Determining Cluster-Cluster Aggregation Kernels
Using Inverse Methods
Peter P. Jones, Supervisors: Dr. Colm Connaughton, Prof. R. C. Ball
Centre for Complexity Science, Zeeman Building, University of Warwick, CV4 7AL, Coventry, UK
Introduction
I Many processes in nature and industry involve large scale coalescence or
aggregation of mass clusters.
K(m,n)
Am + An −−−→ Am+n
I Such processes can be accurately modelled by a mean-field approximation
(MFA) if spatial correlations are absent or can be factored out.
I As there are few analytic solutions to the MFA, determination of the
aggregation rate function, or kernel, K, of the MFA is often via a mixture
of experimentation, simulation, and experienced guesswork.
I Existing inverse methods for aggregation processes may rely on further
assumptions about K, and these assumptions might not hold.
I Also, it might not be feasible to carefully control natural processes.
Aims
I In this research we seek to determine the extent to which key inverse
methods can be held to apply when the key assumptions above cannot be
guaranteed for an observed aggregation process.
I Specifically, we seek to determine the extent to which the existence of
noise and spatial correlations in an aggregation process can be ‘folded
back’ into an estimation of a kernel function that could then be used as a
phenomenological surrogate within the MFA model.
I Such a phenomenological model might then prove useful for enhancing the
accuracy of some models of larger processes that include aggregation
processes.
Inverse Methods
I At their heart, most inverse methods work by ‘least-squares’ minimizing the
square of the normed error of a linear mapping, y = Ak, until a suitable fit
k for the observed data y is found, if this is feasible.
I However, inverse problems are often ill-posed.
I Regularisation
. Ill-posedness can be overcome to some degree by adding a regularising
term to that penalises solutions with large fluctuations.
J(k) = min kAk − yk21 + λreg kkk22
k
. As a first step, we implemented this approach as used specifically for the
derivation of aggregation kernels[3] when the SH applies.
. However, the recommended regularisation did not work, and
experimentation led us to use[1]:
!
Y
λreg w(k) = λreg log
(|ki| + 1)i
i
Results
I Decay Problem: For suitable values of the regularisation parameter, λreg ,
we retrieved good approximations to input kernels K(x, y) = 12 (xξ + yξ ),
ξ ∈ {0.0, 0.25, 0.50, 0.75, 1.0} from simulated data with some
inherent noise. The most interesting cases are shown below. (Black lines
represent the known input kernels.)
1.2
2.5
1.e-10
1.082636733874055e-9
0
I To model a system with finite maximum mass size, noise, mass injection
and fragmentation, additional terms are needed. Fragmentation might need
to be modelled using a separate kernel, and the SCE might need to be
written in a stochastic form.
1.0
1.2689610031679235e-7
2.0
1.3738237958832649e-6
0.000014873521072935142
0.0001610262027560943
0.8
0.0017433288221999925
0.04175318936560414
1.5
0.20433597178569488
1.0000000000000036
0.6
K
Exact
K
The Smoluchowski Coalescence Equation
I Without noise or injected mass, the MFA for a pure aggregation process is
the Smoluchowski Coalescence Equation (SCE):
Z m
∂N(m, t)
1
=
K(m − m1, m1)N(m − m1, t)N(m, t)dm1
∂t
2 0
Z ∞
−
K(m, m1)N(m1, t)N(m, t)dm1
1.1721022975334811e-8
1.e-10
0.4
1.1721022975334811e-8
1.2689610031679235e-7
0.0001610262027560943
0.0017433288221999925
0.20433597178569488
1.0000000000000036
Exact
0.0
1.0
0.8
0.6
0.4
0.2
0.0
0.2
12
14
log(N(m, t)*s(t)^2)
log(N(m, t)*s(t)^2)
10
8
6
4
0.4
2.0
1.5
1.0
0.5
log(m/s(t))
0.0
0.5
1.0
1.5
Decay problem scaled data collapse for mass
distributions with K(x, y) = (xy)1/4.
0.6
0.4
Kernel edge estimates recovered for the case
ξ = 0.25.
1.e-10
1.082636733874055e-9
1.082636733874055e-9
1.1721022975334811e-8
0.4
1.1721022975334811e-8
2.0
1.3738237958832649e-6
1.2689610031679235e-7
1.3738237958832649e-6
0.000014873521072935142
0.000014873521072935142
0.0001610262027560943
0.0001610262027560943
0.0017433288221999925
2.5
0.2
2.5
1.e-10
1.2689610031679235e-7
3.0
0.0
Kernel diagonal estimates recovered for the case
ξ = 0.25.
4.0
3.5
0.2
Log z
0.0017433288221999925
1.5
0.018873918221351028
0.018873918221351028
0.20433597178569488
0.20433597178569488
1.0000000000000036
1.0000000000000036
Exact
K
Exact
1.0
1.5
1.0
0.5
0.5
0.0
0.0
0.5
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.5
1.2
0.4
1.0
0.8
Log z
1.e-10
1.082636733874055e-9
1.082636733874055e-9
1.1721022975334811e-8
0.2
0.4
1.1721022975334811e-8
5
1.2689610031679235e-7
1.2689610031679235e-7
1.3738237958832649e-6
0.000014873521072935142
0.000014873521072935142
0.0001610262027560943
4
0.0017433288221999925
0.018873918221351028
0.0001610262027560943
0.0017433288221999925
0.018873918221351028
0.09236708571873892
0.09236708571873892
1.0000000000000036
3
Exact
1.5
1.0000000000000036
Exact
K
K
0.0
6
1.e-10
1.3738237958832649e-6
2.0
0.2
Kernel diagonal estimates recovered for the case
ξ = 0.50.
3.5
2.5
0.4
Log z
Kernel edge estimates recovered for the case
ξ = 0.50.
3.0
0.6
2
1.0
1
0.5
0.0
0
1.0
0.5
0.0
0.5
Kernel edge estimates recovered for the case
ξ = 0.75.
1
1.5
1.0
0.5
0.0
0.5
Log z
Kernel diagonal estimates recovered for the case
ξ = 0.75.
Further Work
I More study of the use of orthogonal function representations.
I Work-in-progress on a more powerful time-dependent approach.
I Application to stationary and non-scaling distributions.
References
10
[1] Colm Connaughton and Peter P. Jones. Some Remarks on the Inverse Smoluchowski Problem for
Cluster-Cluster Aggregation. In J. Phys.: Conf Ser., number 012005 in 333, 2011.
8
2.5
0.8
12
2
03.0
0.2
1.0
Log z
K
16
0.04175318936560414
0.0
Log z
14
0.000014873521072935142
0.5
0.5
1.5
Generating Fiducial Data
I In order to test the accuracy and reliability of suitable inverse methods we
have created a Monte Carlo simulation to integrate the SCE, tested with
kernel functions for which the analytic solutions of the SCE are known. The
simulation can then be used to generate data from more general kernels.
1.3738237958832649e-6
0.2
2.0
The Scaling Hypothesis
I Analytic solutions to the SCE are often only possible because of assuming
that the Scaling Hypothesis (SH) holds[2]. The SH asserts that all
solutions that start from reasonable initial conditions eventually approach a
scale invariant solution at large times.
I A solution is of scaling form given a function of the typical mass, s(t) such
that:
m
C
N(m, t) =
Φ
2
s(t)
s(t)
I If scaling holds, then this property can be exploited in inverse methods.
However, there are some aggregation processes where neither scaling nor
mean-field assumptions hold.
1.082636733874055e-9
1.0
65
4
3
1
2
log(m/s(t))
0
1
2
Scaled data collapse for the stationary distribution
with injection, with K(x, y) = 12 (x1/4 + y1/4)
[2] F. Leyvraz. Scaling theory and exactly solved models in the kinetics of irreversible aggregation. Phys. Reports,
383(2):95–212, 2003.
[3] H. Wright and D. Ramkrishna. Solutions of Inverse Problems in Population Balances - I. Aggregation
Kinetics. Computers Chem. Eng., 16(12):1019–1038, 1992.
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