A Continuous Model of Computation Real-world Phenomena Computer Simulation Mathematical Model 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. A central dogma of computer science is that the Turing machine model is the appropriate abstraction of a digital computer. Physicists who've thought about this also seem to favor the Turing machine model. For example, Roger Penrose [1] devotes some 60 pages to a description of this abstract model of computation and its implications. I will argue that physicists should consider the real-number model of computation as more appropriate and useful for scientific computation. First, I'll introduce the four " worlds " that will play a role; see Figure 1. Above the horizontal line are two real worlds; the world of physical phenomena and the computer world, where simulations are performed. Below the horizontal line are two formal worlds; a mathematical model of the physical phenomenon and a model of computation which is an abstraction of a physical computer. We get to choose both the mathematical model and the model of computation. What type of models should we choose?