Large Deviations Results for Non-Gradient Systems
Christian Cofoid
Boston College
cofoid@bc.edu
September 16, 2015
Christian Cofoid (Boston College)
Brown REU 2015
September 16, 2015
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Overview
1
Introduction
2
The Problem
3
Summer Work
4
Future Work
Christian Cofoid (Boston College)
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Introduction
Introduction
2
Consider the one dimensonal potential well given by V (x) = x 2 − 1 ,
and the ODE
dV
ẋ = −
.
dx
This ODE has two asymptotically stable fixed points at x = ±1 and a
saddle point at x = 0. Add dampened ”white noise” to the ODE, giving
the one-dimensional SDE:
dx(t) = −∇V (x)dt + εdW (t),
where ε 1, and W (t) is a canonical Weiner process. More generally, we
will consider the n-dimensional SDE
dX (t) = −∇V (X )dt + εdW (t).
Christian Cofoid (Boston College)
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(1)
September 16, 2015
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Introduction
Plot and Contour for n = 2.
2
Two Dimensional Gradient System with Noise: epsilon = 0.5
1.5
1
y
0.5
0
-0.5
-1
-1.5
-2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x
Figure: V (x, y ) = x 4 − 4xy + 2y 2
Christian Cofoid (Boston College)
Figure: 50 paths following
dx(t) = −∇V (x)dt + εdW (t),
ε = 0.5.
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Introduction
Current Knowledge
We can compute the expected time of escape τΩ from a domain of
attraction Ω centered about a stable fixed point. Let x ∗ ∈ Rn be the
stable fixed point and z ∗ ∈ Rn be the saddle point. Then, the expected
time of escape is approximated by the Eyring-Kramers law [Berg11];
s
| det (∇2 V (z ∗ )) |
2 (V (z ∗ ) − V (x ∗ ))
2π
exp
,
E [τΩ ] '
|λ1 (z ∗ ) |
det (∇2 V (x ∗ ))
ε2
where λ1 (z ∗ ) is the negative eigenvalue of the Hessian ∇2 V (z ∗ ).
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Introduction
Current Knowledge (cont’d)
A large deviation principle is a relation stating that for sufficiently small ε,
the probability of the sample paths xt being close to a function ϕ(t) is
given by
P ((xt ' ϕ(t), 0 ≤ t ≤ T ) ' e −I (ϕ)/2ε ,
where I (ϕ) is called the rate functional. It is known that
1
I (ϕ) =
2
Z
T
||ϕ̇(t) − f (ϕ(t))|| dt.
(2)
0
With some theoretical legwork, a large deviations principle tells us that if a
particle is going to escape Ω, it will do so in a ”straight shot” along the
heteroclinic orbit.
Christian Cofoid (Boston College)
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September 16, 2015
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The Problem
The Problem
If we add a small perturbation µg (X ), µ 1, and g is non-gradient to
equation (1), we obtain a non-gradient system
dX (t) = −∇V (X )dt + µg (X )dt + εdW (t).
(3)
The question of interest is; what is the expected time and expected path
of escape from Ω centered about an asymptotically stable fixed point p in
a perturbed system such as this?
Christian Cofoid (Boston College)
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September 16, 2015
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Summer Work
Work Done This Summer
1
We have found a technique that seems promising, called importance
sampling. Basically, we change the distribution from which we sample
N (0, 1) from one normal distribution to one with a larger mean,
increasing the probability of realizing large amounts of noise, and thus
the time needed to escape Ω. We then correct for this bias to find the
unbiased expected time of escape.
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Future Work
Future Work
1
We would like to conclude that our algorithm confirms the theoretical
results for gradient systems.
2
We would then like to apply this to non-gradient systems, and
conduct importance sampling on the expected path of escape.
Christian Cofoid (Boston College)
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