In this paper, we use the Bayesian inversion approach to study the data assimilation problem for a family of tumor growth models described by porous-medium type equations. The models contain uncertain parameters and are indexed by a physical parameter $m$, which characterizes the constitutive relation between density and pressure. Based on these models, we employ the Bayesian inversion framework to infer parametric and nonparametric unknowns that affect tumor growth from noisy observations of tumor cell density. We establish the well-posedness and the stability theories for the Bayesian inversion problem and further prove the convergence of the posterior distribution in the so-called incompressible limit, $m \rightarrow \infty$. Since the posterior distribution across the index regime $m\in[2,\infty)$ can thus be treated in a unified manner, such theoretical results also guide the design of the numerical inference for the unknown. We propose a generic computational framework for such inverse problems, which consists of a typical sampling algorithm and an asymptotic preserving solver for the forward problem. With extensive numerical tests, we demonstrate that the proposed method achieves satisfactory accuracy in the Bayesian inference of the tumor growth models, which is uniform with respect to the constitutive relation.

翻译：本文采用贝叶斯反演方法，研究一类由多孔介质型方程描述的肿瘤生长模型的数据同化问题。该模型包含不确定参数，并由物理参数$m$索引，表征密度与压力之间的本构关系。基于这些模型，我们运用贝叶斯反演框架，从含噪声的肿瘤细胞密度观测数据中推断影响肿瘤生长的参数性和非参数性未知量。我们建立了该贝叶斯反演问题的适定性与稳定性理论，并进一步证明了后验分布在本征不可压缩极限（$m \to \infty$）下的收敛性。由于索引范围$m \in [2, \infty)$内的后验分布可以统一处理，这些理论结果也为未知量的数值推断设计提供了指导。我们提出了一种针对此类反问题的通用计算框架，该框架包含典型的采样算法和用于正问题求解的渐近保持求解器。通过大量数值实验，我们证明所提方法在肿瘤生长模型的贝叶斯推断中实现了令人满意的精度，且该精度相对于本构关系具有一致性。