The heterogeneous multiscale method (HMM)

There has been a lot of interest on HMM (the heterogeneous multiscale method) and the ``equation-free'' approach to multiscale modeling. Many people have asked the questions: What are they exactly? What are the similarities and differences? Are they really useful and what are their potential uses? To address these questions, we have to put ourselves in the context of the releveant literature that already existed before HMM and ``equation-free'' were developed. Most people do not seem to be aware of the fact that there had been many successful algorithms for capturing the macroscale behavior of a system with the help of microscale models before HMM and ``equation-free''. Achi Brandt also proposed general strategies for developing such algorithms, by going between macro and micro states/models using interpolation and restriction operators, and performing microscopic simulations on small windows for short times. These histories are rarely discussed. In addition, it has been difficult to isolate what exactly ``equation-free'' is.

Several years have passed. There is now a large volume of literature on both HMM and ``equation-free''. It is perhaps a good time to start addressing the questions raised above.

This page provides some of the background materials as well as some personal perspectives on these matters. Major players on the ``equation-free'' approach have also been invited to provide their perspectives. These will be posted once they are received.

Section 1. Personal perspectives:

• W. E, HMM and the ``equation-free'' approach to multiscale modeling, preprint.
• W. E, Comments on the papers posted.

Section 2. Some classical examples:

• R. Car and M. Parrinello, Unified approach for molecular dynamics and density functional theory, Phys. Rev. Lett., vol. 55, no. 22, pp. 2471-2474, 1985.
• J. Knap and M. Ortiz, An analysis of the quasicontinuum method, J. Mech. Phys. Solids., vol. 49, no. 9, pp. 1899-1923, 2001.
The original quasicontinuum method uses the Cauchy-Born rule in the local region. This version uses cluster-based summation rule, and is more closely related to the topic discussed here (not just in philosophy, but also in technical detail).
• S. M. Deshpande, Boltzmann schemes for continuum gas dynamics, 1992.
There are many important papers on kinetic schemes for gas dynamics. This paper was selected at the recommendation of some experts in this area.
• W. W. Grabowski and P. K. Smolarkiewicz, CRCP: a cloud resolving conection parametrization for modeling the tropical convecting atmosphere, Physica D, vol. 133, 171-178, 1999.

Section 3. Brandt's review:

• A. Brandt, Multiscale scientific computation: Review 2000, preprint, 2000, 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.
This review outlines a general approach, now called ``systematic upscaling'', for multiscale, multi-physics problems, based on the ``interpolation-equilibration-restriction'' strategy of multi-grid method. It goes beyond the classical multi-grid method in many different ways. Particularly relevant points to this discussion are: It aims at capturing the large scale properties for cases when closed form macroscopic equations are not available, and it tries to reduce computational cost by performing microscopic simulations on small windows with few sweeps. This is a long article. The reader might want to consulte Section 14 for an example. Even though the overall philosophy was clearly laid out and many examples are discussed, the algorithmic details are rather unclear.

Section 4. Some HMM papers:

• W. E and B. Engquist, The heterogeneous multiscale methods, Comm. Math. Sci., vol. 1, no. 1, pp. 87-132, 2003.
• W. E and B. Engquist, Multiscale modeling and computation, Notice of the Amer. Math. Soc., vol. 50, no. 9, pp. 1062-1070, 2003.
• E. Vanden-Eijnden, Numerical techniques for multi-scale dynamical systems with stochastic effects, Comm. Math. Sci., vol. 1, no. 2, pp. 385-391, 2003.
• W. E, B. Engquist, X. Li, W. Ren and E. Vanden-Eijnden, Heterogeneous multiscale methods: A review, Comm. Comput. Phys., vol. 2, no. 3, pp. 367-450, 2007.

Section 5. Some ``equation-free'' papers:

• I. G. Kevrekidis, C. W. Gear, J. M. Hyman, P. G. Kevrekidis, O. Runborg and C. Theodoropoulos, Equation-free, coarse-grained multiscale computation: Enabling microscopic simulators to perform system-level analysis, Comm. Math. Sci., vol. 1, no. 4, pp. 715-762, 2003.
• G. Hummer and I. G. Kevrekidis, Coarse molecular dynamics of a peptide fragment: Free energy, kinetics, and long-time dynamics computations , J. Chem. Phys., vol. 118, no. 23, pp. 10762-10773.
• 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, 084106, 2006.
This is a very interesting application of multiscale modeling. However, the approach used is mainly the classical ``equation-based'' precomputing technique: A simple form of Fokker-Planck equation is assumed to hold. The coefficients are precomputed using microscopic models. The Fokker-Planck equation is then used to compute free energy profile and first-passage times. Many ``equation-free'' papers are of this style. This is the standard sequential coupling approach that has been used for very long time. Calling this approach ``equation-free'' can only lead to confusion. This is not an issue of semantics -- it illustrates the kind of difficulty one faces when trying to understand what exactly ``equation-free'' is supposed to be.
• J. Li, P. G. Kevrekidis, C. W. Gear and I. G. Kevrekidis, Deciding the nature of the coarse equations through microscopic simulations: The baby-bathwater scheme, SIAM Review, vol. 49, no. 3, pp. 469-487, 2007.
• G. Samaey, D. Roose and I. G. Kevrekidis, Finite difference patch dynamics for advection homogenization problems, preprint.

Section 6. A specific example where the issue of HMM and ``equation-free'' was discussed: Is this HMM or ``equation-free''?

• E. Vanden-Eijnden, Numerical techniques for multi-scale dynamical systems with stochastic effects, Comm. Math. Sci., vol. 1, no. 2, pp. 385-391, 2003.
• D. Givon, I. G. Kevrekidis and R. Kupferman, Strong convergence of projective integration schemes for singularly perturbed stochastic differential systems, Comm. Math. Sci., vol. 4, no. 4, pp. 707-729, 2006.
• E. Vanden-Eijnden, On HMM-like integrators and projective integration methods for systems with multiple time scales , Comm. Math. Sci., vol. 5, 498-505, 2007.

Section 7. An attempt to understand ``equation-free'':

• W. E and E. Vanden-Eijnden, Some critical issues for the "equation-free" approach to multiscale modeling, preprint.