Nonlinear least-squares method for the inverse droplet coagulation problem

If the rates, $K(x,y)$, at which particles of size $x$ coalesce with particles of size $y$ is known, then the mean-field evolution of the particle size distribution of an ensemble of irreversibly coalescing particles is described by the Smoluchowski equation. We study the corresponding inverse problem which aims to determine the coalescence rates $K(x,y)$ from measurements of the particle size distribution. We assume that $K(x,y)$ is a homogeneous function of its arguments, a case which occurs commonly in practice. The problem of determining $K(x,y)$, a function to two variables, then reduces to the simpler problem of determining a function of a single variable plus two exponents, $\mu$ and $\nu$, which characterize the scaling properties of $K(x,y)$. The price of this simplification is that the resulting least-squares problem is nonlinear in the exponents $\mu$ and $\nu$. We demonstrate the effectiveness of the method on a selection of coalescence problems arising in polymer physics, cloud science, and astrophysics. The applications include examples in which the particle size distribution is stationary owing to the presence of sources and sinks of particles and examples in which the particle size distribution is undergoing self-similar relaxation in time.

Figure 1

Shape functions, as defined by Eqs. (5) and (6), for the Brownian coagulation kernel, Eq. (8); the astrophysical coalescence kernel, Eq. (9); the differential sedimentation kernel, Eq. (10); and the nonlinear shear velocity kernel, Eq. (11).

Figure 2

An example of scaling for a time-dependent solution of Eq. (1). The main panel shows the time evolution of the particle size distribution for the kernel $K(m_1,m_2)=\sqrt{m_1}+\sqrt{m_2}$. The inset panel shows the scaling function, $\Phi \left(z\right)$, obtained by rescaling the data according to Eq. (24).

Figure 3

Robustness of the nonlinear minimization problem, Eq. (29), to noise in the input data. Estimated exponents are shown as a function of the noise level for the model kernel, Eq. (3), with true exponents $\nu =-\mu =1/3$ (solid lines) and $M=250$. Each data point was obtained from an ensemble of 25 independent realizations of the noise. Error bars correspond to the 95$\%$ confidence interval.

Figure 4

Numerical solution of the stationary inverse problem for the model kernel, Eq. (3), having $\nu =-\mu =1/3$. The maximum mass was $M=250$ and $n+1=4$ Fourier coefficients were used. Symbols denote the numerical solution of the inverse problem and the solid line denotes the true values. Numerical values of the parameters were $\nu =0.3333$, $\mu =-0.3332$, $g=0.9933$, $a_0=0.9933$, $a_1=0.0268$, $a_2=-0.0153$, and $a_3=-0.0006$.

Figure 5

Illustrative example of how the numerical solution of the stationary inverse problem degrades if the number of Fourier coefficients, $n+1$, is too high. Results are shown for the model kernel, Eq. (3), having $\nu =-\mu =1/3$ and $M=250$ with $n+1=1$, 3, 5, and 7. Symbols denote the numerical solution of the inverse problem and the solid line denotes the true values. For clarity, only the transverse slice, $K(M-m+1,m)$, is shown.

Figure 6

Panels (a), (b), and (c) show the numerical solutions for the stationary inverse problems for the model kernels given in Eqs. (8), (9), and (11) respectively. Symbols denote the numerical solution of the inverse problem and solid lines denote the true values. In all cases, the maximum mass was $M=250$ and $n+1=4$ Fourier coefficients were used.

Figure 7

Numerical solution for the stationary inverse problem (symbols) for the differential sedimentation kernel, Eq. (10). Solid lines denote the true values. The maximum mass was $M=250$ and a fully nonlinear estimation algorithm with $n+1=4$ Fourier coefficients was used as described in the text.

Figure 8

Input data for the time-dependent inverse problem in the case of the Brownian coagulation kernel, Eq. (8): time evolution of the size distribution. Inset: Data collapse obtained by rescaling these data according to Eq. (24).

Figure 9

Measurements of $\lambda =\mu +\nu$ from the time dependence of the typical size, $s\left(t\right)$, for the model kernel, Eq. (3) with $\nu =\lambda$ and $\mu =0$, for several values of $\lambda$. Values in parentheses in the legend are the estimates of $\lambda$ obtained from the slopes of the fitted solid lines.

Figure 10

Solution of the time-dependent problem with the sum kernel, $K(m_1,m_2)=m_1+m_2$, using the scaling function obtained from the exact solution, Eq. (42). Symbols denote the numerical solution of the inverse problem and the solid line denotes the true values. The number of discretisation points used was $Z=250$ and $n+1=4$ Fourier coefficients were used. The numerical values of the parameters returned by the inverse algorithm were $\nu =1.0841$, $\mu =-0.0810$, $g=0.8073$, $a_0=1.0032$, $a_1=-0.0133$, $a_2=-0.0024$ and $a_3=-0.036$.

Figure 11

Solution of the time-dependent problem with Brownian coagulation kernel, Eq. (8), based on the data collapse shown in Fig. 8. Symbols denote the numerical solution of the inverse problem and the solid line denotes the true values. The number of discretization points used was $Z=250$, and $n+1=4$ Fourier coefficients were used. Numerical results have been rescaled by a factor of 1.8 coming from the separation constant (see text).