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In this paper, we consider a feasible primal–dual interior point method for linear semidefinite programming problem ( S D P ) based on Alizadeh–Haeberly–Overton ( AHO ) direction (Monteiro, 1997). Firstly, and by a new and simple technique, we establish the existence and uniqueness of optimal solution of the perturbed problem ( S D P ) μ and its convergence to optimal solution of ( S D P ) . Next, we present new different alternatives to calculate the displacement step. After, we establish the convergence of the obtained algorithm and we show that its complexity is O ( n ln [ ε − 1 ( 〈 X 0 , S 0 〉 ) ] ) . Finally, we present some numerical simulations which show the effectiveness of the algorithm developed in this work. [ABSTRACT FROM AUTHOR]

In this paper, a fast second-order finite-difference time-domain (FDTD) algorithm based on the recently proposed FL 2 - 1 σ formula and a weighted approach is presented for solving the time fractional Maxwell's system. Firstly, we construct a discretization to the Caputo derivative term, and then propose a fast second-order fully discrete FDTD algorithm by applying this discretization. And the detailed algorithm implementation process is given. Finally, numerical examples with analytic solutions on 2D and 3D domains are provided to verify the accuracy and efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Existing principal curve algorithms have some drawbacks such as time consuming and narrow application scope in practice, since these algorithms are mainly based on global optimization. In this paper, we present the concept of Constraint Local Principal Curve (CLPC), which uses local optimization methods and restricts the principal curve with two fixed endpoints to reduce the computational complexity. In addition, we propose three CLPC algorithms by Local Optimization and Adaptive Radius to expand the range of applications and increase the solution quality. The first algorithm, i.e., CLPCg is based on greedy thinking. The second algorithm, i.e., CLPCs uses one dimensional search and the last algorithm CLPCc combines the greedy thinking and one dimensional search. Then, we define six performance indices to evaluate the performance of the CLPC algorithms. Finally, we present some numerical experiments with three simulation data sets and two GPS measured data sets in both highway and railway. The results indicate that all of the three CLPC algorithms can obtain high-accuracy data from multiple low-accuracy data efficiently. The CLPC algorithms can improve the accuracy and computational speed compared with the existing K-segment principal curve (KPC) algorithm. In addition, CLPCc outperforms CLPCg and CLPCs according to the comprehensive experiments while CLPCg runs much faster than other ones. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of accelerating the numerical simulation of time dependent problems by time domain decomposition. The available algorithms enabling such decompositions present severe efficiency limitations and are an obstacle for the solution of large scale and high dimensional problems. Our main contribution is the improvement of the parallel efficiency of the parareal in time method. The parareal method is based on combining predictions made by a numerically inexpensive solver (with coarse physics and/or coarse resolution) with corrections coming from an expensive solver (with high-fidelity physics and high resolution). At convergence, the algorithm provides a solution that has the fine solver's high-fidelity physics and high resolution. In the classical version, the fine solver has a fixed high accuracy which is the major obstacle to achieve a competitive parallel efficiency. In this paper, we develop an adaptive variant that overcomes this obstacle by dynamically increasing the accuracy of the fine solver across the parareal iterations. We theoretically show that the parallel efficiency becomes very competitive in the ideal case where the cost of the coarse solver is small, thus proving that the only remaining factors impeding full scalability become the cost of the coarse solver and communication time. The developed theory has also the merit of setting a general framework to understand the success of several extensions of parareal based on iteratively improving the quality of the fine solver and re-using information from previous parareal steps. We illustrate the actual performance of the method in stiff ODEs, which are a challenging family of problems since the only mechanism for adaptivity is time and efficiency is affected by the cost of the coarse solver. [ABSTRACT FROM AUTHOR]