scholarly journals ISOGEOMETRIC COLLOCATION METHODS

2010 ◽  
Vol 20 (11) ◽  
pp. 2075-2107 ◽  
Author(s):  
F. AURICCHIO ◽  
L. BEIRÃO DA VEIGA ◽  
T. J. R. HUGHES ◽  
A. REALI ◽  
G. SANGALLI
Keyword(s):  
Theoretical Analysis ◽  
Isogeometric Analysis ◽  
Computational Cost ◽  
Basis Functions ◽  
Three Dimensions ◽  
Collocation Methods ◽  
One Dimensional ◽  
Numerical Tests ◽  
Low Computational Cost ◽  
Theoretical Results

We initiate the study of collocation methods for NURBS-based isogeometric analysis. The idea is to connect the superior accuracy and smoothness of NURBS basis functions with the low computational cost of collocation. We develop a one-dimensional theoretical analysis, and perform numerical tests in one, two and three dimensions. The numerical results obtained confirm theoretical results and illustrate the potential of the methodology.

scholarly journals Stable Filtering Procedures for Nodal Discontinuous Galerkin Methods

2021 ◽  
Vol 87 (1) ◽  
Author(s):  
Jan Nordström ◽  
Andrew R. Winters
Keyword(s):  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Finite Difference Methods ◽  
Basis Functions ◽  
Galerkin Methods ◽  
Theoretical Discussion ◽  
Numerical Tests ◽  
Dg Methods ◽  
Theoretical Results ◽  
Filtering Procedure

AbstractWe prove that the most common filtering procedure for nodal discontinuous Galerkin (DG) methods is stable. The proof exploits that the DG approximation is constructed from polynomial basis functions and that integrals are approximated with high-order accurate Legendre–Gauss–Lobatto quadrature. The theoretical discussion re-contextualizes stable filtering results for finite difference methods into the DG setting. Numerical tests verify and validate the underlying theoretical results.

On the Use of Symmetries in Building Surrogate Models

2019 ◽  
Vol 141 (6) ◽  
Author(s):  
M. Giselle Fernández-Godino ◽  
S. Balachandar ◽  
Raphael T. Haftka
Keyword(s):  
Linear Regression ◽  
Uncertainty Propagation ◽  
Computational Cost ◽  
Surrogate Models ◽  
Black Box ◽  
Basis Functions ◽  
Regression Kriging ◽  
Challenges And Opportunities ◽  
Designs Of Experiments ◽  
Low Computational Cost

When simulations are expensive and multiple realizations are necessary, as is the case in uncertainty propagation, statistical inference, and optimization, surrogate models can achieve accurate predictions at low computational cost. In this paper, we explore options for improving the accuracy of a surrogate if the modeled phenomenon presents symmetries. These symmetries allow us to obtain free information and, therefore, the possibility of more accurate predictions. We present an analytical example along with a physical example that has parametric symmetries. Although imposing parametric symmetries in surrogate models seems to be a trivial matter, there is not a single way to do it and, furthermore, the achieved accuracy might vary. We present four different ways of using symmetry in surrogate models. Three of them are straightforward, but the fourth is original and based on an optimization of the subset of points used. The performance of the options was compared with 100 random designs of experiments (DoEs) where symmetries were not imposed. We found that each of the options to include symmetries performed the best in one or more of the studied cases and, in all cases, the errors obtained imposing symmetries were substantially smaller than the worst cases among the 100. We explore the options for using symmetries in two surrogates that present different challenges and opportunities: Kriging and linear regression. Kriging is often used as a black box; therefore, we consider approaches to include the symmetries without changes in the main code. On the other hand, since linear regression is often built by the user; owing to its simplicity, we consider also approaches that modify the linear regression basis functions to impose the symmetries.

scholarly journals Characterization of Hydraulic Power in Free-Stream Installations

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Olivier Cleynen ◽  
Stefan Hoerner ◽  
Dominique Thévenin
Keyword(s):  
Open Channel ◽  
Flow Model ◽  
Computational Cost ◽  
Characteristic Parameter ◽  
Operating Speed ◽  
Improve Performance ◽  
Hydraulic Power ◽  
One Dimensional ◽  
Low Computational Cost

The performance of open-channel hydropower devices can be optimized by maximizing the product of their load, hydraulic, and generator efficiencies. The maximum hydraulic power theoretically available must be defined according to the operational scenario retained for the device of interest. In the case of a device operating within a wide, unobstructed channel, the existence of a maximum hydraulic power and the operating speed required to reach it are first predicted using a one-dimensional flow model. This model is then extended to account for the effect of device ducting. As a result, given the available surface level drop and a single duct characteristic parameter, the model predicts the optimum device operating speed, whether the duct can improve performance, and the relative duct size which maximizes the installation’s power density, all at a very low computational cost.

scholarly journals Numerical Solution of Linear Volterra Integral Equation Systems of Second Kind by Radial Basis Functions

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 223
Author(s):  
Pedro González-Rodelas ◽  
Miguel Pasadas ◽  
Abdelouahed Kouibia ◽  
Basim Mustafa
Keyword(s):  
Integral Equation ◽  
Radial Basis Functions ◽  
Volterra Integral Equation ◽  
Computational Cost ◽  
Basis Functions ◽  
Numerical Examples ◽  
Acceptable Accuracy ◽  
Convergence Results ◽  
Radial Basis ◽  
Low Computational Cost

In this paper we propose an approximation method for solving second kind Volterra integral equation systems by radial basis functions. It is based on the minimization of a suitable functional in a discrete space generated by compactly supported radial basis functions of Wendland type. We prove two convergence results, and we highlight this because most recent published papers in the literature do not include any. We present some numerical examples in order to show and justify the validity of the proposed method. Our proposed technique gives an acceptable accuracy with small use of the data, resulting also in a low computational cost.

scholarly journals Optimal 1-NN prototypes for pathological geometries

2021 ◽  
Vol 7 ◽  
pp. e464
Author(s):  
Ilia Sucholutsky ◽  
Matthias Schonlau
Keyword(s):  
Theoretical Analysis ◽  
Heuristic Algorithms ◽  
Computational Cost ◽  
Optimal Number ◽  
Training Data ◽  
Nearest Neighbour ◽  
Original Performance ◽  
Instance Based Learning ◽  
Nearest Neighbour Classifier ◽  
Theoretical Results

Using prototype methods to reduce the size of training datasets can drastically reduce the computational cost of classification with instance-based learning algorithms like the k-Nearest Neighbour classifier. The number and distribution of prototypes required for the classifier to match its original performance is intimately related to the geometry of the training data. As a result, it is often difficult to find the optimal prototypes for a given dataset, and heuristic algorithms are used instead. However, we consider a particularly challenging setting where commonly used heuristic algorithms fail to find suitable prototypes and show that the optimal number of prototypes can instead be found analytically. We also propose an algorithm for finding nearly-optimal prototypes in this setting, and use it to empirically validate the theoretical results. Finally, we show that a parametric prototype generation method that normally cannot solve this pathological setting can actually find optimal prototypes when combined with the results of our theoretical analysis.

scholarly journals Space-time spectral collocation algorithm for solving time-fractional Tricomi-type equations

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 269-280 ◽  
Author(s):  
M.A. Abdelkawy ◽  
Engy A. Ahmed ◽  
Rubayyi T. Alqahtani
Keyword(s):  
Numerical Algorithm ◽  
Jacobi Polynomials ◽  
High Accuracy ◽  
Basis Functions ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
One Dimensional ◽  
Numerical Tests ◽  
Shifted Jacobi Polynomials ◽  
Derivatives Of

AbstractWe introduce a new numerical algorithm for solving one-dimensional time-fractional Tricomi-type equations (T-FTTEs). We used the shifted Jacobi polynomials as basis functions and the derivatives of fractional is evaluated by the Caputo definition. The shifted Jacobi Gauss-Lobatt algorithm is used for the spatial discretization, while the shifted Jacobi Gauss-Radau algorithmis applied for temporal approximation. Substituting these approximations in the problem leads to a system of algebraic equations that greatly simplifies the problem. The proposed algorithm is successfully extended to solve the two-dimensional T-FTTEs. Extensive numerical tests illustrate the capability and high accuracy of the proposed methodologies.

scholarly journals DFT-based investigation of the electronic structure of a double-stranded AC B-DNA dim

2013 ◽  
Vol 16 ◽  
pp. 117-122
Author(s):  
Emerson Rengifo ◽  
Gustavo Murillo
Keyword(s):  
Electronic Structure ◽  
Density Functional ◽  
Computational Cost ◽  
Semiempirical Method ◽  
Basis Sets ◽  
Functional Theory ◽  
One Dimensional ◽  
Exchange Correlation ◽  
Dna Dodecamer ◽  
Low Computational Cost

Calculations of the electronic structure of a stacked dimmer sequence from the D(GCAAACGTTTGC)2 B-DNA dodecamer resolved in a PDB file 1HQ7 are performed within density functional theory. Seeking to understand the minimum level of theory that yields a reliable description for these systems, the basis sets 6-31g*, 6-31g*+BSSE, 6-311g*, 6-311g**, 6-311++g** along with the B3LYP and PBE0 exchange-correlation functionals were employed. These results are then used to implement a one dimensional model of long stacked systems to obtain a new semiempirical method that can be employed at low computational cost.

scholarly journals Family of simultaneous methods with corrections for approximating zeros of analytic functions

Filomat ◽  
2015 ◽  
Vol 29 (10) ◽  
pp. 2217-2225
Author(s):  
Lidija Rancic
Keyword(s):  
Analytic Functions ◽  
Computational Cost ◽  
Convergence Properties ◽  
Simultaneous Methods ◽  
Good Convergence ◽  
Basic Family ◽  
Complex Zeros ◽  
Low Computational Cost ◽  
Theoretical Results ◽  
Methods Numerical

A family of accelerated iterative methods for the simultaneous approximation of complex zeros of a class of analytic functions is proposed. Considered analytic functions have only simple zeros inside a simple smooth closed contour in the complex plane. It is shown that the order of convergence of the basic family can be increased from four to five and six using Newton?s and Halley?s corrections, respectively. The improved convergence is achieved on the account of additional calculations of low computational cost, which significantly increases the computational efficiency of the accelerated methods. Numerical examples demonstrate a good convergence properties, fitting very well theoretical results.

Transversely Stable Extended Kalman Filters for Systems on Manifolds in Euclidean Spaces

2021 ◽  
Vol 143 (6) ◽  
Author(s):  
Jae-Hyeon Park ◽  
Karmvir Singh Phogat ◽  
Whimin Kim ◽  
Dong Eui Chang
Keyword(s):  
Kalman Filter ◽  
Euclidean Space ◽  
Extended Kalman Filter ◽  
Kalman Filters ◽  
Computational Cost ◽  
Three Dimensions ◽  
Extended Kalman Filters ◽  
Body Attitude ◽  
Open Set ◽  
Low Computational Cost

Abstract In this article, we devise a variant of the extended Kalman filter that can be generally applied to systems on manifolds with simplicity and low computational cost. We extend a given system on a manifold to an ambient open set in Euclidean space and modify the system such that the extended system is transversely stable on the manifold. Then, we apply the standard extended Kalman filter derived in Euclidean space to the modified dynamics. This method is efficient in terms of computation and accurate in comparison with the standard extended Kalman filter. It has the merit that we can apply various Kalman filters derived in Euclidean space including extended Kalman filters for state estimation for systems defined on manifolds. The proposed method is successfully applied to the rigid body attitude dynamics whose configuration space is the special orthogonal group in three dimensions.

scholarly journals Formulação nodal aplicada à problemas de transporte bidimensional em geometria cartesiana

2015 ◽  
Vol 11 (8) ◽  
Author(s):  
Carlos Eduardo Souza Ferreira ◽  
Leonardo Ramos Emmendorfer ◽  
João Francisco Prolo Filho
Keyword(s):  
Neutron Transport ◽  
Computational Cost ◽  
Coupled System ◽  
Two Dimensional ◽  
Discrete Ordinate ◽  
One Dimensional ◽  
Scalar Fluxes ◽  
Cartesian Geometry ◽  
Transport Problems ◽  
Low Computational Cost

<div><p class="SPabstract">Neste trabalho, uma formulação nodal é proposta para o tratamento de uma classe de problemas de transporte de nêutrons, em geometria cartesiana bidimensional. Através do processo de integração, equações unidimensionais são obtidas, reescrevendo o modelo em termos de quantidades médias. A resolução das equações integradas é feita através de uma versão do método de Ordenadas Discretas Analítico (ADO), onde também são obtidas soluções explicitas, analíticas em termos das variáveis espaciais, através de um código de fácil implementação. Pode-se destacar também como vantagens desta formulação a versatilidade na escolha da quadratura e o baixo custo computacional, uma vez que esquemas iterativos não são necessários tampouco a subdivisão do domínio em células. Para lidar como os termos do contorno que surgem no processo, propõe-se aqui uma representação por constantes, de forma que as equações nas variáveis x e y são tratadas através de um sistema acoplado. Resultados obtidos por esta formulação são apresentados, bem como alguns perfis de fluxos escalares. </p></div><p><strong>Nodal formulation applied to two-dimensional transport problems in Cartesian geometry.</strong></p><p> In this paper, a nodal formulation is proposed for the treatment of a class of neutron transport problems in two-dimensional Cartesian geometry. By the integration process, one-dimensional equations are obtained, rewriting the model in terms of average quantities. The resolution of the integrated equations is made using a version of the Analytical Discrete Ordinate method (ADO), where also be obtained explicit solutions, analytical in terms of spatial variables, through an easy implementation code. It can also highlight as advantages of this formulation the versatility of the quadrature choice and the low computational cost, since iterative schemes are not needed either subdividing the domain in cells. To deal with the contour terms that arise in the process, is proposed here a representation by constants, so that the equations in the variables x and y are treated through a coupled system. Results obtained by this formulation are presented, as well as some profiles of scalar fluxes. </p>

Sign in / Sign up

Export Citation Format

Share Document