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Author: pancetta

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-      <span id="HuangEtAl2024b">Y.-Y. Huang, P. Y. Fung, S. Y. Hon, and X.-L. Lin, “An efficient preconditioner for evolutionary partial differential equations with θ-method in time discretization,” arXiv:2408.03535v1 [math.NA], 2024 [Online]. Available at: <a href="http://arxiv.org/abs/2408.03535v1" target="_blank">http://arxiv.org/abs/2408.03535v1</a></span>
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-            <pre class="abstract">@unpublished{HuangEtAl2024b,
-  author = {Huang, Yuan-Yuan and Fung, Po Yin and Hon, Sean Y. and Lin, Xue-Lei},
-  howpublished = {arXiv:2408.03535v1 [math.NA]},
-  title = {An efficient preconditioner for evolutionary partial differential equations with $θ$-method in time discretization},
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-            In this study, the θ-method is used for discretizing a class of evolutionary partial differential equations. Then, we transform the resultant all-at-once linear system and introduce a novel one-sided preconditioner, which can be fast implemented in a parallel-in-time way. By introducing an auxiliary two-sided preconditioned system, we provide theoretical insights into the relationship between the residuals of the generalized minimal residual (GMRES) method when applied to both one-sided and two-sided preconditioned systems. Moreover, we show that the condition number of the two-sided preconditioned matrix is uniformly bounded by a constant that is independent of the matrix size, which in turn implies that the convergence behavior of the GMRES method for the one-sided preconditioned system is guaranteed. Numerical experiments confirm the efficiency and robustness of the proposed preconditioning approach.
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