Estimating the division rate and kernel in the fragmentation equation

Abstract

We consider the fragmentation equation $\frac{\partial }{\partial t}f(t,x)=--B(x)f(t,x)+\int \_{y=x}^{y=\infty }k(y,x)B(y)f(t,y)dy,$ and address the question of estimating the fragmentation parameters-i.e. the division rate $B(x)$ and the fragmentation kernel $k(y,x)$-from measurements of the size distribution $f(t,$\times$)$ at various times. This is a natural question for any application where the sizes of the particles are measured experimentally whereas the fragmentation rates are unknown, see for instance (Xue, Radford, Biophys. Journal, 2013) for amyloid fibril breakage. Under the assumption of a polynomial division rate $B(x)=\alpha x^{\gamma }$ and a self-similar fragmentation kernel $k(y,x)=\frac{1}{y}k\_0(x/y)$, we use the asymptotic behaviour proved in (Escobedo, Mischler, Rodriguez-Ricard, Ann. IHP, 2004) to obtain uniqueness of the triplet $(\alpha ,\gamma ,k\_0)$ and a representation formula for $k\_0$. To invert this formula, one of the delicate points is to prove that the Mellin transform of the asymptotic profile never vanishes, what we do through the use of the Cauchy integral.

References in corpus (8)

• Eigenelements of a General Aggregation-Fragmentation Model
• On the Inverse Problem for a Size-Structured Population Model
• Numerical Solution of an Inverse Problem in Size-Structured Population Dynamics
• Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations
• Nonparametric estimation of the division rate of a size-structured population
• Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates
• Nonparametric estimation of the fragmentation kernel based on a PDE stationary distribution approximation
• Inverse Problems for a Class of Conditional Probability Measure-Dependent Evolution Equations

Cited by in corpus (1)

• Nonparametric estimation of the fragmentation kernel based on a PDE stationary distribution approximation