Journal articles: 'Nonlinear convolution'

1

Garcia, Hernando, and Ramki Kalyanaraman. "Convolution theorem for nonlinear optics." Applied Physics Letters 91, no. 11 (September 10, 2007): 111114. http://dx.doi.org/10.1063/1.2780082.

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ORAVECZ, FERENC. "THE NUMBER OF PURE CONVOLUTIONS ARISING FROM CONDITIONALLY FREE CONVOLUTION." Infinite Dimensional Analysis, Quantum Probability and Related Topics 08, no. 03 (September 2005): 327–55. http://dx.doi.org/10.1142/s0219025705002001.

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We show that there are eight special cases of the conditionally free convolution of Bożejko, Leinert and Speicher with the property that in the corresponding moment-cumulant formula no nontrivial weights appear. All the eight convolutions are given. These include the free, the boolean and the Fermi convolutions, another special case of the bold t-free convolution and four more convolution laws that were not treated before.

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Navascués, María, Ram N. Mohapatra, and Arya K. B. Chand. "Some properties of the fractal convolution of functions." Fractional Calculus and Applied Analysis 24, no. 6 (November 22, 2021): 1735–57. http://dx.doi.org/10.1515/fca-2021-0075.

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Abstract We consider the fractal convolution of two maps f and g defined on a real interval as a way of generating a new function by means of a suitable iterated function system linked to a partition of the interval. Based on this binary operation, we consider the left and right partial convolutions, and study their properties. Though the operation is not commutative, the one-sided convolutions have similar (but not equal) characteristics. The operators defined by the lateral convolutions are both nonlinear, bi-Lipschitz and homeomorphic. Along with their self-compositions, they are Fréchet differentiable. They are also quasi-isometries under certain conditions of the scale factors of the iterated function system. We also prove some topological properties of the convolution of two sets of functions. In the last part of the paper, we study stability conditions of the dynamical systems associated with the one-sided convolution operators.

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Arabadzhyan, L. G., and N. B. Engibaryan. "Convolution equations and nonlinear functional equations." Journal of Soviet Mathematics 36, no. 6 (March 1987): 745–91. http://dx.doi.org/10.1007/bf01085507.

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Looney, Carl G. "Nonlinear Rule-based Convolution for Refocusing." Real-Time Imaging 6, no. 1 (February 2000): 29–37. http://dx.doi.org/10.1006/rtim.1998.0154.

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KRYSTEK, ANNA DOROTA, and ŁUKASZ JAN WOJAKOWSKI. "ASSOCIATIVE CONVOLUTIONS ARISING FROM CONDITIONALLY FREE CONVOLUTION." Infinite Dimensional Analysis, Quantum Probability and Related Topics 08, no. 03 (September 2005): 515–45. http://dx.doi.org/10.1142/s0219025705002104.

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We define two families of deformations of probability measures depending on the second free cumulants and the corresponding new associative convolutions arising from the conditionally free convolution. These deformations do not commute with dilation of measures, which means that the limit theorems cannot be obtained as a direct application of the theorems for the conditionally free case. We calculate the general form of the central and Poisson limit theorems. We also find the explicit form for three important examples.

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Stašová, Ol’ga, and Zuzana Krivá. "Regularized Coherence Enhancing Filtering." Tatra Mountains Mathematical Publications 72, no. 1 (December 1, 2018): 107–21. http://dx.doi.org/10.2478/tmmp-2018-0024.

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Abstract The paper deals with the nonlinear tensor diffusion which yields a coherence improvement. It is very appropriate for images with flow-like structures. Two convolutions are used in the construction of diffusion tensor for such a model, see [Drblíková, O.—Mikula, K.: Convergence analysis of finite volume scheme for nonlinear tensor anisotropic diffusion in image processing, SIAM J. Numer. Anal. 46 (2007), 37–60], [Weickert, J.: Coherence-enhancing diffusion filtering, Int. J. Comput. Vis. 31 (1999), 111–127]. In this paper we introduce the third supplemental convolution in order to enhance the diffusion strategy. First, we briefly present the classical coherence enhancing model and explain our modification. Then the discrete scheme is provided. The core of the paper consists in numerical experiments. Benefits of the additional convolution are discussed and illustrated in the figures.

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Hu, Xiao, Daheng Zhang, Ruijun Tan, and Qian Xie. "Controlled Cooling Temperature Prediction of Hot-Rolled Steel Plate Based on Multi-Scale Convolutional Neural Network." Metals 12, no. 9 (August 30, 2022): 1455. http://dx.doi.org/10.3390/met12091455.

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Controlled cooling technology is widely used in hot-rolled steel plate production lines. The final cooling temperature directly affects the microstructure and properties of steel plates, but cooling and heat transfer constitutes a nonlinear process, which is difficult to be accurately described using a mathematical model. In order to improve the accuracy of the controlled cooling temperature, a multi-scale convolutional neural network is used to predict the final cooling temperature. Convolution kernels with different sizes are introduced in the layer of a multi-scale convolutional neural network. This structure can simultaneously extract the feature information of different sizes and improve the perceptual power of the network model. The measured steel plate thickness, speed, header flow, and other variables are taken as input. The final cooling temperature is taken as the output and predicted using a multi-scale convolutional neural network. The results show that the multi-scale convolution neural network prediction model has strong generalization and nonlinear fitting ability. Compared with the traditionally structured BP neural network and convolution neural network (CNN), the mean square error (MSE) of the multi-scale convolutional neural network decreased by 24.7% and 12.2%, the mean absolute error (MAE) decreased by 19.6% and 7.97%, and the coefficient of determination (R2) improved by 4.26% and 2.65%, respectively. The final cooling temperature traditional structure by the multi-scale CNN agreed with the actual temperature within ±10% error bands. As the prediction accuracy improved, the multi-scale CNN can be effectively applied to hot-rolled steel plate production.

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Bushell, P. J., and W. Okrasinski. "Nonlinear Volterra Integral Equations with Convolution Kernel." Journal of the London Mathematical Society s2-41, no. 3 (June 1990): 503–10. http://dx.doi.org/10.1112/jlms/s2-41.3.503.

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MYDLARCZYK, W., and W. OKRASINSKI. "NONLINEAR VOLTERRA INTEGRAL EQUATIONS WITH CONVOLUTION KERNELS." Bulletin of the London Mathematical Society 35, no. 04 (June 9, 2003): 484–90. http://dx.doi.org/10.1112/s0024609303002170.

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Yakubov, A. Ya. "On nonlinear Volterra equations of convolution type." Differential Equations 45, no. 9 (September 2009): 1326–36. http://dx.doi.org/10.1134/s0012266109090080.

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von Wolfersdorf, Lothar, and Jaan Janno. "On a Class of Nonlinear Convolution Equations." Zeitschrift für Analysis und ihre Anwendungen 14, no. 3 (1995): 497–508. http://dx.doi.org/10.4171/zaa/635.

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13

Askhabov, S. N. "Nonlinear convolution-type equations in Lebesgue spaces." Mathematical Notes 97, no. 5-6 (May 2015): 659–68. http://dx.doi.org/10.1134/s0001434615050016.

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Lin, Yuanhua, and Liping He. "Existence of Traveling Wave Fronts for a Generalized Nonlinear Schrodinger Equation." Advances in Mathematical Physics 2022 (August 16, 2022): 1–6. http://dx.doi.org/10.1155/2022/9638150.

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In the presented paper, a generalized nonlinear Schr o dinger equation without delay convolution kernel and with special delay convolution kernel is investigated. By using the geometric singular perturbation theory, the existence of traveling wave fronts is proved. Firstly, we show that such traveling wave fronts exist without delay by non-Hamiltonian qualitative analysis. Then, for the generalized nonlinear Schr o dinger equation with a special local strong delay convolution kernel, the desired heteroclinic orbit is obtained by using the Fredholm theory.

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Jiang, Chao, Canchen Jiang, Dongwei Chen, and Fei Hu. "Densely Connected Neural Networks for Nonlinear Regression." Entropy 24, no. 7 (June 25, 2022): 876. http://dx.doi.org/10.3390/e24070876.

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Densely connected convolutional networks (DenseNet) behave well in image processing. However, for regression tasks, convolutional DenseNet may lose essential information from independent input features. To tackle this issue, we propose a novel DenseNet regression model where convolution and pooling layers are replaced by fully connected layers and the original concatenation shortcuts are maintained to reuse the feature. To investigate the effects of depth and input dimensions of the proposed model, careful validations are performed by extensive numerical simulation. The results give an optimal depth (19) and recommend a limited input dimension (under 200). Furthermore, compared with the baseline models, including support vector regression, decision tree regression, and residual regression, our proposed model with the optimal depth performs best. Ultimately, DenseNet regression is applied to predict relative humidity, and the outcome shows a high correlation with observations, which indicates that our model could advance environmental data science.

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Liu, Shao Gang, and Qiu Jin. "Tolerance Analysis Method Using Improved Convolution Method." Applied Mechanics and Materials 271-272 (December 2012): 1463–66. http://dx.doi.org/10.4028/www.scientific.net/amm.271-272.1463.

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Convolution method is studied to analyze statistical tolerance for linear dimension chain and nonlinear dimension chain. Hybrid convolution method is proposed, which is the integration of analytical convolution and numerical convolution. In order to reduce the algorithm errors, improved convolution method is proposed. Comparing with other statistical tolerance analysis methods, this method is faster and accurate. At last, an example is used to demonstrate the method proposed in this paper.

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Yin, Jian Jun, and Jian Qiu Zhang. "Convolution PHD Filtering for Nonlinear Non-Gaussian Models." Advanced Materials Research 213 (February 2011): 344–48. http://dx.doi.org/10.4028/www.scientific.net/amr.213.344.

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A novel probability hypothesis density (PHD) filter, called the Gaussian mixture convolution PHD (GMCPHD) filter was proposed. The PHD within the filter is approximated by a Gaussian sum, as in the Gaussian mixture PHD (GMPHD) filter, but the model may be non-Gaussian and nonlinear. This is implemented by a bank of convolution filters with Gaussian approximations to the predicted and posterior densities. The analysis results show the lower complexity, more amenable for parallel implementation of the GMCPHD filter than the convolution PHD (CPHD) filter and the ability to deal with complex observation model, small observation noise and non-Gaussian noise of the proposed filter over the existing Gaussian mixture particle PHD (GMPPHD) filter. The multi-target tracking simulation results verify the effectiveness of the proposed method.

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Banjai, Lehel, and Christian Lubich. "Runge–Kutta convolution coercivity and its use for time-dependent boundary integral equations." IMA Journal of Numerical Analysis 39, no. 3 (June 7, 2018): 1134–57. http://dx.doi.org/10.1093/imanum/dry033.

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Abstract A coercivity property of temporal convolution operators is an essential tool in the analysis of time-dependent boundary integral equations and their space and time discretizations. It is known that this coercivity property is inherited by convolution quadrature time discretization based on A-stable multistep methods, which are of order at most 2. Here we study the question as to which Runge–Kutta-based convolution quadrature methods inherit the convolution coercivity property. It is shown that this holds without any restriction for the third-order Radau IIA method, and on permitting a shift in the Laplace domain variable, this holds for all algebraically stable Runge–Kutta methods and hence for methods of arbitrary order. As an illustration the discrete convolution coercivity is used to analyse the stability and convergence properties of the time discretization of a nonlinear boundary integral equation that originates from a nonlinear scattering problem for the linear wave equation. Numerical experiments illustrate the error behaviour of the Runge–Kutta convolution quadrature time discretization.

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Huang, Feizhen, Jinfang Zeng, Yu Zhang, and Wentao Xu. "Convolutional recurrent neural networks with multi-sized convolution filters for sound-event recognition." Modern Physics Letters B 34, no. 23 (April 25, 2020): 2050235. http://dx.doi.org/10.1142/s0217984920502358.

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Sound-event recognition often utilizes time-frequency analysis to produce an image-like spectrogram that provides a rich visual representation of original signal in time and frequency. Convolutional Neural Networks (CNN) with the ability of learning discriminative spectrogram patterns are suitable for sound-event recognition. However, there is relatively little effort that CNN makes full use of the important temporal information. In this paper, we propose MCRNN, a Convolutional Recurrent Neural Networks (CRNN) architecture for sound-event recognition, the letter “M” in the name “MCRNN” of our model denotes the multi-sized convolution filters. Richer features are extracted by using several different convolution filter sizes at the last convolution layer. In addition, cochleagram images are used as the input layer of the network, instead of the traditional spectrogram image of a sound signal. Experiments on the RWCP dataset shows that the recognition rate of the proposed method achieved 98.4% in clean conditions, and it robustly outperforms the existing methods, the recognition rate increased by 0.9%, 1.9% and 10.3% in 20 dB, 10 dB and 0 dB signal-to-noise ratios (SNR), respectively.

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Askhabov, Sultan Nazhmudinovich. "Nonlinear convolution type integral equations in complex spaces." Ufimskii Matematicheskii Zhurnal 13, no. 1 (2021): 17–30. http://dx.doi.org/10.13108/2021-13-1-17.

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Filippucci, Roberta, and Marius Ghergu. "Higher order evolution inequalities with nonlinear convolution terms." Nonlinear Analysis 221 (August 2022): 112881. http://dx.doi.org/10.1016/j.na.2022.112881.

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Askhabov, Sultan N. "Nonlinear convolution integro-differential equation with variable coefficient." Fractional Calculus and Applied Analysis 24, no. 3 (June 1, 2021): 848–64. http://dx.doi.org/10.1515/fca-2021-0036.

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Abstract For an integro-differential equation of the convolution type defined on the half-line [0, ∞) with a power nonlinearity and variable coefficient, we use the weight metrics method to prove a global theorem on the existence and uniqueness of a solution in the cone of nonnegative functions in the space C[0, ∞). It is shown that the solution can be found by a successive approximation method of the Picard type; an estimate for the rate of convergence of the approximations is produced. We present sharp two-sided a-priori estimates for the solutions. These estimates enable us to construct a complete metric space which is invariant under the nonlinear convolution operator considered here and to prove that the equation induced by this operator has a unique solution in this space as well as in the class of all non-negative continuous functions vanishing at the origin. Such equations with operators of fractional calculus as the Riemann-Liouville, Erdélyi-Kober, Hadamard fractional integrals arise, in particular, when describing the process of propagation of shock waves in gas-filled pipes, solving the problem about heating a half-infinite body in a nonlinear heat-transfer process, considering models of population genetics, and elsewhere.

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Comets, F., Th Eisele, and M. Schatzman. "On secondary bifurcations for some nonlinear convolution equations." Transactions of the American Mathematical Society 296, no. 2 (February 1, 1986): 661. http://dx.doi.org/10.1090/s0002-9947-1986-0846602-7.

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Erraoui, Mohamed, Habib Ouerdiane, and José Luís da Silva. "Stochastic Convolution-Type Heat Equations with Nonlinear Drift." Stochastic Analysis and Applications 25, no. 1 (January 2007): 237–54. http://dx.doi.org/10.1080/07362990600753478.

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Hairer, E., Ch Lubich, and M. Schlichte. "Fast Numerical Solution of Nonlinear Volterra Convolution Equations." SIAM Journal on Scientific and Statistical Computing 6, no. 3 (July 1985): 532–41. http://dx.doi.org/10.1137/0906037.

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SORAVIA, PIERPAOLO. "ON NONLINEAR CONVOLUTION AND UNIQUENESS OF VISCOSITY SOLUTIONS." Analysis 20, no. 4 (December 2000): 373–86. http://dx.doi.org/10.1524/anly.2000.20.4.373.

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Wu, Teng, and Ahsan Kareem. "A nonlinear convolution scheme to simulate bridge aerodynamics." Computers & Structures 128 (November 2013): 259–71. http://dx.doi.org/10.1016/j.compstruc.2013.06.004.

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Zolfaghari, Reza. "Lubich-Collocation Method for Solving a System of Nonlinear Integral Equations of Convolution Type." International Journal of Applied Physics and Mathematics 4, no. 2 (2014): 121–25. http://dx.doi.org/10.7763/ijapm.2014.v4.267.

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Wang, Zheng-Xin. "Nonlinear Grey Prediction Model with Convolution Integral NGMC(1,n)and Its Application to the Forecasting of China’s Industrial SO2Emissions." Journal of Applied Mathematics 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/580161.

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The grey prediction model with convolution integral GMC (1,n) is a multiple grey model with exact solutions. To further improve prediction accuracy and describe better the relationship between cause and effect, we introduce nonlinear parameters into GMC (1,n) model and additionally apply a convolution integral to produce an improved forecasting model here designated as NGMC (1,n). The model solving process applied the least-squares method to evaluate the structure parameters of the model: convolution was used to obtain an exact solution with this improved grey model. The nonlinear optimisation took the parameters as the decision variables with the objective of minimising forecasting errors. The GMC (1, 2) and NGMC (1, 2) models were used to predict China’s industrial SO2emissions from the basis of the economic output level as the influencing factor. Results indicated that NGMC (1, 2) can effectively describe the nonlinear relationship between China’s economic output and SO2emissions with an improved accuracy over current GMC (1, 2) models.

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Wan, Renzhuo, Chengde Tian, Wei Zhang, Wendi Deng, and Fan Yang. "A Multivariate Temporal Convolutional Attention Network for Time-Series Forecasting." Electronics 11, no. 10 (May 10, 2022): 1516. http://dx.doi.org/10.3390/electronics11101516.

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Multivariate time-series forecasting is one of the crucial and persistent challenges in time-series forecasting tasks. As a kind of data with multivariate correlation and volatility, multivariate time series impose highly nonlinear time characteristics on the forecasting model. In this paper, a new multivariate time-series forecasting model, multivariate temporal convolutional attention network (MTCAN), based on a self-attentive mechanism is proposed. MTCAN is based on the Convolution Neural Network (CNN) model, using 1D dilated convolution as the basic unit to construct asymmetric blocks, and then, the feature extraction is performed by the self-attention mechanism to finally obtain the prediction results. The input and output lengths of this network can be determined flexibly. The validation of the method is carried out with three different multivariate time-series datasets. The reliability and accuracy of the prediction results are compared with Long Short-Term Memory (LSTM), Gated Recurrent Unit (GRU), Convolutional Long Short-Term Memory (ConvLSTM), and Temporal Convolutional Network (TCN). The prediction results show that the model proposed in this paper has significantly improved prediction accuracy and generalization.

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Toft, Joachim, Karoline Johansson, Stevan Pilipović, and Nenad Teofanov. "Sharp convolution and multiplication estimates in weighted spaces." Analysis and Applications 13, no. 05 (June 29, 2015): 457–80. http://dx.doi.org/10.1142/s0219530514500523.

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We establish sharp convolution and multiplication estimates in weighted Lebesgue, Fourier Lebesgue and modulation spaces. We cover, especially some results in [L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations (Springer, Berlin, 1997); S. Pilipović, N. Teofanov and J. Toft, Micro-local analysis in Fourier Lebesgue and modulation spaces, II, J. Pseudo-Differ. Oper. Appl.1 (2010) 341–376]. The results are also related to some results by Iwabuchi in [T. Iwabuchi, Navier–Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices, J. Differential Equations248 (2010) 1972–2002].

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Guo, Jianxin, Zhen Wang, and Shanwen Zhang. "FESSD: Feature Enhancement Single Shot MultiBox Detector Algorithm for Remote Sensing Image Target Detection." Electronics 12, no. 4 (February 14, 2023): 946. http://dx.doi.org/10.3390/electronics12040946.

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Automatic target detection of remote sensing images (RSI) plays an important role in military surveillance and disaster monitoring. The core task of RSI target detection is to judge the target categories and precise location. However, the existing target detection algorithms have limited accuracy and weak generalization capability for RSI with complex backgrounds. This study presents a novel feature enhancement single shot multibox detector (FESSD) algorithm for remote sensing target detection to achieve accurate detection of different categories targets. The FESSD introduces feature enhancement module and attention mechanism into the convolution neural networks (CNN) model, which can effectively enhance the feature extraction ability and nonlinear relationship between different convolution features. Specifically, the feature enhancement module is used to extract the multi-scale feature information and enhance the model nonlinear learning ability; the self-learning attention mechanism (SAM) is used to expand the convolution kernel local receptive field, which makes the model extract more valuable features. In addition, the nonlinear relationship between different convolution features is enhanced using the feature pyramid attention mechanism (PAM). The experimental results show that the mAP value of the proposed method reaches 81.9% and 81.2% on SD-RSI and DIOR datasets, which is superior to other compared state-of-the-art methods.

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He, Chu, Zishan Shi, Tao Qu, Dingwen Wang, and Mingsheng Liao. "Lifting Scheme-Based Deep Neural Network for Remote Sensing Scene Classification." Remote Sensing 11, no. 22 (November 13, 2019): 2648. http://dx.doi.org/10.3390/rs11222648.

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Recently, convolutional neural networks (CNNs) achieve impressive results on remote sensing scene classification, which is a fundamental problem for scene semantic understanding. However, convolution, the most essential operation in CNNs, restricts the development of CNN-based methods for scene classification. Convolution is not efficient enough for high-resolution remote sensing images and limited in extracting discriminative features due to its linearity. Thus, there has been growing interest in improving the convolutional layer. The hardware implementation of the JPEG2000 standard relies on the lifting scheme to perform wavelet transform (WT). Compared with the convolution-based two-channel filter bank method of WT, the lifting scheme is faster, taking up less storage and having the ability of nonlinear transformation. Therefore, the lifting scheme can be regarded as a better alternative implementation for convolution in vanilla CNNs. This paper introduces the lifting scheme into deep learning and addresses the problems that only fixed and finite wavelet bases can be replaced by the lifting scheme, and the parameters cannot be updated through backpropagation. This paper proves that any convolutional layer in vanilla CNNs can be substituted by an equivalent lifting scheme. A lifting scheme-based deep neural network (LSNet) is presented to promote network applications on computational-limited platforms and utilize the nonlinearity of the lifting scheme to enhance performance. LSNet is validated on the CIFAR-100 dataset and the overall accuracies increase by 2.48% and 1.38% in the 1D and 2D experiments respectively. Experimental results on the AID which is one of the newest remote sensing scene dataset demonstrate that 1D LSNet and 2D LSNet achieve 2.05% and 0.45% accuracy improvement compared with the vanilla CNNs respectively.

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Kheshgi, Haroon S., and Benjamin S. White. "Modelling ocean carbon cycle with a nonlinear convolution model." Tellus B: Chemical and Physical Meteorology 48, no. 1 (January 1996): 3–12. http://dx.doi.org/10.3402/tellusb.v48i1.15660.

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KHESHGI, HAROON S., and BENJAMIN S. WHITE. "Modelling ocean carbon cycle with a nonlinear convolution model." Tellus B 48, no. 1 (February 1996): 3–12. http://dx.doi.org/10.1034/j.1600-0889.1996.00002.x.

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Bodizs, Arpad, Ferenc Szeifert, and Tibor Chovan. "Convolution Model Based Predictive Controller for a Nonlinear Process." Industrial & Engineering Chemistry Research 38, no. 1 (January 1999): 154–61. http://dx.doi.org/10.1021/ie980338q.

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Hu, H., and J. H. Tang. "A convolution integral method for certain strongly nonlinear oscillators." Journal of Sound and Vibration 285, no. 4-5 (August 2005): 1235–41. http://dx.doi.org/10.1016/j.jsv.2004.11.023.

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Askhabov, S. N. "Approximate Solution of Nonlinear Discrete Equations of Convolution Type." Journal of Mathematical Sciences 201, no. 5 (August 19, 2014): 566–80. http://dx.doi.org/10.1007/s10958-014-2012-y.

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Askhabov, Sultan Nazhmudinovich, and Akhmed Lechaevich Dzhabrailov. "Approximate solutions of nonlinear convolution type equations on segment." Ufimskii Matematicheskii Zhurnal 5, no. 2 (2013): 3–11. http://dx.doi.org/10.13108/2013-5-2-3.

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Gomez, Carlos, Humberto Prado, and Sergei Trofimchuk. "Separation dichotomy and wavefronts for a nonlinear convolution equation." Journal of Mathematical Analysis and Applications 420, no. 1 (December 2014): 1–19. http://dx.doi.org/10.1016/j.jmaa.2014.05.064.

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Huang, Meixiang, Chongfei Huang, Jing Yuan, and Dexing Kong. "A Semiautomated Deep Learning Approach for Pancreas Segmentation." Journal of Healthcare Engineering 2021 (July 2, 2021): 1–10. http://dx.doi.org/10.1155/2021/3284493.

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Accurate pancreas segmentation from 3D CT volumes is important for pancreas diseases therapy. It is challenging to accurately delineate the pancreas due to the poor intensity contrast and intrinsic large variations in volume, shape, and location. In this paper, we propose a semiautomated deformable U-Net, i.e., DUNet for the pancreas segmentation. The key innovation of our proposed method is a deformable convolution module, which adaptively adds learned offsets to each sampling position of 2D convolutional kernel to enhance feature representation. Combining deformable convolution module with U-Net enables our DUNet to flexibly capture pancreatic features and improve the geometric modeling capability of U-Net. Moreover, a nonlinear Dice-based loss function is designed to tackle the class-imbalanced problem in the pancreas segmentation. Experimental results show that our proposed method outperforms all comparison methods on the same NIH dataset.

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Martindale, John, Jason Berwick, Chris Martin, Yazhuo Kong, Ying Zheng, and John Mayhew. "Long Duration Stimuli and Nonlinearities in the Neural–Haemodynamic Coupling." Journal of Cerebral Blood Flow & Metabolism 25, no. 5 (February 9, 2005): 651–61. http://dx.doi.org/10.1038/sj.jcbfm.9600060.

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Recent studies have shown that the haemodynamic responses to brief (<2 secs) stimuli can be well characterised as a linear convolution of neural activity with a suitable haemodynamic impulse response. In this paper, we show that the linear convolution model cannot predict measurements of blood flow responses to stimuli of longer duration (>2 secs), regardless of the impulse response function chosen. Modifying the linear convolution scheme to a nonlinear convolution scheme was found to provide a good prediction of the observed data. Whereas several studies have found a nonlinear coupling between stimulus input and blood flow responses, the current modelling scheme uses neural activity as an input, and thus implies nonlinearity in the coupling between neural activity and blood flow responses. Neural activity was assessed by current source density analysis of depth-resolved evoked field potentials, while blood flow responses were measured using laser Doppler flowmetry. All measurements were made in rat whisker barrel cortex after electrical stimulation of the whisker pad for 1 to 16 secs at 5 Hz and 1.2 mA (individual pulse width 0.3 ms).

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G. GAL, SORIN, and IONUT T. IANCU. "Quantitative approximation by nonlinear convolution operators of Landau-Choquet type." Carpathian Journal of Mathematics 36, no. 3 (September 30, 2020): 415–22. http://dx.doi.org/10.37193/cjm.2020.03.09.

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By using the concept of Choquet nonlinear integral with respect to a monotone set function, we introduce the nonlinear convolution operators of Landau-Choquet type, with respect to a family of submodular set functions. Quantitative approximation results in terms of the modulus of continuity are obtained with respect to some particular possibility measures. For some subclasses of functions we prove that these Landau-Choquet type operators can have essentially better approximation properties than their classical correspondents.

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Zhemchuzhnikov, Dmitrii, Ilia Igashov, and Sergei Grudinin. "6DCNN with Roto-Translational Convolution Filters for Volumetric Data Processing." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 4 (June 28, 2022): 4707–15. http://dx.doi.org/10.1609/aaai.v36i4.20396.

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In this work, we introduce 6D Convolutional Neural Network (6DCNN) designed to tackle the problem of detecting relative positions and orientations of local patterns when processing three-dimensional volumetric data. 6DCNN also includes SE(3)-equivariant message-passing and nonlinear activation operations constructed in the Fourier space. Working in the Fourier space allows significantly reducing the computational complexity of our operations. We demonstrate the properties of the 6D convolution and its efficiency in the recognition of spatial patterns. We also assess the 6DCNN model on several datasets from the recent CASP protein structure prediction challenges. Here, 6DCNN improves over the baseline architecture and also outperforms the state of the art.

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Huang, Qingqing, Di Wu, Hao Huang, Yan Zhang, and Yan Han. "Tool Wear Prediction Based on a Multi-Scale Convolutional Neural Network with Attention Fusion." Information 13, no. 10 (October 18, 2022): 504. http://dx.doi.org/10.3390/info13100504.

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Compared with traditional machine learning algorithms, the convolutional neural network (CNN) has an excellent automatic feature learning ability and can complete the nonlinear representation from original data input to output by itself. However, the CNN does not sufficiently mine the tool wear information contained in the multi-sensor data due to disregard of the differences in the contribution of different features when extracting features. In this paper, a tool wear prediction method based on a multi-scale convolutional neural network with attention fusion is proposed, which fuses the tool wear degradation information collected by different types of sensors. In the multi-scale convolution module, convolution kernels with different sizes are used to extract the degradation information of different scales in the wear information, and then the attention fusion module is constructed to fuse the multi-scale feature information. Finally, the mapping between tool wear and multi-sensor data is realized through the feature information obtained by residual connection and full connection layer. By comparing the multi-scale convolutional neural network with different attention mechanisms, the experiments demonstrated the effectiveness and superiority of the proposed method.

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Li, Yishi, Kunran Xu, Rui Lai, and Lin Gu. "Towards an Effective Orthogonal Dictionary Convolution Strategy." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 2 (June 28, 2022): 1473–81. http://dx.doi.org/10.1609/aaai.v36i2.20037.

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Orthogonality regularization has proven effective in improving the precision, convergence speed and the training stability of CNNs. Here, we propose a novel Orthogonal Dictionary Convolution Strategy (ODCS) on CNNs to improve orthogonality effect by optimizing the network architecture and changing the regularized object. Specifically, we remove the nonlinear layer in typical convolution block “Conv(BN) + Nonlinear + Pointwise Conv(BN)”, and only impose orthogonal regularization on the front Conv. The structure, “Conv(BN) + Pointwise Conv(BN)”, is then equivalent to a pair of dictionary and encoding, defined in sparse dictionary learning. Thanks to the exact and efficient representation of signal with dictionaries in low-dimensional projections, our strategy could reduce the superfluous information in dictionary Conv kernels. Meanwhile, the proposed strategy relieves the too strict orthogonality regularization in training, which makes hyper-parameters tuning of model to be more flexible. In addition, our ODCS can modify the state-of-the-art models easily without any extra consumption in inference phase. We evaluate it on a variety of CNNs in small-scale (CIFAR), large-scale (ImageNet) and fine-grained (CUB-200-2011) image classification tasks, respectively. The experimental results show that our method achieve a stable and superior improvement.

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Wang, Xiaomin. "A Coiflets-Based Wavelet Laplace Method for Solving the Riccati Differential Equations." Journal of Applied Mathematics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/257049.

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A wavelet iterative method based on a numerical integration by using the Coiflets orthogonal wavelets for a nonlinear fractional differential equation is proposed. With the help of Laplace transform, the fractional differential equation was converted into equivalent integral equation of convolution type. By using the wavelet approximate scheme of a function, the undesired jump or wiggle phenomenon near the boundary points was avoided and the expansion constants in the approximation of arbitrary nonlinear term of the unknown function can be explicitly expressed in finite terms of the expansion ones of the approximation of the unknown function. Then a numerical integration method for the convolution is presented. As an example, an iterative method which can solve the singular nonlinear fractional Riccati equations is proposed. Numerical results are performed to show the efficiency of the method proposed.

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Chiang, I. F., and S. T. Noah. "A Convolution Approach for the Transient Analysis of Locally Nonlinear Rotor Systems." Journal of Applied Mechanics 57, no. 3 (September 1, 1990): 731–37. http://dx.doi.org/10.1115/1.2897084.

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A computationally efficient convolution method, based on discretized impulse response and transition matrix integral formulations, is developed for the transient analysis of complex linear structures interacting through strong local nonlinearities. In the formulation, the coupling forces due to the nonlinearities are treated as external forces acting on the coupled subsystems. Iteration is utilized to determine their magnitudes at each time increment. The method is applied to a generic rotor-housing model representing a turbopump of a space shuttle main engine (SSME). In that model, the local nonlinearity is due to clearances between the rotor bearing outer races and the carrier attached to the housing. As compared to the fourth-order Runge-Kutta numerical integration methods, the convolution approach proved more efficient and robust for the same accuracy requirement. This is due to the closed-form formulation of the convolution approach which allows for the use of relatively larger time increments and for a reduction in the roundoff errors.

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Lindsay, J. Martin, and Adam G. Skalski. "Quantum Stochastic Convolution Cocycles II." Communications in Mathematical Physics 280, no. 3 (April 22, 2008): 575–610. http://dx.doi.org/10.1007/s00220-008-0465-x.

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Hilal, Eman M. A., and Tarig M. Elzaki. "Solution of Nonlinear Partial Differential Equations by New Laplace Variational Iteration Method." Journal of Function Spaces 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/790714.

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The aim of this study is to give a good strategy for solving some linear and nonlinear partial differential equations in engineering and physics fields, by combining Laplace transform and the modified variational iteration method. This method is based on the variational iteration method, Laplace transforms, and convolution integral, introducing an alternative Laplace correction functional and expressing the integral as a convolution. Some examples in physical engineering are provided to illustrate the simplicity and reliability of this method. The solutions of these examples are contingent only on the initial conditions.

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Journal articles on the topic 'Nonlinear convolution'

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