The Inverse Smoluchowski Problem, Particles In Turbulence 2011, Potsdam, March 17 2011
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This document summarizes Colm Connaughton's presentation on solving the inverse Smoluchowski problem to determine particle collision kernels from observed cluster size distributions. It describes how the forward problem maps kernels to distributions but the inverse problem is ill-posed. Tikhonov regularization is used to obtain approximate kernel reconstructions from numerical solutions with known test kernels, demonstrating partial success in reconstructing kernel features despite ill-posedness. Future work aims to address limitations and applicability to real problems.
The Inverse Smoluchowski Problem, Particles In Turbulence 2011, Potsdam, March 17 2011
- 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 http://www.slideshare.net/connaughtonc
- 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) http://www.slideshare.net/connaughtonc
- 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 http://www.slideshare.net/connaughtonc
- 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. http://www.slideshare.net/connaughtonc
- 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. http://www.slideshare.net/connaughtonc
- 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. http://www.slideshare.net/connaughtonc
- 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. http://www.slideshare.net/connaughtonc
- 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. http://www.slideshare.net/connaughtonc
- 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. http://www.slideshare.net/connaughtonc
- 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. http://www.slideshare.net/connaughtonc
- 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? http://www.slideshare.net/connaughtonc