Anderson加速算法研究及应用

It is necessary to solve all kinds of complex nonlinear equations in the numerical simulations of many important application areas, such as inertial confinement fusion (ICF). The solution for these nonlinear equations dominants the whole time cost of the numerical simulations. At present, the main method for solving nonlinear equations in ICF numerical simulations is Picard iteration. Picard method is a kind of fixed point iteration. This kind of method has many advantages, including easily implemented, etc. However, the convergence rate of the method is slow. Therefore, the key problems concerning the efficiency of ICF numerical simulation is how to improve the convergence rate of Picard method. Recently, one kind of acceleration method for fixed point iteration — Anderson acceleration attracts much attention. Anderson acceleration has been used successfully in some related areas. However, it is difficult to apply this algorithm directly to ICF numerical simulation because of robustness, applicability and some other problems. In this project, some research about Anderson acceleration’s robustness and applicability will be done by combining some important applications including ICF. At the same time, by basing on Anderson acceleration algorithm, some new iterative methods and preconditioning techniques will be developed. It is expected that, with the implementation of the project, the convergence rate of related iterations in ICF numerical simulation can be accelerated dramatically so that the whole efficiency of the numerical simulation can be improved effectively.

在ICF等许多重要应用领域的数值模拟中，需要求解各类复杂的非线性方程组。这些方程组的求解占据数值模拟绝大部分时间开销。目前，在ICF等数值模拟应用中，求解各类非线性方程组以Picard方法为主。该方法属于不动点迭代，具有程序实现简单等优点，但其收敛速度较慢。如何提高Picard方法的收敛速度，成为影响ICF等数值模拟效率的关键。近期，关于不动点迭代过程的一类加速算法—Anderson加速引起广泛关注。Anderson加速已在相关领域得到成功应用。然而，由于健壮性和适应性等问题，难以将该算法直接有效地应用于ICF等数值模拟。本项目将结合ICF等数值模拟应用，开展Anderson加速算法的健壮性和适应性研究。同时，将以该算法为基础，研究和发展新的迭代算法和预处理方法。期望通过本项目实施，能够显著加速ICF等数值模拟应用中相关迭代收敛速度，从而有效提高数值模拟效率。

在ICF等许多重要应用领域的数值模拟中，广泛应用Picard迭代和源迭代等各类不动点迭代方法求解相关非线性问题。提高不动点迭代的收敛性和收敛速度，对于提高实际应用数值模拟效率具有重要意义。Anderson加速是提高不动点迭代收敛速度的一类算法。实际应用中的很多特点，包括强非线性以及多物理耦合等，使得Anderson加速算法难以直接应用于实际问题的求解。本项目以提高不动点迭代的收敛速度为目标，开展Anderson加速算法的健壮性提升、新迭代方法发展以及Anderson加速算法的实际应用等方面的研究。首先，在Anderson加速算法的健壮性改进方面，通过在Anderson加速算法中引入物理约束条件，并在算法中增加矩阵条件数的监控与残差向量的自适应调整功能，获得了健壮性更高的Anderson加速算法。其次，通过将Anderson加速方法与分裂迭代方法结合，发展得到了三类新的迭代方法。再次，以改进的Anderson加速算法为基础，研制了并行Anderson加速算法软件包，并实现了软件包与两个并行自适应软件框架的无缝对接。最后，将Anderson加速算法应用于三温辐射能量方程、中子输运方程和辐射输运耦合物质热传导方程等实际问题的求解，取得了较好的加速效果。针对部分实际模型可以加速约20倍。