液滴凝聚反问题的非线性最小二乘法
A nonlinear least squares method for the inverse droplet coagulation problem
摘要：
如果$x$尺寸的颗粒与$y$尺寸的颗粒聚结的速率$K(x,y)$已知，则不可逆聚结颗粒系综的颗粒尺寸分布的平均场演化由Smoluchowski方程描述。我们研究了相应的反问题，其目的是确定的聚结率，$K(x,y)$从测量的粒度分布。我们假设$K(x,y)$是其参数的齐次函数，这种情况在实践中经常发生。确定两个变量的函数$K(x,y)$的问题则简化为确定单个变量加上两个指数$μ$和$ν$的函数的更简单的问题，这证明了$K(x,y)$的标度性质。这种简化的代价是所得到的最小二乘问题在指数$μ$和$ν$中是非线性的。我们证明了选择的聚合物物理，云科学和天体物理学中出现的聚结问题的方法的有效性。这些应用包括其中颗粒尺寸分布由于存在颗粒的源和汇而静止的示例，以及其中颗粒尺寸分布在时间上经历自相似松弛的示例。
英文摘要：
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 a simpler problem of determining a function of a single variable plus two exponents, $μ$ and $ν$, which characterise the scaling properties of $K(x,y)$. The price of this simplification is that the resulting least squares problem is nonlinear in the exponents $μ$ and $ν$. 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.
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