Cellular Computing and Least Squares for Partial Differential Problems Parallel Solving

Ever since von Neumann [Burks 1969], the question of modeling continuous physics with a discrete set of cellular automata has been raised, whether they handle discrete or continuous values. Many answers have been brought forth through, for instance, the work of Stephen Wolfram [Wolfram 1983] summarized in a recent book [Wolfram 2002]. This problem has been mostly tackled by rightfully considering that modeling physics through Newton and Leibniz calculus is fundamentally different from a discrete modelisation as implied by automata. Indeed, the former implies that physics is considered continuous either because materials and fields are considered continuous in classical physics or because quantum physics wave functions are themselves continuous. On the contrary, modeling physics through automata implies modeling on a discrete basis, in which a unit element called a cell, interacts with its surroundings according to a given law derived from local physics considerations. Such discrete automaton based models have been successfully applied to various applications ranging from reaction-diffusion systems [Weimar and Boon 1994] to forest fires [Drossel and Schwabl 1992], through probably one of the most impressive achievements: the Lattice Gas Automata [Rothman and Zaleski 1994], where atoms