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Register a new userA modified generalized shift-splitting preconditioner for nonsymmetric saddle point problems
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For the nonsymmetric saddle point problems with nonsymmetric positive definite (1,1) parts, the modified generalized shift-splitting (MGSSP) preconditioner as well as the MGSSP iteration method are derived in this paper, which generalize the MSSP preconditioner and the MSSP iteration method newly developed by Huang and Su (J. Comput. Appl. Math. 2017), respectively. The convergent and semi-convergent analysis of the MGSSP iteration method are presented, and we prove that this method is unconditionally convergent and semi-convergent. In addition, some spectral properties of the preconditioned matrix are carefully analyzed. Numerical results demonstrate the robustness and effectiveness of the MGSSP preconditioner and the MGSSP iteration method, and also illustrate that the MGSSP iteration method outperforms the GSS and GMSS iteration methods, and the MGSSP preconditioner is superior to the shift-splitting (SS), generalized SS (GSS), modified SS (MSS) and generalized MSS (GMSS) preconditioners for the GMRES method for solving the nonsymmetric saddle point problems.
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The worst situation in computing the minimal nonnegative solution of a nonsymmetric algebraic Riccati equation associated with an M-matrix occurs when the corresponding linearizing matrix has two very small eigenvalues, one with positive and one with negative real part. When both these eigenvalues are exactly zero, the problem is called critical or null recurrent. While in this case the problem is ill-conditioned and the convergence of the algorithms based on matrix iterations is slow, there exist some techniques to remove the singularity and transform the problem to a well-behaved one. Ill-conditioning and slow convergence appear also in close-to-critical problems, but when none of the eigenvalues is exactly zero the techniques used for the critical case cannot be applied. In this paper, we introduce a new method to accelerate the convergence properties of the iterations also in close-to-critical cases, by working on the invariant subspace associated with the problematic eigenvalues as a whole. We present a theoretical analysis and several numerical experiments which confirm the efficiency of the new method.
In this paper we prove a new abstract stability result for perturbed saddle-point problems based on a norm fitting technique. We derive the stability condition according to Babuv{s}kas theory from a small inf-sup condition, similar to the famous Ladyzhenskaya-Babuv{s}ka-Brezzi (LBB) condition, and the other standard assumptions in Brezzis theory, in a combined abstract norm. The construction suggests to form the latter from individual {it fitted} norms that are composed from proper seminorms. This abstract framework not only allows for simpler (shorter) proofs of many stability results but also guides the design of parameter-robust norm-equivalent preconditioners. These benefits are demonstrated on mixed variational formulations of generalized Poisson, Stokes, vector Laplace and Biots equations.
In this paper, two types of Schur complement based preconditioners are studied for twofold and block tridiagonal saddle point problems. One is 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 discuss different preconditioners incorporating the exact Schur complements. It is shown that some of them will lead to positive stable preconditioned systems. Our theoretical analysis is instructive for devising various exact and inexact preconditioners, as well as iterative solvers for many twofold and block tridiagonal saddle point problems.
In this paper, we focus on solving a class of constrained non-convex non-concave saddle point problems in a decentralized manner by a group of nodes in a network. Specifically, we assume that each node has access to a summand of a global objective function and nodes are allowed to exchange information only with their neighboring nodes. We propose a decentralized variant of the proximal point method for solving this problem. We show that when the objective function is $rho$-weakly convex-weakly concave the iterates converge to approximate stationarity with a rate of $mathcal{O}(1/sqrt{T})$ where the approximation error depends linearly on $sqrt{rho}$. We further show that when the objective function satisfies the Minty VI condition (which generalizes the convex-concave case) we obtain convergence to stationarity with a rate of $mathcal{O}(1/sqrt{T})$. To the best of our knowledge, our proposed method is the first decentralized algorithm with theoretical guarantees for solving a non-convex non-concave decentralized saddle point problem. Our numerical results for training a general adversarial network (GAN) in a decentralized manner match our theoretical guarantees.
We provide a systematic way to design computable bilinear forms which, on the class of subspaces $W^* subseteq V$ that can be obtained by duality from a given finite dimensional subspace $W$ of an Hilbert space $V$, are spectrally equivalent to the scalar product of $V$. Such a bilinear form can be used to build a stabilized discretization algorithm for the solution of an abstract saddle point problem allowing to decouple, in the choice of the discretization spaces, the requirements related to the approximation from the inf-sup compatibility condition, which, as we show, can not be completely avoided.
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