Acta Numerica (2012), pp. 1–87
c
Cambridge University Press, 2012
doi:10.1017/S09624929XXXXXXXX Printed in the United Kingdom
The heterogeneous multiscale method
∗
Assyr Abdulle
ANMC, Mathematics Section,
École Polytechnique Fédérale de Lausanne,
Lausanne, Switzerland
E-mail: assyr.abdulle@epfl.ch
Weinan E
Beijing International Center for Mathematical Research,
Peking University,
Beijing, China
and
Department of Mathematics and PACM,
Princeton University,
Princeton, USA
E-mail: weinan@math.princeton.edu
Björn Engquist
Department of Mathematics,
University of Texas,
Austin, USA
E-mail: engquist@ices.utexas.edu
Eric Vanden-Eijnden
Courant Institute of Mathematical Sciences,
New York University,
New York, USA
E-mail: eve2@cims.nyu.edu
The heterogeneous multiscale method (HMM), a general framework for de-
signing multiscale algorithms, is reviewed. Emphasis is given to the error
analysis that comes naturally with the framework. Examples of finite ele-
ment and finite difference HMM are presented. Applications to dynamical
systems and stochastic simulation algorithms with multiple time scales, spall
fracture and heat conduction in microprocessors are discussed.
∗
Colour online for monochrome figures available at journals.cambridge.org/anu.
2 A. Abdulle, Weinan E, B. Engquist and E. Vanden-Eijnden
CONTENTS
1 Introduction 2
2 The HMM framework 4
3 ODEs and dynamical systems 16
4 Finite element HMM 37
5 Finite volume methods 67
6 Conclusion 78
References 80
1. Introduction
The heterogeneous multiscale method (HMM) is a general framework for
designing multiscale algorithms for a wide variety of applications (E and
Engquist 2002, E and Engquist 2003, E et al. 2007a). The word ‘hetero-
geneous’ was used in order to emphasize the fact that the algorithm may
involve macro and micro models of very different natures: for example, the
micro model may come from molecular dynamics and the macro model may
come from continuum theory. In fact, at a very rough level, HMM can be
thought of as a way of blending together models of very heterogeneous types.
Most problems that we encounter in nature have a multiscale character.
The multiscale character can occur in a variety of ways. Take, for example,
problems from materials science, where many properties, such as conductiv-
ity, have a multiscale nature. This is the case for composites. It could also
be that the material can be viewed at different levels of detail: as a continu-
ous medium, in which case one naturally applies the principles of continuum
mechanics, or at the atomic scale, in which case one naturally applies var-
ious atomistic models of molecular dynamics or quantum mechanics. Each
viewpoint has its merits and drawbacks. Continuum models are quite effi-
cient but sometimes their accuracy is inadequate, particularly when defects
are involved. Atomic models are typically more accurate, but much less effi-
cient. This situation is not limited to materials science but is quite common
in most areas of science and engineering. One of the main motivations for
multiscale modelling is to develop models that have accuracy close to that
of microscopic models and efficiency close to that of macroscopic models.
From the viewpoint of numerical algorithms, we are interested in extract-
ing useful information from the microscopic model, which in principle has the
required accuracy. If we use the traditional viewpoint, then we would have
to solve the microscopic model in full detail, which is practically impossible
for engineering applications. In terms of computational complexity, the best
one can do with such an approach is to have linear scaling algorithms: the
complexity scales as the number of microscopic degrees of freedom. How-
ever, in many cases, we are not interested in the full microscopic solution or
The heterogeneous multiscale method 3
we cannot afford the cost of computing it. Instead, we are only interested in
the behaviour of some macroscopic variables, or the microscopic behaviour
in very small parts of the system, for example near defects. The question
is: Can we develop much more efficient algorithms, such as sublinear scal-
ing algorithms, that would give us such information? To develop these new
types of algorithms, we have to compress not only the operators, as has been
done in multigrid methods, but also the variables. We have to be content
with getting information about only a subset of the system variables. These
types of algorithms cannot be completely general: one has to explore special
features of the problem in order to construct such algorithms.
From the viewpoint of analysis, many analytical techniques have been
developed in order to derive simplified models. Examples include averag-
ing methods, homogenization methods, matched asymptotics, WKB meth-
ods, geometric optics approximations, and renormalization group methods
(E 2011). The principles of such techniques are quite general, but in practice
they only give us explicit analytical models in very limited situations. In
other situations, it is natural to ask whether one can devise efficient compu-
tational techniques based on these principles. This is the case that we are
interested in, and it was one of the main motivations for developing HMM.
This was the background against which HMM was proposed. Of course,
prior to HMM, there were already many techniques of a similar spirit in
many different fields. Well-known examples include:
• Car–Parrinello molecular dynamics, in which electronic structure mod-
els are used together with molecular dynamics to predict the dynamics
of nuclei (Car and Parrinello 1985),
• the quasicontinuum method, in which atomistic models are used to
analyse the mechanical deformation of crystalline solids (Tadmor, Ortiz
and Phillips 1996),
• superparametrization models, in which cloud-resolving models are used
to capture large-scale tropical dynamics of the atmosphere (Grabow-
ski 2001, Xing, Grabowski and Majda 2009).
HMM was proposed as a general framework that can be used for a variety of
problems. It was not the only attempt. Other notable examples include the
extended multigrid method and the equation-free approach (Brandt 2002,
Kevrekidis et al. 2003). A common theme of these approaches is that the
microscopic models are used throughout the computational process. These
should be contrasted with techniques such as model reduction methods,
wavelet-based homogenization and variational multiscale methods, in which
the microscale model is only used at the beginning of the computation to
obtain compressed effective operators.
In spite of competing efforts, HMM was the only general attempt based
on a top-down philosophy, which at the time was not the most popular
4 A. Abdulle, Weinan E, B. Engquist and E. Vanden-Eijnden
viewpoint. In fact, in the early days of multiscale modelling, most efforts
were devoted to a bottom-up approach, seeking strategies that would give us
the information needed by working only with the microscale model, without
any prior information about the system at the macroscale. This certainly
sounds very attractive, and may at first sight seem the only correct approach.
In one way, a key insight of HMM was the recognition that the bottom-up
approach is bound to have technical difficulties, and will for some time be
limited to rather simple applications. One can appreciate such difficulties
by noticing the fact that, even if the effective macroscale model is explic-
itly available, designing stable and accurate numerical algorithms for such
macroscale models is still a non-trivial task. Important constraints, such
as conservation properties or upwinding, have to be implemented in order
to guarantee that the algorithms give rise to the correct numerical solu-
tions. Implementing such constraints at the level of microscale models, in
the absence of any explicit knowledge about the macroscale model, seems
to be next to impossible. Therefore compromises are necessary: for many
problems we have to guess the form of the macroscale model to start from.
Fortunately, in many cases we do have some prior knowledge of the macro-
scale behaviour of the system under consideration, and this knowledge is
often sufficient for us to make an adequate guess.
Since multiscale modelling is a vast subject, touching almost all aspects of
modelling, we will not be able to do justice to all the work that has been done
on this subject. Instead we will focus on HMM. For a general introduction
to multiscale modelling, we refer to the monograph by E (2011).
2. The HMM framework
2.1. The main components of HMM
We will use U to describe the set of macroscopic variables, and u the set of
microscopic variables. They are related by
U = Q(u), (2.1)
where Q is called the compression operator. Any operator that reconstructs
u from U is called a reconstruction operator:
u = R(U). (2.2)
For consistency, Q and R should satisfy the relation
Q(R(U)) = U. (2.3)
In HMM, we assume that we have an incomplete macroscale model to
begin with:
F (U, D)=0. (2.4)
The heterogeneous multiscale method 5
reconstruction
compression
constraints
data estimation
U
uf(u, d)=0
F (U, D)=0
Figure 2.1. Schematics of the HMM framework.
Here D represents the missing part of the model. For example, if this is a
model in continuum mechanics, then D might be the constitutive relation
for the stress. If it is a model for molecular dynamics, then D might be
the inter-atomic forces. If it is a model for heat conduction in composite
materials, then D might be the macroscale effective conductivity tensor.
HMM proceeds by estimating the missing data on the fly using the micro-
scale model, at each location where some missing data is needed. To do
this, the microscale model has to be constrained so that its macrostate is
the same as the macrostate we are interested in, that is,
f(u, d(U )) = 0. (2.5)
Here d(U) represents the constraint for the microscale model. For example,
if the microscale model is the canonical ensemble of molecular dynamics, d
might be the average density, momentum and energy.
If we use H and h to denote the macro and micro numerical parameters,
such as mesh size, one can write HMM abstractly in the following form:
F
H
(U
H
,D
H
(u
h
)) = 0,
f
h
(u
h
,d
h
(U
H
)) = 0.
(2.6)
In practical terms, the basic components of HMM are as follows.
1 A macroscopic solver. Based on knowledge of the macroscale behaviour
of the system, we make an assumption about the form of the macroscale
model, for which we select a suitable macroscale solver. For example, if
we are dealing with a variational problem, we may use a finite element
method as the macroscale solver.
2 A procedure for estimating the missing macroscale data D using the
microscale model. This is typically done in two steps.
(a) Constrained microscale simulation. At each point where macroscale
data are needed, perform a series of microscopic simulations which
are constrained so that they are consistent with the local value of
the macro variable.
![Figure 5.1. A schematic illustration of the numerical procedure. Starting from piecewise constant solutions unk , one integrates (5.6) in time and in the cell xk, xk+1. The time step Δt is chosen in such a way that the waves coming from xk+1/2 will not reach xk, and thus for t ∈ [tn, tn+1], u(xk, t) = u n k . If f(u) at xk is found to be unknown, we perform an MD simulation using unk to invoke and restrict the microscopic process. The required flux is then extracted from the simulation and the integration is completed. Analogously one can embed the MD simulation in higher-order macro schemes or higher dimensions.](/figures/figure-5-1-a-schematic-illustration-of-the-numerical-118y1m9b.webp)