标题： 用于由参数半线性抛物反应扩散方程建模的肿瘤生长不确定性量化的小样本蒙特卡罗方法 显示英文标题

标题： Quasi-Monte Carlo methods for uncertainty quantification of tumor growth modeled by a parametric semi-linear parabolic reaction-diffusion equation

摘要： 我们研究了准蒙特卡罗（QMC）方法在一类用于模拟肿瘤生长的半线性抛物反应-扩散偏微分方程中的应用。 肿瘤生长的数学模型在本质上主要是现象学的，捕捉肿瘤对周围健康组织的浸润、现有肿瘤的增殖以及患者对治疗（如化疗和放疗）的反应。 患者间的显著差异、疾病的固有异质性、稀疏且噪声的数据收集以及模型的不足都会导致模型参数中存在显著的不确定性。 这些不确定性能够通过模型高效传播以计算感兴趣的量（QoIs）至关重要，这些量反过来可用于指导临床决策。 我们表明，QMC方法在计算有意义的QoIs的期望值方面是成功的。 为该模型开发了适定性结果，并用于展示均匀随机场情况下的理论误差界。 理论上的线性误差率优于标准蒙特卡罗方法，并通过数值验证。 还提供了针对对数正态随机场的有希望的计算结果，促使进一步的理论发展。 显示英文摘要

摘要： We study the application of a quasi-Monte Carlo (QMC) method to a class of semi-linear parabolic reaction-diffusion partial differential equations used to model tumor growth. Mathematical models of tumor growth are largely phenomenological in nature, capturing infiltration of the tumor into surrounding healthy tissue, proliferation of the existing tumor, and patient response to therapies, such as chemotherapy and radiotherapy. Considerable inter-patient variability, inherent heterogeneity of the disease, sparse and noisy data collection, and model inadequacy all contribute to significant uncertainty in the model parameters. It is crucial that these uncertainties can be efficiently propagated through the model to compute quantities of interest (QoIs), which in turn may be used to inform clinical decisions. We show that QMC methods can be successful in computing expectations of meaningful QoIs. Well-posedness results are developed for the model and used to show a theoretical error bound for the case of uniform random fields. The theoretical linear error rate, which is superior to that of standard Monte Carlo, is verified numerically. Encouraging computational results are also provided for lognormal random fields, prompting further theoretical development.