A Tensor Decomposition Algorithm for Large ODEs with Conservation Laws
A Tensor Decomposition Algorithm for Large ODEs with Conservation Laws
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DOI：
10.1515/cmam-2018-0023
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发表时间：
2019-01-01
影响因子：
1.3
通讯作者：
Dolgov, Sergey, V
中科院分区：
文献类型：
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作者：
Dolgov, Sergey, V
我们提出了一个算法的解决方案的高维演化方程（常微分方程和离散的时间相关的PUE）的张量列车（TT）分解，假设的解决方案和右手边的常微分方程允许这样的分解与低存储。一个线性常微分方程，离散通过一步或切比雪夫微分计划，变成一个大的线性系统。张量分解允许使用交替最小二乘算法的扩展同时求解该系统的多个时间点。这种方法计算一个简化的TT模型的解决方案，但与传统的离线在线减少计划，解决原来的大问题是从来没有需要。相反，该方法解决了一系列简化的Galerkin问题，由于右侧的TT分解，可以有效地设置。减少系统允许快速估计的时间离散化误差，因此适应的时间步长。此外，通过线性不变量的生成向量对近似子空间的扩展和对欧氏范数的修正，可以使约化模型中的守恒律得到精确的保持。在与运输和化学主方程的数值实验中，我们证明了新方法比传统的时间步进和随机模拟算法更快，而不变量被保留到机器精度，而不管TT近似精度。
We propose an algorithm for solution of high-dimensional evolutionary equations (ODEs and discretized time-dependent PUEs) in the Tensor Train (TT) decomposition, assuming that the solution and the right-hand side of the ODE admit such a decomposition with a low storage. A linear ODE, discretized via one-step or Chebyshev differentiation schemes, turns into a large linear system. The tensor decomposition allows to solve this system for several time points simultaneously using an extension of the Alternating Least Squares algorithm. This method computes a reduced TT model of the solution, but in contrast to traditional offline-online reduction schemes, solving the original large problem is never required. Instead, the method solves a sequence of reduced Galerkin problems, which can be set up efficiently due to the TT decomposition of the right-hand side. The reduced system allows a fast estimation of the time discretization error, and hence adaptation of the time steps. Besides, conservation laws can be preserved exactly in the reduced model by expanding the approximation subspace with the generating vectors of the linear invariants and correction of the Euclidean norm. In numerical experiments with the transport and the chemical master equations, we demonstrate that the new method is faster than traditional time stepping and stochastic simulation algorithms, whereas the invariants are preserved up to the machine precision irrespectively of the TT approximation accuracy.