Numerical Algorithms and Computations

Random directed polymers still present a number of open questions and this gives impulse to attack them from a numerical viewpoint. The onedimensional character of the problems allows the numerical treatment of very large systems. 

however, have to stress that the chapter focuses only on computations with a constant look at rigorous results. The computation is mainly seen as a tool for getting ideas on what to prove and how to prove it and therefore it rarely pushes much beyond the boundary of what has been rigorously understood. In Section 9.4 we will refer to works pushing further, trying for example to estimate the critical exponents of the various models, but this will not be considered in detail. 

One of the main aims of this chapter is to explain that one can assert localization by numerical computation in an almost rigorous way, while asserting delocalization is essentially out of control. 

---

DPMs offer a viable tool to study the macroscopic behavior of assemblies of particles and originate from MD methods. Initiated in the 1950s by Alder and Wainwright (1957), MD is by now a well-developed method with thousands of papers published in the open literature on just the technical and numerical aspects. A thorough discussion of MD techniques can be found in the book by Allen and Tildesley (1990), where the details of both numerical algorithms and computational tricks are presented. Also, Frenkel and Smit (1996) provide a comprehensive introduction to the recipes of classical MD with emphasis on the physics underlying these methods. Nearly all techniques developed for MD can be directly applied to discrete particles models, except the formulation of particle-particle interactions. Based on the mechanism of particle-particle interaction, a granular system may be modeled either as hard-spheres or as soft-spheres. ... 

Selim Senkan is noted for his work in environmental engineering, and particularly for his work in the reaction rates of chlorinated hydrocarbons. He writes in Detailed Chemical Kinetic Mechanisms on the impact of efficient numerical algorithms and computational quantum mechanics on the prediction of reaction mechanisms and rates. 

Algorithmic and computational solutions for model (or design) equations, combined with chemical/biological modeling, are the main subjects of this book. We shall learn that the complexities for generally nonlinear chemical/biological systems force us to use mainly numerical techniques, rather than being able to find analytical solutions. 

As a matter of illustration, let us write in detail the numerical algorithm for computing ER(p,p ) with the PCM model (1.30) and (1.31) and the Galerkin approximation with P0 planar boundary elements ... 

It is generally the case that advances in numerical algorithms, and their associated applications, have been greatly accelerated when general-use software becomes widely available for use by others. For example, the late John A. Pople was recognized by the Nobel Prize in Chemistry in 1998 for development of computational methods in quantum chemistry. It was his development of the software Gaussian that implemented those computational methods that resulted in a significant research impact. 

Francois M, Shyy W (2003) Computations of drop dynamics with the immersed boundary method, part 1 numerical algorithm and buoyancy-induced effect. Numerical Heat Transfer, Part B 44 101-118... 

Eor complex reaction and reactor models, the sensitivity analysis and the parameter estimation bv optimization are comppter-time consuming and call for more efficient algorithms and computers. Here clearly, any improvement in the speed of such computations is desirable, and even necessary, for the practical use of fundamental models. The requirements of speedness would be rather increased if a fundamental model, instead of a black box, were used for optimal control purposes. So, we think that supercomputers will be more and more useful for solving the numerical problems involved in the mechanistic noodelling of complex gas phase reactions. 

---