Accuracy Improvement of Approximate Solutions Generated by the Method for Solving Saddle Point Problems Using Block Structure

This paper focuses on saddle point problems with a 2-by-2 block coefficient matrix. When the number of columns in the upper-right block and the number of rows in the lower-left block of the coefficient matrix is large, the convergence behavior of Krylov subspace methods for the saddle point problems tends to be poor even if the upper-left block is a well-conditioned matrix. In this paper, an efficient approach for solving the saddle point problems using block structure of the problems is proposed. The most time-consuming part of our proposed approach is the solution of a linear system with multiple right-hand sides. To solve the linear system with multiple right-hand sides efficiently, we propose to apply Block Krylov subspace methods to this linear system. Numerical experiments show that the proposed approach with Block Krylov subspace methods can solve the saddle point problems more efficiently than the conventional approach in terms of the number of iterations and the computation time.

Recently, by combining the preconditioned accelerated Hermitian and skew-Hermitian splitting (PAHSS) and the parameterized Uzawa (PU) methods, Zheng and Ma (Appl Math Comput 273, 217–225, 2016b) presented the PAHSS–PU method for saddle point problems. By adding a block lower triangular matrix to the coefficient matrix on two sides of the first equation of the PAHSS–PU iterative scheme, the modified PAHSS–PU (MPAHSS–PU) iteration method is proposed in this paper, which has a faster convergence rate than the PAHSS–PU one. Furthermore, changing the position of the parameters in the MPAHSS–PU method, we develop another new method referred to as the modified PPHSS-SOR (MPPHSS-SOR) iteration method for solving saddle point problems. We provide the convergence properties of the MPAHSS–PU and the MPPHSS-SOR iteration methods, which show that the new methods are convergent if the related parameters satisfy suitable conditions. Meanwhile, practical ways to choose iteration parameters for the proposed methods are developed. Finally, numerical experiments demonstrate that the MPAHSS–PU and the MPPHSS-SOR methods outperform some existing ones both on the number of iterations and the computational times.

In this paper, the nonlinear inexact Uzawa (NIU) methods for saddle point problems are studied. A further improved convergence result of the NIU algorithm for stabilized saddle point problems is presented, which results in much smaller convergence factor. This result can also be viewed as a generalization of some previous work for classical saddle point problems. Based on this result, we generalize some adaptive versions of the NIU algorithm with variable iteration parameters for classical saddle point problems to solve stabilized saddle point problems. Convergence analyses of these algorithms are presented. It shows that they converge under similar conditions as those in classical cases, which are more practical and without estimates on the extreme eigenvalues of the involved preconditioned systems. Numerical experiments are given to demonstrate the effectiveness of these algorithms over some original NIU algorithms.

In this paper we consider additive Schwarz-type iteration methods for saddle point problems as smoothers in a multigrid method. Each iteration step of the additive Schwarz method requires the solutions of several small local saddle point problems. This method can be viewed as an additive version of a (multiplicative) Vanka-type iteration, well-known as a smoother for multigrid methods in computational fluid dynamics. It is shown that, under suitable conditions, the iteration can be interpreted as a symmetric inexact Uzawa method. In the case of symmetric saddle point problems the smoothing property, an important part in a multigrid convergence proof, is analyzed for symmetric inexact Uzawa methods including the special case of the additive Schwarz-type iterations. As an example the theory is applied to the Crouzeix-Raviart mixed finite element for the Stokes equations and some numerical experiments are presented.

Modified Uzawa methods for saddle point problems

On unsymmetric block overrelaxation-type methods for saddle point problems

Block Krylov subspace methods are efficient iterative methods for solving linearsystems with multiple right-hand sides. The Block BiCGGR method is one of the Block Krylov subspace methods, and can generate highly accurate approximatesolutions. However, when the number of right-hand sides is large, this method becomes numerically unstable. In this paper, the Block BiCGGR2 method is proposed. This method is a numerically stable and highly accurate method than the Block BiCGGR method and the Block BiCGSTAB method. Numerical results illustrate the robustness and the accuracy of the approximate solutions of the Block BiCGGR2 method.

Domain decomposition based preconditioner combined local low-rank approximation with global corrections

Contemporary large-scale Partial Differential Equation (PDE) simulations usually require the solution of large and sparse linear systems. Moreover, it is often needed to solve these linear systems with different or multiple Right-Hand Sides (RHSs). In this paper, various strategies will be presented to extend the scalability of existing multigrid or domain decomposition linear solvers using appropriate recycling strategies or block methods—i.e., by treating multiple right-hand sides simultaneously. The scalability of this work is assessed by performing simulations on up to 8,192 cores for solving linear systems arising from various physical phenomena modeled by Poisson's equation, the system of linear elasticity, or Maxwell's equation. This work is shipped as part of on open-source software, readily available and usable in any C/C++, Python, or Fortran code. In particular, some simulations are performed on top of a well-established library, PETSc, and it is shown how our approaches can be used to decrease time to solution down by 30%.

For Hermitian positive definite linear systems and eigenvalue problems, the eigCG algorithm is a memory efficient algorithm that solves the linear system and simultaneously computes some of its eigenvalues. The algorithm is based on the Conjugate-Gradient (CG) algorithm, however, it uses only a window of the vectors generated by the CG algorithm to compute approximate eigenvalues. The number and accuracy of the eigenvectors can be increased by solving more right-hand sides. For Hermitian systems with multiple right-hand sides, the computed eigenvectors can be used to speed up the solution of subsequent systems. The algorithm was tested on Lattice QCD problems by solving the normal equations and was shown to give large speed up factors and to remove the critical slowing down as we approach light quark masses. Here, an extension to the non-symmetric case based on the two-sided Lanczos algorithm is given. The new algorithm is tested on Lattice QCD problems and is shown to give promising results. We also study the removal of the critical slowing down and compare results with those of the eigCG algorithm. We also discuss the case when the system is gamma5-Hermitian.

In this paper two classes of iterative methods for saddle point problems are considered: inexact Uzawa algorithms and a class of methods with symmetric preconditioners. In both cases the iteration matrix can be transformed to a symmetric matrix by block diagonal matrices, a simple but essential observation which allows one to estimate the convergence rate of both classes by studying associated eigenvalue problems. The obtained estimates apply for a wider range of situations and are partially sharper than the known estimates in literature. A few numerical tests are given which confirm the sharpness of the estimates.

We design and analyze multigrid methods for the saddle point problems resulting from Raviart–Thomas–Nedelec mixed finite element methods for the Darcy system in porous media flow. Uniform convergence of the W-cycle algorithm in a nonstandard energy norm is established. Extensions to general second order elliptic problems are also addressed.

In this paper, wavelet techniques are employed for the fast numerical solution of a control problem governed by an elliptic boundary value problem with boundary control. A quadratic cost functional involving natural norms of the state and the control is to be minimized. Firstly the constraint, the elliptic boundary value problem, is formulated in an appropriate weak form that allows to handle varying boundary conditions explicitly: the boundary conditions are treated by Lagrange multipliers, leading to a saddle point problem. This is combined with a fictitious domain approach in order to cover also more complicated boundaries. Deviating from standard approaches, we then use (biorthogonal) wavelets to derive an equivalent infinite discretized control problem which involves only e2-norms and -operators. Classical methods from optimization yield the corresponding optimality conditions in terms of two weakly coupled (still infinite) saddle point problems for which a unique solution exists. For deriving finite-dimensional systems which are uniformly invertible, stability of the discretizations has to be ensured. This together with the e2-setting circumvents the problem of preconditioning: all operators have uniformly bounded condition numbers independent of the discretization. In order to numerically solve the resulting (finite-dimensional) linear system of the weakly coupled saddle point problems, a fully iterative method is proposed which can be viewed as an inexact gradient scheme. It consists of a gradient algorithm as an outer iteration which alternatingly picks the two saddle point problems, and an inner iteration to solve each of the saddle point problems, exemplified in terms of the Uzawa algorithm. It is proved here that this strategy converges, provided that the inner systems are solved sufficiently well. Moreover, since the system matrix is well-conditioned, it is shown that in combination with a nested iteration strategy this iteration is asymptotically optimal in the sense that it provides the solution on discretization level J with an overall amount of arithmetic operations that is proportional to the number of unknows NJ on that level. Finally, numerical results are provided.

This paper focuses on total least squares (TLS) problems $AX\approx B$ with multiple right-hand sides. Existence and uniqueness of a TLS solution for such problems was analyzed in the paper [I. Hnětynková et al., SIAM J. Matrix Anal. Appl., 32, 2011, pp. 748--770]. For TLS problems with single right-hand sides the paper [C. C. Paige and Z. Strakoš, SIAM J. Matrix Anal. Appl., 27, 2006, pp. 861--875] showed how necessary and sufficient information for solving $Ax\approx b$ can be revealed from the original data through the so-called core problem concept. In this paper we present a theoretical study extending this concept to problems with multiple right-hand sides. The data reduction we present here is based on the singular value decomposition of the system matrix $A$. We show minimality of the reduced problem; in this sense the situation is analogous to the single right-hand side case. Some other properties of the core problem, however, cannot be extended to the case of multiple right-hand sides.

Efficient approximate solution of sparse linear systems