面向GMRES的不完全分解预条件器学习
Learning incomplete factorization preconditioners for GMRES
AI总结：
针对传统不完全LU分解预条件器计算耗时、对超参数敏感的问题，提出用图神经网络学习近似矩阵分解，设计可逆保证激活函数与稀疏输出架构，经实验验证可有效减少GMRES迭代次数、改善谱特性。
摘要：
稀疏矩阵的不完全LU分解作为预条件器被广泛用于Krylov子空间方法中，以加速线性系统的求解。但遗憾的是，预条件器本身的计算可能耗时较长，且对超参数敏感。对此，我们用图神经网络替代人工设计的算法，训练其直接近似矩阵分解。为将神经网络的输出用作预条件器，我们提出了一种输出激活函数，可保证预测得到的分解是可逆的。此外，采用图神经网络架构还能确保输出本身是稀疏的，这从计算角度来看是理想的特性。我们对用于训练该学习型预条件器的不同损失函数进行了理论分析与实证评估，证明其在合成数据上能有效减少GMRES迭代次数并改善谱特性。代码可在https://github.com/paulhausner/neural-incomplete-factorization获取。
英文摘要：
Incomplete LU factorizations of sparse matrices are widely used as preconditioners in Krylov subspace methods to speed up solving linear systems. Unfortunately, computing the preconditioner itself can be time-consuming and sensitive to hyper-parameters. Instead, we replace the hand-engineered algorithm with a graph neural network that is trained to approximate the matrix factorization directly. To apply the output of the neural network as a preconditioner, we propose an output activation function that guarantees that the predicted factorization is invertible. Further, applying a graph neural network architecture allows us to ensure that the output itself is sparse which is desirable from a computational standpoint. We theoretically analyze and empirically evaluate different loss functions to train the learned preconditioners and show their effectiveness in decreasing the number of GMRES iterations and improving the spectral properties on synthetic data. The code is available at https://github.com/paulhausner/neural-incomplete-factorization.
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