Nonlinear parallel computing

The implementation of molecular dynamics simulations on parallel computers needs a method that distributes over the processors both the evaluation of pair interactions and the integration of particle motions. The force terms involved in integrating the set of coupled differential equations (Newton s equations) characteristic of any MD simulation are typically nonlinear functions of the distance between pairs of atoms and may be either long-range or short-range. We use this attribute of the force terms in detailing the parallel algorithmic work conducted to date. 

While SHAKE/RATTLE is a robust procedure the sequential, iterative character of nonlinear solvers needed for its implementation can lead inefficiencies when it is implemented on a parallel computer. Therefore, a number of the alternative methods avoid iteration. We briefly summarize a few of the popular alternative schemes here. 

The exceptional computational abilities of the human brain have motivated the concept of an NN. The brain can perform certain types of computation, such as perception, pattern recognition, and motor control, much faster than existing digital computers (Haykin, 2009). The operation of the human brain is complex and nonlinear and involves massive parallel computation. Its computations are performed using structural constituents called neurons and the synaptic interconnections between them (that is, a neural network), The development of artificial neural networks is an admittedly approximate attempt to mimic this biological neural network, in order to achieve some of its computational advantages. 

Bucy, R.S. and Senne, K.D., "Nonlinear Filtering Algorithms for Parallel and Pipe Line Machines". Proceedings of the Meeting on Parallel Mathematics - Parallel Computers, Munich March 1976, North Holland Press, Amsterdam. 

Moreover, the harmonic function h is not continuous across the discontinuities of N (this follows from (4.1.9), (4.1.7), and (4.1.3)). The magnitude of the appropriate jumps in h is a nonlinear function of the local values of ip and h themselves, so that h cannot be computed separately from

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Early applications of MPC took place in the 1970s, mainly in industrial contexts, but only later MPC became a research topic. One of the first solid theoretic formulations of MPC is due to Richalet et al. [53], who proposed the so-called Model Predictive Heuristic Control (MPHC). MPHC uses a linear model, based on the impulse response and, in the presence of constraints, computes the process input via a heuristic iterative algorithm. In [23], the Dynamic Matrix Control (DMC) was introduced, which had a wide success in chemical process control both impulse and step models are used in DMC, while the process is described via a matrix of constant coefficients. In later formulations of DMC, constraints have been included in the optimization problem. Starting from the late 1980s, MPC algorithms using state-space models have been developed [38, 43], In parallel, Clarke et al. used transfer functions to formulate the so-called Generalized Predictive Control (GPC) [19-21] that turned out to be very popular in chemical process control. In the last two decades, a number of nonlinear MPC techniques has been developed [34,46, 57],... 

S. A. Zenios and M. C. Pinar, SIAM J. Sci. Statist. Comput., 13 (Sept. 1992). Parallel Block-Partitioning of Truncated Newton for Nonlinear Network Optimization. 

The coupled code developed by Steefel and Lasaga (1994) for multicomponent reactive transport with kinetics of precipitation and dissolution of minerals has been developed further into the OS3D/GIMRT code (Steefel and Yabusaki, 1996). This model has been applied to reaction fronts in fracture-dominated flow systems (Steefel and Lichtner, 1998). Eurther developments for nonuniform velocity helds by Yabusaki et al. (1998) required the use of massively parallel processing computers, although ... the accuracy of the numerical formulation coupling the nonlinear processes becomes difficult to verify. ... 

For physically relevant states, the propagation and collision rules for the behavior of such a set of cells as time goes on may mirror what would happen with a physical system. This is why cellular automata are appealing. Another advantage is that due to the locality mentioned above, the relevant computer programs may be effectively parallelized, which usually significantly speeds up computations. The most interesting cellular automata are those for which the rules are of a nonlinear character (cf. Chapter 15). 

Series/Parallel Scan with time delay and integration remains the principal approach to advanced thermal imaging systems. However, for applications where only a small number of resolution elements are needed, two-dimensional staring detector arrays with CCD or CID readout are being considered [8.106]. This does away with the scanner and a focal optics used with conventional systems. However, to compensate for nonuniformities, both dc offset and gain correction must be made on a pixel by pixel basis. Detector responsivity and readout nonlinearities will increase the number of computations needed for sufficient correction and only experience with the stability of different types of arrays will determine how often the correction algorithms must be calibrated [8.107,108]. 

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