Optimized Splitting Methods Research Articles

Improving Efficiency of Passive RFID Tag Anti-Collision Protocol Using Dynamic Frame Adjustment and Optimal Splitting.

Radio frequency identification is a wireless communication technology, which enables data gathering and identifies recognition from any tagged object. The number of collisions produced during wireless communication would lead to a variety of problems including unwanted number of iterations and reader-induced idle slots, computational complexity in terms of estimation as well as recognition of the number of tags. In this work, dynamic frame adjustment and optimal splitting are employed together in the proposed algorithm. In the dynamic frame adjustment method, the length of frames is based on the quantity of tags to yield optimal efficiency. The optimal splitting method is conceived with smaller duration of idle slots using an optimal value for splitting level , where (M > 2), to vary slot sizes to get the minimal identification time for the idle slots. The application of the proposed algorithm offers the advantages of not going for the cumbersome estimation of the quantity of tags incurred and the size (number) of tags has no effect on its performance efficiency. Our experiment results show that using the proposed algorithm, the efficiency curve remains constant as the number of tags varies from 50 to 450, resulting in an overall theoretical gain in the efficiency of 0.032 compared to system efficiency of 0.441 and thus outperforming both dynamic binary tree slotted ALOHA (DBTSA) and binary splitting protocols.

Fourier-splitting methods for the dynamics of rotating Bose–Einstein condensates

We present a new method to propagate rotating Bose–Einstein condensates subject to explicitly time-dependent trapping potentials. Using algebraic techniques, we combine Magnus expansions and splitting methods to yield any order methods for the multivariate and nonautonomous quadratic part of the Hamiltonian that can be computed using only Fourier transforms at the cost of solving a small system of polynomial equations. The resulting scheme solves the challenging component of the (nonlinear) Hamiltonian and can be combined with optimized splitting methods to yield efficient algorithms for rotating Bose–Einstein condensates. The method is particularly efficient for potentials that can be regarded as perturbed rotating and trapped condensates, e.g., for small nonlinearities, since it retains the near-integrable structure of the problem. For large nonlinearities, the method remains highly efficient if higher order p > 2 is sought. Furthermore, we show how it can be adapted to the presence of dissipation terms. Numerical examples illustrate the performance of the scheme.

An efficient algorithm based on splitting for the time integration of the Schrödinger equation

We present a practical algorithm based on symplectic splitting methods intended for the numerical integration in time of the Schrödinger equation when the Hamiltonian operator is either time-independent or changes slowly with time. In the later case, the evolution operator can be effectively approximated in a step-by-step manner: first divide the time integration interval in sufficiently short subintervals, and then successively solve a Schrödinger equation with a different time-independent Hamiltonian operator in each of these subintervals. When discretized in space, the Schrödinger equation with the time-independent Hamiltonian operator obtained for each time subinterval can be recast as a classical linear autonomous Hamiltonian system corresponding to a system of coupled harmonic oscillators. The particular structure of this linear system allows us to construct a set of highly efficient schemes optimized for different precision requirements and time intervals. Sharp local error bounds are obtained for the solution of the linear autonomous Hamiltonian system considered in each time subinterval. Our schemes can be considered, in this setting, as polynomial approximations to the matrix exponential in a similar way as methods based on Chebyshev and Taylor polynomials. The theoretical analysis, supported by numerical experiments performed for different time-independent Hamiltonians, indicates that the new methods are more efficient than schemes based on Chebyshev polynomials for all tolerances and time interval lengths. The algorithm we present automatically selects, for each time subinterval, the most efficient splitting scheme (among several new optimized splitting methods) for a prescribed error tolerance and given estimates of the upper and lower bounds of the eigenvalues of the discretized version of the Hamiltonian operator.

Discrete and Continuous, Probabilistic Anticipation for Autonomous Robots in Urban Environments

This paper develops a probabilistic anticipation algorithm for dynamic objects observed by an autonomous robot in an urban environment. Predictive Gaussian mixture models are used due to their ability to probabilistically capture continuous and discrete obstacle decisions and behaviors; the predictive system uses the probabilistic output (state estimate and covariance) of a tracking system and map of the environment to compute the probability distribution over future obstacle states for a specified anticipation horizon. A Gaussian splitting method is proposed based on the sigma-point transform and the nonlinear dynamics function, which enables increased accuracy as the number of mixands grows. An approach to caching elements of this optimal splitting method is proposed, in order to enable real-time implementation. Simulation results and evaluations on data from the research community demonstrate that the proposed algorithm can accurately anticipate the probability distributions over future states of nonlinear systems.

Efficient algorithms for rigid body integration using optimized splitting methods and exact free rotational motion

The goal of this note on Hamiltonian splitting methods, an established technique to derive stable and accurate integration schemes in molecular dynamics, is to comment on the best combination of optimized splitting and gradient methods that avoids splitting the kinetic propagator.