1. The inverse Smoluchowski problem for
cluster-cluster aggregation
Colm Connaughton
Mathematics Institute and Centre for Complexity Science,
University of Warwick, UK
Joint work with Robin Ball and Peter Jones.
Particles in turbulence
University of Potsdam
17 March 2011
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2. Cluster aggregation and Smoluchowski equation
Physical picture:
Large "cloud" of particles moving around (eg by
turbulence).
Particles merge irreversibly on contact.
Rate of merging of particles with masses, m1 and m2 is
K (m1 , m2 ). Kernel K (m1 , m2 ) encodes microphysics.
Size distribution, Nm (t), is the average density of clusters of
mass m at time t.
Smoluchowski equation :
m
∂t Nm (t) = dm1 K (m1 , m − m1 )Nm1 (t)Nm−m1 (t)
0
∞
− 2Nm (t) dm1 K (m, m1 )Nm1 (t)
0
(Smoluchowski, 1916)
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3. Equivalent formulation
It is convenient to work with the cumulative cluster size
distribution: m
Fm (t) = m1 Nm1 (t)dm1 .
0
The usual cluster size distribution is
1 ∂Fm (t)
Nm (t) = .
m ∂m
In terms of Fm (t) we have:
Equivalent Smoluchowski equation:
m ∞
dFm2 (t)
∂t Fm (t) = − dFm1 (t) K (m1 , m2 )
0 m−m1 m2
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4. Scaling Solutions of the Smoluchowski equation
In many applications kernel
is a homogeneous function:
K (am1 , am2 ) = aγ K (m1 , m2 )
Resulting cluster size
distributions exhibit
self-similarity.
Self-similar solutions have the form
m
Fm (t) ∼ s(t)a F (z) z=
s(t)
where s(t) is the typical cluster size and a is a dynamical
scaling exponent. The scaling function, F (z), determines the
shape of the cluster size distribution.
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5. The inverse Smoluchowski problem
Forward problem: given kernel, K (m1 , m2 ), compute the size
distribution, Fm (t).
Inverse problem: given observations of the size distribution,
Fm (t), compute the kernel, K (m1 , m2 ) (Wright and
Ramakrishna, 1992).
Inverse problem is useful because:
Kernel may not be known.
May help in building models and guiding micro-physics
theory.
Quantifies the sensitivity of the size distribution to
variations in the kernel.
but
The inverse problem is typically ill-posed.
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6. Ill-posedness at the discrete level
Assume scaling. Then scaling function, F (z), must satisfy:
z ∞
dF dF (z2 )
z =− dF (z1 ) K (z1 , z2 ).
dz 0 z−z1 z2
Linear in K (z1 , z2 ).
Assume we have measurements of the scaling function,
F (z), at N discrete z-points.
Discretises to a set of N linear equations for the N 2 values
of the K (z1 , z2 ) on the discretisation points:
b = S k.
This system is enormously under-determined ⇒ one can
find many solutions but they are all entirely determined by
the noise in the data.
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7. Tikhonov Regularisation (Ridge regression)
One way of dealing with under-determinedness is to solve a
minimization problem. The estimated kernel is:
kest = arg min |S k − b|2 + λ |k|2 .
k
Noise-dominated solutions have to compete against the
regularization term λ |k|2 . The trick is to choose the "best" value
of the regularization parameter, λ.
A rational approach to
determining λ is provided
by an “L-curve”. Plot the
size of the solution, |k|, as
a function of the residual,
|S k − b| (Hansen 1992).
"Best" values of λ are near
the kink in the curve.
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8. Does it work? Numerical solution of the inverse
problem with known kernel
Constant kernel case K (z1 , z2 ) = 1
Diagonal of reconstructed kernel. L-curve.
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9. Does it work? Numerical solution of the inverse
problem with known kernel
Sum kernel case K (z1 , z2 ) = 1 (z1 + z2 )
2
Diagonal of reconstructed kernel. L-curve.
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10. Does it work? Numerical solution of the inverse
problem with known kernel
1 √ √
Sqrt sum kernel case K (z1 , z2 ) = 2 ( z1 + z2 )
Diagonal of reconstructed kernel. L-curve.
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11. Conclusions and Future work
Although the inverse Smoluchowski problem is ill-posed,
some features of the collision kernel can be reconstructed
from measurements of the size distribution.
We have demonstrated proof-of-concept but much remains to
be investigated:
Allow more flexibility in the class of potential kernels.
What can we do without assuming scaling?
Can we handle gelling kernels?
Does the method break entirely if we add a source of
monomers, fragmentation, condensation?
Noisy data?
Is it useful for real-world problems?
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