Homotopy Methods for Computing Bifurcations and Multiple Solutions of Nonlinear Partial Differential Equations with Biological Applications

计算非线性偏微分方程的分岔和多重解的同伦方法及其生物学应用

基本信息

批准号：
1818769
负责人：
Wenrui Hao
金额：
$ 10万
依托单位：
Pennsylvania State Univ University Park
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2022-01-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1818769&HistoricalAwards=false
关键词：
Homotopy Methods Computing Bifurcations Multiple

项目摘要

Many mathematical models of natural phenomena, e.g., biology, physics and materials science, involve nonlinear systems of partial differential equations (PDEs). Traditional numerical methods focus on the unique time-marching solution and are very hard to capture these solution structures such as bifurcations, multiple steady states, and the structural stability. This project aims to address these challenges by developing efficient computational methods to explore bifurcations and multiple solutions of nonlinear PDEs.The aim of this project is to develop efficient numerical methods to study the bifurcations and multiple solutions of nonlinear systems of PDEs. Two novel numerical techniques are proposed to study nonlinear systems of PDEs. First, an adaptive homotopy tracking with bifurcation detection algorithm will be designed for computing bifurcation points and bifurcation solution branches based on adaptive tracking, endgame technique, and inflation process. Second, a homotopy method with optimal basis approximation will be developed to compute multiple solutions of nonlinear systems by optimizing the solution basis in order to minimize the size of the discretized polynomial system. Homotopy method will be employed to solve this bootstrapped discretized polynomial system. The project will also focus on demonstrating and verifying efficiency/reliability of these proposed numerical methods on the real biological problems which have attracted extensive mathematical studies since the models were proposed.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

许多自然现象的数学模型，例如生物学，物理和材料科学，都涉及偏微分方程（PDE）的非线性系统。传统的数值方法着眼于独特的时间摩擦解决方案，并且很难捕获这些解决方案结构，例如分叉，多个稳态和结构稳定性。该项目旨在通过开发有效的计算方法来探索非线性PDE的分叉和多种解决方案来应对这些挑战。该项目的目的是开发有效的数值方法来研究PDE的非线性系统的分叉和多个解决方案。提出了两种新型的数值技术来研究PDE的非线性系统。首先，具有分叉检测算法的自适应同拷贝跟踪将设计用于基于自适应跟踪，最终游戏技术和通货膨胀过程的分叉点和分叉解决方案分支。其次，将开发具有最佳基础近似值的同质方法，以通过优化解决方案基础来最大程度地减少离散多项式系统的大小来计算非线性系统的多个解决方案。同型方法将用于解决此自举离散的多项式系统。该项目还将着重于证明和验证这些提出的数值方法对实际生物学问题的效率/可靠性，这些方法自从提出模型以来就吸引了广泛的数学研究。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估来通过评估来获得支持的值得支持的。