随机系数的多项式混沌展开及其在张量核心格式中的随机偏微分方程求解
Polynomial Chaos Expansion of random coefficients and the solution of stochastic partial differential equations in the Tensor Train format
AI总结：
本文利用张量核心分解构建随机场的张量积多项式混沌展开，用于求解随机椭圆扩散PDE，并计算感兴趣量。提出新的块张量核心交叉算法，以高效近似多项式混沌展开。
摘要：
我们应用张量核心（TT）分解来构建随机场的张量积多项式混沌展开（PCE），以求解具有随机伽辽金离散化的随机椭圆扩散偏微分方程，并计算一些感兴趣量（均值、方差、超越概率）。我们假设随机扩散系数是高斯随机场的平滑变换。在这种情况下，PCE由一个复杂的公式给出，缺乏解析的TT表示。为了数值地构建其TT近似，我们开发了新的块TT交叉算法，这是一种通过少量PCE公式评估来计算整个TT分解的方法。新方法在概念上类似于TT格式中的自适应交叉近似，但当多个张量必须存储在相同的TT表示中时（如PCE的情况）更有效。此外，我们展示了如何组装随机伽辽金矩阵，并计算椭圆方程的解及其后处理，保持在TT格式中。我们比较了该技术与传统稀疏多项式混沌和蒙特卡洛方法。在张量积多项式混沌中，每个随机变量的多项式次数独立地被限制。这提供了比稀疏多项式集或蒙特卡洛方法更高的精度，但张量积集的基数随随机变量数量呈指数增长。然而，当PCE系数以TT格式隐式近似时，与完整张量积多项式集的计算成为可能。在数值实验中，我们确认了新方法在广泛参数范围内具有竞争力，特别是在需要高精度和高多项式次数的情况下。
英文摘要：
We apply the Tensor Train (TT) decomposition to construct the tensor product Polynomial Chaos Expansion (PCE) of a random field, to solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization, and to compute some quantities of interest (mean, variance, exceedance probabilities). We assume that the random diffusion coefficient is given as a smooth transformation of a Gaussian random field. In this case, the PCE is delivered by a complicated formula, which lacks an analytic TT representation. To construct its TT approximation numerically, we develop the new block TT cross algorithm, a method that computes the whole TT decomposition from a few evaluations of the PCE formula. The new method is conceptually similar to the adaptive cross approximation in the TT format, but is more efficient when several tensors must be stored in the same TT representation, which is the case for the PCE. Besides, we demonstrate how to assemble the stochastic Galerkin matrix and to compute the solution of the elliptic equation and its post-processing, staying in the TT format. We compare our technique with the traditional sparse polynomial chaos and the Monte Carlo approaches. In the tensor product polynomial chaos, the polynomial degree is bounded for each random variable independently. This provides higher accuracy than the sparse polynomial set or the Monte Carlo method, but the cardinality of the tensor product set grows exponentially with the number of random variables. However, when the PCE coefficients are implicitly approximated in the TT format, the computations with the full tensor product polynomial set become possible. In the numerical experiments, we confirm that the new methodology is competitive in a wide range of parameters, especially where high accuracy and high polynomial degrees are required.
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