Jie Liu

Bio: Jie Liu is an academic researcher from National University of Defense Technology. The author has contributed to research in topics: Parallel algorithm & Cost efficiency. The author has an hindex of 7, co-authored 12 publications receiving 128 citations.

An efficient parallel solution for Caputo fractional reaction—diffusion equation

Abstract: The computational complexity of Caputo fractional reaction—diffusion equation is $$O(MN^2)$$ O ( M N 2 ) compared with $$O(MN)$$ O ( M N ) of traditional reaction—diffusion equation, where $$M$$ M , $$N$$ N are the number of time steps and grid points. A efficient parallel solution for Caputo fractional reaction—diffusion equation with explicit difference method is proposed. The parallel solution, which is implemented with MPI parallel programming model, consists of three procedures: preprocessing, parallel solver and postprocessing. The parallel solver involves the parallel tridiagonal matrix vector multiplication, vector vector addition and constant vector multiplication. The sum of constant vector multiplication is optimized. As to the authors' knowledge, this is the first parallel solution for Caputo fractional reaction—diffusion equation. The experimental results show that the parallel solution compares well with the analytic solution. The parallel solution on single Intel Xeon X5540 CPU runs more than three times faster than the serial solution on single X5540 CPU core, and scales quite well on a distributed memory cluster system.

Computational Challenge of Fractional Differential Equations and the Potential Solutions: A Survey

Abstract: We present a survey of fractional differential equations and in particular of the computational cost for their numerical solutions from the view of computer science. The computational complexities of time fractional, space fractional, and space-time fractional equations are O(N2M), O(NM2), and O(NM(M

A domain decomposition method for time fractional reaction-diffusion equation.

Abstract: The computational complexity of one-dimensional time fractional reaction-diffusion equation is compared with for classical integer reaction-diffusion equation. Parallel computing is used to overcome this challenge. Domain decomposition method (DDM) embodies large potential for parallelization of the numerical solution for fractional equations and serves as a basis for distributed, parallel computations. A domain decomposition algorithm for time fractional reaction-diffusion equation with implicit finite difference method is proposed. The domain decomposition algorithm keeps the same parallelism but needs much fewer iterations, compared with Jacobi iteration in each time step. Numerical experiments are used to verify the efficiency of the obtained algorithm.

A Parallel Algorithm for the Two-Dimensional Time Fractional Diffusion Equation with Implicit Difference Method

Abstract: algorithm for 2D-TFDE and give an in-depth discussion about this algorithm. A task distribution model and data layout with virtual boundary are designed for this parallel algorithm. The experimental results show that the parallel algorithm compares well with the exact solution. The parallel algorithm on single Intel Xeon X5540 CPU runs 3.16–4.17 times faster than the serial algorithm on single CPU core. The parallel efficiency of 81 processes is up to 88.24% compared with 9 processes on a distributed memory cluster system. We do think that the parallel computing technology will become a very basic method for the computational intensive fractional applications in the near future.

Solving the Caputo Fractional Reaction-Diffusion Equation on GPU

Abstract: We present a parallel GPU solution of the Caputo fractional reaction-diffusion equation in one spatial dimension with explicit finite difference approximation. The parallel solution, which is implemented with CUDA programming model, consists of three procedures: preprocessing, parallel solver, and postprocessing. The parallel solver involves the parallel tridiagonal matrix vector multiplication, vector-vector addition, and constant vector multiplication. The most time consuming loop of vector-vector addition and constant vector multiplication is optimized and impressive performance improvement is got. The experimental results show that the GPU solution compares well with the exact solution. The optimized GPU solution on NVIDIA Quadro FX 5800 is 2.26 times faster than the optimized parallel CPU solution on multicore Intel Xeon E5540 CPU.

An implicit RBF meshless approach for time fractional diffusion equations

Abstract: This paper aims to develop an implicit meshless approach based on the radial basis function (RBF) for numerical simulation of time fractional diffusion equations. The meshless RBF interpolation is firstly briefed. The discrete equations for two-dimensional time fractional diffusion equation (FDE) are obtained by using the meshless RBF shape functions and the strong-forms of the time FDE. The stability and convergence of this meshless approach are discussed and theoretically proven. Numerical examples with different problem domains and different nodal distributions are studied to validate and investigate accuracy and efficiency of the newly developed meshless approach. It has proven that the present meshless formulation is very effective for modeling and simulation of fractional differential equations.

Solution of Multi-Term Time-Fractional PDE Models Arising in Mathematical Biology and Physics by Local Meshless Method

Abstract: Fractional differential equations depict nature sufficiently in light of the symmetry properties which describe biological and physical processes. This article is concerned with the numerical treatment of three-term time fractional-order multi-dimensional diffusion equations by using an efficient local meshless method. The space derivative of the models is discretized by the proposed meshless procedure based on the multiquadric radial basis function though the time-fractional part is discretized by Liouville–Caputo fractional derivative. The numerical results are obtained for one-, two- and three-dimensional cases on rectangular and non-rectangular computational domains which verify the validity, efficiency and accuracy of the method.

A new approach for the nonlinear fractional optimal control problems with external persistent disturbances

Abstract: The aim of this manuscript is to investigate an efficient iterative approach for the nonlinear fractional optimal control problems affected by the external persistent disturbances. For this purpose, first the internal model principle is employed to transform the fractional dynamic system with disturbance into an undisturbed system with both integer- and fractional-order derivatives. The necessary optimality conditions are then reduced into a sequence of linear algebraic equations by using a series expansion approach and the Grunwald–Letnikov approximation for the fractional derivatives. The convergence of the latter sequence to the optimal solution is also studied. In addition, an iterative algorithm designing the suboptimal control law is presented. Numerical simulations confirm that the new approach is efficient to reject the external disturbance and provides satisfactory results compared to the other existing methods.

The TianHe-1A Supercomputer: Its Hardware and Software

Abstract: This paper presents an overview of TianHe-1A (TH-1A) supercomputer, which is built by National University of Defense Technology of China (NUDT). TH-1A adopts a hybrid architecture by integrating CPUs and GPUs, and its interconnect network is a proprietary high-speed communication network. The theoretical peak performance of TH-1A is 4700 TFlops, and its LINPACK test result is 2566 TFlops. It was ranked the No. 1 on the TOP500 List released in November, 2010. TH-1A is now deployed in National Supercomputer Center in Tianjin and provides high performance computing services. TH-1A has played an important role in many applications, such as oil exploration, weather forecast, bio-medical research.