论文标题
高维非线性PDE的动态张量近似
Dynamic tensor approximation of high-dimensional nonlinear PDEs
论文作者
论文摘要
我们提出了一种基于功能张量分解和动态张量近似的新方法，以计算高维时依赖性非线性偏微分方程（PDE）的解。动态近似的想法是将PDE解的时间导数投射到每次低级功能张量歧管的切线空间上。可以通过最大程度地减少切线空间上的凸能功能来计算这种投影。这个最小化问题产生了独特的最佳速度向量，该速度向量使我们能够在恒定等级的张量歧管上向前集成PDE。在实际可分开的希尔伯特空间中定义的初始/边界值问题的情况下，此过程以一维时间依赖性PDES的耦合系统的形式为张量模式产生进化方程。我们将动态张量近似应用于具有非恒定漂移和扩散系数的四维Fokker-Planck方程，并证明了其在预测统计平衡方面的放松方面的准确性。
We present a new method based on functional tensor decomposition and dynamic tensor approximation to compute the solution of a high-dimensional time-dependent nonlinear partial differential equation (PDE). The idea of dynamic approximation is to project the time derivative of the PDE solution onto the tangent space of a low-rank functional tensor manifold at each time. Such a projection can be computed by minimizing a convex energy functional over the tangent space. This minimization problem yields the unique optimal velocity vector that allows us to integrate the PDE forward in time on a tensor manifold of constant rank. In the case of initial/boundary value problems defined in real separable Hilbert spaces, this procedure yields evolution equations for the tensor modes in the form of a coupled system of one-dimensional time-dependent PDEs. We apply the dynamic tensor approximation to a four-dimensional Fokker-Planck equation with non-constant drift and diffusion coefficients, and demonstrate its accuracy in predicting relaxation to statistical equilibrium.
