experience with this type of problem.
A form of deterministic cellular automata, the lattice gas method for solving partial differential equations, is successful in solving hydrodynamics problems (Doolen, 1991). With lattice gas methods, simulations of fluid turbulence are available that provide direct checks of many approximate theories. Calculations of fluid flowing through a porous material, a problem important for oil recovery from sandstone, have also been successful. These methods have been applied to the study of phase-separation and chemical-reaction kinetics.
A strength of these methods is the ease with which they can handle complicated boundaries and boundary conditions. Because of the ability to use millions of cells, these methods have the potential of more easily tracking fronts and thus may be a considerable asset in solving moving interface and free-boundary problems associated with solidification and grain growth. The methods also adapt well to parallel computers; specialized computers that perform just this type of calculation could be designed.
A basic problem with lattice gas methods is determining whether the rules specified are sufficient to ensure that the calculated phenomena correspond to physical phenomena. To what level of accuracy do lattice gas methods solve the Navier-Stokes equations, the diffusion equation, and so forth? For fluid problems, the answer appears to be on relatively firm ground. This correspondence follows from analyses analogous to those used in statistical mechanics for determining macroscopic behavior in the kinetic theory of gases, from which the Navier-Stokes equations for a collection of interacting particles can be derived. For other physical systems, more analysis and understanding are needed.
Many of the concepts and methods of probability and statistics used in materials science are products of mathematical sciences research done much earlier in this century or previous centuries. Yet, probability and statistics sport many newer concepts and methods that may influence the analysis and design of materials experiments; see, for example, Chapters 9 and 10 in National Research Council (1991d). What impact might methods of nonparametric function estimation, modeling, and simulation have on the interpretation and analysis of materials experiments and simulations? Recently, Bayesian methods of statistical analysis have been used successfully in several areas of materials science, including optimization of the design of neutron scattering interferometers (Sivia et al., 1990), analysis of reflectivity data (Sivia et al., 1991), and solving ill-posed problems associated with the use of quantum Monte Carlo calculations (Gubernatis et al., 1991). The saying "With enough parameters one can fit anything" recedes into the background as these methods estimate not only what fit to the parameters is most probable but also what is the most probable number of parameters needed (Sivia and Carlile, 1992).
The ubiquitous need for nonlinear optimization has been highlighted several times in this report. Nonlinear optimization is also used in comparing theory with experiment (see, for example, the Modeling Protein Structure and Dynamics section of Chapter 3). Such optimization problems are especially challenging because of noise and the incomplete nature of the data.