The paper discusses equation-free approaches. Though the applications are many, the authors focus on a few. Also, many multiscale methods are not mentioned, so it is focused on a class of methods. The authors can mention this in the beginning. I recommend the paper for publication.
Reviewer B (User:): Multiscale modeling
One main theme of multiscale modeling is to develop efficient numerical algorithms for capturing the macroscale behavior of a complex systems through coupling between the macroscopic and microscopic models. This is necessary since in many cases, one either does not have an explicit macroscale model or the available macroscopic models are not accurate enough. It might also be quite feasible since very often the macroscopic and microscopic scales in the systems are quite disparate, and this separation of scales can be exploited for numerical use [2, 4].
There are two kinds of philosophies for developing such algorithms: the top-down and the bottom-up philosophy. The top-down philosophy starts with a macroscopic model and tries to enrich the macroscopic model using information supplied by the microscopic model. The bottom-up philosophy starts with a microscopic model, and tries to systematically coarse-grain the microscopic model or develops efficient numerical algorithms for the microscopic model to mimic the macroscopic behavior of the system. The heterogeneous multiscale method (HMM)  is an example of the first type. Equation-free modeling is an example of the second type. In HMM, one starts with a preconceived form of the macroscopic equation with missing information such as the constitutive relation, the missing information is computed on the fly using the microscopic model. Scale separation can be used in a natural way when estimating the needed macroscale information from the microscopic model: One only has to simulate the microscopic model on macroscopically small but microscopically large temporal/spatial domains in order to get a sufficiently accurate estimate of the needed data. In the equation-free approach, one starts with a microscopic model, through interpolation in space and extrapolation in time, one can use results obtained from simulating the microscopic model on an array of small domains to mimic the behavior of a system at macroscopic scales. Each methodology has its merits and drawbacks. For HMM, clearly if one starts with a mis-conceived form of the macroscopic equation, one is likely going to get wrong results. In equation-free, it is very difficult to guarantee accuracy and stability at the macroscale using microscopic simulations on small temporal/spatial domains.
It should be emphasized that even though HMM and equation-free have emerged as the leading candidates of general frameworks for developing multiscale algorithms, many other proposals have been made in the literature, some earlier than HMM and equation-free. For example, Brandt proposed to extend the multi-grid method to deal with problems for which one does not have a reliable macroscopic model but only a microscopic model. Brandt also noted that one can take advantage of the scale separation by limiting the multi-grid sweeps to small temporal/spatial domains . In many ways, the lifting-equilibration-restriction-extrapolation-based projective integrators and patch dynamics, two main components of the equation-free approach, can be considered as an extension of Brandt's multi-grid algorithm by adding an extrapolation step after the usual prolongation-equilibration-restriction multi-grid approach. Other successful multiscale algorithms such as the quasi-continuum method, the kinetic scheme, and the Car-Parrinello method, etc , also use similar ideas in one way or another.
The set of techniques developed within the equation-free framework can be useful for systems whose macroscopic behavior is relatively simple. For example, for stiff ordinary differential equations (ODE) in which the fast component of the dynamics relaxes quickly to a slow manifold, projective integrators can be quite effective. Of course there are more efficient stiff ODE solvers, but the projective integrators have the advantage of being simple and easy to implement.
However, when there are nontrivial relaxational modes or nontrivial spatial processes happening in the system, the equation-free approch has significant difficulties. For example, if the system has temporal oscillatory modes or if the macroscopic behavior has both deterministic and stochastic components, then it is very difficult to use the kind of extrapolation-based projective integrators. If the macroscopic behavior has spatial singularities such as shocks, it is very difficult to construct stable patch dynamics.
A word of caution: The article uses the term ``equation-free modeling at two different levels: It is used to refer to the general practice of performing simulations without using an explicit closed-form equation and it is used to refer to the set of techniques developed under the equation-free umbrella, namely, coarse bifurcation, projective integration, and patch dynamics. This kind of change of concept has caused substantial confusion about what the ``equation-free approach really is. If it is understood in the broad terms, then it would include all the work done in multiscale modeling since the very reason for the need of multiscale modeling is the lack of explicit closed form macroscopic models. Even in classical modeling, it is often the case that some of the parameters, such as diffusion coefficients or equations of state, are not readily available and have to be computed from microscopic models. Indeed some of the papers have referred to this kind of classical parameter passing or sequential multiscale modeling techniques as ``equation-free modeling . It would have been much better for the community if the article had made an effort to clarify this point, namely to distinguish the general issue of performing simulation without explicit closed-form models and the specific methodologies proposed under the ``equation-free umbrella.
Weinan E, Princeton University
1. A. Brandt, ``Multiscale scientific computation: Review 2001, in Multiscale and Multiresolution Methods: Theory and Applications, Yosemite Educational Symposium Conf. Proc., 2000, Lecture Notes in Comp. Sci. and Engrg., T.J. Barth, et.al (eds.), vol. 20, pp. 3--96, Springer-Verlag, 2002.
2. W. E, Principles of Multiscale Modeling, Cambridge University Press, www.cambridge.org/978117096547, to appear.
3. W. E and B. Engquist, ``The heterogeneous multi-scale methods, Comm. Math. Sci., vol. 1, pp. 87--133, 2003.
4. W. E and B. Engquist, ``Multiscale modeling and computation, Notices of the Amer. Math. Soc., vol. 50, no. 9, pp. 1062--1070, 2003.
5. R. Erban, I. G. Kevrekidis, D. Adalsteinsson and T. C. Elston, ``Gene regulatory networks: A coarse-grained, equation-free approach to multiscale computations, J. Chem. Phys., vol. 124, pp. 084106-17, 2006.