We propose a variant of the generalized minimal residual (GMRES) method for solving linear systems of equations with one or multiple right-hand sides. Our method is based on the idea of the enlarged Krylov subspace to reduce communication. It can be interpreted as a block GMRES method. Hence, we are interested in detecting inexact breakdowns. We introduce a strategy to perform the test of detection. Furthermore, we propose a technique for deflating eigenvalues that has two benefits. The first advantage is to avoid the plateau of convergence after the end of a cycle in the restarted version. The second is to have very fast convergence when solving the same system with different right-hand sides, each given at a different time (useful in the context of a constrained pressure residual preconditioner). We test our method with these deflation techniques on academic test matrices arising from solving linear elasticity and convection–diffusion problems as well as matrices arising from two real-life applications, seismic imaging and simulations of reservoirs. With the same memory cost we obtain a saving of up to |$50 \%$| in the number of iterations required to reach convergence with respect to the original method.

中文翻译：

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扩大的GMRES用于求解具有一个或多个右侧的线性系统

我们提出了广义最小残差（GMRES）方法的一种变体，用于求解具有一个或多个右侧的方程式的线性系统。我们的方法基于扩大Krylov子空间以减少通信的思想。可以将其解释为块GMRES方法。因此，我们有兴趣检测不精确的故障。我们介绍一种执行检测测试的策略。此外，我们提出了一种压缩特征值的技术，它有两个好处。第一个优点是避免重新启动版本的周期结束后收敛的平稳期。第二是在求解具有不同右侧的相同系统时，具有非常快的收敛性，每个右侧在不同的时间给出（在受约束的压力余量预处理器的情况下很有用）。我们使用这些放气技术对解决线性弹性和对流扩散问题的学术测试矩阵以及两种实际应用，地震成像和储层模拟产生的矩阵测试了我们的方法。在相同的内存成本下，我们最多可以节省| $ 50 \％$ | 相对于原始方法而言，达到收敛所需的迭代次数。