The inverse Smoluchowski problem forcluster-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 turbulenceUniversity of Potsdam

17 March 2011

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Cluster aggregation and Smoluchowski equation

Physical picture:Large "cloud" of particles moving around (eg byturbulence).Particles merge irreversibly on contact.Rate of merging of particles with masses, m1 and m2 isK (m1,m2). Kernel K (m1,m2) encodes microphysics.

Size distribution, Nm(t), is the average density of clusters ofmass m at time t .

Smoluchowski equation :

∂tNm(t) =

∫ m

0dm1K (m1,m −m1)Nm1(t)Nm−m1(t)

− 2Nm(t)∫ ∞

0dm1K (m,m1)Nm1(t)

(Smoluchowski, 1916)

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Equivalent formulation

It is convenient to work with the cumulative cluster sizedistribution:

Fm(t) =∫ m

0m1 Nm1(t)dm1.

The usual cluster size distribution is

Nm(t) =1m∂Fm(t)∂m

.

In terms of Fm(t) we have:

Equivalent Smoluchowski equation:

∂tFm(t) = −∫ m

0dFm1(t)

∫ ∞m−m1

dFm2(t)m2

K (m1,m2)

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Scaling Solutions of the Smoluchowski equation

In many applications kernelis a homogeneous function:

K (am1,am2) = aγ K (m1,m2)

Resulting cluster sizedistributions exhibitself-similarity.

Self-similar solutions have the form

Fm(t) ∼ s(t)a F (z) z =m

s(t)

where s(t) is the typical cluster size and a is a dynamicalscaling exponent. The scaling function, F (z), determines theshape of the cluster size distribution.

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The inverse Smoluchowski problem

Forward problem: given kernel, K (m1,m2), compute the sizedistribution, Fm(t).Inverse problem: given observations of the size distribution,Fm(t), compute the kernel, K (m1,m2) (Wright andRamakrishna, 1992).

Inverse problem is useful because:Kernel may not be known.May help in building models and guiding micro-physicstheory.Quantifies the sensitivity of the size distribution tovariations in the kernel.

butThe inverse problem is typically ill-posed.

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Ill-posedness at the discrete level

Assume scaling. Then scaling function, F (z), must satisfy:

zdFdz

= −∫ z

0dF (z1)

∫ ∞z−z1

dF (z2)

z2K (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 N2 valuesof the K (z1, z2) on the discretisation points:

b = S k.

This system is enormously under-determined⇒ one canfind many solutions but they are all entirely determined bythe noise in the data.

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Tikhonov Regularisation (Ridge regression)

One way of dealing with under-determinedness is to solve aminimization problem. The estimated kernel is:

kest = arg mink

{|S k− b|2 + λ |k|2

}.

Noise-dominated solutions have to compete against theregularization term λ |k|2. The trick is to choose the "best" valueof the regularization parameter, λ.

A rational approach todetermining λ is providedby an “L-curve”. Plot thesize of the solution, |k|, asa function of the residual,|S k− b| (Hansen 1992)."Best" values of λ are nearthe kink in the curve.

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Does it work? Numerical solution of the inverseproblem with known kernel

Constant kernel case K (z1, z2) = 1

Diagonal of reconstructed kernel. L-curve.

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Does it work? Numerical solution of the inverseproblem with known kernel

Sum kernel case K (z1, z2) =12(z1 + z2)

Diagonal of reconstructed kernel. L-curve.

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Does it work? Numerical solution of the inverseproblem with known kernel

Sqrt sum kernel case K (z1, z2) =12(√

z1 +√

z2)

Diagonal of reconstructed kernel. L-curve.

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Conclusions and Future work

Although the inverse Smoluchowski problem is ill-posed,some features of the collision kernel can be reconstructedfrom measurements of the size distribution.

We have demonstrated proof-of-concept but much remains tobe 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 ofmonomers, fragmentation, condensation?Noisy data?Is it useful for real-world problems?

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