Abstract. Size-structured population models provide a popular means to mathematically describe phenomena such as bacterial aggregation, schooling fish, and planetesimal evolution. For parameter estimation, generalized sensitivity functions (GSFs) provide a tool that quantifies the impact of data from specific regions of the experimental domain. These functions help identify the most relevant data subdomains, which enhances the optimization of experimental design. To our knowledge, GSFs have not been used in the partial differential equation (PDE) realm, so we provide a novel PDE extension of the discrete and continuous ordinary differential equation ( 1. Introduction. General structured population models provide a link from the individuals in a population to the population processes [18,19,37]. A popular example, size-structured population models describe the distribution of individuals throughout varying size classes [13,16]. Typical ODE based population models make a number of simplifying assumptions, a major one of which presumes homogeneity of the individuals' physical structure across the entire population. One effort to relax the homogeneity assumption resulted in the creation of age-structured population models which account for the effects of differing ages amongst the individuals comprising the population. Unfortunately, for some systems, age does not comprise the most influential physical attribute, but in many of these cases, size-structured population models do provide an adequate structuring of the population [14].Size-structured population models often include an unknown parameter (or a set of unknown parameters). The value of this parameter is estimated via the inverse problem of parameter estimation based on experimental data. With a goal of optimizing the experiments, we seek to sample from domains which contain the most relevant information regarding the parameter estimation. Generalized sensitivity functions provide a tool which quantifies the importance of specific regions of a domain to the parameter of interest. Previous studies, such as cardiovascular regulation [8,21,22], HIV modeling [15], and HTLV-1 transactivation simulation [12], have applied the generalized sensitivity functions to ordinary differential equations. We denote these ODE-based GSFs as OGSFs. With our emphasis on size-structured population models, the primary goal of our work is to extend the concepts of OGSFs to the application of generalized sensitivity functions to PDEs, which we denote as PGSFs.Thomaseth and Cobelli introduced the concept of OGSFs in [36] 1 andBatzel et al. recast the OGSFs into a probabilistic setting [9]. In a series of studies, Banks et al. [4,6,5] further develop the OGSF concept.. In particular, the work by Banks, Dediu, and Ernstberger [5] compares traditional sensitivity functions (TSFs) with OGSFs (in the context of general nonlinear ODEs) and highlights the potential utili-

Researchers have employed variations of the Smoluchowski coagulation equation to model a wide variety of both organic and inorganic phenomena and with relatively few known analytical solutions, numerical solutions play an important role in studying this equation. In this article, we consider numerical approximations, focusing on how different discretization schemes impact the accuracy of approximate solution moments. Pursuing the eventual goal of comparing simulated solutions to experimental data, we must carefully choose the numerical method most appropriate to the type of data we attain. Within this context, we compare and contrast the accuracy and computational cost of a finite element approach and a finite volume-based scheme.Our study provides theoretical and numerical evidence that the finite element approach achieves much more accuracy when the system aggregates slowly, and it does so with much less computation cost. Conversely, the finite volume method is slightly more accurate approximating the zeroth moment when the system aggregates quickly and is much more accurate approximating the first moment in general.Lastly, our study also provides numerical evidence that the finite element method (conventionally considered first order) actually belongs to a class of discontinuous Galerkin methods that exhibit superconvergence, or second order in our case.

Size-structured population models provide a popular means to mathematically describe phenomena such as bacterial aggregation, schooling fish, and planetesimal evolution. For parameter estimation, generalized sensitivity functions (GSFs) provide a tool that quantifies the impact of data from specific regions of the experimental domain. These functions help identify the most relevant data subdomains, which enhances the optimization of experimental design. To our knowledge, GSFs have not been used in the partial differential equation (PDE) realm, so we provide a novel PDE extension of the discrete and continuous ordinary differential equation (ODE) concepts of Thomaseth and Cobelli and Banks et al. respectively. We analyze the GSFs in the context of size-structured population models, and specifically analyze the Smoluchowski coagulation equation to determine the most relevant time and volume domains for three, distinct aggregation kernels. Finally, we provide evidence that parameter estimation for the Smoluchowski coagulation equation does not require post-gelation data.