Biomedical subjects

Siu A Chin

Publications and source records attributed to Siu A Chin.

5 recordsLinked to original sources

PubMed2005

Fourth-order algorithms for solving the imaginary-time Gross-Pitaevskii equation in a rotating anisotropic trap.

By implementing the exact density matrix for the rotating anisotropic harmonic trap, we derive a class of very fast and accurate fourth-order algorithms for evolving the Gross-Pitaevskii equation in imaginary time. Such fourth-order algorithms are possible only with the use of forward, positive time step factorization schemes. These fourth-order algorithms converge at time-step sizes an order-of-magnitude larger than conventional second-order algorithms. Our use of time-dependent factorization schemes provides a systematic way of devising algorithms for solving this type of nonlinear equations.

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PubMed2005

Forward symplectic integrators and the long-time phase error in periodic motions.

We show that when time-reversible symplectic algorithms are used to solve periodic motions, the energy error after one period is generally two orders higher than that of the algorithm. By use of correctable algorithms, we show that the phase error can also be eliminated two orders higher than that of the integrator. The use of fourth order forward time step integrators can result in sixth order accuracy for the phase error and eighth order accuracy in the periodic energy. We study the one-dimensional harmonic oscillator and the two-dimensional Kepler problem in great detail, and compare the effectiveness of some recent fourth order algorithms.

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PubMed2005

Structure of positive decompositions of exponential operators.

The solution of many physical evolution equations can be expressed as an exponential of two or more operators acting on initial data. Accurate solutions can be systematically derived by decomposing the exponential in a product form. For time-reversible equations, such as the Hamilton or the Schrödinger equation, it is immaterial whether or not the decomposition coefficients are positive. In fact, most symplectic algorithms for solving classical dynamics contain some negative coefficients. For time-irreversible systems, such as the Fokker-Planck equation or the quantum statistical propagator, only positive-coefficient decompositions, which respect the time-irreversibility of the diffusion kernel, can yield practical algorithms. These positive time steps only, forward decompositions, are a highly effective class of factorization algorithms. This work presents a framework for understanding the structure of these algorithms. By a suitable representation of the factorization coefficients, we show that specific error terms and order conditions can be solved analytically. Using this framework, we can go beyond the Sheng-Suzuki theorem and derive a lower bound for the error coefficient e(VTV). By generalizing the framework perturbatively, we can further prove that it is not possible to have a sixth-order forward algorithm by including only the commutator [VTV] tripple bond [V, [T,V]]. The pattern of these higher-order forward algorithms is that in going from the (2n)th to the (2n+2)th order, one must include a different commutator [V T(2n-1)V] in the decomposition process.

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PubMed2004

Quantum statistical calculations and symplectic corrector algorithms.

The quantum partition function at finite temperature requires computing the trace of the imaginary time propagator. For numerical and Monte Carlo calculations, the propagator is usually split into its kinetic and potential parts. A higher-order splitting will result in a higher-order convergent algorithm. At imaginary time, the kinetic energy propagator is usually the diffusion Green's function. Since diffusion cannot be simulated backward in time, the splitting must maintain the positivity of all intermediate time steps. However, since the trace is invariant under similarity transformations of the propagator, one can use this freedom to "correct" the split propagator to higher order. This use of similarity transforms classically gives rise to symplectic corrector algorithms. The split propagator is the symplectic kernel and the similarity transformation is the corrector. This work proves a generalization of the Sheng-Suzuki theorem: no positive time step propagators with only kinetic and potential operators can be corrected beyond second order. Second-order forward propagators can have fourth-order traces only with the inclusion of an additional commutator. We give detailed derivations of four forward correctable second-order propagators and their minimal correctors.

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PubMed2004

Dynamical multiple-time stepping methods for overcoming resonance instabilities.

Current molecular dynamics simulations of biomolecules using multiple time steps to update the slowly changing force are hampered by instabilities beginning at time steps near the half period of the fastest vibrating mode. These "resonance" instabilities have became a critical barrier preventing the long time simulation of biomolecular dynamics. Attempts to tame these instabilities by altering the slowly changing force and efforts to damp them out by Langevin dynamics do not address the fundamental cause of these instabilities. In this work, we trace the instability to the nonanalytic character of the underlying spectrum and show that a correct splitting of the Hamiltonian, which renders the spectrum analytic, restores stability. The resulting Hamiltonian dictates that in addition to updating the momentum due to the slowly changing force, one must also update the position with a modified mass. Thus multiple-time stepping must be done dynamically.

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