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

许多自然现象的数学模型，例如，生物学、物理学和材料科学涉及偏微分方程（PDE）的非线性系统。传统的数值方法只关注时间推进解的唯一性，很难捕捉到解的分叉、多稳态、结构稳定等结构。本项目旨在通过发展有效的计算方法来研究非线性偏微分方程的分支和多解，以解决这些挑战。本项目的目的是发展有效的数值方法来研究非线性偏微分方程系统的分支和多解。提出了两种新的数值方法来研究偏微分方程的非线性系统。首先，基于自适应同伦跟踪、endgame技术和膨胀过程，设计了一种带分支检测的自适应同伦跟踪算法，用于计算分支点和分支解分支。其次，同伦方法与最佳基近似将发展到计算多个解决方案的非线性系统，通过优化的解决方案的基础，以最小化的离散多项式系统的大小。同伦方法将被用来解决这个自举离散多项式系统。该项目还将重点展示和验证效率/可靠性的真实的生物学问题，吸引了广泛的数学研究，因为models.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.