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1. Schur Complement Based Preconditioners for Twofold and Block Tridiagonal Saddle Point Problems

   https://doi.org/10.4208/cmr.2023-0051  

   Cai, Mingchao; Ju, Guoliang; Li, Jingzhi (May 2024, Communications in Mathematical Research)

   In this paper, we consider using Schur complements to design preconditioners for twofold and block tridiagonal saddle point problems. One type of the preconditioners are based on the nested (or recursive) Schur complement, the other is based on an additive type Schur complement after permuting the original saddle point systems. We analyze different preconditioners incorporating the exact Schur complements. We show that some of them will lead to positively stable preconditioned systems if proper signs are selected in front of the Schur complements. These positive-stable preconditioners outperform other preconditioners if the Schur complements are further approximated inexactly. Numerical experiments for a 3-field formulation of the Biot model are provided to verify our predictions. 

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2. Parameter-robust preconditioners for Biot’s model

   https://doi.org/10.1007/s40324-023-00336-2  

   Rodrigo, Carmen; Gaspar, Francisco J.; Adler, James; Hu, Xiaozhe; Ohm, Peter; Zikatanov, Ludmil (August 2023, SeMA Journal)

   Abstract This work presents an overview of the most relevant results obtained by the authors regarding the numerical solution of the Biot’s consolidation problem by preconditioning techniques. The emphasis here is on the design of parameter-robust preconditioners for the efficient solution of the algebraic system of equations resulting after proper discretization of such poroelastic problems. The classical two- and three-field formulations of the problem are considered, and block preconditioners are presented for some of the discretization schemes that have been proposed by the authors for these formulations. These discretizations have been proved to be well-posed with respect to the physical and discretization parameters, what provides a framework to develop preconditioners that are robust with respect to such parameters as well. In particular, we construct both norm-equivalent (block diagonal) and field-of-value-equivalent (block triangular) preconditioners, which are proved to be parameter-robust. The theoretical results on this parameter-robustness are demonstrated by considering typical benchmark problems in the literature for Biot’s model. 

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3. Non-stationary Structure-Preserving Preconditioning for Image Restoration

   https://doi.org/10.1007/978-3-030-32882-5_3  

   Dell'Acqua, P.; Donatelli, M.; Reichel, L. (October 2019, Computational Methods for Inverse Problems in Imaging)

   Non-stationary regularizing preconditioners have recently been proposed for the acceleration of classical iterative methods for the solution of linear discrete ill-posed problems. This paper explores how these preconditioners can be combined with the flexible GMRES iterative method. A new structure-respecting strategy to construct a sequence of regularizing preconditioners is proposed. We show that flexible GMRES applied with these preconditioners is able to restore images that have been contaminated by strongly non-symmetric blur, while several other iterative methods fail to do this. 

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4. Effectiveness and robustness revisited for a preconditioning technique based on structured incomplete factorization

   https://doi.org/10.1002/nla.2294  

   Xin, Zixing; Xia, Jianlin; Cauley, Stephen; Balakrishnan, Venkataramanan (March 2020, Numerical Linear Algebra with Applications)

   Summary In this work, we provide new analysis for a preconditioning technique called structured incomplete factorization (SIF) for symmetric positive definite matrices. In this technique, a scaling and compression strategy is applied to construct SIF preconditioners, where off‐diagonal blocks of the original matrix are first scaled and then approximated by low‐rank forms. Some spectral behaviors after applying the preconditioner are shown. The effectiveness is confirmed with the aid of a type of two‐dimensional and three‐dimensional discretized model problems. We further show that previous studies on the robustness are too conservative. In fact, the practical multilevel version of the preconditioner has a robustness enhancement effect, and is unconditionally robust (or breakdown free) for the model problems regardless of the compression accuracy for the scaled off‐diagonal blocks. The studies give new insights into the SIF preconditioning technique and confirm that it is an effective and reliable way for designing structured preconditioners. The studies also provide useful tools for analyzing other structured preconditioners. Various spectral analysis results can be used to characterize other structured algorithms and study more general problems. 

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5. Comparisons of some iterative algorithms for Biot equations

   Mingchao, Cai; Guoping, Zhang (January 2017, International journal of evolution equations)

   In this paper, we aim at solving the Biot model under stabilized finite element discretizations. To solve the resulting generalized saddle point linear systems, some iterative methods are proposed and compared. In the first method, we apply the GMRES algorithm as the outer iteration. In the second method, the Uzawa method with variable relaxation parameters is employed as the outer iteration method. In the third approach, Uzawa method is treated as a fixed-point iteration, the outer solver is the so-called Anderson acceleration. In all these methods, the inner solvers are preconditioners for the generalized saddle point problem. In the preconditioners, the Schur complement approximation is derived by using Fourier analysis approach. These preconditioners are implemented exactly or inexactly. Extensive experiments are given to justify the performance of the proposed preconditioners and to compare all the algorithms. 

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