Smooth Kernel Research Articles

Nonlinear Systems of Volterra Equations with Piecewise Smooth Kernels: Numerical Solution and Application for Power Systems Operation

The evolutionary integral dynamical models of storage systems are addressed. Such models are based on systems of weakly regular nonlinear Volterra integral equations with piecewise smooth kernels. These equations can have non-unique solutions that depend on free parameters. The objective of this paper was two-fold. First, the iterative numerical method based on the modified Newton–Kantorovich iterative process is proposed for a solution of the nonlinear systems of such weakly regular Volterra equations. Second, the proposed numerical method was tested both on synthetic examples and real world problems related to the dynamic analysis of microgrids with energy storage systems.

On the Lagrangian Trajectories for the One-Dimensional Euler Alignment Model without Vacuum Velocity

A well-known result of Carrillo, Choi, Tadmor, and Tan [1] states that the 1D Euler Alignment model with smooth interaction kernels possesses a “critical threshold” criterion for the global existence or finite-time blowup of solutions, depending on the global nonnegativity (or lack thereof) of the quantity e 0 =∂ x u 0 +ϕ*ρ 0 . In this note, we rewrite the 1D Euler Alignment model as a first-order system for the particle trajectories in terms of a certain primitive ψ 0 of e 0 ; using the resulting structure, we give a complete characterization of global-in-time existence versus finite-time blowup of regular solutions that does not require a velocity to be defined in the vacuum. We also prove certain upper and lower bounds on the separation of particle trajectories, valid for smooth and weakly singular kernels, and we use them to weaken the hypotheses of Tan [25] sufficient for the global-in-time existence of a solution in the weakly singular case, when the order of the singularity lies in the range s∈(0,1 2) and the initial data is critical.

BROWNIAN MOTION MINUS THE INDEPENDENT INCREMENTS: REPRESENTATION AND QUEUING APPLICATION

This paper relaxes assumptions defining multivariate Brownian motion (BM) to construct processes with dependent increments as tractable models for problems in engineering and management science. We show that any Gaussian Markov process starting at zero and possessing stationary increments and a symmetric smooth kernel has a parametric kernel of a particular form, and we derive the unique unbiased, jointly sufficient, maximum-likelihood estimators of those parameters. As an application, we model a single-server queue driven by such a process and derive its transient distribution conditional on its history.

On the properties of hermite series based distribution function estimators

Hermite series based distribution function estimators have recently been applied in the context of sequential quantile estimation. These distribution function estimators are particularly useful because they allow the online (sequential) estimation of the full cumulative distribution function. This is in contrast to the empirical distribution function estimator and smooth kernel distribution function estimator which only allow sequential cumulative probability estimation at particular values on the support of the associated density function. Hermite series based distribution function estimators are well suited to the settings of streaming data, one-pass analysis of massive data sets and decentralised estimation. In this article we study these estimators in a more general context, thereby redressing a gap in the literature. In particular, we derive new asymptotic consistency results in the mean squared error, mean integrated squared error and almost sure sense. We also present novel Bias-robustness results for these estimators. Finally, we study the finite sample performance of the Hermite series based estimators through a real data example and simulation study. Our results indicate that in the general (non-sequential) context, the Hermite series based distribution function estimators are inferior to smooth kernel distribution function estimators, but may remain compelling in the context of sequential estimation of the full distribution function.

Adverse events reporting in stage III NSCLC trials investigating surgery and radiotherapy.

BackgroundCurrent treatment options for stage III non-small cell lung cancer (NSCLC) consist of different combinations of chemotherapy, surgery, radiotherapy and immunotherapy. Treatment choices are highly individual decisions, in which adverse events (AEs) are relevant for decision-making. This study aims to analyse reporting of AEs in prospective stage III NSCLC trials, focussing on trials including radiotherapy and/or surgery.MethodsPubMed was searched for prospective studies dealing with stage III NSCLC from January 1987 to April 2019. Meta-analyses were screened as a positive control. Pearson's Chi-squared test and smooth kernel distribution were used to estimate distributions. Data was resampled using bootstrapping.ResultsOut of 1193 initially identified studies, 119 met the inclusion criteria. Of these, 31 had a surgical procedure in any study arm. Grade 3 and 4 AEs were reported in 94.12% and 92.44% of the included studies, respectively. Reporting of grade 5 AEs was provided in 87.39% of cases. Grade 1 and 2 AEs were less commonly reported at 53.78% and 63.03%, respectively. One study did not mention any AEs. Of the 31 treatment arms including any form of surgery, AEs were not reported in 10. Overall, 231 different AE items were reported, only 18 of them were included in at least 20% of the analysed studies.ConclusionOverall, AE reporting in stage III NSCLC was inconsistent and inhomogeneous. Studies including surgical study arms often reported only treatment-related deaths in regards of surgical AEs. Underreporting of AEs prohibits the extraction of patient-relevant information for decision-making and represents a suboptimal use of invested resources.

The Fine Error Estimation of Collocation Methods on Uniform Meshes for Weakly Singular Volterra Integral Equations

It is well known that for the second-kind Volterra integral equations (VIEs) with weakly singular kernel, if we use piecewise polynomial collocation methods of degree m to solve it numerically, due to the weak singularity of the solution at the initial time $$t = 0$$ , only $$1 - \alpha $$ global convergence order can be obtained on uniform meshes, comparing with m global convergence order for VIEs with smooth kernel. However, in this paper, we will see that at mesh points, the convergence order can be improved, and it is better and better as n increasing. In particular, 1 order can be recovered for $$m = 1$$ at the endpoint. Some superconvergence results are obtained for iterated collocation methods, and a representative numerical example is presented to illustrate the obtained theoretical results.

Emphysema quantification using low-dose computed tomography with deep learning-based kernel conversion comparison.

This study determined the effect of dose reduction and kernel selection on quantifying emphysema using low-dose computed tomography (LDCT) and evaluated the efficiency of a deep learning-based kernel conversion technique in normalizing kernels for emphysema quantification. A sample of 131 participants underwent LDCT and standard-dose computed tomography (SDCT) at 1- to 2-year intervals. LDCT images were reconstructed with B31f and B50f kernels, and SDCT images were reconstructed with B30f kernels. A deep learning model was used to convert the LDCT image from a B50f kernel to a B31f kernel. Emphysema indices (EIs), lung attenuation at 15th percentile (perc15), and mean lung density (MLD) were calculated. Comparisons among the different kernel types for both LDCT and SDCT were performed using Friedman's test and Bland-Altman plots. All values of LDCT B50f were significantly different compared with the values of LDCT B31f and SDCT B30f (p < 0.05). Although there was a statistical difference, the variation of the values of LDCT B50f significantly decreased after kernel normalization. The 95% limits of agreement between the SDCT and LDCT kernels (B31f and converted B50f) ranged from - 2.9 to 4.3% and from - 3.2 to 4.4%, respectively. However, there were no significant differences in EIs and perc15 between SDCT and LDCT converted B50f in the non-chronic obstructive pulmonary disease (COPD) participants (p > 0.05). The deep learning-based CT kernel conversion of sharp kernel in LDCT significantly reduced variation in emphysema quantification, and could be used for emphysema quantification. • Low-dose computed tomography with smooth kernel showed adequate performance in quantifying emphysema compared with standard-dose CT. • Emphysema quantification is affected by kernel selection and the application of a sharp kernel resulted in a significant overestimation of emphysema. • Deep learning-based kernel normalization of sharp kernel significantly reduced variation in emphysema quantification.

Mean-square stability and convergence of a split-step theta method for stochastic Volterra integral equations

In this paper, a split-step θ method is introduced for solving stochastic Volterra integral equations with general smooth kernels. First, the mean-square boundedness and convergence properties of the numerical solution are analyzed. In particular, when the kernel function in the stochastic integral term satisfies a certain condition, the method can achieve superconvergence. Then, the mean-square stability of the method with respect to a convolution test equation is studied. A recurrence relation is found and mean-square stability regions are given by using root locus method. In particular, when the test equation degrades to the deterministic case, the new method with θ≥1∕2 is V0-stable (i.e., it can preserve the stability of the convolution test equation), which is superior to the stochastic θ method. Finally, some numerical experiments are given to verify the theoretical results.

CT kernel conversions using convolutional neural net for super-resolution with simplified squeeze-and-excitation blocks and progressive learning among smooth and sharp kernels

PurposeComputed tomography (CT) volume sets reconstructed with different kernels are helping to increase diagnostic accuracy. However, several CT volumes reconstructed with different kernels are difficult to sustain, due to limited storage and maintenance issues. A CT kernel conversion method is proposed using convolutional neural networks (CNN). MethodsA total of 3289 CT images from ten patients (five men and five women; mean age, 63.0 ± 8.6 years) were obtained in May 2016 (Somatom Sensation 16, Siemens Medical Systems, Forchheim, Germany). These CT images were reconstructed with various kernels, including B10f (very smooth), B30f (medium smooth), B50f (medium sharp), and B70f (very sharp) kernels. Smooth kernel images were converted into sharp kernel images using super-resolution (SR) network with Squeeze-and-Excitation (SE) blocks and auxiliary losses, and vice versa. In this study, the single-conversion model and multi-conversion model were presented. In case of the single-conversion model, for the one corresponding output image (e.g., B10f to B70), SE-Residual blocks were stacked. For the multi-conversion model, to convert an image into several output images (e.g., B10f to B30f, B50f, and B70f, and vice versa), progressive learning (PL) was employed by calculating auxiliary losses in every four SE-Residual blocks. Through auxiliary losses, the model could learn mutual relationships between different kernel types. The conversion quality was evaluated by the root-mean-square-error (RMSE), structural similarity (SSIM) index and mutual information (MI) between original and converted images. ResultsThe RMSE (SSIM index , MI) of the multi-conversion model was 4.541 ± 0.688 (0.998 ± 0.001 , 2.587 ± 0.137), 27.555 ± 5.876 (0.944 ± 0.021 , 1.735 ± 0.137), 72.327 ± 17.387 (0.815 ± 0.053 , 1.176 ± 0.096), 8.748 ± 1.798 (0.996 ± 0.002 , 2.464 ± 0.121), 9.470 ± 1.772 (0.994 ± 0.003 , 2.336 ± 0.133), and 9.184 ± 1.605 (0.994 ± 0.002 , 2.342 ± 0.138) in conversion between B10f–B30f, B10f–B50f, B10f–B70f, B70f–B50f, B70f–B30f, and B70f–B10f, respectively, which showed significantly better image quality than the conventional model. ConclusionsWe proposed deep learning-based CT kernel conversion using SR network. By introducing simplified SE blocks and PL, the model performance was significantly improved.

Density-adaptive kernel based efficient reranking approaches for person reidentification

Person reidentification (ReID) refers to the task of verifying the identity of a pedestrian observed from nonoverlapping views in a surveillance camera network. It has recently been validated that reranking can achieve remarkable performance improvements in person ReID systems. However, current reranking approaches either require feedback from users or suffer from burdensome computational costs. In this paper, we propose to exploit a density-adaptive smooth kernel technique to achieve efficient and effective reranking. Specifically, we adopt a smooth kernel function to formulate the neighbor relationships among data samples with a density-adaptive parameter. Based on this new formulation, we present two simple yet effective reranking methods, termed inverse density-adaptive kernel based reranking (inv-DAKR) and bidirectional density-adaptive kernel based reranking (bi-DAKR), in which the local density information in the vicinity of each gallery sample is elegantly exploited. Moreover, we extend the proposed inv-DAKR and bi-DAKR methods to incorporate the available extra probe samples and demonstrate that when and why these extra probe samples are able to improve the local neighborhood and thus further refine the ranking results. Extensive experiments are conducted on six benchmark datasets, including: PRID450s, VIPeR, CUHK03, GRID, Market-1501 and Mars. The experimental results demonstrate that our proposals are effective and efficient.

Superconvergence results for the nonlinear Fredholm–Hemmerstein integral equations of second kind

The multi-projection methods for solving the Fredholm-Hammerstein integral equation is proposed in this paper. We obtain the similar super-convergence results as in Mandal and Nelakanti (J Comput Appl Math 319:423–439, 2017) with a smooth kernel using piecewise polynomials of degree $$\le r-1,$$ i.e., for both the multi-Galerkin and multi-collocation methods have order of convergence $$\mathcal O (h^{3r})$$ in uniform norm, where h is the norm of the partition. We have also considered iterated version of these methods and prove that both iterated multi-Galerkin and iterated multi-collocation methods have order of convergence $$\mathcal O(h^{4r})$$ in uniform norm. Numerical examples are given to illustrate the theoretical results.

A fully discrete boundary integral method based on embedding into a periodic box

This paper shows how numerical methods on a regular grid in a box can be used to generate numerical schemes for problems in general smooth domains contained in the box with no need for a domain specific discretization (other than a list of points on it). The focus is mainly on spectral discretizations due to their ability to accurately resolve the interaction of finite order distributions (generalized functions) and smooth functions. Mimicking the analytical structure of the relevant (pseudodifferential) operators leads to viable and accurate numerical representations and algorithms. An important byproduct of the structural insights gained in the process is the introduction of smooth kernels (at the discrete level) to replace classical singular kernels which are typically used in the (numerical) representations of the solution. The new kernel representations yield enhanced numerical resolution.

A stable kernel-based technique for solving linear Fredholm integral equations of the second kind and its applications

Meshless methods based on infinitely smooth radial kernels have the potential to provide spectrally accurate function approximations with enormous geometric flexibility in any number of dimensions. The highest accuracy can often be obtained when the shape parameter in the basis function is small. But as the shape parameter goes to zero, the standard RBF interpolant matrix becomes severely ill-conditioned. The ill-conditioning can be reduced using alternate bases. One of these alternative bases is the Hilbert–Schmidt SVD basis. The Hilbert–Schmidt SVD method offers a stable mechanism for converting a set of near-flat kernels with scattered centres to a well-conditioned base for exactly the same space. In this paper, we apply the Gaussian Hilbert–Schmidt SVD basis functions method for solving the linear Fredholm integral equations of the second kind. The method estimates the solution by the discrete collocation method based on Gaussian Hilbert–Schmidt SVD basis functions constructed on a set of disordered data. This approach reduces the solution of the problem under study to the solution of a linear system of algebraic equations. Also, the convergence of the proposed approach is analyzed. Lastly, several numerical experiments are presented to test the stability and accuracy of the proposed method.

Application of Hilbert–Schmidt SVD approach to solve linear two-dimensional Fredholm integral equations of the second kind

Meshfree techniques based on infinitely smooth radial kernels have the great potential to provide spectrally accurate function approximations with irregular domain in high dimensions. The maximum accuracy can mostly be found when the RBF shape parameter is small, i.e., when the radial kernel is relatively smooth. However, as the shape parameter goes to zero, the standard RBF interpolant matrix will be very ill-conditioned. The ill-conditioning can be alleviated using alternate bases. One of these alternative bases is the Hilbert–Schmidt SVD basis. The Hilbert–Schmidt SVD approach suggests a stable mechanism for replacing a set of near-flat kernels with scattered centres to a well-conditioned base for exactly the same space. In this work, the Gaussian Hilbert–Schmidt SVD basis functions method is presented to numerically solve the linear two-dimensional Fredholm integral equations of the second kind. The method estimates the solution by the discrete collocation method based on Gaussian Hilbert–Schmidt SVD basis functions constructed on a set of scattered points. The emerged integrals in the scheme are approximately computed by the Gauss–Legendre quadrature rule. This approach reduces the problem under study to a linear system of algebraic equations which can be solved easily via applying an appropriate numerical technique. Also, the convergence of the proposed approach is established. Finally, numerical results are compared with standard RBF method to indicate the accuracy and efficiency of the suggested approach.

Modeling tissue perfusion in terms of 1d-3d embedded mixed-dimension coupled problems with distributed sources

We present a new method for modeling tissue perfusion on the capillary scale. The microvasculature is represented by a network of one-dimensional vessel segments embedded in the extra-vascular space. Vascular and extra-vascular space exchange fluid over the vessel walls. This exchange is modeled by distributed sources using smooth kernel functions for the extra-vascular domain. It is shown that the proposed method may significantly improve the approximation of the exchange flux, in comparison with existing methods for mixed-dimension embedded problems. Furthermore, the method exhibits better convergence rates of the relevant quantities due to the increased regularity of the extra-vascular pressure solution. Numerical experiments with a vascular network from the rat cortex show that the error in the approximation of the exchange flux for coarse grid resolutions may be decreased by a factor of 3. This may open the way for computing on larger network domains, where a fine grid resolution cannot be achieved in practical simulations due to constraints in computational resources, for example in the context of uncertainty quantification.

A consistent parallel isotropic unstructured mesh generation method based on multi-phase SPH

In this paper, we propose a consistent parallel unstructured mesh generator based on a multi-phase SPH method. A set of physics-motivated modeling equations are developed to achieve the targets of domain decomposition, communication volume optimization and high-quality unstructured mesh generation simultaneously. A unified density field is defined as the target function for both partitioning the geometry and distributing the mesh-vertexes. A multi-phase Smoothing Particle Hydrodynamics (SPH) method is employed to solve the governing equations. All the optimization targets are achieved implicitly and consistently by the particle relaxation procedure without constructing triangulation/tetrahedralization explicitly. The target of communication reduction is achieved by introducing a surface tension model between distinct partitioning sub-domains, which are characterized by colored SPH particles. The resulting partitioning diagram features physically localized sub-domains and optimized interface communication. The target of optimizing the mesh quality is achieved by introducing a tailored equation-of-state (EOS) and a smooth isotropic kernel function. The mesh quality near the interface of neighboring sub-domains is improved by gradually removing the surface-tension force once a steady state is achieved. The proposed method is developed basing on a new parallel environment for multi-resolution SPH to exploit both coarse- and fine-grained parallelism. A set of benchmarks are conducted to verify that all the optimization targets are achieved consistently within the current framework.

The Optimization of Mastoid CT Image Using Windows and Kernel Reconstructions

CT scan can be used to show the anatomical and pathological evaluation of Mastoid bones where an X-ray across it to create a cross-sectional image with an advanced computer. This imaging modality allows the radiologist to look at different levels of the bone ridge behind the ear. In our hospital, this procedure needs to be improved. Radiographer could make optimization by adjusting windows and developing kernel to maintain the image quality. This study aimed to obtain the optimum image of Mastoid bones, using variations of window and kernel reconstruction. The study was descriptive quantitative with an experimental approach. It resulted in eight images of two windowing levels (sinus and inner ear) and four kernel variations (smooth, medium, sharp, and ultra-sharp). Three radiologists evaluated the injury, bleedin, and soft tissue abnormalities images. The result showed that all window settings are acceptable. Kernel reconstructions have no different anatomical image information in soft tissue, Internal Auditory Canal, and External Auditory Canal. There is a difference for overall anatomical information of Mastoid bones (p value<0.05). Highest values of mean rank are obtained from sharp and ultra-sharp. Our recommendations are using the H.20s smooth kernel for soft tissue abnormalities and H.70s sharp kernel for fracture and bleeding cases.

On the Spectral Property of Kernel-Based Sensor Fusion Algorithms of High Dimensional Data

We apply local laws of random matrices and free probability theory to study the spectral properties of two kernel-based sensor fusion algorithms, nonparametric canonical correlation analysis (NCCA) and alternating diffusion (AD), for two simultaneously recorded high dimensional datasets under the null hypothesis. The matrix of interest is the product of the kernel matrices associated with the databsets, which may not be diagonalizable in general. We prove that in the regime where dimensions of both random vectors are comparable to the sample size, if NCCA and AD are conducted using a smooth kernel function, then the first few nontrivial eigenvalues will converge to real deterministic values provided the datasets are independent Gaussian random vectors. Toward the claimed result, we also provide a convergence rate of eigenvalues of a kernel affinity matrix.

Convergence analysis of Galerkin and multi-Galerkin methods on unbounded interval using Hermite polynomials

We consider in this paper the Galerkin and multi-Galerkin methods and their iterated versions to solve the linear Fredholm integral equation of the second kind on the real line with a sufficiently smooth kernels, using Hermite polynomials as basis functions. We obtain optimal convergence results in iterated Galerkin method in weighted L2−norm. We also discuss multi-Galerkin methods and we obtain the superconvergence results in both multi-Galerkin and iterated multi-Galerkin methods. Numerical results are presented to confirm the theoretical results.

Atangana-Baleanu fractional framework of reproducing kernel technique in solving fractional population dynamics system

Abstract In this article, a class of population growth model, the fractional nonlinear logistic system, is studied analytically and numerically. This model is investigated by means of Atangana-Baleanu fractional derivative with a non-local smooth kernel in Sobolev space. Existence and uniqueness theorem for the fractional logistic equation is provided based on the fixed-point theory. In this orientation, two numerical techniques are implemented to obtain the approximate solutions; the reproducing-kernel algorithm is based on the Schmidt orthogonalization process to construct a complete normal basis, while the successive substitution algorithm is based on an appropriate iterative scheme. Convergence analysis associated with the suggested approaches is provided to demonstrate the applicability theoretically. The impact of the fractional derivative on population growth is discussed by a class of nonlinear logistical models using the derivatives of Caputo, Caputo-Fabrizio, and Atangana-Baleanu. Using specific examples, numerical simulations are presented in tables and graphs to show the effect of the fractional operator on the population curve as . The present results confirm the theoretical predictions and depict that the suggested schemes are highly convenient, quite effective and practically simplify computational time.