Abstract In a previous work, embedded ensemble propagation was proposed to improve the efficiency of sampling-based uncertainty quantification methods of computational models on emerging computational architectures. It consists of simultaneously evaluating the model for a subset of samples together, instead of evaluating them individually. A first approach introduced to solve parametric linear systems with ensemble propagation is ensemble reduction. In Krylov methods for example, this reduction consists in coupling the samples together using an inner product that sums the sample contributions. Ensemble reduction has the advantages of being able to use optimized implementations of BLAS functions and having a stopping criterion which involves only one scalar. However, the reduction potentially decreases the rate of convergence due to the gathering of the spectra of the samples. In this paper, we investigate a second approach: ensemble propagation without ensemble reduction in the case of GMRES. This second approach solves each sample simultaneously but independently to improve the convergence compared to ensemble reduction. This raises two new issues which are solved in this paper: the fact that optimized implementations of BLAS functions cannot be used anymore and that ensemble divergence, whereby individual samples within an ensemble must follow different code execution paths, can occur. We tackle those issues by implementing a high-performing ensemble GEMV and by using masks. The proposed ensemble GEMV leads to a similar cost per GMRES iteration for both approaches, i.e. with and without reduction. For illustration, we study the performances of the new linear solver in the context of a mesh tying problem. This example demonstrates improved ensemble propagation speed-up without reduction.

中文翻译：

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具有嵌入式集成传播的 GMRES，用于计算模型不确定性量化中参数线性系统的有效解决方案

摘要 在之前的工作中，提出了嵌入式集成传播，以提高新兴计算架构上计算模型的基于采样的不确定性量化方法的效率。它包括同时评估样本子集的模型，而不是单独评估它们。引入解决具有集成传播的参数线性系统的第一种方法是集成减少。例如，在 Krylov 方法中，这种减少包括使用对样本贡献求和的内积将样本耦合在一起。集成归约的优点是能够使用 BLAS 函数的优化实现，并且具有仅涉及一个标量的停止标准。然而，由于样本光谱的聚集，这种减少可能会降低收敛速度。在本文中，我们研究了第二种方法：在 GMRES 的情况下没有集成减少的集成传播。与集成减少相比，第二种方法同时但独立地求解每个样本以提高收敛性。这提出了本文解决的两个新问题：无法再使用 BLAS 函数的优化实现这一事实，并且可能发生集成发散，即集成中的各个样本必​​须遵循不同的代码执行路径。我们通过实施高性能集成 GEMV 和使用掩码来解决这些问题。建议的集成 GEMV 导致两种方法的每次 GMRES 迭代的成本相似，即有和没有减少。为了说明，我们在网格绑定问题的上下文中研究了新的线性求解器的性能。这个例子展示了改进的集成传播加速而不减少。