The integration algorithms

The European Research Council and the Swedish Research Council partially support the current research. 

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Fig. 6. Multiple threads in NAMD 2 allow the integration algorithm to be expressed sequentially as a single function. This function, shown illegibly at left, runs in sequencer threads associated with home patches. A similar function running in a controller thread on the master node communicates with the sequencers to deal with output and global calculations. Compute objects execute in the larger stack space of each node s main thread.

By ensuring that the first derivative is zero at the endpoints the force also approaches zero smoothly. A continuous second derivative is required to ensure that the integration algorithm works properly. If the switch function is assumed to take the following form ... 

Prepare a molecule for a molecular dynamics simulation. If the forces on atoms are too large, the integration algorithm may fail during a molecular dynamics calculation. 

It is tempting to get an improved value x 0) by extrapolation of x 2h) and x h) to h = 0. However, in general, if this improved value is introduced into the integration algorithm, the properties of stability disappear. Thus, this extrapolation process can only be used at the end of computations and is called global extrapolation (or passive extrapolation). 

As the integration algorithms were different between the two applications, an early decision in the migration was taken to avoid using the peak area calculations as a comparator between the... 

Once the initial state of the system has been set up, the equations of motion (e.g., in the form of the Newton s equations) are solved numerically by means of the finite difference approach. Known atomic positions, velocities, and forces at time t are used to obtain the coordinates and velocities at time t + At, after which the procedure is repeated. The value of the time step that can be used depends on the specific model of the system and the accuracy of the integration algorithm. In general, the time step should be smaller than t/lO, where t is the minimum characteristic time in the system, e.g., the period of the highest frequency vibration. 

To avoid the need for special procedures and modification of the integration algorithm, delays may be modeled by using rational approximations, e.g., Fade functions or multiple first-order lags in series. Experimentation suggests that 10 series lags is adequate for most applications, so this is used as a default. The approximation should be checked by comparison with a more detailed model where it is believed to be particularly significant. 

There is additional information stored with a link, e.g. its state and information about possible rule applications. This information is needed by the integration algorithm but not for the definition of integration rules. 

An example of a triple rule is given in Fig. 3.30. The rule refers to the running example to be used for the integration algorithm, namely the creation of connections as introduced in Fig. 3.28 a). Here, the triple rule is presented as a graph transformation in PROGRES [414] syntax. 

In the optimized version of the integration algorithm, the context check is per-... 

Both implementation approaches apply the integration algorithm presented in Subsect. 3.2.4. They differ in how the algorithm is executed. 

We provide a precise formal specification of the integration algorithm,. In... 

Transformations between documents are urgently needed, not only in chemical engineering. They have to be incremental, interactive, and bidirectional. Additionally, transformation rules are most likely ambiguous. There are a lot of transformation approaches and consistency checkers with repair actions that can be used for transformation as well, but none of them fulfills all of these requirements. Especially, the detection of conflicts between ambiguous rules is not supported. We address these requirements with the integration algorithm described in this contribution. 

All extensions affect the integration algorithm as well as the derivation of forward, backward, and correspondence analysis rules. For instance, the existence of paths can be checked, but not all paths can be created in a deterministic way. Thus, the constructs either have to be restricted, or generated forward, backward, and correspondence analysis rules have to be manually post-processed. 

Another way to view MD simulation is as a technique to probe the atomic positions and momenta that are available to a molecular system under certain conditions. In other words, MD is a statistical mechanics method that can be used to obtain a set of configurations distributed according to a certain statistical ensemble. The natural ensemble for MD simulation is the microcanonical ensemble, where the total energy E, volume V, and amount of particles N (NVE) are constant. Modifications of the integration algorithm also allow for the sampling of other ensembles, such as the canonical ensemble (NVT) with constant temperature... 

Equations (20.67) and (20.68) may now be integrated using the integration algorithm of the main simulation. 

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