Doumic M, Tine Léon M
INRIA Paris-Rocquencourt, EPI BANG, Domaine de Voluceau, 78153, Le Chesnay Cedex, France.
J Math Biol. 2013 Jul;67(1):69-103. doi: 10.1007/s00285-012-0553-6. Epub 2012 Jun 5.
Growth-fragmentation equations arise in many different contexts, ranging from cell division, protein polymerization, neurosciences etc. Direct observation of temporal dynamics being often difficult, it is of main interest to develop theoretical and numerical methods to recover reaction rates and parameters of the equation from indirect observation of the solution. Following the work done in Perthame and Zubelli (Inverse Probl 23:1037-1052, 2007) and Doumic et al. (2009) for the specific case of the cell division equation, we address here the general question of recovering the fragmentation rate of the equation from the observation of the time-asymptotic solution, when the fragmentation kernel and the growth rates are fully general. We give both theoretical results and numerical methods, and discuss the remaining issues.
生长-碎裂方程出现在许多不同的情境中，涵盖细胞分裂、蛋白质聚合、神经科学等领域。由于直接观测时间动态往往很困难，因此开发理论和数值方法以从解的间接观测中恢复方程的反应速率和参数成为主要研究兴趣。继佩尔塔梅和祖贝利（《反问题》23:1037 - 1052，2007年）以及杜米克等人（2009年）针对细胞分裂方程的特定情况所做的工作之后，在此我们探讨当碎裂核和生长速率完全通用时，从时间渐近解的观测中恢复方程碎裂速率的一般问题。我们给出了理论结果和数值方法，并讨论了遗留问题。
