Pseudospectral methods and iterative solvers for optimization problems from multiscale particle dynamics
Pseudospectral methods and iterative solvers for optimization problems from multiscale particle dynamics
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DOI：
10.1007/s10543-022-00928-w
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发表时间：
2020-09
影响因子：
1.5
通讯作者：
Mildred Aduamoah;B. Goddard;J. Pearson;Jonna C. Roden
中科院分区：
文献类型：
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作者：
Mildred Aduamoah;B. Goddard;J. Pearson;Jonna C. Roden
我们推导出新的算法，用于描述多尺度粒子动力学的偏微分方程约束的优化问题，包括代表粒子之间相互作用的非局部积分项。特别是，我们调查的问题，控制作为一个平流“流”向量或源项的偏微分方程，约束是配备有边界条件的狄利克雷或无通量型。在推导出这些问题的连续一阶最优性条件后，我们通过与统计力学的计算方法建立联系，推导出空间和时间变量的伪谱方法，并利用现有不动点方法的变体以及最近开发的牛顿-克雷洛夫计划来解决所产生的系统。数值实验表明，我们的方法的有效性的一系列的问题设置，边界条件，以及正则化和模型参数，在二维和三维。一个关键的贡献是提供软件，允许离散化和解决一系列的优化问题的微分方程描述粒子动力学的约束。
We derive novel algorithms for optimization problems constrained by partial differential equations describing multiscale particle dynamics, including non-local integral terms representing interactions between particles. In particular, we investigate problems where the control acts as an advection ‘flow’ vector or a source term of the partial differential equation, and the constraint is equipped with boundary conditions of Dirichlet or no-flux type. After deriving continuous first-order optimality conditions for such problems, we solve the resulting systems by developing a link with computational methods for statistical mechanics, deriving pseudospectral methods in space and time variables, and utilizing variants of existing fixed-point methods as well as a recently developed Newton–Krylov scheme. Numerical experiments indicate the effectiveness of our approach for a range of problem set-ups, boundary conditions, as well as regularization and model parameters, in both two and three dimensions. A key contribution is the provision of software which allows the discretization and solution of a range of optimization problems constrained by differential equations describing particle dynamics.