scholarly journals An advanced meshless approach for the high-dimensional multi-term time-space-fractional PDEs on convex domains

2021 ◽  
Author(s):  
X. G. Zhu ◽  
Y. F. Nie ◽  
J. G. Wang ◽  
Z. B. Yuan
Keyword(s):  
High Dimensional ◽  
Convex Domains ◽  
Time Space ◽  
Fractional Pdes

scholarly journals Large random simplicial complexes, I

2016 ◽  
Vol 08 (03) ◽  
pp. 399-429 ◽  
Author(s):  
A. Costa ◽  
M. Farber
Keyword(s):  
Probability Measure ◽  
Simplicial Complex ◽  
Parameter Space ◽  
Simplicial Complexes ◽  
High Dimensional ◽  
Topological Properties ◽  
Convex Domains ◽  
Dimensional Parameter ◽  
Random Simplicial Complexes ◽  
Simply Connected

In this paper we introduce and develop the multi-parameter model of random simplicial complexes with randomness present in all dimensions. Various geometric and topological properties of such random simplicial complexes are characterised by convex domains in the high-dimensional parameter space (rather than by intervals, as in the usual one-parameter models). We find conditions under which a multi-parameter random simplicial complex is connected and simply connected. Besides, we give an intrinsic characterisation of the multi-parameter probability measure. We analyse links of simplexes and intersections of multi-parameter random simplicial complexes and show that they are also multi-parameter random simplicial complexes.

scholarly journals A High-Dimensional Video Sequence Completion Method with Traffic Data Completion Generative Adversarial Networks

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Lan Wu ◽  
Tian Gao ◽  
Chenglin Wen ◽  
Kunpeng Zhang ◽  
Fanshi Kong
Keyword(s):  
Intelligent Transportation Systems ◽  
Spatial Information ◽  
Transportation Systems ◽  
Video Data ◽  
Generative Adversarial Networks ◽  
High Dimensional ◽  
Traffic Data ◽  
Adversarial Network ◽  
Time Space ◽  
Data Completion

The lack of traffic data is a bottleneck restricting the development of Intelligent Transportation Systems (ITS). Most existing traffic data completion methods aim at low-dimensional data, which cannot cope with high-dimensional video data. Therefore, this paper proposes a traffic data complete generation adversarial network (TDC-GAN) model to solve the problem of missing frames in traffic video. Based on the Feature Pyramid Network (FPN), we designed a multiscale semantic information extraction model, which employs a convolution mechanism to mine informative features from high-dimensional data. Moreover, by constructing a discriminator model with global and local branch networks, the temporal and spatial information are captured to ensure the time-space consistency of consecutive frames. Finally, the TDC-GAN model performs single-frame and multiframe completion experiments on the Caltech pedestrian dataset and KITTI dataset. The results show that the proposed model can complete the corresponding missing frames in the video sequences and achieve a good performance in quantitative comparative analysis.

Seismic data interpolation without iteration using t-x-y streaming prediction filter with varying smoothness

Geophysics ◽  
2021 ◽  
pp. 1-57
Author(s):  
Yang Liu ◽  
Geng WU ◽  
Zhisheng Zheng
Keyword(s):  
Iterative Methods ◽  
Seismic Data ◽  
Convex Sets ◽  
Interpolation Method ◽  
Computational Cost ◽  
Economic Factor ◽  
High Dimensional ◽  
Data Interpolation ◽  
Time Space ◽  
Nonstationary Field

Although there is an increase in the amount of seismic data acquired with wide-azimuth geometry, it is difficult to achieve regular data distributions in spatial directions owing to limitations imposed by the surface environment and economic factor. To address this issue, interpolation is an economical solution. The current state of the art methods for seismic data interpolation are iterative methods. However, iterative methods tend to incur high computational cost which restricts their application in cases of large, high-dimensional datasets. Hence, we developed a two-step non-iterative method to interpolate nonstationary seismic data based on streaming prediction filters (SPFs) with varying smoothness in the time-space domain; and we extended these filters to two spatial dimensions. Streaming computation, which is the kernel of the method, directly calculates the coefficients of nonstationary SPF in the overdetermined equation with local smoothness constraints. In addition to the traditional streaming prediction-error filter (PEF), we proposed a similarity matrix to improve the constraint condition where the smoothness characteristics of the adjacent filter coefficient change with the varying data. We also designed non-causal in space filters for interpolation by using several neighboring traces around the target traces to predict the signal; this was performed to obtain more accurate interpolated results than those from the causal in space version. Compared with Fourier Projection onto a Convex Sets (POCS) interpolation method, the proposed method has the advantages such as fast computational speed and nonstationary event reconstruction. The application of the proposed method on synthetic and nonstationary field data showed that it can successfully interpolate high-dimensional data with low computational cost and reasonable accuracy even in the presence of aliased and conflicting events.

GENERAL PLANNING METHOD FOR MACHINE COORDINATION AND RENDEZVOUS

1994 ◽  
Vol 04 (04) ◽  
pp. 351-377 ◽  
Author(s):  
KAREN I. TROVATO
Keyword(s):  
A Priori ◽  
Optimal Strategies ◽  
Planning Method ◽  
High Dimensional ◽  
A Priori Information ◽  
Time Space ◽  
Rendezvous Problem ◽  
Priori Information ◽  
Planning Problems ◽  
General Method

This paper describes a general method for representing and solving planning problems. The framework has well-defined subcomponents to simplify the problem using transforms, define costs over permissible motions, define illegal regions in the transformed space, and efficiently find optimal motions. The method fully exploits any a priori information and also provides a method to augment this information efficiently at runtime. The method is powerful in that it can be used in arbitrarily high dimensional spaces, and has been used to solve non-holonomic problems efficiently and with ease. Examples will be given for moving robotic equipment optimally while avoiding obstacles, for automatically maneuvering a vehicle around obstacles, and for determining alternative rendezvous locations for machinery based on the separate constraints (time, space, fuel, etc.) of each. The rendezvous problem will also give optimal strategies, and precise paths.

Quantized-TT-Cayley Transform for Computing the Dynamics and the Spectrum of High-Dimensional Hamiltonians

2011 ◽  
Vol 11 (3) ◽  
pp. 273-290 ◽  
Author(s):  
Ivan Gavrilyuk ◽  
Boris Khoromskij
Keyword(s):  
High Dimensional ◽  
Time Interval ◽  
Matrix Product ◽  
Cayley Transform ◽  
Time And Space ◽  
Transform Method ◽  
Time Space ◽  
Long Time ◽  
Asymptotic Complexity ◽  
Space Separation

Abstract In the present paper, we propose and analyse a class of tensor methods for the efficient numerical computation of the dynamics and spectrum of high-dimensional Hamiltonians. We focus on the complex-time evolution problems. We apply the quantized-TT (QTT) matrix product states type tensor approximation that allows to represent N-d tensors generated by the grid representation of d-dimensional functions and operators with log-volume complexity, O(d log N), where N is the univariate discretization parameter in space. Making use of the truncated Cayley transform method allows us to recursively separate the time and space variables and then introduce the efficient QTT representation of both the temporal and the spatial parts of the solution to the high-dimensional evolution equation. We prove the exponential convergence of the m-term time-space separation scheme and describe the efficient tensor-structured preconditioners for the arising system with multidimensional Hamiltonians. For the class of "analytic" and low QTT-rank input data, our method allows to compute the solution at a fixed point in time t=T>0 with an asymptotic complexity of order O(d log N ln^q (1/ε)), where ε>0 is the error bound and q is a fixed small number. The time-and-space separation method via the QTT-Cayley-transform enables us to construct a global m-term separable (x,t)-representation of the solution on a very fine time-space grid with complexity of order O(dm^4 log N_t log N), where N_t is the number of sampling points in time. The latter allows efficient energy spectrum calculations by FFT (or QTT-FFT) of the autocorrelation function computed on a sufficiently long time interval [0,T]. Moreover, we show that the spectrum of the Hamiltonian can also be represented by the poles of the t-Laplace transform of a solution. In particular, the approach can be an option to compute the dynamics and the spectrum in the time-dependent molecular Schrödinger equation.

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