A Continuous Model of Computation

Although the Turing-machine model of computation is widely used in computer science it is fundamentally inadequate as a foundation for the theory of modern scientific computation. The real-number model is described as an alternative. Physicists often choose continuous mathematical models for problems ranging from the dynamical systems of classical physics to the operator equations and path integrals of quantum mechanics.These mathematical models use the real or complex number fields and we argue that the real-number model of computation should be used in the study of the computational complexity of continuous mathematical models. The study of continuous complexity is called information-based complexity. In this expository article we apply information-based complexity to topics such as breaking the curse of dimensionality, approximating the calculation of path integrals, and solving ill-posed problems. Precise formulations of these ideas may be found in J. F. Traub and A. G. Werschulz, "Complexity and Information", Cambridge University Press, 1998.