Weinan E. Principles of Multiscale Modeling

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Chapter 11

Some Pe rspectives

Despite the explosive growth of activities on multiscale modeling in recent years, this

is still a largely open territory. Many important issues are not clearly understood. Let

us mention a few problems that are of general interest.

1. Better understanding of the renormalization group analysis. The renormalization

group method is one of the most popular analytical techniques in physics for han-

dling multiscale problems. However, at a more rigorous level, it is largely a mystery.

Since the renormalization group analysis carried out in the physics literature typi-

cally involves drastic truncations, it is very desirable to have a b et ter understanding

of the nature of the error caused by these truncations.

2. Systematic analysis of the memory eﬀects using the Mori-Zwanzig formalism. As

we discussed in Chapter 2, the Mori-Zwanzig formalism is a systematic way of

eliminating variables in a system. However, it often leads to very complicated

models with memory terms. Therefore it is natural to ask: How do we make

systematic approximations for the memory eﬀects? We will discuss one example

below for the case when the interaction involving the eliminated variables is linear.

However, it would be very useful to develop more general strategies. In particular,

it would be of interest to use this kind of approximation techniques to assess the

accuracy of memory-dependent hydrodynamic mod els of polymeric ﬂuids such as

the Maxwell models [3].

3. Numerical analysis for the eﬀect of noise. It was possible to carry out the kind of

analysis of the quasi-continuum method presented in Chapter 7 because the eﬀect of

481

482 CHAPTER 11. SOME PERSPECTIVES

noise can b e ignored. This is not the case for most multiscale problems. Indeed, as

we ind icated in the notes in Chapter 7, one major question in formulating coupled

atomistic-continuum models is whether we should add noise terms to the continuum

models. Therefore, it is very important to develop a better framework for assessing

the numerical error in the presence of noise.

4. Systematic analysis of the error in QM-MM. From a numerical analysis viewpoint,

one major diﬃculty is associated with the fact that we do not have an obvious

small parameter (such as grid size, or ratio of time scales) with which we can speak

about the order of accuracy, or assess the accuracy asymptotically.

5. Adaptive algorithms. Adaptivity is crucial for the success of multi-scale algorithms.

In fact, multiscale, multi-physics modeling introduces another dimension for adap-

tivity: Besides traditional issu es such as adaptively selecting the grid sizes and

algorithms, one also has the issue of adaptively selecting the models, the coarse-

grained variables, and the partition of the set of slow and fast comp onents of the

dynamics.

Most of these problems are geared toward a better or more rigorous understanding

of things. This is natural since as we said in the beginning, one main motivation for

multiscale modeling is to put more rogor into modeling.

There are many other problems, of course. Instead of going through more problems,

we will provide some personal perspective on problems without scale separation.

We have seen throughout t his book that scale separation is a key property that one

can use in order to derive reduced models or develop sublinear scaling algorithms. Unfor-

tunately many multiscale problems we encounter in practice do not have this property.

A most well-known example is fully developed turbulent ﬂows [20].

The ﬁrst thing one should ask for this kind of problems is whether there are other

special features that one can take advantage of. We have discussed the case when the

problem has (statistically) self-similar behavior. In this case, one can either use the

renormalization group method discussed in Section 2.6, or HMM as we illustrated in the

notes of Chapter 6. This allows us to obtain reduced models or develop sublinear scaling

algorithms.

Fully developed turbulent ﬂ ows are expected to be approximately self-similar in a

statistical sense, but due to intermittency, one does n ot expect self-similarity to hold

483

exactly. Characterizing the kind of more sophisticated scaling behavior in turbulent

ﬂows is still among the most important grand challenges in classical physics [20].

This brings us to the inevitable question: What if the problem does not have any

special features that we can take advantage of? One answer to this question is simply

that there is not much one can do and one just has to solve the original microscale

problem by brute force. There are no alternatives even if one is only interested in the

macroscale behavior of the solution.

However, it is often more productive to formulate a diﬀerent question: What is the

best appproximate solution one can get, given the maximum cost one can aﬀord? The

answer to this question goes under the general umbrella of variational model reduction.

This is consistent with the philosophy of optimal prediction advocated by Chorin et al.

[4, 5] and can indeed be regarded as being a poorman’s version of optimal prediction.

We will discuss some general guidelines for this kind of problems. A theoretical

guideline is the Mori-Zwanzig formalism. As was explained in Section 2.7, this is a

universal tool for eliminating degrees of freedom for any kind of problems. However, the

complexity of the problem is not substantially reduced unless we make approximations.

Therefore in practice, the task is to ﬁnd the best approximation within a speciﬁed class

of models.

11.0.1 Variational model reduction

There are two main ingredients in variational model reduction [12]:

1. The class of models in which the reduced model lies, including the choice of the

reduced (or coarse-grained) variables.

2. An objective function with which we can discuss optimality.

The class of models can be:

1. Some speciﬁc form of models, as in the case of HMM (see Section 6.8). For example,

if the microscale model is the Boltzmann equation, we may consider the class of

models that are nonlinear conservation laws for the 0th, 1st, 2nd and 3rd order

moments, with ﬂuxes depending only on the local values of these moments.

484 CHAPTER 11. SOME PERSPECTIVES

2. Models of limited complexity. For example, if the microscale model is given by

a large N × N matrix A, we may consider the class of rank-M matrices, where

M ≪ N.

The objective function is usually some measurement of the error for a class of solutions

to the original microscale model. Obviously the quality of the reduced model crucially

depends on these two factors.

After these ingredients are speciﬁed, we still have the practical problem of how to

obtain the r educed model. We will illustrate these practical issues using the example of

operator compression [10]: Given a large N × N symmetric positive deﬁnite matrix A,

and a number M which is much smaller than N, we would like to ﬁnd the rank-M matrix

that best approximates A, in the sense that

I(B) = kA

−1

− B

∗

(11.0.1)

is minimized. Here B is a N × N matrix whose rank is no more than M, B

∗

is its

pseudo-inverse.

It is well known that if A has the diagonalized form:

A = O

ΛO (11.0.2)

where O is an N ×N orthogonal matrix, Λ = diag(λ

, λ

, ··· , λ

), and λ

≤ λ

≤ ··· ≤

, then the optimal approximation is given by

= O

O (11.0.3)

where Λ

= diag(λ

, λ

, ··· , λ

, 0, ··· , 0). A

can also be regarded as the projection of

A onto its eigen-subspace that corresponds to its M smallest eigenvalues. We will denote

this subspace by V

To ﬁnd A

, the most obvious approach is to diagonalize A. This is a general solution.

However, its computational complexity scales as N

or at least M

N [14, 8].

In many cases, A comes from the discretization of some diﬀerential operator which

is local, or it is t he generator of some Markov chain whose dynamics is mostly local. In

this case, it might be possible to develop much more eﬃcient algorithms, as was shown

in [10]. The key idea is to choose a localized basis for V

in order to represent A

. This

is possible in many situations of practical interest.

485

Let us ﬁrst look at a simple example, the one-dimensional Laplace operator with

periodic boundary condition with period 2π. Let M = 2n + 1, where n is an integer.

The eigen-subspace of interest is

= span

−inx

, e

−i(n−1)x

, ··· , e

inx

(11.0.4)

This expression also provides a basis for V

. Obviously, these b asis functions are non-

local.

One can also choose other basis sets. For example, if we let

(x) =

k=−n

ik(x−x

)

, j = −n, −n + 1, ··· , n (11.0.5)

where x

= 2jπ/(2n + 1), then {u

} forms a basis of V

. This is called the Dirichlet

basis since they correspond to th e Dirichlet kernel in Fourier series. One feature of this

basis set is that the basis functions decay away from their centers. Therefore we should

be able to obtain a set of approximate basis functions by truncating the functions {u

}

away from the centers.

However, the Dirichlet kernel does not decay very fast – it has a slowly decaying

oscillatory tail. To understand this, we can view the Dirichlet kernel as the projection

of δ-funct ion, which is the most localized function, to the subspace V

: Let δ(x) =

∞

k=−∞

ikx

, its projection to V

(x) = D

(x) =

k=−n

ikx

sin(n +

sin

, −π ≤ x ≤ π

which is the Dirichlet kernel. The slow decay of the oscillatory tail of D

is a result of

the Gibbs phenomenon caused by the lack of smoothness of δ(x). An obvious idea for

obtaining basis functions with better decay properties is to use ﬁltering [15]. We deﬁne

(x) =

k=−n





ikx

, (11.0.6)

where σ(·) is a ﬁlter. It is well-known that the decay properties of δ

depends on the

smoothness of the ﬁlter σ. The smoother the ﬁlter, the faster the decay of δ

. See [15]

or [10] for the precise formulation of these statements.

486 CHAPTER 11. SOME PERSPECTIVES

This discussion only applies to the particular example we just discussed. In the

general case, one obvious way of getting basis functions with fast decay properties is to

solve the optimization problem:

min

ψ∈V

, ψ6=0

B[ψ] =

w(x)|ψ(x)|

|ψ(x)|

, (11.0.7)

where w(·) is an even, nonnegative weight function. By centering w at various locations,

one obtains a set of basis functions that decay away from the centers. The rate of decay

depends on the choice of w. Such ideas have been explored in [19] for electronic structure

analysis where w(x) = x

is used. Oth er examples of weight functions include:

1. w(x) = x

where p is an integer [10].

2. w(x) = 1 − ξ

[−1,1]

(x) where ξ

[−1,1]

is the indicator function of the set [−1, 1] [13].

It is natural to ask how w determines the decay property of the basis functions selected

this way. This question was addressed in [10] for the example discussed above. It is shown

that the decay rate of the basis function is determined by how fast w(x) approaches 0 as

x → 0.

Lemma. Assume that w is smooth in (−π, π). If w

(2l)

(0) is the ﬁrst nonzero term in

(2m)

(0) | m ∈ N}, then

|ψ(x)| ≤

|nx|

l+1

, x ∈ (−π, π), x 6= 0, (11.0.8)

where C is a constant independent of n and x.

This lemma suggests that if we take w(x) = x

, then the larger the value of p, the

faster the basis functions decay. However, choosing large values of p is bad for numerical

stability and accuracy, since the condition number of the problem (11.0.7) increases very

fast as a function of p [10]. Therefore, the best weight function has to b e a compromise

between the decay rate and numerical stability. Numerical results suggest that the weight

functions x

and x

are good choices.

Qualitatively similar results are expected to hold for more general cases, such as the

case when V

is a high dimensional eigen-subspace of some elliptic diﬀerential operator.

But this has not been investigated.

The standard numerical technique for ﬁnding eigen-subspaces is the subspace itera-

tion method [14, 21]. This is a generalization of the power method for computing leading