The Experts below are selected from a list of 459231 Experts worldwide ranked by ideXlab platform
Hong Wang - One of the best experts on this subject based on the ideXlab platform.
-
an hp galerkin method with fast solution for linear peridynamic models in one dimension
Computers & Mathematics With Applications, 2017Co-Authors: Aijie Cheng, Hong WangAbstract:The Computational Work and memory requirement are bottlenecks for Galerkin finite element methods for peridynamic models because of their non-locality. In this paper, fast Galerkin and hp-Galerkin finite element methods are introduced and analyzed to solve a steady-state peridynamic model. We present a fast solution technique to accelerate non-square Toeplitz matrixvector multiplications arising from piecewise-linear, piecewise-quadratic and piecewise-cubic Galerkin methods. This fast solution technique is based on a fast Fourier transform and depends on the special structure of coefficient matrices, and it helps to reduce the Computational Work from O(N3) required by traditional methods to O(Nlog2N) and the memory requirement from O(N2) to O(N) without using any lossy compression, where N is the number of unknowns. The peridynamic model admits solutions having jump discontinuities. For problems with discontinuous solutions, we therefore introduce a piecewise-constant Galerkin method and give an h- and p-refinement algorithm. Then, we develop fast hp-Galerkin methods based on hybrid piecewise-constant/piecewise-linear and piecewise-constant/piecewise-quadratic finite element approximations. The new method reduces the Computational Work from O(N3) required by the traditional methods to O(Nlog2N) and the memory requirement from O(N2)to O(N).
-
a preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh
Journal of Computational Physics, 2015Co-Authors: Hong WangAbstract:Numerical methods for fractional differential equations generate full stiffness matrices, which were traditionally solved via Gaussian type direct solvers that require O ( N 3 ) of Computational Work and O ( N 2 ) of memory to store where N is the number of spatial grid points in the discretization. We develop a preconditioned fast Krylov subspace iterative method for the efficient and faithful solution of finite volume schemes defined on a locally refined composite mesh for fractional differential equations to resolve boundary layers of the solutions. Numerical results are presented to show the utility of the method.
-
a fast and faithful collocation method with efficient matrix assembly for a two dimensional nonlocal diffusion model
Computer Methods in Applied Mechanics and Engineering, 2014Co-Authors: Hong Wang, Hao TianAbstract:Abstract The peridynamic theory provides an appropriate description of the deformation of a continuous body involving discontinuities or other singularities, which cannot be described properly by the classical theory of solid mechanics. However, the operator in the peridynamic theory is nonlocal, so the resulting numerical methods generate dense or full coefficient matrices which require O ( N 2 ) of memory where N is the number of unknowns in the discretized system. Gaussian types of direct solvers, which were traditionally used to solve these problems, require O ( N 3 ) of operations. Furthermore, due to the singularity of the kernel in the peridynamic model, the evaluation and assembly of the coefficient matrix can be very expensive. Numerous numerical experiments have shown that in many practical simulations the evaluation and assembly of the coefficient matrix often constitute the main Computational cost! The significantly increased Computational Work and memory requirement of the peridynamic model over those for the classical partial differential equation models severely limit their applications, especially in multiple space dimensions. We develop a fast and faithful collocation method for a two-dimensional nonlocal diffusion model, which can be viewed as a scalar-valued version of a peridynamic model, without using any lossy compression, but rather, by exploiting the structure of the coefficient matrix. The new method reduces the evaluation and assembly of the coefficient matrix by O ( N ) , reduces the Computational Work from O ( N 3 ) required by the traditional methods to O ( N log 2 N ) and the memory requirement from O ( N 2 ) to O ( N ) . Numerical results are presented to show the utility of the fast method.
-
fast alternating direction finite difference methods for three dimensional space fractional diffusion equations
Journal of Computational Physics, 2014Co-Authors: Hong Wang, Ning DuAbstract:Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second-order diffusion equations. Because of the nonlocal property of fractional differential operators, numerical methods for space-fractional diffusion equations generate dense or even full coefficient matrices with complicated structures. Traditionally, these methods were solved via Gaussian elimination, which requires Computational Work of O ( N 3 ) per time step and O ( N 2 ) of memory to store where N is the number of spatial grid points in the discretization. The significant Computational Work and memory requirement of these methods makes a numerical simulation of three-dimensional space-fractional diffusion equations Computationally prohibitively expensive. We present an alternating-direction implicit (ADI) finite difference formulation for space-fractional diffusion equations in three space dimensions and prove its unconditional stability and convergence rate provided that the fractional partial difference operators along x-,?y-,?z-directions commute. We base on the ADI formulation to develop a fast iterative ADI finite difference method, which has a Computational Work count of O ( N log N ) per iteration at each time step and a memory requirement of O ( N ) . We also develop a fast multistep ADI finite difference method, which has a Computational Work count of O ( N log 2 N ) per time step and a memory requirement of O ( N log N ) . Numerical experiments of a three-dimensional space-fractional diffusion equation show that these both fast methods retain the same accuracy as the regular three-dimensional implicit finite difference method, but have significantly improved Computational cost and memory requirement. These numerical experiments show the utility of the fast method.
-
a fast finite difference method for two dimensional space fractional diffusion equations
SIAM Journal on Scientific Computing, 2012Co-Authors: Hong Wang, Treena BasuAbstract:Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second-order diffusion equations. Because of the nonlocal property of fractional differential operators, numerical methods for space-fractional diffusion equations generate complicated dense or full coefficient matrices. Consequently, these numerical methods were traditionally solved by Gaussian elimination, which requires Computational Work of $O(N^3)$ per time step and $O(N^2)$ of memory, where $N$ is the number of spatial grid points in the discretization. The significant Computational Work and memory requirement of the numerical methods impose a serious challenge for the numerical simulation of two- and especially three-dimensional space-fractional diffusion equations. We develop a fast and yet accurate solution method for the implicit finite difference discretization of space-fractional diffusion equations in two space dimensions by carefully analyzing the structure of the coefficient ma...
Ning Du - One of the best experts on this subject based on the ideXlab platform.
-
fast alternating direction finite difference methods for three dimensional space fractional diffusion equations
Journal of Computational Physics, 2014Co-Authors: Hong Wang, Ning DuAbstract:Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second-order diffusion equations. Because of the nonlocal property of fractional differential operators, numerical methods for space-fractional diffusion equations generate dense or even full coefficient matrices with complicated structures. Traditionally, these methods were solved via Gaussian elimination, which requires Computational Work of O ( N 3 ) per time step and O ( N 2 ) of memory to store where N is the number of spatial grid points in the discretization. The significant Computational Work and memory requirement of these methods makes a numerical simulation of three-dimensional space-fractional diffusion equations Computationally prohibitively expensive. We present an alternating-direction implicit (ADI) finite difference formulation for space-fractional diffusion equations in three space dimensions and prove its unconditional stability and convergence rate provided that the fractional partial difference operators along x-,?y-,?z-directions commute. We base on the ADI formulation to develop a fast iterative ADI finite difference method, which has a Computational Work count of O ( N log N ) per iteration at each time step and a memory requirement of O ( N ) . We also develop a fast multistep ADI finite difference method, which has a Computational Work count of O ( N log 2 N ) per time step and a memory requirement of O ( N log N ) . Numerical experiments of a three-dimensional space-fractional diffusion equation show that these both fast methods retain the same accuracy as the regular three-dimensional implicit finite difference method, but have significantly improved Computational cost and memory requirement. These numerical experiments show the utility of the fast method.
Reza Abazari - One of the best experts on this subject based on the ideXlab platform.
-
exact solutions for non linear schrodinger equations by differential transformation method
Journal of Applied Mathematics and Computing, 2011Co-Authors: A Borhanifar, Reza AbazariAbstract:In this paper, we implemented relatively new, exact series method of solution known as the differential transform method for solving linear and non-linear Schrodinger equations with initial profile. The method can easily be applied to many linear and non-linear problems and is capable of reducing the size of Computational Work. Exact solutions can also be achieved by the known forms of the series solutions. The results obtained are in good agreement with the exact solution. These results show that the technique introduced here is accurate and easy to apply.
Kaixin Wang - One of the best experts on this subject based on the ideXlab platform.
-
an o n log2n alternating direction finite difference method for two dimensional fractional diffusion equations
Journal of Computational Physics, 2011Co-Authors: Hong Wang, Kaixin WangAbstract:Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the second-order diffusion equations. Because of the nonlocal property of fractional differential operators, the numerical methods for fractional diffusion equations often generate dense or even full coefficient matrices. Consequently, the numerical solution of these methods often require Computational Work of O(N^3) per time step and memory of O(N^2) for where N is the number of grid points. In this paper we develop a fast alternating-direction implicit finite difference method for space-fractional diffusion equations in two space dimensions. The method only requires Computational Work of O(N log^2N) per time step and memory of O(N), while retaining the same accuracy and approximation property as the regular finite difference method with Gaussian elimination. Our preliminary numerical example runs for two dimensional model problem of intermediate size seem to indicate the observations: To achieve the same accuracy, the new method has a significant reduction of the CPU time from more than 2 months and 1 week consumed by a traditional finite difference method to 1.5h, using less than one thousandth of memory the standard method does. This demonstrates the utility of the method.
Alfredo Remon - One of the best experts on this subject based on the ideXlab platform.
-
using hybrid cpu gpu platforms to accelerate the computation of the matrix sign function
International Conference on Parallel Processing, 2009Co-Authors: Peter Benner, Pablo Ezzatti, Enrique S Quintanaorti, Alfredo RemonAbstract:We investigate the numerical computation of the matrix sign function of large-scale dense matrices. This is a common task in various application areas. The main Computational Work in Newton's iteration for the matrix sign function consits of matrix inversion. Therefore, we investigate the performance of two approaches for matrix inversion based on Gaussian (LU factorization) and Gauss-Jordan eliminations. The target architecture is a current general-purpose multi-core processor connected to a graphics processor. Parallelism is extracted in both processors by linking sequential versions of the codes with multithreaded implementations of BLAS. Our results on a system with two Intel Quad-Core processors and an NVIDIA Tesla C1060 illustrate the performance and scalability attained by the codes on this system.
