带参数的心电学偏微分方程的模型降阶算法研究及其应用

Computational modeling of cardiac electrophysiology using the monodomain/bidomain partial differential equations is a bio-physically based and well-established field. However, the large-scale computational complexity hinders the effective application of numerical simulation for improved diagnosis and prognosis of cardiac arrhythmia. Model order reduction is a powerful technique to reduce large-scale into small-scale for improving computational efficiency. This project seeks to address several open problems in reduced order modeling of monodomain/bidomain equations, that arise from the shock characteristics of solutions and the high sensitivity to model parameters. The purpose of this project includes the derivation of an a posteriori error estimator for the reduced monodomain equation and developing the greedy reduced basis algorithm; the development of preconditioned iterative solvers for the reduced monodomain/bidomain equations, so as to eliminate the limitation due to shock characteristics; using dynamic mode decomposition (DMD) to efficiently build the adaptive reduced basis and applying it to optimal control problems constrained by the monodomain equation; to study the inherent relation between DMD and exponential time differencing. This research, on the one hand, can make ideas of real-time simulation more realistic. On the other hand, it could also provide new ideas for the reduced order modeling of other partial differential equations with similar characteristics.

基于monodomain/bidomain偏微分方程的心电传导数值模拟是一个有生理学基础且发展成熟的领域，但超大规模的计算复杂度阻碍了数值仿真在心脏疾病预控中的有效应用。模型降阶快速算法是将大规模数值计算降阶成小规模以提高计算效率的有力工具。本项目致力解决心电传导方程的模型降阶算法中的几个公开问题，这些难题源于心电传导方程解的激波特性和对模型参数的高度敏感性。项目具体目的包括分析monodomain方程降阶解的后验误差估计，建立缩减基贪婪算法；对降阶模型研究预条件迭代解法，从而去掉解的激波特性带来的局限性；借助动态模式分解法（DMD）快速构造自适应缩减基，应用到以monodomain方程为约束条件的最优控制中；研究DMD与指数时间差分算法的内在联系。本研究一方面可以把很多实时模拟的设想从不可能变成可能，另一方面也将为其它类似方程的模型降阶算法研究提供新的思路。

本项目主要研究了心血管疾病预控方面相关的若干数值计算问题，包括模拟心血管流渗透的带参数的Stokes-Darcy模型、带参数的Navier-Stokes方程的快速算法设计和分析. 研究目标是提高心血管流模拟中数值计算的有效性、准确性和高效性，以更好地辅助针对病人个案的心血管疾病预控. 目前发表的四项工作从不同角度实现了该目标. 具体而言，第一项工作针对带参数的Stokes-Darcy血流渗透模型设计了一种快速解耦和集成算法，该工作受到了学术同行关注；第二项研究对心血管流体力学模型中带参数的Navier-Stokes方程提出了两种二阶、稳定的标性辅助变量(SAV) 集成算法，并利用两种稳定子提高了已有SAV算法的准确性；第三项研究对带参数的Stokes-Darcy模型设计了两种新的无条件稳定的解耦-集成算法，该算法具有普适性，便于推广到其他类似的耦合模型。最后一项工作是对Stokes-Darcy模型中不同的方程采用不同的时间步长来离散以提高计算效率, 这里仍然考虑的是变参数的情况，证明了算法的长时间稳定性。