Efficient Preconditioners for Solving Twofold and Block Tridiagonal Saddle Point Problems

Temel Kavramlar

The authors propose and analyze Schur complement-based preconditioners for efficiently solving twofold and block tridiagonal saddle point problems, demonstrating that positively stable preconditioners can outperform other alternatives when the Schur complements are further approximated inexactly.

Özet

The paper focuses on designing and analyzing Schur complement-based preconditioners for efficiently solving twofold and block tridiagonal saddle point problems.
For the twofold saddle point problem, the authors consider two approaches:

Nested (or recursive) Schur complement-based preconditioners: The authors show that these preconditioners lead to preconditioned systems satisfying a polynomial equation of degree 3 (block triangular) or 6 (block diagonal).
Additive-type Schur complement-based preconditioners: The authors demonstrate that these preconditioners yield preconditioned systems satisfying a polynomial equation of degree 2 (block triangular) or 4 (block diagonal).

The authors generalize their analysis to n-tuple block tridiagonal saddle point problems. They prove that judiciously selecting the signs in front of the Schur complements in the preconditioners results in positively stable preconditioned systems. These positively stable preconditioners are shown to outperform other preconditioners when the Schur complements are further approximated inexactly.
Numerical experiments for a 3-field formulation of the Biot model are provided to verify the superior performance of the positively stable preconditioners compared to other alternatives.

İstatistikler

The authors assume that the system matrix A and the Schur complements are invertible.
The authors assume that A1, A2, and A3 can be non-symmetric or negative definite.

Alıntılar

"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."

Önemli Bilgiler Şuradan Elde Edildi

Schur complement based preconditioners for twofold and block tridiagonal saddle point problems

by Mingchao Cai... : arxiv.org 04-05-2024

https://arxiv.org/pdf/2108.08332.pdf

Daha Derin Sorular

How can the proposed preconditioners be extended to handle more general saddle point problems beyond the block tridiagonal structure

The proposed preconditioners based on nested Schur complements can be extended to handle more general saddle point problems by considering different structures of the system matrices. For saddle point problems that do not fit the block tridiagonal form, the nested Schur complement-based preconditioners can still be applied by appropriately defining the Schur complements and the block structures. By adjusting the recursive relationships and the block decomposition, the preconditioners can be adapted to accommodate various configurations of saddle point systems. This flexibility allows the approach to be utilized for a broader range of saddle point problems, providing a versatile framework for preconditioning strategies in numerical computations.

What are the potential limitations or drawbacks of the nested Schur complement-based preconditioners compared to the additive-type Schur complement-based preconditioners

While nested Schur complement-based preconditioners offer advantages in terms of stability and performance, they may have certain limitations compared to additive-type Schur complement-based preconditioners. One potential drawback is the increased computational complexity associated with the recursive nature of nested Schur complements. The iterative calculations involved in the nested approach can lead to higher computational costs, especially for large-scale systems. Additionally, the nested structure may introduce more dependencies among the blocks, potentially impacting the scalability and efficiency of the preconditioning method. In contrast, additive-type Schur complement-based preconditioners offer a more straightforward and direct approach to constructing preconditioners, which can sometimes result in simpler implementations and lower computational overhead.

Can the insights from this work be applied to develop efficient preconditioners for other types of large-scale linear systems beyond saddle point problems

The insights gained from this work on Schur complement-based preconditioners for saddle point problems can be applied to develop efficient preconditioners for a wide range of large-scale linear systems beyond saddle point problems. The concept of using Schur complements to design preconditioners can be extended to other types of linear systems, such as symmetric positive definite matrices, general nonsymmetric matrices, and systems arising from various applications in computational science and engineering. By adapting the principles of Schur complement-based preconditioning and leveraging the stability and performance analysis techniques presented in the study, researchers and practitioners can explore the application of these preconditioning strategies to diverse classes of linear systems encountered in numerical simulations, optimization, and scientific computing.