基于MapReduce的高瘦奇异值分解模型降阶
Model Reduction with MapReduce-enabled Tall-and-Skinny Singular Value Decomposition
AI总结：
本文提出一种基于参数化偏微分方程解的降阶模型计算方法，利用参数快照矩阵的奇异值分解近似参数化解的奇异值展开，通过选择子集的右奇异向量插值得到降阶模型系数，并利用参数梯度选择子集以生成均值预测和预测协方差，用于评估降阶模型的置信度。
摘要：
我们提出了一种计算参数化偏微分方程解降阶模型的方法。关键的分析工具是参数化解的奇异值展开，我们用参数快照矩阵的奇异值分解来近似。为了在新参数上评估降阶模型，我们插值一部分右奇异向量以生成降阶模型的系数。我们采用一种新的方法来选择这个子集，该方法利用右奇异向量的参数梯度将展开项分成均值预测和预测协方差——类似于高斯过程近似。协方差作为降阶模型的置信度度量。我们通过随机介质中热传导的参数研究验证了降阶模型的有效性。高保真模拟产生超过4TB的数据；我们使用可扩展的MapReduce/Hadoop实现计算奇异值分解并评估降阶模型。我们比较了我们的方法与标量响应面在温度剖面测量集上的准确性，并发现我们的模型更好地捕捉了参数空间中的尖锐局部特征。
英文摘要：
We present a method for computing reduced-order models of parameterized partial differential equation solutions. The key analytical tool is the singular value expansion of the parameterized solution, which we approximate with a singular value decomposition of a parameter snapshot matrix. To evaluate the reduced-order model at a new parameter, we interpolate a subset of the right singular vectors to generate the reduced-order model's coefficients. We employ a novel method to select this subset that uses the parameter gradient of the right singular vectors to split the terms in the expansion yielding a mean prediction and a prediction covariance---similar to a Gaussian process approximation. The covariance serves as a confidence measure for the reduce order model. We demonstrate the efficacy of the reduced-order model using a parameter study of heat transfer in random media. The high-fidelity simulations produce more than 4TB of data; we compute the singular value decomposition and evaluate the reduced-order model using scalable MapReduce/Hadoop implementations. We compare the accuracy of our method with a scalar response surface on a set of temperature profile measurements and find that our model better captures sharp, local features in the parameter space.
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