Weinan E. Principles of Multiscale Modeling

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2.7. THE MORI-ZWANZIG FORMALISM 87

Substituting this into the ﬁrst equation, we obtain

= A

p + A

(t−τ)

p(τ) dτ + A

q(0)

= A

p +

K(t − τ)p(τ) dτ + f(t),

where K(t) = A

is the memory kernel and f(t) = A

q(0) is the so-called

noise term, if we think of q(0) as a random variable. Note that it do es make sense to

think of q(0) as a random variable since we are not really interested in the details of q.

Our next example is the original model considered by Zwanzig [?]. Consider a system

with the Hamiltonian:

H(x, y, p, q) =

+ U(x) +

k=1



−



One may think of (x, y) as the variables that describe the phase space of a distinguished

particle, while {(q

, p

)} describes other particles that provide the thermal bath. We will

use the Mori-Zwanzig formalism to integrate out the variables {(q

, p

)}. The Hamilton’s

equations are

˙x = y, (2.7.3)

˙y = −U

′

(x) +

−

x), (2.7.4)

˙q

= p

, (2.7.5)

˙p

= −ω

+ γ

x. (2.7.6)

Integrating the equations for q

and p

, we get

(t) = q

(0) cos ω

t +

(0)

sin ω

t −

sin(ω

(t − τ ))x(τ) dτ.

Therefore, the equations for x and y become

˙x = y,

˙y = −U

′

(x) −

cos(ω

(t − τ ))y(τ ) dτ



(0) −

x(0)) cos ω

t +

(0)

sin ω



88 CHAPTER 2. ANALYTICAL METHODS

Again, this has the form of a generalized Langevin equation with memory and noise

terms. Under suitable assumptions on the initial values q

(0) and p

(0), k = 1, ··· , N,

the noise term will be Gaussian.

Our next example is a general procedure for eliminating variables in linear elliptic

equations. Consider a linear diﬀerential equation

Lu = f (2.7.7)

Here L is some nice operator, f is the forcing function. Let H be a Hilbert space which

contains both u and f, and let H

be a closed subspace of H, containing only the coarse-

scale component of the functions in H. Let P be the projection operator to H

and

Q = I −P : Functions in H can be decomposed into the form:

v = v

+ v

′

, v ∈ H, v

∈ H

, v

′

∈ H

⊥

(2.7.8)

Here H

⊥

is the orthogonal complement of H

, v

and v

′

are respectively the coarse and

ﬁne scale part of v.

= P v, v

′

= Qv (2.7.9)

We can write (2.7.7) equivalently as:

P LP u + P LQu = P f (2.7.10)

QLP u + QLQu = Qf (2.7.11)

Assuming that QLQ is invertible on H

⊥

, we can write abstractly an equation for u

= P u

[?]:

LP u =

f (2.7.12)

where

L = PLP − P LQ(QLQ)

−1

QLP,

f = P f − P LQ(QLQ)

−1

Qf (2.7.13)

This is an abstract upscaling procedure, and (2.7.12) is an abstract form of an upscaled

equation. Note that even if the original equation was local, say an elliptic diﬀerential

equation, (2.7.12) is in general nonlocal. Its eﬀectiveness depends on how to choose

and whether the operator

L can be simpliﬁed. We will return to this issue later

in Chapter 8 when we d iscuss upscaling of elliptic diﬀerential equations with multiscale

coeﬃcients.

We now explain the general Mori-Zwanzig formalism. The main steps are as follows

[?]:

2.7. THE MORI-ZWANZIG FORMALISM 89

1. Divide the variables into two sets: A set of retained variables and a set of variables

to be eliminated.

2. Given the time history of the retained variables and the initial condition for the

variables to be eliminated, one can ﬁnd the solution for the variables to b e elimi-

nated.

3. Substituting these solutions into the equation for the retained variables, one obtains

the generalized Langevin equation for the retained variables, if we regard the initial

condition for the eliminated variables as being random.

We now show how these steps can be implemented [?].

Consider a general nonlinear system

= f(x) (2.7.14)

Let ϕ

be a function of x. Deﬁne the Liouville operator L:

(Lϕ

)(x) = (f · ∇ϕ

)(x) (2.7.15)

Let ϕ(t) = ϕ

(x(t)) where x(t) is the solution of (2.7.14) at time t. Using the semi-group

notation, we can write ϕ(t) as

ϕ(t) = e

(2.7.16)

Now let x = (p, q) where q is the variable to be eliminated and p is the retained vari-

able (see for example [?]). We will put ourselves in a setting where the initial conditions

of q are random. Deﬁne the projection operator (as a conditional expectation):

P g = E(g|q(0)) (2.7.17)

where g is a function of x, and let Q = I −P . Let ϕ be a function of p only, then

ϕ(t) = e

Lϕ(0) = e

P Lϕ(0) + e

QLϕ(0) (2.7.18)

Using Dyson’s formula:

= e

tQL

(t−s)L

P Le

sQL

ds (2.7.19)

90 CHAPTER 2. ANALYTICAL METHODS

we get

ϕ(t) = e

P Lϕ(0) +

(t−s)L

K(s)ds + R(t) (2.7.20)

where

R(t) = e

tQL

QLϕ(0), K(t) = P LR(t). (2.7.21)

This is the desired model for the retained variables. As one can see, this is not in the

form of an equation for p directly. Instead, it is an equation for any functions of p.

The term R(t) is usually regarded as a noise term since it is related to the initial

conditions of q which are random. The term involving K(t) is a memory term and is

often dissipative in nature, as can b e seen from the examples discussed earlier. Therefore

(2.7.20) is often regarded as a generalized Langevin equation.

2.8 Notes

This chapter is a crash course in the analytical techniques used in multiscale problems.

Many textbooks have been written on some of the topics d iscussed here. Our treatment

is certainly not thorough, but we have attempted to cover the basic issues and ideas.

The techniques discussed here are largely considered to be understood, except for the

renormalization group method and the Mori-Zwanzig formalism. While many physicists

use renormalization group analysis at ease, mathematical understanding of this technique

is still at its infancy. This is quite a concern, since after all, the renormalization group

analysis is a mathematical technique. Our treatment of the two-dimensional Ising model

has largely followed that of [?]. One can clearly see that drastic approximations seem to

be n ecessary in order to obtain some numbers that can be compared with experiments,

or in this case, the exact solution of Onsager.

For some progress on the renormalization group analysis of one-dimensional models

and the Feigenbaum universality, we refer to [?].

The issue with the Mori-Zwanzig formalism is quite diﬀerent. Here the question is:

What should we do with it? Mori-Zwanzig formalism is formally exact, but the resulting

model is often extremely complicated. Clearly we need to make approximations but

what is not so clear is how one should make approximation and how accurate these

approximations are. We will return to this issue at the end of the book.

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Chapter 3

Classical Multiscale Algorithms

In this chapter, we discuss some of the classical algorithms that have been developed

using multiscale ideas. Unlike the algorithms that we discuss in the second half of this

volume, the methods presented here are developed for the purpose of resolving the ﬁne

scale details of the solutions. For this reason, the best we can hope for from the view-

point of computational complexity is linear scaling. This should be contrasted with the

sublinear scaling algorithms that we will discuss in the second half of this volume.

We have chosen the following examples:

1. The multigrid method.

2. The fast multi-pole method.

3. Domain decomposition methods.

4. Adaptive mesh reﬁnement.

5. Multi-resolution representation.

A lot has been written on these topics, including some very nice monographs (see for

example, [1, 13, 37]). It is not our intention to discuss them in great detail. Our focus

will be on the main ideas and the general aspects that are relevant to the overall theme

of this volume.

96 CHAPTER 3. CLASSICAL MULTISCALE ALGORITHMS

3.1 Multigrid method

Although its basic ideas were initiated in the 1960’s [22], multigrid method only

became a practical tool in the late 1970’s. Initially, multigrid method was developed

as a tool for solving the algebraic equations that arise from discretizing PDEs [11, 27].

Later, it was extended to other contexts such as the Monte Carlo methods and molecular

dynamics [12, 24]. In particular, Brandt proposed to extend the multigrid method as a

tool for capturing the macroscale behavior of multiscale or multi-physics problems [12],

a topic that we will take up in Chapter 6. Here we will focus on the classical multigrid

method.

Consider the following problem:

−△u(x) = f (x), x ∈ Ω (3.1.1)

with the boundary condition u|

∂Ω

= 0. The standard procedure for solving such a PDE

is:

1. First, discretize the PDE using, for example, ﬁnite diﬀerence or ﬁnite element

methods. This gives rise to a linear system of equations of the form:

= b

(3.1.2)

where h is the typical grid size used in the discretization.

2. Solve the linear system (3.1.2) using some direct or iterative methods.

The rate of convergence of iterative methods depends on t he condition number of the

matrix A

, deﬁned as κ(A

) = ||A

||A

−1

. Here ||A

is the l

norm of the matrix

. In most cases th e condition number of A

scales as:

κ(A

) ∼ h

−2

(3.1.3)

This suggests that as h becomes smaller, the iterative methods also converge more slowly.

It is easy to understand why this has to be the case: The ﬁnite diﬀerence or ﬁnite element

operators are local, b ut the solution operator (the Green’s function for (3.1.1)) is global.

If one makes an error at one corner of the domain, it aﬀects the solution at all other

places. If the iteration method is local as most iterative methods are, in order to remove