Towards more reliable extrapolation algorithms with applications to organic chemistry
One of the main objectives of science and engineering is to predict the results of different situations. For example, in Newton's mechanics, we want to predict the positions and velocities of different objects (e.g., planets) at future moments of time.^ In this thesis, as a case study, we take the problem of predicting the properties of new chemical substances. One of the main objectives of chemistry is to design new molecules (and, more generally, new chemical compounds) which are useful for various practical tasks. New substances have already resulted in new materials for buildings and for spaceships, new explosives and new fuels, new medicines, etc. New compounds are being designed and tested all the time.^ For example, this is how new medicines are designed: a large number of different promising substances are synthesized and tested, but only a few turn out to be practically useful. Synthesizing a new compound is often difficult and time-consuming. It is therefore desirable to predict the properties of new compounds, so as to filter out the ones which do not have the desired properties.^ In physics, usually, we know the exact equations that describe the objects of interest, and we know how to solve these equations. This is the case for Newton's mechanics. In such situations, we face a purely mathematical problem: to solve these equations and thus compute the value y of the desired characteristic based on the known values of the parameters x1,…,xn that describe the given objects.^ In many other application areas, we either do not know the equations, or the equations are so complex that we do not know how to solve them. For example, in chemistry, in principle, we can use the equations of quantum mechanics to describe an arbitrary chemical substance, but in practice, especially for organic substances, these equations are too complex to solve.^ In the situations in which we do not know the equations - or we do not know how to solve the equations - the prediction problem takes the following form: (1) we know the values of a quantity v(a) for some objects a, and (2) we want to predict the values of this quantity for some other objects a'.^ There are many examples of successful predictions in science. In many cases, to solve a new prediction problem, researchers use ideas which are specific for this problem. In addition to problem-specific predictions, there exist successful prediction algorithms. Most of these algorithms are heuristic - in the sense that they are empirically successful, but since they do not have any domain-related theoretical justification, there is no guarantee that they will work in other situations as well.^ It is therefore desirable to provide more justified extrapolation algorithms - e.g., by providing a solid justification for the existing heuristic techniques. In numerical mathematics, more justified algorithms are often called more reliable. In this thesis, we provide a theoretical justification for an important class of heuristic extrapolation algorithms - algorithms based on partially ordered sets (posets). This justification makes these algorithms more reliable in the sense of numerical mathematics.^
Chemistry, Analytical|Computer Science
Nava, Jaime, "Towards more reliable extrapolation algorithms with applications to organic chemistry" (2009). ETD Collection for University of Texas, El Paso. AAI1468957.