Numerical modelling, error estimation and applications

空间现象的数学模型，如流体动力学和心脏电生理学中使用的那些，导致偏微分方程组(PDE)。通常有不同的模型来分析相同的现象。很难确定哪个模型可以提供有效的结果，与更复杂的模型相比，简化模型的解有多远，以及在什么情况下建模误差较小。此外，相关的偏微分方程解的解析解很少。人们必须依靠数值方法来解决偏微分方程组问题，从而产生所谓的“数值模型”。从数值模型获得的准确预测是基于控制建模误差的能力，例如通过适当地设置模型中的参数，以及能够提供具有受控精度和可管理的计算资源的解的高效数值方法。虽然已经有误差估计可以用来提高数值解的精度，但对于时间相关和复杂的非线性问题，仍然需要研究如何处理数值误差。类似地，可用于控制建模误差的误差估计并不常见。 提出的研究计划的目标是在心脏电生理学和生物流体动力学中开发现实、可靠和有效的数值模型。现实主义将来自于适当模型的选择和开发、已发表的生物物理知识、基于患者的几何学和包含子组件模型。可靠性将基于对数值模式中来自不同来源的误差的估计器，以及对这些估计器的有效利用，以改进数值方法和解的精度。这些误差源包括建模误差、时间和空间离散化引起的误差以及线性和非线性系统的近似解。效率将来自于选择和开发数值方法，这些方法考虑到PDE模型的具体特征，在空间和时间上是自适应的，并与最新的迭代求解器相结合。除了培训3名硕士和8名博士生外，该研究计划还将大大促进普遍存在的对流-扩散-反应系统的数值方法的进步，并从更实用的角度开发高效和准确的心脏和呼吸道数值模型。

Mathematical models for spatial phenomena such as those used in fluid dynamics and cardiac electrophysiology lead to partial differential equations (PDE). Usually there is a diversity of models to analyze the same phenomena. It is hard to decide which model can provide valid results, how far is the solution of a simplified model compared to a more complex model, and under what circumstances the modelling error is small. Moreover, the associated PDE can rarely be solved analytically. One must rely on numerical methods for solving the PDE problems, leading to so-called "numerical models". Accurate predictions obtained from numerical models are based on the ability to control the modelling error, for instance by properly setting parameters in the model, and also on efficient numerical methods that can provide solutions with controlled accuracy and manageable computational resources. While there are error estimates available for improving the accuracy of numerical solutions, research is still needed for handling the numerical error for time-dependent and complex nonlinear problems. Similarly, error estimates that can be used to control the modelling error are not as commonly available. The objectives of the proposed research program are to develop realistic, reliable and efficient numerical models in cardiac electrophysiology and bio-fluid dynamics. Realism will come from the selection and development of appropriate models, published biophysical knowledge, patient-based geometries and the inclusion of sub-component models. Reliability will be based on estimators for the error coming from different sources in numerical models and on the effective use of these estimators to improve numerical methods and the accuracy of solutions. These sources of error include the modelling error, the error resulting from discretization in time and space, and from the approximate solution of linear and nonlinear systems. Efficiency will come from the selection and development of numerical methods that accounts for specific features of the PDE models, are adaptive in space and time, and coupled with up-to-date iterative solvers. Aside from training 3 MSc and 8 PhD students, the research program will significantly contribute to the advancement of numerical methods for prevalent advection-diffusion-reaction systems and from a more practical standpoint to the development efficient and accurate numerical models of the heart and airways.