ПРОСТРАНСТВЕННО-ВРЕМЕННОЙ СЕТКИ ПРИ МАТЕМАТИЧЕСКОМ МОДЕЛИРОВАНИИ НАГРЕВА ЗАГОТОВОК В ПРОМЫШЛЕННЫХ ПЕЧАХ

The efficiency of two numerical methods for solving multidimensional problems of the theory of thermal conductivity ‒ the fractional steps method and the Liebman method - is evaluated. The efficiency of these methods is compared for the operating conditions of industrial furnaces by the example of calculating the symmetrical heating of a two-dimensional cylinder and a three-dimensional plate made of materials with different thermophysical properties (corundum, ceramic brick and carbon steel). A two-dimensional axisymmetric temperature field for a cylinder and a three-dimensional temperature field for a workpiece in the form of a parallelepiped were found by the grid method under boundary conditions of the II and III genera. The difference approximation of differential equations and boundary conditions is performed by the control volume method according to an implicit finite difference scheme. The developed algorithms for solving multidimensional problems of internal heat transfer by fractional steps and the Liebman method are implemented in the form of computer programs in the Object Pascal programming environment. When comparing the effectiveness of solving multidimensional problems with these methods, the criterion of the effectiveness of difference schemes (CERS) proposed by V.V. Bukhmirov and T.E. Sozinova was used as an optimization criterion. Nomograms are constructed to select the optimal parameters of the space-time grid and the best numerical method for solving multidimensional problems for a specific process of heating (cooling) a solid body

An efficient numerical method on modified space-time sparse grid for time-fractional diffusion equation with nonsmooth data.

In this paper, we focus on developing a high efficient algorithm for solving d-dimension time-fractional diffusion equation (TFDE). For TFDE, the initial function or source term is usually not smooth, which can lead to the low regularity of exact solution. And such low regularity has a marked impact on the convergence rate of numerical method. In order to improve the convergence rate of the algorithm, we introduce the space-time sparse grid (STSG) method to solve TFDE. In our study, we employ the sine basis and the linear element basis for spatial discretization and temporal discretization, respectively. The sine basis can be divided into several levels, and the linear element basis can lead to the hierarchical basis. Then, the STSG can be constructed through a special tensor product of the spatial multilevel basis and the temporal hierarchical basis. Under certain conditions, the function approximation on standard STSG can achieve the accuracy order with degrees of freedom (DOF) for and DOF for , where J denotes the maximal level of sine coefficients. However, if the solution changes very rapidly at the initial moment, the standard STSG method may reduce accuracy or even fail to converge. To overcome this, we integrate the full grid into the STSG, and obtain the modified STSG. Finally, we obtain the fully discrete scheme of STSG method for solving TFDE. The great advantage of the modified STSG method can be shown in the comparative numerical experiment.

A note on the spectral analysis of matrix sequences via GLT momentary symbols: from all-at-once solution of parabolic problems to distributed fractional order matrices

The first focus of this paper is the characterization of the spectrum and the singular values of the coefficient matrix stemming from the discretization of a parabolic diffusion problem using a space-time grid and secondly from the approximation of distributed-order fractional equations. For this purpose we use the classical GLT theory and the new concept of GLT momentary symbols. The first permits us to describe the singular value or eigenvalue asymptotic distribution of the sequence of the coefficient matrices. The latter permits us to derive a function that describes the singular value or eigenvalue distribution of the matrix of the sequence, even for small matrix sizes, but under given assumptions. The paper is concluded with a list of open problems, including the use of our machinery in the study of iteration matrices, especially those concerning multigrid-type techniques.

A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions. An alternative approach

A numerical technique is devised to solve a problem of the steady laminar fluid motion in a straight plane hard-walled duct with two local axisymmetric rectangular constrictions. It uses the stream function, the vorticity and the pressure as the basic variables, has the second order of accuracy in the spatial and the first order of accuracy in the temporal coordinates, provides high stability of a solution, and needs significantly less computational time to obtain a result compared to appropriate techniques available in a scientific literature. The technique consists in: a) introducing the stream function and the vorticity, and subsequent transiting from the non-dimensional governing relations for the fluid velocity and the pressure to the corresponding non-dimensional relations for the stream function, the vorticity and the pressure; b) deriving their discrete counterparts in the nodes of the chosen space-time computational grid; c) integrating the systems of linear algebraic equations obtained after making the discretization. The discretization is based on applying appropriate differencing schemes to the terms of the equations for the basic variables. These are the two-point temporal onward difference for the unsteady term of the vorticity equation, as well as the two-point backward differences (for its convective term) and the five-point approximations (for its diffusive term and for the Poisson’s equations for the stream function and the pressure) in the axial and cross-flow coordinates. As for the velocity components, the appropriate central differences are applied to discretize their expressions. The above-mentioned systems of linear algebraic equations for the stream function and the pressure are integrated by the iterative successive over-relaxation method. The algebraic relation for the vorticity does not need application of any method to be solved, because it is a computational scheme to find this quantity based on the known magnitudes computed at the previous instant of time.

ADER scheme for incompressible Navier-Stokes equations on overset grids with a compact transmission condition

A space-time Finite Volume method is devised to simulate incompressible viscous flows in an evolving domain. Inspired by the ADER method (based on a Finite-Element-prediction/Finite-Volume-correction approach), the Navier-Stokes equations are discretized onto a space-time overset grid which is able to take into account both the shape of a possibly moving object and the evolution of the domain. A compact transmission condition is employed in order to mutually exchange information from one mesh to the other. The resulting method is second order accurate in space and time for both velocity and pressure. The accuracy and efficiency of the method are tested through reference simulations.

Robot Path Planning Method Based on Indoor Spacetime Grid Model

In the context of digital twins, smart city construction and artificial intelligence technology are developing rapidly, and more and more mobile robots are performing tasks in complex and time-varying indoor environments, making, at present, the unification of modeling, dynamic expression, visualization of operation, and wide application between robots and indoor environments a pressing problem to be solved. This paper presents an in-depth study on this issue and summarizes three major types of methods: geometric modeling, topological modeling, and raster modeling, and points out the advantages and disadvantages of these three types of methods. Therefore, in view of the current pain points of robots and complex time-varying indoor environments, this paper proposes an indoor spacetime grid model based on the three-dimensional division framework of the Earth space and innovatively integrates time division on the basis of space division. On the basis of the model, a dynamic path planning algorithm for the robot in the complex time-varying indoor environment is designed, that is, the Spacetime-A* algorithm (STA* for short). Finally, the indoor spacetime grid modeling experiment is carried out with real data, which verifies the feasibility and correctness of the spacetime relationship calculation algorithm encoded by the indoor spacetime grid model. Then, experiments are carried out on the multi-group path planning algorithms of the robot under the spacetime grid, and the feasibility of the STA* algorithm under the indoor spacetime grid and the superiority of the spacetime grid are verified.

Tensor-Based Sparse Recovery Space-Time Adaptive Processing for Large Size Data Clutter Suppression in Airborne Radar

Sparse recovery space-time adaptive processing (SR-STAP) can achieve an ideal clutter suppression with very few training samples, however, its application faces two challenges: (i) severe gird mismatch effect; (ii) large time-resources requirement. In practice, a coarse space-time grids will bring a serious mismatch between the true clutter points and the divided grids, which leads to a significant performance degradation of clutter suppression. Although the high-resolution mesh can effectively reduce the grid mismatch effect, its cost is huge computational load. Thus, it is meaningful to reduce the large-scale dictionary operation complexity while maintaining suboptimal clutter suppression performance for SR-STAP when applying in real airborne radar system. This paper proposed a tensor-based SR-STAP scheme aims at large-scale dictionary application. In the proposed framework, traditional vector-based operations are replaced by their corresponding low-complexity tensor representation. As a result, a large-scale matrix operation can be degraded into multiple small-scale matrix calculation, thus the huge computational loading can be saved in recovery. A comparison of tensor-based SR-STAP and traditional vector-based SR-STAP in large-scale dictionary application is also exhaustive discussed here. Based on this framework, a tensor-based sparse Bayesian learning and its fast matrix-realization form are developed. A series of carefully designed numerical simulation and measurement experiments indicate that the significant advantages of the tensor-based SR-STAP whether in performance or computation loading.

A new entropy-variable-based discretization method for minimum entropy moment approximations of linear kinetic equations

In this contribution we derive and analyze a new numerical method for kinetic equations based on a variable transformation of the moment approximation. Classical minimum-entropy moment closures are a class of reduced models for kinetic equations that conserve many of the fundamental physical properties of solutions. However, their practical use is limited by their high computational cost, as an optimization problem has to be solved for every cell in the space-time grid. In addition, implementation of numerical solvers for these models is hampered by the fact that the optimization problems are only well-defined if the moment vectors stay within the realizable set. For the same reason, further reducing these models by, e.g., reduced-basis methods is not a simple task. Our new method overcomes these disadvantages of classical approaches. The transformation is performed on the semi-discretized level which makes them applicable to a wide range of kinetic schemes and replaces the nonlinear optimization problems by inversion of the positive-definite Hessian matrix. As a result, the new scheme gets rid of the realizability-related problems. Moreover, a discrete entropy law can be enforced by modifying the time stepping scheme. Our numerical experiments demonstrate that our new method is often several times faster than the standard optimization-based scheme.

Refined range analysis of early warning and pre-control based on multi-element spatiotemporal clustering

There are three parts in public security big data analysis when early warning and pre-control oriented: refined analysis scope, visualization of temporal and spatial characteristics of analysis scope, and subspace emergency analysis model for early warning and controlling. These three parts are interrelated, and complete the early warning and pre-control analysis of emergency iteratively. The exploration method of refined analysis range is analysed in the paper, which is based on the calculation and comparison of multi-element spatiotemporal clustering results. This research adopts a grid division strategy to convert heterogeneous data into a normalized space-time grid, in order to improve the versatility of the framework. And the estimation of grid records without observed values using kernel density estimation method. The added clustering ensemble is to improve the stability of clustering results. It can avoid the influence of uncertain factors on the clustering results, as well as improving the real-time performance of clustering algorithm, by the optimization of clustering ensemble parameters, and the clustering ensemble deployed on Hadoop platform.

The study of a continuous Galerkin method for Sobolev equation with space-time variable coefficients

In this article, we will apply a space-time continuous Galerkin (CG) method to solve the numerical solution of Sobolev equation with space-time variable coefficients. Both the spatial and temporal variables of this method are discretized by finite element (FE) method, and hence it can easily attain the high order precision both in space and time directions as well as have the good stability. In addition, the variable spatial grid structures from one time span to the next and the time steps are permitted for this method, which are more suitable to design adaptive algorithm on unstructured mesh. We detailedly give the well-posed analysis of the numerical solution and the a priori error estimation in L∞(H1) norm without any restrictions on the space-time grid ratio. Finally, some numerical experiments are showed to conform the convergence orders and validate the high efficiency for the method investigated here.

Quantum control using quantum memory

We propose a new quantum numerical scheme to control the dynamics of a quantum walker in a two dimensional space–time grid. More specifically, we show how, introducing a quantum memory for each of the spatial grid, this result can be achieved simply by acting on the initial state of the whole system, and therefore can be exactly controlled once for all. As example we prove analytically how to encode in the initial state any arbitrary walker’s mean trajectory and variance. This brings significantly closer the possibility of implementing dynamically interesting physics models on medium term quantum devices, and introduces a new direction in simulating aspects of quantum field theories (QFTs), notably on curved manifold.

Chronotopic information interaction: integrating temporal and spatial structure for historical indexing and interactive search

Abstract Domain-based learning and research are important applications driving the development of exploratory search systems. A wealth of historical information about events from around the world resides within documents on the web, yet contemporary search engines do not take advantage of the closely integrated temporal and spatial information found within these web pages for indexing and design of search user interfaces. This gap limits the use of the web as a resource for historical and geohistorical information seeking. In this article, we propose chronotopic information interaction as a new interaction concept for web search that explicitly links temporal and spatial entities to keywords using a space–time grid index and a paired search user interface. The space–time grid index allows different modes of interaction between spatial, temporal, and keyword-based views in the search user interface. We demonstrate the use of the space–time grid index and chronotopic information interaction concept with the development of Pteraform, a prototype of a search engine that enables users to explore information in the English version of Wikipedia through a geohistorical lens.

A velocity control strategy for collision avoidance of autonomous agricultural vehicles

Collision avoidance ability is very important for autonomous agricultural vehicles, but the influence of different obstacles in agricultural environment is rarely taken into account. In this paper, a velocity control strategy for collision avoidance was proposed to adjust the velocity of autonomous agricultural vehicles according to the movement state and dangerous degree of the obstacles and the distance between the obstacles and the vehicles, thus to improve intelligence and safety of the vehicles. The control strategy involved two steps: collision prediction in dynamic environments with an improved obstacle space–time grid map, and velocity generator for collision avoidance with a cloud model. Simulations were conducted on the obstacle collision prediction and the designed cloud generator for velocity control respectively. Simulation results show that the proposed strategy can effectively predict collision with anti-disturbance ability for threat-free obstacles and rapid and accurate velocity output. And it realizes the real-time operation in dynamic environments with an average time of 0.2 s to predict collision. Additionally, field experiments including five trial schemes were performed to test the proposed velocity control strategy on an agricultural robot, where a haystack, a tractor and walking persons were regarded as static or dynamic obstacles. The results of the field experiments show that the proposed velocity control strategy has strong feasibility and effectiveness.

Numerical analysis of a continuous Galerkin method for damped sine‐Gordon equation

AbstractIn this article, we discuss the numerical solution for the two‐dimensional (2‐D) damped sine‐Gordon equation by using a space–time continuous Galerkin method. This method allows variable time steps and space mesh structures and its discrete scheme has good stability which are necessary for adaptive computations on unstructured grids. Meanwhile, it can easily get the higher‐order accuracy in both space and time directions. The existence and uniqueness to the numerical solution are strictly proved and a priori error estimate in maximum‐norm is given without any space–time grid conditions attached. Also, we prove that if the mesh in each time level is generated in a reasonable way, we can get the optimal order of convergence in both temporal and spatial variables. Finally, the convergence rates are presented and analyzed by some numerical experiments to illustrate the validity of the scheme.

Source mechanism identification using regional waveform inversion approach, case study: July 7, 2019 Molucca Sea earthquake

Molucca Sea is a seismically active area in eastern Indonesia. An earthquake occurred near to Ternate City, Province of North Maluku (M6.8: depth 29 km) on July 7, 2019. To investigate the detail about the mechanism of the earthquake, we analyzed the moment tensor of the earthquake by applying the regional waveform inversion. We used three components waveform broadband data from 18 station of IA-net seismic network in this study. We carried out the deviatoric mode to determine the double couple and compensated linear vector dipole (CLVD) component of the earthquake. The position and origin time of the earthquake were calculated by a space-time grid search in vertical and lateral positions. The frequency band of 0.01 – 0.023 Hz is used in the inversion process to reduce the instrument low-frequency disturbance and the effect of inaccurate velocity model for the synthetic seismogram. The moment tensor inversion result shows that the source mechanism of the earthquake is transpressional fault. This result agrees well with the tectonic setting of the study area.

Adaptive multilevel space-time-stepping scheme for transport in heterogeneous porous media (ADM-LTS)

We present ADM-LTS, an adaptive multilevel space-time-stepping scheme for transport in heterogeneous porous media. At each time step, firstly, the flow (pressure) solution is obtained. Then, the transport equation is solved using the ADM-LTS method, which consists of two stages. In the first stage, an initial solution is obtained by imposing the coarsest space-time grid. This initial solution is then improved, in the second stage, by imposing a space-time adaptive grid on the cells where the solution does not satisfy the desired quality. The quality control is based on error estimators with user-defined threshold values. The time-integration procedure, in which the coarsest-scale solution provides local flux boundary conditions for sub-domains with local time refinement, is strictly mass conservative. In addition, the method employs space-time fine grid cells only at the moving saturation fronts. In order to ensure local mass conservation at all levels, finite-volume restriction operators and unity prolongation operators are developed. Several numerical experiments have been performed to analyze the efficiency and accuracy of the proposed ADM-LTS method for both homogeneous and heterogeneous permeability fields on two and three dimensional domains. The results show that the method provides accurate solutions, at the same time it maintains the computational efficiency. The ADM-LTS implementation is publicly available at https://gitlab.com/darsim2simulator.

Maximal discrete sparsity in parabolic optimal control with measures

<p style='text-indent:20px;'>We consider variational discretization [<xref ref-type="bibr" rid="b18">18</xref>] of a parabolic optimal control problem governed by space-time measure controls. For the state discretization we use a Petrov-Galerkin method employing piecewise constant states and piecewise linear and continuous test functions in time. For the space discretization we use piecewise linear and continuous functions. As a result the controls are composed of Dirac measures in space-time, centered at points on the discrete space-time grid. We prove that the optimal discrete states and controls converge strongly in <inline-formula><tex-math id="M1">\begin{document}$ L^q $\end{document}</tex-math></inline-formula> and weakly-<inline-formula><tex-math id="M2">\begin{document}$ * $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{M} $\end{document}</tex-math></inline-formula>, respectively, to their smooth counterparts, where <inline-formula><tex-math id="M4">\begin{document}$ q \in (1,\min\{2,1+2/d\}] $\end{document}</tex-math></inline-formula> is the spatial dimension. Furthermore, we compare our approach to [<xref ref-type="bibr" rid="b8">8</xref>], where the corresponding control problem is discretized employing a discontinuous Galerkin method for the state discretization and where the discrete controls are piecewise constant in time and Dirac measures in space. Numerical experiments highlight the features of our discrete approach.

Capabilities and Limitations of Mathematical Models in Ecological Safety Forecasting

The evolution of the mathematical models of ecosystems over the past century was reviewed, starting from the simplest schemes of biological oxygen uptake (Streeter–Phelps, 1925) and ending with modern space-time models including dozens of factors and important components. The principal shortcomings and limitations of statistical models were discussed. For models based on the systems of differential or differential-algebraic equations, it was demonstrated that increased computational power and more detailed treatment do not improve the prognostic qualities of the model. When choosing a space-time grid for solving the problems by the finite difference method, the kinetic picture for the significant components is more important than their spatial distribution, which asymptotically tends toward normal. The efficiency of the Crank–Nicolson scheme based on the tridiagonal elimination method was demonstrated once again, although it does not guarantee a stable forecast.