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Summary: This paper presents some recent advances for parallel‐in‐time methods applied to linear elasticity. With recent computer architecture changes leading to stagnant clock speeds, but ever increasing numbers of cores, future speedups will be available through increased concurrency. Thus, sequential algorithms, such as time stepping, will suffer a bottleneck. This paper explores multigrid reduction in time (MGRIT) for an important application area, linear elasticity. Previously, efforts at parallel‐in‐time for elasticity have experienced difficulties, for example, the beating phenomenon. As a result, practical parallel‐in‐time algorithms for this application area currently do not exist. This paper proposes some solutions made possible by MGRIT (e.g., slow temporal coarsening and FCF‐relaxation) and, more importantly, a different formulation of the problem that is more amenable to parallel‐in‐time methods. Using a recently developed convergence theory for MGRIT and Parareal, we show that the changed formulation of the problem avoids the instability issues and allows the reduction of the error using two temporal grids. We then extend our approach to the multilevel case, where we demonstrate how slow temporal coarsening improves convergence. The paper ends with supporting numerical results showing a practical algorithm enjoying speedup benefits over the sequential algorithm. [ABSTRACT FROM AUTHOR]

The ChebFilterCG algorithm, proposed by Golub, Ruiz, and Touhami [SIAM J. Matrix Anal. Appl. 29 (2007), pp. 774-795] is an iterative method that combines Chebyshev filter and conjugate gradient for solving symmetric positive definite linear systems with multiple right-hand sides. The Chebyshev filter is used to produce initial residuals rich in eigenvectors corresponding to the smallest eigenvalues, which are then used in the initial phase of the conjugate gradient. This paper presents a convergence analysis of ChebFilterCG. In particular, it is shown theoretically and numerically that the algorithm yields an approximation of the invariant subspace associated with the smallest eigenvalues that can be recycled for solving several linear systems with the same matrix and different right-hand sides. A refined error bound when solving these systems is also given. The choice and influence of the Chebyshev filtering steps is discussed. Numerical experiments are described to illustrate that the Chebyshev filter does not degrade the distribution of the smallest eigenvalues and highlight the effect of rounding errors when large outlying eigenvalues are present. Finally, it is shown that the method may become more effective when an additional Chebyshev filtering step is used in the initialization phase of ChebFilterCG. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of weakly nonlinear complementarity problems arising from the discretization of free boundary problems. By reformulating the complementarity problems as implicit fixed-point equations based on splitting of the system matrices, we propose a class of modulus-based matrix splitting algorithms. We show their convergence by assuming that the system matrix is positive definite. Moreover, we give several kinds of typical practical choices of the modulus-based matrix splitting iteration methods based on the different splitting of the system matrix. Numerical experiments on two model problems are presented to illustrate the theoretical results and examine the numerical effectiveness of our modulus-based matrix splitting algorithms. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

A typical approach to decrease computational costs and memory requirements of classical algebraic multigrid methods is to replace a conservative coarsening algorithm and short-distance interpolation on a fixed number of fine levels by an aggressive coarsening with a long-distance interpolation. Although the quality of the resulting algebraic multigrid grid preconditioner often deteriorates in terms of convergence rates and iteration counts of the preconditioned iterative solver, the overall performance can improve substantially. We investigate here, as an alternative, a possibility to replace the classical aggressive coarsening by aggregation, which is motivated by the fact that the convergence of aggregation methods can be independent of the problem size provided that the number of levels is fixed. The relative simplicity of aggregation can lead to improved solution and setup costs. The numerical experiments show the relevance of the proposed combination on both academic and benchmark problems in reservoir simulation from oil industry. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY A squared Smith type algorithm for solving large-scale discrete-time Stein equations is developed. The algorithm uses restarted Krylov spaces to compute approximations of the squared Smith iterations in low-rank factored form. Fast convergence results when very few iterations of the alternating direction implicit method are applied to the Stein equation beforehand. The convergence of the algorithm is discussed and its performance is demonstrated by several test examples. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]