"Other work has attempted to build multiple time step (MTS) integrators for MD that allow for time steps of differing lengths according to how rapidly a given type of interaction is evolving in time. The prototypical algorithm is the Verlet-l/r-RESPA/Impulse integrator (Grubmuller et al., 1991; Humphreys et al., 1994), which splits the forces into fast (short-range) and slow (long-range) components and evaluates the former more frequently than the latter. The ratio between frequencies of evaluation of the long-range forces (outer step-size) and short-range forces (inner step-size) measures the gain in simulation time and will be further referred to as 'the step-size ratio'.
"FIG. 2 is a diagramatic explanation of the MTS idea, which splits the forces in a system into bonded 'fast' forces and long range non bonded 'slow' forces (which tend to be non-linear), evaluating the slow forces less frequently. For this, multiple timestepping integrators are required to solve modified ODEs (Ordinary Differential Equations).
"For biomolecular applications, the computational complexity of the fast short-range force evaluations scales linearly in the number of atoms in the system, N, while it scales quadratically in N for the slow long-range force evaluations. Furthermore, while the short-range fast forces are easy to compute in parallel, long-range slow forces require global data communication and hence are more difficult to parallelize efficiently. Therefore, in theory, MTS methods can dramatically speed up MD simulations by reducing the number of expensive slow force evaluations.
"Although in theory MTS methods can dramatically speed up MD simulations by reducing the number of expensive slow force evaluations, in practice, however, impulse MTS methods such as the popular Verlet-l/r-RESPA suffer from severe resonance instabilities that limit practical performance gain (Ma et al., 2003; Izaguirre et al., 2001). For solvated biomolecular systems, for example, the stability limit means that the 'the step-size ratio' becomes equal to .about.4. Performance of impulse MTS methods was recently improved in the Langevin stabilized MTS methods (Izaguirre et al., 2001) by reducing resonance induced instabilities through the introduction of mollified MTS methods by Izaguirre et al., 1999 and weak coupling to a stochastic heat bath (Langevin dynamics) (Izaguirre et al., 2001) to weaken non-linear instabilities. This allowed an increase of the step-size ratio to 6-12 for solvated biomolecular systems (Izaguirre et al., 1999, 2001).
"In addition to the limitations which are specific to each listed method, all those methods have a common drawback--they do not exactly sample from the target temperature even if the simulations are stable and are subject to a thermostat (Pastor et al., 1988; Bond and Leimkuhler, 2007). This error can be controlled with a loss of computational efficiency by increasing the frequency of slow force updates.
"It is desirable to provide a method and apparatus for simulation which overcome or at least mitigate some of the disadvantages of the prior art."
In addition to obtaining background information on this patent, VerticalNews editors also obtained the inventors' summary information for this patent: "Embodiments of the invention provide a computer-implemented method of simulating behaviour of a thermodynamic system over time, the thermodynamic system having potential energy that can be split into more quickly varying parts and more slowly varying parts and having a state described by collective vectors of position and momentum at any given time, the method comprising a momentum refreshment process and a conservative dynamics process, wherein the momentum refreshment process comprises carrying out an operation of mixing the collective momentum vector with a noise vector and carrying out an acceptance/rejection operation; the conservative dynamics process comprises applying a multiple time stepping conservative dynamics operation to a current state, in which operation calculations for forces corresponding to more slowly varying energy parts in the thermodynamic system undergo an averaging procedure and are carried out at a larger time step than calculations for forces corresponding to more quickly varying energy parts; and carrying out an acceptance/rejection operation; and wherein the acceptance/rejection operations are based on an approximation of the system energy expressed using shadow Hamiltonians and comprise accepting a current state or returning a replacement state.
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