论文标题
非线性fokker-plank方程的微麦克罗随机galerkin方法，随机输入
Micro-macro stochastic Galerkin methods for nonlinear Fokker-Plank equations with random inputs
论文作者
论文摘要
非线性福克 - 普兰克方程在建模大型相互作用粒子的系统中起着重要作用，在描述现实世界现象从经典领域（例如流体和等离子体）到社会和生物动态的现实现象。它们的数学表述通常是要面对具有显着的随机成分的物理力，或者与生活在随机环境中的颗粒可以通过实验数据推导出来，因此可以推导出不确定性依赖性均衡状态。在这项工作中，为了解决有效求解随机福克 - 普兰克系统的问题，我们将通过基于随机盖尔金方法的微麦克罗方法来构建一种新的平衡保存方案。与基础Fokker-Planck模型未知数的参数空间中直接应用随机的Galerkin投影相反，所得的数值方法可导致对不确定性依赖性大的时间行为的高度准确描述。提出了在社会和生命科学集体行为的背景下进行的几项数值测试，以评估当前方法对标准方法的有效性。
Nonlinear Fokker-Planck equations play a major role in modeling large systems of interacting particles with a proved effectiveness in describing real world phenomena ranging from classical fields such as fluids and plasma to social and biological dynamics. Their mathematical formulation has often to face with physical forces having a significant random component or with particles living in a random environment which characterization may be deduced through experimental data and leading consequently to uncertainty-dependent equilibrium states. In this work, to address the problem of effectively solving stochastic Fokker-Planck systems, we will construct a new equilibrium preserving scheme through a micro-macro approach based on stochastic Galerkin methods. The resulting numerical method, contrarily to the direct application of a stochastic Galerkin projection in the parameter space of the unknowns of the underlying Fokker-Planck model, leads to highly accurate description of the uncertainty dependent large time behavior. Several numerical tests in the context of collective behavior for social and life sciences are presented to assess the validity of the present methodology against standard ones.
