Polynomial Basis Research Articles

Neutron Multiplicity Counting Distribution Reconstruction from Moments Using Meixner Polynomial Expansion and N-Forked Branching Approximations

Methods used to infer nuclear parameters from neutron count statistics fall into two categories depending on whether they use moments or count number probabilities. As probabilities are in general more difficult to calculate, we are interested here in the reconstruction of distributions from their lower-order moments. For this, we explore two approaches. The first one relies on a generalization of the two-forked branching correlation (quadratic) approximation used in the PMZBB and Poisson radical distributions, and the second one is founded on the expansion of the distribution on a Meixner discrete orthogonal polynomial base.

Pastro polynomials and Sobolev-type orthogonal polynomials on the unit circle based on a q-difference operator

ABSTRACT Applications of orthogonal polynomials on the unit circle have attracted the attention of many researchers in recent years. Pastro polynomials, which are basic hypergeometric polynomials, are known to be biorthogonal polynomials on the unit circle. However, with special choice of parameters they provide one of the nicest examples of orthogonal polynomials on the unit circle. Our objective here is to consider some properties of three sequences of polynomials which are related to these Pastro orthogonal polynomials on the unit circle by a q-difference operator. We also provide information regarding connection formulas, bounds for the connection coefficients as well as outer relative asymptotics associated with these sequences of polynomials.

An interpretable transformer network for the retinal disease classification using optical coherence tomography

Retinal illnesses such as age-related macular degeneration and diabetic macular edema will lead to irreversible blindness. With optical coherence tomography (OCT), doctors are able to see cross-sections of the retinal layers and provide patients with a diagnosis. Manual reading of OCT images is time-consuming, labor-intensive and even error-prone. Computer-aided diagnosis algorithms improve efficiency by automatically analyzing and diagnosing retinal OCT images. However, the accuracy and interpretability of these algorithms can be further improved through effective feature extraction, loss optimization and visualization analysis. In this paper, we propose an interpretable Swin-Poly Transformer network for performing automatically retinal OCT image classification. By shifting the window partition, the Swin-Poly Transformer constructs connections between neighboring non-overlapping windows in the previous layer and thus has the flexibility to model multi-scale features. Besides, the Swin-Poly Transformer modifies the importance of polynomial bases to refine cross entropy for better retinal OCT image classification. In addition, the proposed method also provides confidence score maps, assisting medical practitioners to understand the models’ decision-making process. Experiments in OCT2017 and OCT-C8 reveal that the proposed method outperforms both the convolutional neural network approach and ViT, with an accuracy of 99.80% and an AUC of 99.99%.

Implementing the method of figurative transformations to minimize partially defined Boolean functions

This paper reports a research that established the possibility of increasing the effectiveness of the method of figurative transformations to minimize partially defined Boolean functions. The method makes it possible, without loss of functionality, to reduce the complexity of the minimization procedure, compared to sorting out binary definitions of partially defined Boolean functions. The interpretation of the result is that the 2-(n, b)-design, 2-(n, x/b)-design systems are a reflection of logical operations. Therefore, the identification of such combinatorial systems in the truth table of logical functions directly and unambiguously establishes the location of logical operations for equivalent transformations of Boolean expressions. This, in turn, implicates an algorithm for simplifying Boolean functions, including partially defined Boolean functions. Thus, the method of figurative transformations simplifies and speeds up the procedure for minimizing partially defined Boolean functions, compared to analogs. This indicates that the visual-matrix form of the analytical method still has the prospect of increasing its hardware capabilities, including in terms of minimizing partially defined Boolean functions. It has been experimentally confirmed that the method of figurative transformations increases the efficiency of minimizing partially defined Boolean functions, compared with analogs, by 100–200 %. There is reason to argue about the possibility of increasing the efficiency of minimizing partially defined Boolean functions in the main and polynomial bases by the specified method. The effectiveness of the method, in particular, is ensured by carrying out all operations of generalized gluing of variables for dead-end disjunctive normal forms (DNF), followed by the use of implicant tables; optimal combination of a sequence of logical operations for gluing variables

Space-time spectral method for the Stokes problem

The Stokes equations are a linearized version of the Navier-Stokes equations and model incompressible viscous fluid flow with low Reynolds numbers. Several spectral methods, exhibiting exponential decay in error when the solution is analytic, are known to solve the steady-state Stokes problem numerically. A common strategy to solve such a problem in the time-dependent case involves extending the spectral scheme in spatial derivatives by implementing a low-order finite difference scheme for the time derivatives. Instead, we implement and analyze a space-time spectral method for the Stokes problem, which converges exponentially in both space and time. This numerical scheme imposes spectral collocation in time and PN−PN−2 spectral Galerkin scheme in space by using a recombined Legendre polynomial basis, resulting in a global spectral operator that is a saddle point matrix. The main objectives of the research are estimating the condition number of the global spectral operators and proving the spectral convergence of this scheme in space and time. The analysis is not quite complete because two of the estimates are based on numerical evidence. However, throughout the project, some intermediate results-such as the 2-norm of the pseudospectral derivative matrix for Chebyshev-Gauss-Lobatto nodes as well as the condition number of the mass matrix and discrete Laplacian for a recombined Legendre basis-were proved to obtain the aforementioned findings. Numerical experiments of this scheme verify the theoretical results. Also, the numerical result of applying this scheme to the Navier-Stokes equations is presented.

A Flux Reconstruction Stochastic Galerkin Scheme for Hyperbolic Conservation Laws

The study of uncertainty propagation poses a great challenge to design high fidelity numerical methods. Based on the stochastic Galerkin formulation, this paper addresses the idea and implementation of the first flux reconstruction scheme for hyperbolic conservation laws with random inputs. High-order numerical approximation is adopted simultaneously in physical and random space, i.e., the modal representation of solutions is based on an orthogonal polynomial basis and the nodal representation is based on solution collocation points. Therefore, the numerical behaviors of the scheme in the (physical-random) phase space can be designed and understood uniformly. A family of filters is developed in multi-dimensional cases to mitigate the Gibbs phenomenon arising from discontinuities in both physical and random space. The filter function is switched on and off by the dynamic detection of discontinuous solutions, and a slope limiter is employed to preserve the positivity of physically realizable solutions. As a result, the proposed method is able to capture the stochastic flow evolution where resolved and unresolved regions coexist. Numerical experiments including a wave propagation, a Burgers’ shock, a one-dimensional Riemann problem, and a two-dimensional shock-vortex interaction problem are presented to validate the current scheme. The order of convergence of the high-order scheme is identified. The capability of the scheme for simulating smooth and discontinuous stochastic flow dynamics is demonstrated. The open-source codes to reproduce the numerical results are available under the MIT license (Xiao et al. in FRSG: stochastic Galerkin method with flux reconstruction. https://github.com/CSMMLab/FRSG, (2021). https://doi.org/10.5281/zenodo.5588317).

Hopf instability of a Rayleigh–Taylor unstable thin film heated from the gas side

A thin liquid film located on the underside of a horizontal solid substrate can be stabilized by the Marangoni effect if the liquid is heated at its free surface. Applying long-wave approximation and projecting the velocity and temperature fields onto a basis of low-order polynomials, we derive a dimension-reduced set of three coupled evolution equations where nonlinearities of both the Navier–Stokes and the heat equation are included. We find that in a certain range of fluid parameters and layer depth, the first bifurcation from the motionless state is oscillatory which sets in with a finite but small wave number. The oscillatory branch is determined using a linear stability analysis of the long-wave model, but also by solving the linearized original hydrodynamic equations. Finally, numerical solutions of the reduced nonlinear model equations in three spatial dimensions are presented.

Multiplierless low‐cost implementation of Hindmarsh–Rose neuron model in case of large‐scale realization

SummaryImplementation of neural networks in case of hardware helps us to understand the different parts of the human brain operation, using artificial intelligence (AI). This paper presents a new model of the Hindmarsh–Rose (HR) Neuron that is based on basic polynomial functions called Nyquist‐look up table‐Hindmarsh–Rose (N‐LUT‐HR) based on an accurate sampling of the original model. The proposed approach is investigated in terms of its digital realization feasibility. According to high matching between the original and proposed terms, it is showed that the new modified model can follow all spiking patterns of primary model with low‐error computations. In hardware case, the proposed and original models are implemented on Xilinx FPGA XC2VP30 chip to validate different aspects of the simulation results. Hardware results demonstrate that our model regenerates the desired patterns in low‐cost and high‐frequency (speed‐up) in comparison with the other similar works. Overall saving in FPGA resources show that this new model is capable of being used in large‐scale networks in case of minimum required resources (FPGA costs). In addition, the analysis of hardware indicates that the new circuits can work in a maximum frequency of 123 MHz with 98.25% saving in FPGA costs (resources utilization of FPGA).

Radial Point Interpolation-Based Error Recovery Estimates for Finite Element Solutions of Incompressible Elastic Problems

Error estimation and adaptive applications help to control the discretization errors in finite element analysis. The study implements the radial point interpolation (RPI)-based error-recovery approaches in finite element analysis. The displacement/pressure-based mixed approach is used in finite element formulation. The RPI approach considers the radial basis functions (RBF) and polynomials basis functions together to interpolate the finite element solutions, i.e., displacement over influence zones to recover the solution errors. The energy norm is used to represent global and local errors. The reliability and effectiveness of RPI-based error-recovery approaches are assessed by adaptive analysis of incompressibility elastic problems including the problem with singularity. The quadrilateral meshes are used for discretization of problem domains. For adaptive improvement of mesh, the square of error equally distributed technique is employed. The computational outcome for solution errors, i.e., error distribution and convergence rate, are obtained for RPI technique-based error-recovery approach employing different radial basis functions (multi quadratic, thin-plate splint), RBF shape parameters, different shapes of influence zones (circular, rectangular) and conventional patches. The error convergence in the original FEM solution, in FEM solution considering influence-zone-based RPI recovery with MQ RBF, conventional patch-based RPI recovery with MQ RBF and conventional patch LS-based error recovery are found as (0.97772, 2.03291, 1.97929 and 1.6740), respectively, for four-node quadrilateral discretization of problem, while for nine-node quadrilateral discretization, the error convergence is (1.99607, 3.53087, 4.26621 and 2.54955), respectively. The study concludes that the adaptive analysis, using error-recovery estimates-based RPI approach, provides results with excellent accuracy and reliability.

PENDETEKSI UJARAN KEBENCIAN PADA PLATFORM MEDIA SOSIAL TWITTER MENGGUNAKAN SUPPORT VECTOR MACHINE

Twitter is one of the world social media giants, which has enormous flow text-based comment in everysecond. There are many types of writing sentiments that users create to discuss something such as famousfigures, companies or politics. One of the types is hate speech. The analysis was carried out using theSupport Vector Machine method as a text analysis model with the help of TF-IDF to assess the weight ofeach word. The experiment was carried out with several types of kernels and resulted in varying degrees ofaccuracy. The types of kernels tested were linear, radial basis function, polynomial and sigmoid with a testdata distribution of 20%, 25% and 30%.

Low-wavenumber compensation with Zernike tilt for non-Kolmogorov turbulence phase screens.

The fast-Fourier-transform-based filtering method for phase screen generation remains popular for numerical simulation of optical propagation through turbulence; however, these screens inherently underrepresent the spectral density at low wavenumbers. Here, the "Z-tilt" approach is explored to augment the spectral density at low wavenumbers by adding a random phase tilt, which is derived from the wavefront phase statistics of a Zernike polynomial basis. This approach is computationally efficient and can be applied to any statistically homogeneous and isotropic refractive index field. An analytic result is provided for the von Kármán spectrum with finite outer scale. In a quantitative comparison with phase screens compensated for using a common subharmonic approach, the Z-tilt method shows the best agreement with the analytical structure function when the outer scale is greater than about three times the screen dimension. For outer scales of the order of the screen dimension, the subharmonic and a modified Z-tilt method give the most accurate results. A propagation simulation demonstrates that the aperture-averaged angle-of-arrival variance is accurately predicted using the Z-tilt method.

Prediction of Food Factory Energy Consumption Using MLP and SVR Algorithms

The industrial sector accounts for a significant proportion of total energy consumption. Factory Energy Management Systems (FEMSs) can be a measure to reduce energy consumption in the industrial sector. Therefore, machine learning (ML)-based electricity and liquefied natural gas (LNG) consumption prediction models were developed using data from a food factory. By applying these models to FEMSs, energy consumption can be reduced in the industrial sector. In this study, the multilayer perceptron (MLP) algorithm was used for the artificial neural network (ANN), while linear, radial basis function networks and polynomial kernels were used for support vector regression (SVR). Variables were selected through correlation analysis with electricity and LNG consumption data. The coefficient of variation of root mean square error (CvRMSE) and coefficient of determination (R2) were examined to verify the prediction performance of the implemented models and validated using the criteria of the American Society of Heating, Refrigerating, and Air-Conditioning Engineers Guideline 14. The MLP model exhibited the highest prediction accuracy for electricity consumption (CvRMSE: 17.35% and R2: 0.84) and LNG consumption (CvRMSE: 12.52% and R2: 0.88). Our findings demonstrate it is possible to attain accurate predictions of electricity and LNG consumption in food factories using relatively simple data.

Triangular recurrences, generalized Eulerian numbers, and related number triangles

Many combinatorial and other number triangles are solutions of recurrences of the Graham–Knuth–Patashnik (GKP) type. Such triangles and their defining recurrences are investigated analytically. They are acted upon by a transformation group generated by two involutions: a left–right reflection and an upper binomial transformation, acting row-wise. The group also acts on the bivariate exponential generating function (EGF) of the triangle. By the method of characteristics, the EGF of any GKP triangle has an implicit representation in terms of the Gauss hypergeometric function. There are several parametric cases when this EGF can be obtained in closed form. One is when the triangle elements are the generalized Stirling numbers of Hsu and Shiue. Another is when they are generalized Eulerian numbers of a newly defined kind. These numbers are related to the Hsu–Shiue ones by an upper binomial transformation, and can be viewed as coefficients of connection between polynomial bases, in a manner that generalizes the classical Worpitzky identity. Many identities involving these generalized Eulerian numbers and related generalized Narayana numbers are derived, including closed-form evaluations in combinatorially significant cases.

Numerical study of the variable-order time-fractional mobile/immobile advection-diffusion equation using direct meshless local Petrov-Galerkin methods

In this paper, we use direct meshless local Petrov-Galerkin (DMLPG) methods for solving the variable-order time-fractional mobile/immobile advection-diffusion equation in two dimensions. The basis of the DMLPG methods is on the generalized moving least-square (GMLS) approximation and local weak form of the equation. Since the GMLS approximation uses basic polynomials as shape functions, there is no need to construct complex shape functions in the moving least-square (MLS) approximation. In addition, the calculation of numerical integrals in the local weak form of the equation is simple. We also give proof of the convergence and stability of the time-discrete scheme. The analytical results show that the semi-discrete time scheme is unconditionally stable and the convergence order is O(Δt2). Moreover, to show the accuracy and efficiency of the methods, we use the regular and irregular domains with distributions of uniform and scattered nodes in numerical examples.

Reconstructing terrestrial water storage anomalies using convolution-based support vector machine

Study regionThe Congo River basin in west-central Africa Study focusTraditional machine learning algorithms are recently being replaced by integrated learning techniques in pattern recognitions and predictions. These updated tools (or techniques), which attempt to explore higher dimensions and uncover hidden patterns in considerably non-linear datasets are the new normal in small, medium and large dataset even in higher orders. This study investigates the performance of a convolution-based support vector machine in hydrological analysis to optimize forecast of terrestrial water storage anomalies (TWSA). The linear, polynomial, and radial basis function (RBF) kernels were explored in reconstructing the three variants of TWSA obtained from the Gravity Recovery and Climate Experiment (GRACE) level 3 products. Using the original input training datasets, we built and trained the convolution-neural network (CNN) which is composed of convolution and fully connected layers, and was integrated into the traditional support vector machine (SVM). The network was trained with twenty-two (22) datasets. The original predictor datasets are composed of hydrological fluxes (HF), climate indices (CI), and sea surface temperature (SST) datasets whose influence on GRACE-TWSA was analyzed in the body of our work. Our results show that the polynomial-kernel based on the convolution-based support vector machine (CSVM) outperformed the other regression models in the reconstruction of all variants of TWSA. The high accuracy achieved using the CSVM demonstrates its promising potential to fill the gaps in the missing GRACE observations through its refined reconstruction capabilities. New hydrological insight for the regionIn this study, (i) SST variants or components (e.g., those of east tropical Atlantic) were leading predictors of TWSA, (ii) increasing polynomial order does not increase accuracy in all polynomial kernel operations, especially in cases of severe non-linearity between predictors and predictands, (iii) for the RBF kernels, intermediate gamma values ranging from 0.05 to 0.5 are ideal for climatic analysis, as very small or very large gamma values will either behave like a linear model or over fit, respectively, and (iv) overall, the polynomial kernels reconstructed and predicted TWSA better than the other kernels. The other conventional machine learning procedures used to compare the fit of the CSVM show significant insight for future TWSA reconstruction processes. Their robustness can be explored by varying the independent variables fed into the machine learning framework and tuning their hyper-parameters to result in better fitting index, which promises to be much more useful than traditional learning methods.

Residual estimation for grid modification in wall-modeled large eddy simulation using unstructured high-order methods

The accuracy and computational cost of a large eddy simulation are highly dependent on the computational grid. Building optimal grids manually from a priori knowledge is not feasible in most practical use cases; instead, solution-adaptive strategies can provide a robust and cost-efficient method to generate a grid with the desired accuracy. We adapt the residual estimation algorithm developed by Toosi and Larsson (2020) for Discontinuous Galerkin Spectral Elements Methods (DGSEM) to guide the grid-adaptation process. The core of the method is the computation of the estimated modeling residual using the polynomial basis functions used in DGSEM, and the averaging of the estimated residual over each element. The final method is assessed in multiple channel flow test cases and for the transonic flow over an airfoil, in both cases making use of mortar interfaces between elements with hanging nodes. The method is found to be robust and reliable, and to provide solutions on grids with significantly fewer elements at comparable accuracy compared to when using human-generated grids.

Deneb: An open-source high-performance multi-physical flow solver based on high-order DRM-DG method

High-order methods are being recognized as powerful tools for handling scale-resolving simulations over complex geometry. However, several obstacles still block their complete applications to practical engineering problems: a compromise between accuracy and efficiency on mixed-curved meshes, inherent vulnerability to numerical oscillations, and lack of open-source high-performance solvers for researchers. To address these issues, we present Deneb, an open-source high-order accurate numerical solver that enables high-performance scale-resolving simulations on PDE-based flow systems. Deneb uses the physical domain-based modal discontinuous Galerkin (DG) method; thus, it can provide an arbitrary high-order accurate solution on mixed-curved meshes and has the potential for handling polyhedral meshes as well. The direct reconstruction method (DRM) efficiently executes the numerical integration of DG volume and surface integrals without accuracy loss on non-affine elements where mapping functions are high-degree. The resulting DRM-DG method eliminates the severe cost of a quadrature-based approach on mixed-curved meshes. Deneb offers explicit and implicit Runge–Kutta methods as well to achieve high-order accuracy in time. In addition, Krylov subspace methods and preconditioners are available for high-performance linear system solving in parallel. Deneb possesses a strong capability to resolve multi-physical shocks without numerical instabilities with the aid of multi-dimensional limiting and artificial viscosity methods. In particular, the hierarchical multi-dimensional limiting process enables efficient computations of supersonic flows without time-step restriction. The current release of Deneb covers the simulations of hypersonic equilibrium and magneto-hydrodynamic flows as well as compressible Navier–Stokes equations, but it has the potential to solve any PDE-based multi-physical flow systems. Several benchmark problems are presented to highlight Deneb's capability to perform scale-resolving and multi-physical flow simulations. A scalability test is also presented to verify the scaling characteristics of Deneb for high-performance computing. Program summaryProgram title: DenebCPC library link to program files:https://doi.org/10.17632/723n5r797n.1Developer's repository link:https://github.com/HojunYouKr/DenebLicensing provisions: BSD-3-ClauseProgramming language: C++17Nature of problem: The physical domain-based modal DG method can achieve the expected order of accuracy with the optimal number of polynomial bases even on non-affine elements. However, the DG method becomes significantly expensive on high-order curved elements when using quadrature rules, blocking its applicability to practical engineering problems. The numerical integration should be much more efficient without compromising accuracy. In addition, the less diffusive nature of high-order methods makes them susceptible to producing spurious numerical oscillations near flow discontinuities, potentially leading to numerical instabilities. Thus, an accurate and robust shock-capturing method is essential to simulate multi-physical flows under compressible regimes. Finally, the solver needs high scalability to perform large-scale computations in parallel.Solution method: DRM is applied to the DG volume and surface integrals to perform efficient numerical integration on non-affine elements without accuracy loss. The resulting method, DRM-DG, provides arbitrary high-order accurate solutions to various PDE-based flow problems on mixed-curved meshes. The solution is also high-order accurate in time due to high-order explicit and implicit Runge–Kutta methods implemented. The external library enables high-performance linear system solving with various preconditioners in parallel. Both multi-dimensional limiting and artificial viscosity methods suppress unwanted subcell oscillations across physical discontinuities. In particular, the limiting methods simulate complex supersonic flows efficiently without time-step restriction. The solver is highly scalable on parallel computing with the aid of non-blocking communications and latency hiding.

A note on exotic integrals

We consider Bernoulli measures μ p \mu _p on the interval [ 0 , 1 ] [0,1] . For the standard Lebesgue measure the digits 0 0 and 1 1 in the binary representation of real numbers appear with an equal probability 1 / 2 1/2 . For the Bernoulli measures, the digits 0 0 and 1 1 appear with probabilities p p and 1 − p 1-p , respectively. We provide explicit expressions for various μ p \mu _p -integrals. In particular, integrals of polynomials are expressed in terms of the determinants of special Hessenberg matrices, which, in turn, are constructed from the Pascal matrices of binomial coefficients. This allows us to find closed-form expressions for the Fourier coefficients of μ p \mu _p in the Legendre polynomial basis. At the same time, the trigonometric Fourier coefficients are values of some special entire functions, which admit explicit infinite product expansions and satisfy interesting properties, including connections with the Stirling numbers and the polylogarithm.

A kernel‐based method for solving the time‐fractional diffusion equation

AbstractIn this paper, we focus on the development and study of a numerical method based on the idea of kernel‐based approximation and finite difference discretization to obtain the solution for the time‐fractional diffusion equation. Using the theory of reproducing kernel, reproducing kernel functions with a polynomial form will be established in polynomial reproducing kernel spaces spanned by the Chebychev basis polynomials. In the numerical method, first the time‐fractional derivative term in the aforementioned equation is approximated by using the finite difference scheme. Then, by the help of collocation method based on reproducing kernel approximation, we will illustrate how to derive the numerical solution in polynomial reproducing kernel space. Finally, to support the accuracy and efficiency of the numerical method, we provide several numerical examples. In numerical experiments, the quality of approximation is calculated by absolute error and discrete error norms.

On the mechanics of droplet surface crater during impact on immiscible viscous liquid pool

We study drop impacts on an immiscible viscous liquid pool and investigate the formation of droplet surface craters using experimental and theoretical analyses. We attribute the formation of air craters to the rapid deceleration of the droplet due to viscous drag force. The droplet response to the external impulsive decelerating force induces oscillatory modes on the surface exposed to the air forming capillary waves that superimpose to form air craters of various shapes and sizes. We introduce a non-dimensional parameter (${\varGamma }$), that is, the ratio of the drag force to the capillary force acting on the droplet. We show that${\varGamma }$is directly proportional to the capillary number. We show that droplets forming air craters of significant depths have${\varGamma }>1$. Further, we demonstrate that Legendre polynomials can locally approximate the central air crater jet profile. We also decipher that the air crater response time scale ($T$) varies as the square root of impact Weber number ($T\sim We^{1/2}$). Further, we generalize the local droplet response with a global response model for low impact energies based on an eigenvalue problem. We represent the penetrating drop as a constrained Rayleigh drop problem with a dynamic contact line. The air–water interface dynamics is analysed using an inviscid droplet deformation model for small deformation amplitudes. The local and global droplet response theories conform with each other and depict that the deformation profiles could be represented as a linear superposition of eigenmodes in Legendre polynomial basis. We unearth that the droplet response in an immiscible impact system differs from the miscible impact systems due to the presence of such a dynamic contact line.