A New Extrapolation Economy Cascadic Multigrid Method for Image Restoration Problems

In this paper, a new extrapolation economy cascadic multigrid method is proposed to solve the image restoration model. The new method combines the new extrapolation formula and quadratic interpolation to design a nonlinear prolongation operator, which provides more accurate initial values for the fine grid level. An edge preserving denoising operator is constructed to remove noise and preserve image edges. The local smoothing operator reduces the influence of staircase effect. The experiment results show that the new method not only improves the computational efficiency but also ensures good recovery quality.

CEMA-LSTM:Enhancing Contextual Feature Correlation for Radar Extrapolation Using Fine-Grained Echo Datasets

Accurate precipitation nowcasting can provide great convenience to the public so they can conduct corresponding arrangements in advance to deal with the possible impact of upcoming heavy rain.Recent relevant research activities have shown their concerns on various deep learning models for radar echo extrapolation,where radar echo maps were used to predict their consequent moment,so as to recognize potential severe convective weather events.However,these approaches suffer from an inaccurate prediction of echo dynamics and unreliable depiction of echo aggregation or dissipation,due to the size limitation of convolution filter,lack of global feature,and less attention to features from previous states.To address the problems,this paper proposes a CEMA-LSTM recurrent unit,which is embedded with a Contextual Feature Correlation Enhancement Block(CEB)and a Multi-Attention Mechanism Block(MAB).The CEB enhances contextual feature correlation and supports its model to memorize significant features for near-future prediction;the MAB uses a position and channel attention mechanism to capture global features of radar echoes.Two practical radar echo datasets were used involving the FREM and CIKM 2017 datasets.Both quantification and visualization of comparative experimental results have demonstrated outperformance of the proposed CEMA-LSTMover recentmodels,e.g.,PhyDNet,MIM and PredRNN++,etc.In particular,compared with the second-rankedmodel,its average POD,FAR and CSI have been improved by 3.87%,1.65%and 1.79%,respectively on the FREM,and by 1.42%,5.60%and 3.16%,respectively on the CIKM 2017.

Extrapolation over temporal knowledge graph via hyperbolic embedding

Predicting potential facts in the future,Temporal Knowledge Graph(TKG)extrapolation remains challenging because of the deep dependence between the temporal association and semantic patterns of facts.Intuitively,facts(events)that happened at different timestamps have different influences on future events,which can be attributed to a hierarchy among not only facts but also relevant entities.Therefore,it is crucial to pay more attention to important entities and events when forecasting the future.However,most existing methods focus on reasoning over temporally evolving facts or mining evolutional patterns from known facts,which may be affected by the diversity and variability of the evolution,and they might fail to attach importance to facts that matter.Hyperbolic geometry was proved to be effective in capturing hierarchical patterns among data,which is considered to be a solution for modelling hierarchical relations among facts.To this end,we propose ReTIN,a novel model integrating real-time influence of historical facts for TKG reasoning based on hyperbolic geometry,which provides low-dimensional embeddings to capture latent hierarchical structures and other rich semantic patterns of the existing TKG.Considering both real-time and global features of TKG boosts the adaptation of ReTIN to the ever-changing dynamics and inherent constraints.Extensive experiments on benchmarks demonstrate the superiority of ReTIN over various baselines.The ablation study further supports the value of exploiting temporal information.

Extrapolation of upwelling irradiance just beneath the ocean surface from the in-water radiometric profile measurements

Accurate measurements of upwelling irradiance just beneath the ocean surface,E_(u)(λ,0^(-)),can be used to calculate ocean optical parameters,and further develop retrieval algorithms for remotely sensing water component concentrations.Due to the effects of sea surface waves,perturbation from instrument platform(ship),and instrument self-shading,E_(u)(λ,0^(-))is often difficult to be accurately measured.This study presents a procedure for extrapolating the E_(u)(λ,0^(-))from the in-water radiometric profile measurements.Using the optical profile data from 13 bands(ranging from 381 to 779 nm)measured by 45 casts in the Ligurian Sea during 2003–2009,the E_(u)(λ,0^(-))was extrapolated from in-water upwelling irradiance measurements between the initial shallow depth,Z_(0),and an optimal bottom depth,Z_(1),by three linear models(linear,2-degree polynomial,and exponential)and two nonlinear models(LOESS and spline).The accumulated errors of extrapolated E_(u)(λ,0^(-))at each wavelength for the five models were calculated.It was found that the optimal Z_(1) depth for the linear and exponential models was at the depth of80%of E_(u)(λ,Z_(0)),50%of E_(u)(λ,Z_(0))for the 2-degree polynomial model,40%of E_(u)(λ,Z_(0))for the LOESS model,and 15%of E_(u)(λ,Z_(0))for the spline model.The extrapolated E_(u)(λ,0^(-))derived from the five models was in good agreement with the calculated true E_(u)(λ,0^(-)).In all bands,the 2-degree polynomial model achieved the highest accuracy,followed by the LOESS model.In the short band of 381–559 nm,the linear and exponential models had the third-best performance,and the spline model performed worst within this range.For the red band of 619–779 nm,the accuracies of the exponential and spline models had the third highest performance,and the linear model produced lowest accuracy.Hence,the 2-degree polynomial model was an optimal procedure for extrapolation of E_(u)(λ,0^(-))from the in-water radiometric profile measurements.

Improved Weather Radar Echo Extrapolation Through Wind Speed Data Fusion Using a New Spatiotemporal Neural Network Model

Weather radar echo extrapolation plays a crucial role in weather forecasting.However,traditional weather radar echo extrapolation methods are not very accurate and do not make full use of historical data.Deep learning algorithms based on Recurrent Neural Networks also have the problem of accumulating errors.Moreover,it is difficult to obtain higher accuracy by relying on a single historical radar echo observation.Therefore,in this study,we constructed the Fusion GRU module,which leverages a cascade structure to effectively combine radar echo data and mean wind data.We also designed the Top Connection so that the model can capture the global spatial relationship to construct constraints on the predictions.Based on the Jiangsu Province dataset,we compared some models.The results show that our proposed model,Cascade Fusion Spatiotemporal Network(CFSN),improved the critical success index(CSI)by 10.7%over the baseline at the threshold of 30 dBZ.Ablation experiments further validated the effectiveness of our model.Similarly,the CSI of the complete CFSN was 0.004 higher than the suboptimal solution without the cross-attention module at the threshold of 30 dBZ.

基于Extrapolation Tikhonov正则化算法的重力数据三维约束反演

通过研究重力数据三维反演解的病态性,利用基于拉格朗日插值方法的Extrapolation Tikhonov正则化方法来解决反演中解的不唯一性和不稳定性问题,该方法最大限度的减少了因正则化参数的引入而在反演结果中介入的误差,同时详细讨论了基于三种选择原则的正则化双参数的具体选择方法,模型试算结果表明,与原Tikhonov方法相比,该方法提高了反演的拟合精度.其次,为了消除核函数随深度增加而快速衰减对反演结果的影响,本文改进了前人的重力数据三维反演深度加权函数,改进后的加权函数与原函数相比能更好的识别异常体底部密度分布特征,对于埋深较深的异常体具有较好的识别效果,更好的解决了由近地面趋肤效应作用引起的密度分布不均的问题.同时,利用上下限约束函数限制每一个立方体的密度差范围,并应用于多组人工合成模型.结果表明:该反演方法能准确地获得正演模型的预设参数范围和位置.

基于Power Extrapolation和Adaptive Method的网页评估新算法

Google的PageRank算法通过对超链接结构的分析,有效地提高了搜索结果的排序质量。PowerExtrapolation算法通过特征值直接求解马尔可夫超链接矩阵的主特征向量,但该算法的迭代次数与参数d的选择密切相关,而参数d的确定目前无明显规律可寻。另一方面,AdaptiveMethod通过将马尔可夫超链接矩阵稀疏化以达到节省迭代时间的目的。文章在PowerExtrapolation算法的基础上引入AdaptiveMethod,实验结果初步证明了新算法可以减少迭代运算的时间。

EXTRAPOLATION OF RF ECHO DATA BASED ON AR MODELING

Autoregressive (AR) modeling is applied to data extrapolation of radio frequency (RF) echo signals, and Burg algorithm, which can be computed in small amount and lead to a stable prediction filter, is used to estimate the prediction parameters of AR modeling. The complex data samples are directly extrapolated to obtain the extrapolated echo data in the frequency domain. The small rotating angle data extrapolation and the large rotating angular data extrapolation are considered separately in azimuth domain. The method of data extrapolation for the small rotating angle is the same as that in frequency domain, while the amplitude samples of large rotating angle echo data are extrapolated to obtain extrapolated echo amplitude, and the complex data of large rotating angle echo samples are extrapolated to get the extrapolated echo phase respectively. The calculation results show that the extrapolated echo data obtained by the above mentioned methods are accurate.

SVD-MPE: An SVD-Based Vector Extrapolation Method of Polynomial Type

An important problem that arises in different areas of science and engineering is that of computing the limits of sequences of vectors , where , N being very large. Such sequences arise, for example, in the solution of systems of linear or nonlinear equations by fixed-point iterative methods, and are simply the required solutions. In most cases of interest, however, these sequences converge to their limits extremely slowly. One practical way to make the sequences converge more quickly is to apply to them vector extrapolation methods. Two types of methods exist in the literature: polynomial type methods and epsilon algorithms. In most applications, the polynomial type methods have proved to be superior convergence accelerators. Three polynomial type methods are known, and these are the minimal polynomial extrapolation (MPE), the reduced rank extrapolation (RRE), and the modified minimal polynomial extrapolation (MMPE). In this work, we develop yet another polynomial type method, which is based on the singular value decomposition, as well as the ideas that lead to MPE. We denote this new method by SVD-MPE. We also design a numerically stable algorithm for its implementation, whose computational cost and storage requirements are minimal. Finally, we illustrate the use of SVD-MPE with numerical examples.

SnO_2-based solid solutions for CH_4 deep oxidation: Quantifying the lattice capacity of SnO_2 using an X-ray diffraction extrapolation method

A series of SnO2‐based catalysts modified by Mn, Zr, Ti and Pb oxides with a Sn/M （M=Mn, Zr, Ti and Pb） molar ratio of 9/1 were prepared by a co‐precipitation method and used for CH4 and CO oxidation. The Mn3＋, Zr4＋, Ti4＋and Pb4＋cations are incorporated into the lattice of tetragonal rutile SnO2 to form a solid solution structure. As a consequence, the surface area and thermal stability of the catalysts are improved. Moreover, the oxygen species of the modified catalysts become easier to be reduced. Therefore, the oxidation activity over the catalysts was improved, except for the one modified by Pb oxide. Manganese oxide demonstrates the best promotional effects for SnO2. Using an X‐ray diffraction extrapolation method, the lattice capacity of SnO2 for Mn2O3 was 0.135 g Mn2O3/g SnO2, which indicates that to form stable solid solution, only 21%Sn4＋cations in the lattice can be maximally replaced by Mn3＋. If the amount of Mn3＋cations is over the capacity, Mn2O3 will be formed, which is not favorable for the activity of the catalysts. The Sn rich samples with only Sn‐Mn solid solution phase show higher activity than the ones with excess Mn2O3 species.

A Composite Approach of Radar Echo Extrapolation Based on TREC Vectors in Combination with Model-Predicted Winds

Extending the lead time of precipitation nowcasts is vital to improvements in heavy rainfall warning, flood mitigation, and water resource management. Because the TREC vector （tracking radar echo by correlation） represents only the instantaneous trend of precipitation echo motion, the approach using derived echo motion vectors to extrapolate radar reflectivity as a rainfall forecast is not satisfactory if the lead time is beyond 30 minutes. For longer lead times, the effect of ambient winds on echo movement should be considered. In this paper, an extrapolation algorithm that extends forecast lead times up to 3 hours was developed to blend TREC vectors with model-predicted winds. The TREC vectors were derived from radar reflectivity patterns in 3 km height CAPPI （constant altitude plan position indicator） mosaics through a cross-correlation technique. The background steering winds were provided by predictions of the rapid update assimilation model CHAF （cycle of hourly assimilation and forecast）. A similarity index was designed to determine the vertical level at which model winds were applied in the extrapolation process, which occurs via a comparison between model winds and radar vectors. Based on a summer rainfall case study, it is found that the new algorithm provides a better forecast.

High-resolution algorithm based on temporal-spatial extrapolation

To enhance the resolution of parameter estimation with limited samples received by a short passive array, an iterative nonparametric algorithm for estimating the frequencies and direction-of-arrivals （DOAs） of signals is proposed. The cost function is constructed using 12-norm Gaussian entropy combined with an additional constraint, 12-norm constraint or linear constraint. By minimizing the cost functions in the temporal and the spatial dimensions using corresponding iteration algorithms respectively, the sparse discrete Fourier transforms （DFTs） of temporal and spatial samples are obtained to represent the extrapolated sequences with much larger sizes than the original samples. Then frequency and angle estimates are obtained by performing the traditional simple methods on the extrapolated sequences. It is shown that the proposed algorithm offers increased resolution and significantly reduced sidelobes compared with the periodogram and beamforming based methods. And it achieves high precision compared with the high-resolution method with lower computational burden. Some numerical simulations and real data processing results are presented to verify the effectiveness of the method.

High-resolution digital beamforming of UWB signals based on Carathéodory representation for delay compensation and array extrapolation

To realize high-resolution digital beamforming(DBF)of ultra-wideband(UWB) signals, we propose a DBF method based on Carath ′eodory representation for delay compensation and array extrapolation. Delay compensation by Carath ′eodory representation could achieve high interpolation accuracy while using the single channel sampling technique. Array extrapolation by Carath ′eodory representation reformulates and extends each snapshot, consequently extends the aperture of the original uniform linear array(ULA) by several times and provides a better realtime performance than the existing aperture extrapolation utilizing vector extrapolation based on the two dimensional autoregressive(2-D AR) model. The UWB linear frequency modulated(LFM) signal is used for simulation analysis. Simulation results demonstrate that the proposed method is featured by a much higher spatial resolution than traditional DBF methods and lower sidelobes than using Lagrange fractional filters.

A reduced-order extrapolation algorithm based on CNLSMFE formulation and POD technique for two-dimensional Sobolev equations

A reduced-order extrapolation algorithm based on Crank-Nicolson least-squares mixed finite element （CNLSMFE） formulation and proper orthogonal decomposition （POD） technique for two-dimensional （2D） Sobolev equations is established. The error estimates of the reduced-order CNLSMFE solutions and the implementation for the reduced-order extrapolation algorithm are provided. A numerical example is used to show that the results of numerical computations are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order extrapolation algorithm is feasible and efficient for seeking numerical solutions to 2D Sobolev equations.

WAVE FIELD EXTRAPOLATION IN VISCOELASTIC MEDIUM AND VARIABLE FOCUS METHOD TO SEPARATE SEISMIC COMPOUND WAVE

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Parameter estimation of GTD model and RCS extrapolation based on a modified 3D-ESPRIT algorithm

The noise robustness and parameter estimation performance of the classical three-dimensional estimating signal parameter via rotational invariance techniques(3D-ESPRIT)algorithm are poor when the parameters of the geometric theory of the diffraction(GTD)model are estimated at low signal-to-noise ratio(SNR).To solve this problem,a modified 3D-ESPRIT algorithm is proposed.The modified algorithm improves the parameter estimation accuracy by proposing a novel spatial smoothing technique.Firstly,we make cross-correlation of the auto-correlation matrices;then by averaging the cross-correlation matrices of the forward and backward spatial smoothing,we can obtain a novel equivalent spatial smoothing matrix.The formula of the modified algorithm is derived and the performance of this improved method is also analyzed.Then we compare root-meansquare-errors(RMSEs)of different parameters and the locating accuracy obtained by different algorithms.Furthermore,radar cross section(RCS)of radar targets is extrapolated.Simulation results verify the effectiveness and superiority of the modified 3DESPRIT algorithm.

Fatigue life evaluation of girth butt weld within welded cast steel joints based on the extrapolation notch stress method

The fatigue life evaluation of the girth butt weld within the welded cast steel joint was studied based on the extrapolation notch stress method.Firstly,the mesh sensitivity of the finite element model of the welded cast steel joint was analyzed to determine the optimal mesh size.Based on the stress field analysis of the finite element model of the welded cast steel joint at the weld toe and weld root,the sharp model of the extrapolation notch stress method was applied to derive the effective notch stress of the rounded model belonging to the effective notch stress method,in which the key problem is to calculate the extrapolation point C,and the extrapolation point C has an exponential function relationship with the geometric parameters of the welded cast steel joint.By setting different values of geometric parameters,the corresponding value of parameter C is calculated,and then the functional relationship between the extrapolation point C and the geometric parameters can be obtained by the multiple linear regression analysis.Meanwhile,the fatigue life evaluation of the girth butt weld within welded cast steel joints based on the effective notch stress was performed according to the guideline recommended by the IIW(International Institute of Welding).The results indicate that the extrapolation notch stress method can effectively simplify the process of calculating the effective notch stress and accurately evaluate the fatigue life of the girth butt weld within welded cast steel joints.

Higher Order Aitken Extrapolation with Application to Converging and Diverging Gauss-Seidel Iterations

In this paper, Aitken’s extrapolation normally applied to convergent fixed point iteration is extended to extrapolate the solution of a divergent iteration. In addition, higher order Aitken extrapolation is introduced that enables successive decomposition of high Eigen values of the iteration matrix to enable convergence. While extrapolation of a convergent fixed point iteration using a geometric series sum is a known form of Aitken acceleration, it is shown that in this paper, the same formula can be used to estimate the solution of sets of linear equations from diverging Gauss-Seidel iterations. In both convergent and divergent iterations, the ratios of differences among the consecutive values of iteration eventually form a convergent (divergent) series with a factor equal to the largest Eigen value of the iteration matrix. Higher order Aitken extrapolation is shown to eliminate the influence of dominant Eigen values of the iteration matrix in successive order until the iteration is determined by the lowest possible Eigen values. For the convergent part of the Gauss-Seidel iteration, further acceleration is made possible by coupling of the extrapolation technique with the successive over relaxation (SOR) method. Application examples from both convergent and divergent iterations have been provided. Coupling of the extrapolation with the SOR technique is also illustrated for a steady state two dimensional heat flow problem which was solved using MATLAB programming.

Methods of Lightning Nowcasting Based on Radar Echo Extrapolation Technology

An improved echo extrapolation technology( MOD-COTREC) was introduced firstly,and then two plans for lightning nowcasting based on MOD-COTREC and both isothermal radar reflectivity and MOD-COTREC were proposed based on the technology. Afterwards,the two plans for lightning nowcasting were tested by a case respectively. It is concluded that during the process of lightning nowcasting singly based on MOD-COTREC,the appearance and disappearance of lightning are not considered,and only lightning position is predicted when lightning density is constant,so the prediction error is big. The plan for lightning nowcasting based on both isothermal radar reflectivity and MOD-COTREC is still at an experimental stage,and the nowcasting products of cloud-to-ground lightning based on the plan are very different from the actual density and position of cloud-to-ground lightning,so it needs to be improved further.

Extrapolation Method for Cauchy Principal Value Integral with Classical Rectangle Rule on Interval

In this paper,the classical composite middle rectangle rule for the computation of Cauchy principal value integral(the singular kernel 1=(x-s))is discussed.With the density function approximated only while the singular kernel is calculated analysis,then the error functional of asymptotic expansion is obtained.We construct a series to approach the singular point.An extrapolation algorithm is presented and the convergence rate of extrapolation algorithm is proved.At last,some numerical results are presented to confirm the theoretical results and show the efficiency of the algorithms.