Kernel Type Research Articles

A new exponential type Kernel integral transform: Khalouta transform and its applications

In this paper, we suggest a new integral transform called the Khalouta transform, which is a generalization of many integral transforms having exponential type kernel. We discuss certain results on the inverse and the existence of this integral transform. We present useful properties of the Khalouta transform and their applications to solve differential equations. Furthermore, we prove the duality between the Khalouta transform and other transforms such as the Laplace-Carson transform, Sumudu transform, ZZ transform, ZMA transform, Elzaki transform, Aboodh transform, Natural transform and Shehu transform. Finally, we ensure the efficiency and accuracy of the Khalouta transform by solving various examples of both ordinary, integro and partial differential equations.

A NEW FORM OF L1-PREDICTOR–CORRECTOR SCHEME TO SOLVE MULTIPLE DELAY-TYPE FRACTIONAL ORDER SYSTEMS WITH THE EXAMPLE OF A NEURAL NETWORK MODEL

In this paper, we derive a new version of L1-Predictor–Corrector (L1-PC) method by using some previously given methods (L1-PC for single delay, PC for non-delay, and decomposition algorithm) to solve multiple delay-type fractional differential equations. The Caputo fractional derivative with singular type kernel is used to establish the results. Some important remarks related to the delay term estimation and error analysis are mentioned. In order to check the accuracy and correctness of our method, we solve a neural network system with two delay parameters. A number of graphs are given to justify the role of delays as well as the accuracy of the algorithm. The given method is fully novel and reliable to solve multiple delay type fractional order systems in Caputo sense.

Solution of Linear Volterra Integral Equation of Second Kind via Rishi Transform

The solution of various problems of engineering and science can easily determined by representing these problems in integral equations. There are numerous analytical and numerical methods which can be used for solving different kinds of integral equations. In this paper, authors used recently developed integral transform “Rishi Transform” for obtaining the analytical solution of linear Volterra integral equation of second kind (LVIESK). For this, the kernel of LVIESK has assumed a convolution type kernel. Five numerical examples are considered for demonstrating the complete procedure of determining the solution. Results of these problems suggest that Rishi transform provides the exact analytical solution of LVIESK without doing complicated calculation work.

HBCA: A Toolchain for High-Accuracy Branch-Fused CNN Accelerator on FPGA with Dual-Decimal-Fused Technique

The programmability of FPGA suits the constantly changing convolutional neural network (CNN). However, several challenges arise when the previous FPGA-based accelerators update CNN. Firstly, although the model of RepVGG can balance accuracy and speed, it solely supports two types of kernels. Meanwhile, 8-bit integer-only quantization of PyTorch which can support various CNNs is seldom successfully supported by the FPGA-based accelerators. In addition, Winograd F(4 × 4, 3 × 3) uses less multiplication, but its transformation matrix contains irregular decimals, which could lead to accuracy problems. To tackle these issues, this paper proposes High-accuracy Branch-fused CNN Accelerator (HBCA): a toolchain and corresponding FPGA-based accelerator. The toolchain proposes inception-based branch–fused technique, which can support more types of kernels. Meanwhile, the accelerator proposes Winograd-quantization dual decimal–fuse techniques to balance accuracy and speed. In addition, this accelerator supports multi-types of kernels and proposes Winograd decomposed-part reuse, multi-mode BRAM & DSP and data reuse to increase power efficiency. Experiments show that HBCA is capable of supporting seven CNNs with different types of kernels and more branches. The accuracy loss is within 0.1% when compared to the quantized model. Furthermore, the power efficiency (GOPS/W) of Inception, ResNet and VGG is up to 226.6, 188.1 and 197.7, which are better than other FPGA-based CNN accelerators.

ASYMPTOTIC NORMALITY SINGLE FUNCTIONAL INDEX QUANTILE REGRESSION UNDER RANDOMLY CENSORED DATA

The main objective of this paper is to estimate non-parametrically the quantiles of a conditional distribution based on the single-index model in the censorship model when the sample is considered as an independent and identically distributed (i.i.d.) random variables. First of all, a kernel type estimator for the conditional cumulative distribution function (cond-cdf) is introduced. Afterwards, we give an estimation of the quantiles by inverting this estimated cond-cdf, the asymptotic properties are stated when the observations are linked with a single-index structure. Finally, a simulation study is carried out to evaluate the performance of this estimate.

Algebraic fibring of a hyperbolic 7‐manifold

AbstractWe show that there is a finite‐volume, hyperbolic 7‐manifold that algebraically fibres with finitely presented kernel of type . This manifold is a finite cover of the one constructed by Italiano–Martelli–Migliorini.

A 2D-GRAPPA Algorithm with a Boomerang Kernel for 3D MRI Data Accelerated along Two Phase-Encoding Directions

For the reconstruction of 3D MRI data that are accelerated along the two phase-encoding directions, the 2D-generalized autocalibrating partially parallel acquisitions (GRAPPA) algorithm can be used to estimate the missing data in the k-space. We propose a new boomerang-shaped kernel based on theoretic and systemic analyses of the shape and dimensions of the kernel. The reconstruction efficiency of the 2D-GRAPPA algorithm with the proposed boomerang-shaped kernel (i.e., boomerang kernel (BK)-2D-GRAPPA) was compared with other 2D-GRAPPA algorithms that utilize different types of kernels (i.e., EX-2D-GRAPPA and SK-2D-GRAPPA) based on computer simulation, phantom and in vivo experiments. The proposed method was validated for different sets of ACS lines with acceleration factors from four to eight and various sizes of the kernels. A quantitative analysis was also performed by comparing the normalized root mean squared error (nRMSE) in the images and the undersampled edges. Computer simulation, in vivo and phantom experiments, and the quantitative analysis, showed that the proposed method could reduce aliasing artifacts without reducing the SNRs of the reconstructed images.

Numerical Solutions of Volterra Integral Equations of Third Kind and Its Convergence Analysis

The current work suggests a method for the numerical solution of the third type of Volterra integral equations (VIEs), based on Lagrange polynomial, modified Lagrange polynomial, and barycentric Lagrange polynomial approximations. To do this, the interpolation of the unknown function is considered in terms of the above polynomials with unknown coefficients. By substituting this approximation into the considered equation, a system of linear algebraic equations is obtained. Then, we demonstrate the method’s convergence and error estimations. The proposed approaches retain the possible singularity of the solution. To the best of the authors’ knowledge, the singularity case has not been addressed by researchers yet. To illustrate the applicability, effectiveness, and correctness of new methods for the proposed integral equation, examples with both types of kernels, symmetric as well as non-symmetric, are provided at the end.

Efficient vectorised kernels for unstructured high-order finite element fluid solvers on GPU architectures in two dimensions

We develop efficient kernels for elemental operators of matrix-free solvers of the Helmholtz equation, which are the core operations for incompressible Navier-Stokes solvers, for use on graphics-processing units (GPUs). Our primary concern in this work is the extension of matrix-free routines to efficiently evaluate this elliptic operator on regular and curvilinear triangular elements in a tensor-product manner. We investigate two types of efficient CUDA kernels for a range of polynomial orders and thus varying arithmetic intensities: the first maps each elemental operation to a CUDA-thread for a completely vectorised kernel, whilst the second maps each element to a CUDA-block for nested parallelism. Our results show that the first option is beneficial for elements with low polynomial order, whereas the second option is beneficial for elements of higher order. The crossover point between these two schemes for the hardware used in this study corresponds to polynomial orders at around P=4−5, depending on element type. For both options, we highlight the importance of the layout of data structures, which necessitates the development of interleaved elemental data for vectorised kernels, and analyse the effect of selecting different memory spaces on the GPU. As the considered kernels are foremost memory-bandwidth bound, we develop kernels for curved elements that trade memory bandwidth against additional arithmetic operations, and demonstrate improved throughput in selected cases. We further compare our optimised CUDA kernels against optimised OpenACC kernels, to contrast the performance between a native and a portable programming model for GPUs.

ENTAIL: yEt aNoTher amyloid fIbrils cLassifier

BackgroundThis research aims to increase our knowledge of amyloidoses. These disorders cause incorrect protein folding, affecting protein functionality (on structure). Fibrillar deposits are the basis of some wellknown diseases, such as Alzheimer, Creutzfeldt–Jakob diseases and type II diabetes. For many of these amyloid proteins, the relative precursors are known. Discovering new protein precursors involved in forming amyloid fibril deposits would improve understanding the pathological processes of amyloidoses.ResultsA new classifier, called ENTAIL, was developed using over than 4000 molecular descriptors. ENTAIL was based on the Naive Bayes Classifier with Unbounded Support and Gaussian Kernel Type, with an accuracy on the test set of 81.80%, SN of 100%, SP of 63.63% and an MCC of 0.683 on a balanced dataset.ConclusionsThe analysis carried out has demonstrated how, despite the various configurations of the tests, performances are superior in terms of performance on a balanced dataset.

TOUCHARD METHOD FOR SOLVING NONLINEAR VOLTERRA INTEGRAL EQUATION OF SECOND TYPE

In this paper, a novel numerical method has been outlined for finding a novel solution to nonlinear Volterra integral (NLVI) equation of second type, besides, two kernel type of this equation. Touchard polynomials (TPs) and to different degrees were used for this purpose. The function of approximation was obtained to derive the technique for solving this type of integral equations. Three numerical examples were provided to demonstrate the importance of the method used and the accuracy of the extracted results. In some examples, the results were compared with those of another method. The MATLAB R2018b program was used to carry out all calculations and generate all graphics. Keywords: Nonlinear Volterra Integral Equation, Numerical Method, Touchard Polynomials, Exact Function, Numerical Solution

Consistency results of the M‐regression function estimator for stationary continuous‐time and ergodic data

This paper is devoted to the study of asymptotic properties of the kernel estimator of the robust regression function for stationary continuous‐time and ergodic data. Such a dependence structure is an alternative to the strong mixing conditions usually assumed in functional time series analysis. More precisely, we consider the kernel type estimator of the robust regression function constructed from the stationary and continuous‐time ergodic data for . Then, we establish the almost sure (with rate) pointwise convergence of this estimator. A simulation study was conducted in order to compare the performance of this method to the classical regression method.

Weighted Lipschitz estimates for commutators of multilinear Calderón–Zygmund operators with Dini type kernels

In this paper, we discuss the properties for commutators and iterated commutators generated by the multilinear ω-CZO and weighted Lipschitz functions on Lebesgue space.

Non-Markovian dynamics of black hole phase transition

We provide a comprehensive study on the non-Markovian dynamics of the black hole phase transitions based on the underlying free energy landscape. By assuming that the transition processes between different black hole states are stochastic, the non-Markovian dynamics of the black hole phase transition is governed by the generalized Langevin equation with the time-dependent friction that represents the memory effect from the effective thermal bath when the timescale of the system is comparable or shorter than the timescale of the effective thermal bath. We consider the first passage problem associated with the black hole phase transitions and derive the analytical expressions of the mean first passage time in the weak, intermediate, and large friction regimes. As the concrete examples, we study the effects of three types of time dependent friction kernel (delta function friction, exponentially decayed friction, and oscillatory friction) on the dynamics of Hawking-Page phase transition and the small/large RNAdS black hole phase transition. we found that there is a kinetic turnover point for each type of friction kernel when the friction strength varies. For the exponentially decayed friction kernel, it is shown that the non-Markovian effect slows down the phase transition dynamics in the weak friction regime and speeds up the transition process in the strong friction regime. For the oscillating decayed friction kernel, we found kinetic resonances when the oscillating frequency of the effective thermal bath is equal to the oscillating frequency of the black hole state in the initial potential well on the free energy landscape.

Solitary Waves of the Perturbed KdV Equation with Nonlocal Effects

In this paper, the Korteweg–de Vries (KdV) equation is considered, which is a shallow water wave model in fluid mechanic fields. First the existence of solitary wave solutions for the original KdV equation and geometric singular perturbation theory are recalled. Then the existence of solitary wave solutions is established for the equation with two types of delay convolution kernels by using the method of dynamical system, especially the geometric singular perturbation theory, invariant manifold theory and Melnikov method. Finally, the asymptotic behaviors of solitary wave solution are discussed by applying the asymptotic theory. Moreover, an interesting result is found for the equation without backward diffusion effect, there is no solitary wave solution in the case of local delay, but there is a solitary wave solution in the case of nonlocal delay.

Constructing an Intelligent Model Based on Support Vector Regression to Simulate the Solubility of Drugs in Polymeric Media

This study constructs a machine learning method to simultaneously analyze the thermodynamic behavior of many polymer-drug systems. The solubility temperature of Acetaminophen, Celecoxib, Chloramphenicol, D-Mannitol, Felodipine, Ibuprofen, Ibuprofen Sodium, Indomethacin, Itraconazole, Naproxen, Nifedipine, Paracetamol, Sulfadiazine, Sulfadimidine, Sulfamerazine, and Sulfathiazole in 1,3-bis[2-pyrrolidone-1-yl] butane, Polyvinyl Acetate, Polyvinylpyrrolidone (PVP), PVP K12, PVP K15, PVP K17, PVP K25, PVP/VA, PVP/VA 335, PVP/VA 535, PVP/VA 635, PVP/VA 735, Soluplus analyzes from a modeling perspective. The least-squares support vector regression (LS-SVR) designs to approximate the solubility temperature of drugs in polymers from polymer and drug types and drug loading in polymers. The structure of this machine learning model is well-tuned by conducting trial and error on the kernel type (i.e., Gaussian, polynomial, and linear) and methods used for adjusting the LS-SVR coefficients (i.e., leave-one-out and 10-fold cross-validation scenarios). Results of the sensitivity analysis showed that the Gaussian kernel and 10-fold cross-validation is the best candidate for developing an LS-SVR for the given task. The built model yielded results consistent with 278 experimental samples reported in the literature. Indeed, the mean absolute relative deviation percent of 8.35 and 7.25 is achieved in the training and testing stages, respectively. The performance on the largest available dataset confirms its applicability. Such a reliable tool is essential for monitoring polymer-drug systems' stability and deliverability, especially for poorly soluble drugs in polymers, which can be further validated by adopting it to an actual implementation in the future.

Reducing the dimensionality of data using tempered distributions

We reformulate unsupervised dimension reduction problem (UDR) in the language of tempered distributions, i.e. as a problem of approximating an empirical probability density function by another tempered distribution, supported in a k-dimensional subspace. We show that this task is connected with another classical problem of data science — the sufficient dimension reduction problem (SDR). In fact, an algorithm for the first problem induces an algorithm for the second and vice versa.In order to reduce an optimization problem over distributions to an optimization problem over ordinary functions we introduce a nonnegative penalty function that “forces” the support of the model distribution to be k-dimensional. Then we present an algorithm for the minimization of the penalized objective, based on the infinite-dimensional low-rank optimization, which we call the alternating scheme. Also, we design an efficient approximate algorithm for a special case of the problem, where the distance between the empirical distribution and the model distribution is measured by Maximum Mean Discrepancy defined by a Mercer kernel of a certain type. We test our methods on four examples (three UDR and one SDR) using synthetic data and standard datasets.

Adaptive Estimation of a Function from its Exponential Radon Transform in Presence of Noise

In this article we propose a locally adaptive strategy for estimating a function from its Exponential Radon Transform (ERT) data, without prior knowledge of the smoothness of functions that are to be estimated. We build a non-parametric kernel type estimator and show that for a class of functions comprising a wide Sobolev regularity scale, our proposed strategy follows the minimax optimal rate up to a \(\log {n}\) factor. We also show that there does not exist an optimal adaptive estimator on the Sobolev scale when the pointwise risk is used and in fact the rate achieved by the proposed estimator is the adaptive rate of convergence.

ReaDy: A ReRAM-Based Processing-in-Memory Accelerator for Dynamic Graph Convolutional Networks

Dynamic graph convolutional networks (DGCNs) have emerged as an effective approach to analyzing graph data that is constantly changing. The typical DGCNs incorporate not only graph convolutional networks (GCNs) to extract the structural information but also with recurrent neural networks (RNNs) to capture the temporal information from evolving graph data. These two alternative execution kernels of DGCNs impose unique architecture challenges for both types of kernels to be implemented efficiently. The presence of complex execution patterns of DGCNs renders existing architectures unsuitable. In this article, we present the first DGCN accelerator with an integrated architecture, named ReaDy, to accelerate DGCNs based on emerging PIM-featured ReRAM architectures. ReaDy is novel with an integrated architecture that enables running the GCN and RNN kernels of DGCNs simultaneously. Specifically, ReaDy is equipped with a redundancy-free scheduling mechanism to alleviate intrinsic dynamic irregularity for the GCN kernel, improving hardware utilization. In addition, ReaDy also includes a locality-aware dataflow strategy to exploit the inherent intervertex data locality for the RNN kernel, reducing superfluous data accesses to vertices and weight parameters. In a holistic view, ReaDy further enhances the entire system via an interkernel pipeline to reduce the off-chip accesses of intermediate results, boosting the overall efficiency of DGCNs significantly. Compared to the state-of-the-art software framework, PyGT, running on Intel Xeon E5-2680v4 CPU and NVIDIA Ampere A100 GPU, ReaDy achieves the average speedups of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$955\times $ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$27.33\times $ </tex-math></inline-formula> , and the average energy savings of 1 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$093\times $ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$80.21\times $ </tex-math></inline-formula> , respectively. In addition, ReaDy outperforms ReFlip-ERA, which is obtained by combining a state-of-the-art GCN accelerator ReFlip and RNN accelerator ERA-LSTM, by an average speedup of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$8.30\times $ </tex-math></inline-formula> and an average energy saving of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$7.29\times $ </tex-math></inline-formula> .

Remaining Useful Life Prediction using Gaussian Process Regression Model

This paper evaluates the merits of a multi-variable Gaussian process regression (GPR) model for remaining useful life (RUL) estimation. The paper presents an optimization method that trains the GPR model to find the best kernel type and hyper-parameter combination. Furthermore, the paper evaluates the performance of the GPR model for small training datasets and with a reduction (missing) of input features. A comparison is made to the multi-layer perceptron (MLP) neural network which forms the basis of deep learning models. To illustrate model performance, an air filter clogging RUL dataset is used. The performance results show that both GPR and MLP models have similar sensitivity to training set size but GPR also computes the uncertainty. Empirically, MLP is more robust to a test set with a missing input while the data suggests that the GPR performs better when the training data also did not include the same input.