The Experts below are selected from a list of 31191 Experts worldwide ranked by ideXlab platform
Alfio Borzì - One of the best experts on this subject based on the ideXlab platform.
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a Multigrid scheme for solving convection diffusion integral optimal control problems
Computing and Visualization in Science, 2019Co-Authors: Duncan Kioi Gathungu, Alfio BorzìAbstract:The fast Multigrid solution of an optimal control problem governed by a convection–diffusion partial-integro differential equation is investigated. This optimization problem considers a cost functional of tracking type and a constrained distributed control. The optimal control sought is characterized by the solution to the corresponding optimality system, which is approximated by a finite volume and quadrature discretization schemes and solved by Multigrid techniques. The proposed Multigrid approach combines a Multigrid method for the governing model with a fast Multigrid integration method. The convergence of this solution procedure is analyzed by local Fourier analysis and validated by results of numerical experiments.
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A Multigrid scheme for solving convection–diffusion-integral optimal control problems
Computing and Visualization in Science, 2019Co-Authors: Duncan Kioi Gathungu, Alfio BorzìAbstract:The fast Multigrid solution of an optimal control problem governed by a convection–diffusion partial-integro differential equation is investigated. This optimization problem considers a cost functional of tracking type and a constrained distributed control. The optimal control sought is characterized by the solution to the corresponding optimality system, which is approximated by a finite volume and quadrature discretization schemes and solved by Multigrid techniques. The proposed Multigrid approach combines a Multigrid method for the governing model with a fast Multigrid integration method. The convergence of this solution procedure is analyzed by local Fourier analysis and validated by results of numerical experiments.
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A FEM-Multigrid Scheme for Elliptic Nash-Equilibrium Multiobjective Optimal Control Problems
Numerical Mathematics-theory Methods and Applications, 2015Co-Authors: Mohammad Tanvir Rahman, Alfio BorzìAbstract:AbstractA finite-element Multigrid scheme for elliptic Nash-equilibrium multiobjective optimal control problems with control constraints is investigated. The Multigrid computational framework implements a nonlinear Multigrid strategy with collective smoothing for solving the multiobjective optimality system discretized with finite elements. Error estimates for the optimal solution and two-grid local Fourier analysis of the Multigrid scheme are presented. Results of numerical experiments are presented to demonstrate the effectiveness of the proposed framework.
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Multigrid Methods for Control-Constrained Elliptic Optimal Control Problems
Numerical Mathematics and Advanced Applications 2009, 2010Co-Authors: Michelle Vallejos, Alfio BorzìAbstract:Multigrid schemes that solve control-constrained elliptic optimal control problems discretized by finite differences are presented. A gradient projection method is used to treat the constraints on the control variable. A comparison is made between two Multigrid methods, the Multigrid for optimization (MGOPT) method and the collective smoothing Multigrid (CSMG) method. To illustrate both techniques, we focus on minimization problems governed by elliptic differential equations with constraints on the control variable.
Jun Zhang - One of the best experts on this subject based on the ideXlab platform.
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sixth order compact scheme combined with Multigrid method and extrapolation technique for 2d poisson equation
Journal of Computational Physics, 2009Co-Authors: Yin Wang, Jun ZhangAbstract:We develop a sixth order finite difference discretization strategy to solve the two dimensional Poisson equation, which is based on the fourth order compact discretization, Multigrid method, Richardson extrapolation technique, and an operator based interpolation scheme. We use Multigrid V-Cycle procedure to build our multiscale Multigrid algorithm, which is similar to the full Multigrid method (FMG). The Multigrid computation yields fourth order accurate solution on both the fine grid and the coarse grid. A sixth order accurate coarse grid solution is computed by using the Richardson extrapolation technique. Then we apply our operator based interpolation scheme to compute sixth order accurate solution on the fine grid. Numerical experiments are conducted to show the solution accuracy and the computational efficiency of our new method, compared to Sun-Zhang's sixth order Richardson extrapolation compact (REC) discretization strategy using Alternating Direction Implicit (ADI) method and the standard fourth order compact difference (FOC) scheme using a Multigrid method.
Michael Maire - One of the best experts on this subject based on the ideXlab platform.
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CVPR - Multigrid Neural Architectures
2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2017Co-Authors: Michael MaireAbstract:We propose a Multigrid extension of convolutional neural networks (CNNs). Rather than manipulating representations living on a single spatial grid, our network layers operate across scale space, on a pyramid of grids. They consume Multigrid inputs and produce Multigrid outputs, convolutional filters themselves have both within-scale and cross-scale extent. This aspect is distinct from simple multiscale designs, which only process the input at different scales. Viewed in terms of information flow, a Multigrid network passes messages across a spatial pyramid. As a consequence, receptive field size grows exponentially with depth, facilitating rapid integration of context. Most critically, Multigrid structure enables networks to learn internal attention and dynamic routing mechanisms, and use them to accomplish tasks on which modern CNNs fail. Experiments demonstrate wide-ranging performance advantages of Multigrid. On CIFAR and ImageNet classification tasks, flipping from a single grid to Multigrid within the standard CNN paradigm improves accuracy, while being compute and parameter efficient. Multigrid is independent of other architectural choices, we show synergy in combination with residual connections. Multigrid yields dramatic improvement on a synthetic semantic segmentation dataset. Most strikingly, relatively shallow Multigrid networks can learn to directly perform spatial transformation tasks, where, in contrast, current CNNs fail. Together, our results suggest that continuous evolution of features on a Multigrid pyramid is a more powerful alternative to existing CNN designs on a flat grid.
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Multigrid Neural Architectures
arXiv: Computer Vision and Pattern Recognition, 2016Co-Authors: Michael MaireAbstract:We propose a Multigrid extension of convolutional neural networks (CNNs). Rather than manipulating representations living on a single spatial grid, our network layers operate across scale space, on a pyramid of grids. They consume Multigrid inputs and produce Multigrid outputs; convolutional filters themselves have both within-scale and cross-scale extent. This aspect is distinct from simple multiscale designs, which only process the input at different scales. Viewed in terms of information flow, a Multigrid network passes messages across a spatial pyramid. As a consequence, receptive field size grows exponentially with depth, facilitating rapid integration of context. Most critically, Multigrid structure enables networks to learn internal attention and dynamic routing mechanisms, and use them to accomplish tasks on which modern CNNs fail. Experiments demonstrate wide-ranging performance advantages of Multigrid. On CIFAR and ImageNet classification tasks, flipping from a single grid to Multigrid within the standard CNN paradigm improves accuracy, while being compute and parameter efficient. Multigrid is independent of other architectural choices; we show synergy in combination with residual connections. Multigrid yields dramatic improvement on a synthetic semantic segmentation dataset. Most strikingly, relatively shallow Multigrid networks can learn to directly perform spatial transformation tasks, where, in contrast, current CNNs fail. Together, our results suggest that continuous evolution of features on a Multigrid pyramid is a more powerful alternative to existing CNN designs on a flat grid.
Yongbin Ge - One of the best experts on this subject based on the ideXlab platform.
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Multigrid method and fourth order compact difference discretization scheme with unequal meshsizes for 3d poisson equation
Journal of Computational Physics, 2010Co-Authors: Yongbin GeAbstract:A fourth-order compact difference discretization scheme with unequal meshsizes in different coordinate directions is employed to solve a three-dimensional (3D) Poisson equation on a cubic domain. Two multgrid methods are developed to solve the resulting sparse linear systems. One is to use the full-coarsening Multigrid method with plane Gauss-Seidel relaxation, which uses line Gauss-Seidel relaxation to compute each planewise solution. The other is to construct a partial semi-coarsening Multigrid method with the traditional point or plane Gauss-Seidel relaxations. Numerical experiments are conducted to test the computed accuracy of the fourth-order compact difference scheme and the computational efficiency of the Multigrid methods with the fourth-order compact difference scheme.
Yin Wang - One of the best experts on this subject based on the ideXlab platform.
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sixth order compact scheme combined with Multigrid method and extrapolation technique for 2d poisson equation
Journal of Computational Physics, 2009Co-Authors: Yin Wang, Jun ZhangAbstract:We develop a sixth order finite difference discretization strategy to solve the two dimensional Poisson equation, which is based on the fourth order compact discretization, Multigrid method, Richardson extrapolation technique, and an operator based interpolation scheme. We use Multigrid V-Cycle procedure to build our multiscale Multigrid algorithm, which is similar to the full Multigrid method (FMG). The Multigrid computation yields fourth order accurate solution on both the fine grid and the coarse grid. A sixth order accurate coarse grid solution is computed by using the Richardson extrapolation technique. Then we apply our operator based interpolation scheme to compute sixth order accurate solution on the fine grid. Numerical experiments are conducted to show the solution accuracy and the computational efficiency of our new method, compared to Sun-Zhang's sixth order Richardson extrapolation compact (REC) discretization strategy using Alternating Direction Implicit (ADI) method and the standard fourth order compact difference (FOC) scheme using a Multigrid method.
