scholarly journals A High Accurate and Stable Legendre Transform Based on Block Partitioning and Butterfly Algorithm for NWP

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 966
Author(s):  
Fukang Yin ◽  
Jianping Wu ◽  
Junqiang Song ◽  
Jinhui Yang
Keyword(s):  
Computational Complexity ◽  
Error Analysis ◽  
Numerical Stability ◽  
High Order ◽  
Numerical Results ◽  
Computational Time ◽  
Legendre Transform ◽  
Practical Application ◽  
Block Partitioning ◽  
Very High

In this paper, we proposed a high accurate and stable Legendre transform algorithm, which can reduce the potential instability for a very high order at a very small increase in the computational time. The error analysis of interpolative decomposition for Legendre transform is presented. By employing block partitioning of the Legendre-Vandermonde matrix and butterfly algorithm, a new Legendre transform algorithm with computational complexity O(Nlog2N /loglogN) in theory and O(Nlog3N) in practical application is obtained. Numerical results are provided to demonstrate the efficiency and numerical stability of the new algorithm.

scholarly journals High-Order Analytical Formulation of Soliton Self-Frequency Shift

2020 ◽  
Author(s):  
Martin Rochette
Keyword(s):  
Computational Complexity ◽  
Frequency Shift ◽  
Wavelength Conversion ◽  
Computation Time ◽  
Supercontinuum Generation ◽  
High Order ◽  
Numerical Results ◽  
Analytical Formulation ◽  
Dispersion Slope ◽  
Propagation Losses

<div>We derive an analytical formulation of the Raman-induced frequency shift experienced by a fundamental soliton. By including propagation losses, self-steepening, and dispersion slope, the resulting formulation is a high-order (HO) extension of the well-known Gordon's formula for soliton self-frequency shift (SSFS). The HO-SSFS formula closely agrees with numerical results of the generalized nonlinear Schrödinger equation, but without the computational complexity and required computation time. The HO-SSFS formula is a useful tool for the design and validation of wavelength conversion systems and supercontinuum generation systems.</div>

scholarly journals A New Numerical Algorithm for Two-Point Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Lihua Guo ◽  
Boying Wu ◽  
Dazhi Zhang
Keyword(s):  
Exact Solution ◽  
Error Analysis ◽  
Numerical Solution ◽  
Boundary Value Problems ◽  
Numerical Algorithm ◽  
Numerical Stability ◽  
Boundary Value ◽  
Numerical Results ◽  
Point Boundary ◽  
Stability And Error Analysis

We present a new numerical algorithm for two-point boundary value problems. We first present the exact solution in the form of series and then prove that then-term numerical solution converges uniformly to the exact solution. Furthermore, we establish the numerical stability and error analysis. The numerical results show the effectiveness of the proposed algorithm.

Bistatic RCS of Electrically-Large Objects Calculations with High-Order Parabolic Equation Method

2013 ◽  
Vol 765-767 ◽  
pp. 557-561 ◽  
Author(s):  
Meng Kong ◽  
Ming Sheng Chen ◽  
Jin Hua Hu ◽  
Xian Liang Wu ◽  
Yuan Cheng
Keyword(s):  
Parabolic Equation ◽  
High Order ◽  
Numerical Results ◽  
Equation Method ◽  
Computational Time ◽  
Two Dimensional ◽  
Parabolic Equation Method ◽  
Order Parabolic Equation ◽  
Low Order ◽  
Crank Nicolson

A high-order parabolic equation (PE) method is applied to calculate the Bistatic RCS of two-dimensional (2-D) Electrically-large Objects. According to the shapes of objexts, Crank-Nicolson and Pade (1,0) schemes are introduced to solve the high-order PE. The numerical results demonstrate that the method not only extends the accurate angle range but also decreases the rotation times and computational time in comparison with the traditional low-order parabolic equation method.

scholarly journals High-Order Analytical Formulation of Soliton Self-Frequency Shift

2020 ◽  
Author(s):  
Martin Rochette
Keyword(s):  
Computational Complexity ◽  
Frequency Shift ◽  
Wavelength Conversion ◽  
Computation Time ◽  
Supercontinuum Generation ◽  
High Order ◽  
Numerical Results ◽  
Analytical Formulation ◽  
Dispersion Slope ◽  
Propagation Losses

<div>We derive an analytical formulation of the Raman-induced frequency shift experienced by a fundamental soliton. By including propagation losses, self-steepening, and dispersion slope, the resulting formulation is a high-order (HO) extension of the well-known Gordon's formula for soliton self-frequency shift (SSFS). The HO-SSFS formula closely agrees with numerical results of the generalized nonlinear Schrödinger equation, but without the computational complexity and required computation time. The HO-SSFS formula is a useful tool for the design and validation of wavelength conversion systems and supercontinuum generation systems.</div>

scholarly journals Computational Complexity Reduction of Neural Networks of Brain Tumor Image Segmentation by Introducing Fermi–Dirac Correction Functions

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 223
Author(s):  
Yen-Ling Tai ◽  
Shin-Jhe Huang ◽  
Chien-Chang Chen ◽  
Henry Horng-Shing Lu
Keyword(s):  
Neural Networks ◽  
Deep Learning ◽  
Computational Complexity ◽  
High Performance ◽  
Low Cost ◽  
Structural Complexity ◽  
Correction Function ◽  
Computational Time ◽  
Learning Methods ◽  
Band Theory

Nowadays, deep learning methods with high structural complexity and flexibility inevitably lean on the computational capability of the hardware. A platform with high-performance GPUs and large amounts of memory could support neural networks having large numbers of layers and kernels. However, naively pursuing high-cost hardware would probably drag the technical development of deep learning methods. In the article, we thus establish a new preprocessing method to reduce the computational complexity of the neural networks. Inspired by the band theory of solids in physics, we map the image space into a noninteraction physical system isomorphically and then treat image voxels as particle-like clusters. Then, we reconstruct the Fermi–Dirac distribution to be a correction function for the normalization of the voxel intensity and as a filter of insignificant cluster components. The filtered clusters at the circumstance can delineate the morphological heterogeneity of the image voxels. We used the BraTS 2019 datasets and the dimensional fusion U-net for the algorithmic validation, and the proposed Fermi–Dirac correction function exhibited comparable performance to other employed preprocessing methods. By comparing to the conventional z-score normalization function and the Gamma correction function, the proposed algorithm can save at least 38% of computational time cost under a low-cost hardware architecture. Even though the correction function of global histogram equalization has the lowest computational time among the employed correction functions, the proposed Fermi–Dirac correction function exhibits better capabilities of image augmentation and segmentation.

VERY HIGH-ORDER SPATIALLY IMPLICIT SCHEMES FOR COMPUTATIONAL ACOUSTICS ON CURVILINEAR MESHES

2001 ◽  
Vol 09 (04) ◽  
pp. 1259-1286 ◽  
Author(s):  
MIGUEL R. VISBAL ◽  
DATTA V. GAITONDE
Keyword(s):  
Three Dimensional ◽  
Radiation Condition ◽  
Higher Order ◽  
High Order ◽  
High Frequencies ◽  
Decomposition Strategies ◽  
The Impact ◽  
Spurious Modes ◽  
Very High ◽  
Computational Strategy

A high-order compact-differencing and filtering algorithm, coupled with the classical fourth-order Runge–Kutta scheme, is developed and implemented to simulate aeroacoustic phenomena on curvilinear geometries. Several issues pertinent to the use of such schemes are addressed. The impact of mesh stretching in the generation of high-frequency spurious modes is examined and the need for a discriminating higher-order filter procedure is established and resolved. The incorporation of these filtering techniques also permits a robust treatment of outflow radiation condition by taking advantage of energy transfer to high-frequencies caused by rapid mesh stretching. For conditions on the scatterer, higher-order one-sided filter treatments are shown to be superior in terms of accuracy and stability compared to standard explicit variations. Computations demonstrate that these algorithmic components are also crucial to the success of interface treatments created in multi-domain and domain-decomposition strategies. For three-dimensional computations, special metric relations are employed to assure the fidelity of the scheme in highly curvilinear meshes. A variety of problems, including several benchmark computations, demonstrate the success of the overall computational strategy.

An accurate double-superposition global–local theory for vibration and bending analyses of cylindrical composite and sandwich shells subjected to thermo-mechanical loads

2011 ◽  
Vol 225 (8) ◽  
pp. 1816-1832 ◽  
Author(s):  
M Shariyat
Keyword(s):  
Vibration Analysis ◽  
Cylindrical Shells ◽  
High Order ◽  
Computational Time ◽  
Local Theory ◽  
Transverse Stress ◽  
Mechanical Loads ◽  
Local Technique ◽  
Sandwich Shells ◽  
First Time

Based on the idea of double superposition, an accurate high-order global–local theoryis proposed for bending and vibration analysis of cylindrical shells subjected to thermo-mechanical loads, for the first time. The theory has many novelties, among them: (1) less computational time due to the use of the global–local technique and matrix formulations; (2) satisfaction of the complete kinematic and transverse stress continuity conditions at the layer interfaces under thermo-mechanical loads; (3) consideration of the transverse flexibility; (4) release of Love–Timoshenko assumption; and (5) capability of investigating the local phenomena. Various comparative examples are included to validate the theory and to examine its accuracy and efficiency.

scholarly journals Comparison of numerical schemes of river flood routing with an inertial approximation of the Saint Venant equations

RBRH ◽  
2018 ◽  
Vol 23 (0) ◽  
Author(s):  
Alice César Fassoni-Andrade ◽  
Fernando Mainardi Fan ◽  
Walter Collischonn ◽  
Artur César Fassoni ◽  
Rodrigo Cauduro Dias de Paiva
Keyword(s):  
Numerical Stability ◽  
Flood Routing ◽  
Computational Time ◽  
Numerical Schemes ◽  
Time Step ◽  
Simple Formulation ◽  
One Dimensional ◽  
Saint Venant Equations ◽  
Original Scheme ◽  
The One

ABSTRACT The one-dimensional flow routing inertial model, formulated as an explicit solution, has advantages over other explicit models used in hydrological models that simplify the Saint-Venant equations. The main advantage is a simple formulation with good results. However, the inertial model is restricted to a small time step to avoid numerical instability. This paper proposes six numerical schemes that modify the one-dimensional inertial model in order to increase the numerical stability of the solution. The proposed numerical schemes were compared to the original scheme in four situations of river’s slope (normal, low, high and very high) and in two situations where the river is subject to downstream effects (dam backwater and tides). The results are discussed in terms of stability, peak flow, processing time, volume conservation error and RMSE (Root Mean Square Error). In general, the schemes showed improvement relative to each type of application. In particular, the numerical scheme here called Prog Q(k+1)xQ(k+1) stood out presenting advantages with greater numerical stability in relation to the original scheme. However, this scheme was not successful in the tide simulation situation. In addition, it was observed that the inclusion of the hydraulic radius calculation without simplification in the numerical schemes improved the results without increasing the computational time.

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