Markovian approximation of underresolved
mechanical systems
Carsten Hartmann (Matheon, FU Berlin)
joint work with A. Stuart (U Warwick), K. Zygaklakis (U
Oxford) and M. Luskin (U Minnesota)
Edinburgh, July 1, 2010
Outline
Motivation
Underresolved mechanical systems
Generalized Langevin equation
Markovian approximation
Balanced truncation
Example: linear chains
Atomistic modelling of materials
I
Equations of motion on Q ⊆ Rn
M q̈ + ∇V (q) = f .
I
Hooke’s law: ∇V (q) = Kq
Boundary conditions, thermostatting
I
Distributed initial conditions
(q, v ) ∼ ρ
I
Dislocations, cracks or defects
But: system is stiff (multiple
time and length scales).
Semi-empirical force fields (ab-initio, tight-binding etc.): Pettifor (U Oxford), Tersoff (IBM), . . . ; Quasi-continuum
methods: Tadmor & Luskin (U Minnesota), Philipps & Ortiz (Caltech), . . . ; HMM (multiphysics, homogenization):
E (Princeton), Brandt (Weizman Inst.), Li (Penn State), Friesecke (TUM), . . .
Wave equation as a paradigm
Harmonic lattice:
mi,j q̈i,j =qi−1,j − 4qi,j + qi+1,j
+ qi,j−1 + qi,j+4
with mi,j > 0, i, j = 1, . . . , n
Ensemble of vibrating strings
Linear wave equation:
µ(x)∂tt q(x, t) = ∆q(x, t)
Coupled oscillators
with mass density µ > 0.
Motivation
Underresolved mechanical systems
Generalized Langevin equation
Markovian approximation
Balanced truncation
Example: linear chains
High-dimensional linear mechanical systems
I
Consider a system assuming states (q, v ) ∈ TQ with energy
1
1
E = v · Mv + q · Kq .
2
2
I
where TQ ∼
= Q × Rn , Q ⊆ Rn and M, K ∈ Rn×n s.p.d.
The system’s dynamics are governed by Newton’s equations
M q̈ + Kq = 0 ,
q(0) = q , q̇(0) = v
with Gaussian initial conditions
(q, v ) ∼ exp(−βE ) ,
I
Suppose n is large.
β = (kT )−1
Spatial decomposition
Still suppose that n is large and pick a
number “relevant” atoms or clusters
thereof (e.g., all the red atoms).
I
Call S = span{v1 , . . . , vd } ⊂ Rn the configuration subspace of
the resolved variables and let the matrix
P = V (V ∗ M −1 V )−1 V ∗ M −1 ,
P2 = P
be the oblique projection onto S ⊂ Rn with PM(I − P) = 0.
I
Given the initial distribution for q and q̇, we see a reduced
description of the dynamics of the low-dimensional variable
x = (V ∗ MV )−1 V ∗ M −1 q ,
x ∈ Rd ⊂ S .
Generalized Langevin equation
I
Invariance of the Gaussian distribution ρ ∝ exp(−βE ) under
the Newtonian dynamics implies that x solves the GLE
Z t
M1 ẍ(t) +
D(t − s)ẋ(s) ds + K1 x(t) = ζ(t) .
0
where ζ is a coloured Gaussian process with covariance
E[ζ(t) ⊗ ζ(s)] = β −1 D(t − s) .
I
Its unique stationary distribution is given by the marginal
ρ̄ ∝ exp(−β Ē ) where
1
1
Ē = ẋ · M1 ẋ + x · K1 x ,
2
2
I
−1 ∗
K1 = K11 − K12 K22
K12
is the free energy of the resolved variables.
No magic, just a bit of linear algebra and Laplace transform.
Ford, Kac & Mazur, J. Math. Phys. 1965; Zwanzig, J. Stat. Phys., 1973; . . .
Motivation
Underresolved mechanical systems
Generalized Langevin equation
Markovian approximation
Balanced truncation
Example: linear chains
The memory kernel
I
Given the initial distribution of the unresolved variables, the
generalized Langevin equation is exact.
I
However the dynamics are no longer Markovian.
I
Even worse, the symmetric memory kernel
−1 ∗
D(t) = K12 cos(M2−1 K22 t)K22
K12 .
depends on the high-dimensional unresolved system.
I
Further notice that D is periodic, i.e., it does not decay (the
period diverges in the thermodynamic limit).
Low-rank Markovian approximation
I
We may still find a low-rank approximant for D on the
interval t ∈ [0, T ] that is of the form
D̃(t) = B exp(−Ct)B ∗
where −C ∈ Rk×k is Hurwitz and k n − d.
I
If we replace D by D̃ and ζ by ζ̃ with covariance β −1 D̃, then
our GLE is equivalent to the augmented system
M1 ẍ + K1 x = Bη
dη = (−C η + B ∗ ẋ) dt +
q
β −1 (C + C ∗ )dW
in Rn+d with random initial conditions η(0) ∼ N (0, β −1 I ).
cf. Kupferman, J. Stat. Phys., 2004
Recasting the memory kernel
I
It is helpful to consider the integral operator
Z t
D(t − s)u(s) ds
(Gu)(t) =
0
as the transfer function (TF) G : L2 [0, T ] → L2 [0, T ] of
ż(t) = Az(t) + Fu(t) ,
z(0) = 0
∗
y (t) = F z(t)
with the 2(n − d) × 2(n − d) coefficient matrices
!
!
1/2
1/2
−1/2 ∗
0 K22 M2−1 K22
K22 K12
A=
.
, F =
−I
0
0
I
Approximating D means approximating the above ODE.
Rational approximation of the memory kernel
I
Consider the finite-time Gramians
Z T
QT =
exp(At)FF ∗ exp(A∗ t) dt
0
Z T
PT =
exp(A∗ t)F ∗ F exp(At) dt
0
on a sufficiently long interval [0, T ] on which D is decaying.
I
The controllability Gramian QT measures to what degree
the kernel is excited by an input u(t) = ẋ(t).
I
The observability Gramian PT measures to what degree the
kernel passes it down to the output (i.e., produces friction).
I
Idea: Compute low-rank factors of the product QT PT .
B.C. Moore, IEEE Trans. Auto. Contr.,1981; K. Glover, Int. J. Control, 1984
Balanced truncation of the memory kernel
Since QT and PT are symmetric positive semi-definite, there exists
a coordinate transformation z 7→ Sz, such that
S −1 QT S −T = diag(Σ1 , Σ2 , 0, 0)
S T PT S = diag(Σ1 , 0, Σ3 , 0)
with Σ1 , Σ2 , Σ3 0 independent of the choice of coordinates.
Theorem (Snazier 2002, H. 2010)
Let Σ1 = diag(σ1 , . . . , σs ) and let G̃ be the TF obtained by
projecting the ODE onto the dominant k < s columns of S. Then
sup
k(G − G̃ )uk[0,T ] < 2(σk+1 + . . . + σs ).
1=kuk[0,T ]
where k · k[0,T ] is the L2 norm on [0, T ].
Snazier et al., Proc. Amer. Control Conf., 2002; H., Zygalakis & Stuart, preprint, 2010
Motivation
Underresolved mechanical systems
Generalized Langevin equation
Markovian approximation
Balanced truncation
Example: linear chains
Recall: discrete wave equation
Harmonic lattice:
mi,j q̈i,j =qi−1,j − 4qi,j + qi+1,j
+ qi,j−1 + qi,j+4
with mi,j > 0, i, j = 1, . . . , n
Ensemble of vibrating strings
Linear wave equation:
µ(x)∂tt q(x, t) = ∆q(x, t)
Coupled oscillators
with mass density µ > 0.
Three heavy particles forming a cluster (d=3)
1.8
1.6
1.4
1.2
mi
1
Chain of n oscillators:
0.8
0.6
0.4
0.2
mi q̈i = qi−1 − 2qi + qi+1
0
0
5
10
15
20
i
First 20 Hankel singular values σi
with

o
nh n i h n i
 5 if i ∈
,
±1
2
2
mi =
 1 else.
2.5
exact
rankï2
rankï4
rankï8
2
D
1.5
1
0.5
0
ï0.5
0
5
10
15
t
20
25
30
Memory kernel and its approximants
Lamb’s system (radiation induced damping)
Particle coupled to a wave (d=1):
0.45
0.4
mi q̈i = qi−1 − 2qi + qi+1
0.35
i
m(100)
0.3
0.25
with mi = 1, i ∈ N
0.2
0.15
0.1
0.05
0
0
2
4
6
8
10
mode #
1
n=40
n=100
n=200
First 10 Hankel SVs for n = 100, d = 1
D(n)
0.5
0
ï0.5
0
20
40
60
80
100
t
Thermodynamic limit of Lamb’s system, τ ∼ n
Lamb, Proc. London Math. Soc., 1900
Memory kernel and its approximants
Conclusions and open problems
I
On finite time intervals [0, T ] the underresolved system can
be nicely embedded into a space of moderate dimension.
I
The choice of T and the stable subspace is critical and by
now requires “probing”.
I
The Hankel norm bound is very rough and an error bound for
the Markovian approximation would be nice.
I
Thermodynamic limit of the Lamb system and relation to
quasi-continuum methods still open?
I
Extension to infinite-dimensional problems.