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，如生物学、物理学和材料科学，都涉及到非线性偏微分方程组。传统的数值方法关注的是唯一的时间推进解，很难捕捉到这些解的结构，如分叉、多稳态和结构稳定性。本项目旨在通过开发有效的计算方法来探索非线性偏微分方程组的分叉和多解来解决这些挑战。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组的分叉和多解。提出了两种新的数值方法来研究非线性偏微分方程组。首先，基于自适应跟踪、终局技术和膨胀过程，设计了一种自适应同伦跟踪和分叉检测算法，用于计算分岔点和分叉解分支。其次，为了最小化离散化的多项式系统的规模，通过优化解基，发展了一种具有最优基逼近的同伦方法来计算非线性系统的多个解。用同伦方法求解这个自举离散化的多项式系统。该项目还将专注于展示和验证这些建议的数值方法在真实生物学问题上的效率/可靠性，自模型提出以来，这些方法吸引了广泛的数学研究。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.