[Paper Review] On a weighted quasi-residual minimization strategy of the QMR method for solving complex symmetric shifted linear systems

Tomohiro Sogabe, Takeo Hoshi+2 more|ArXiv.org|Feb 16, 2009

Matrix Theory and Algorithms4 citations

TL;DR

This paper proposes a weighted quasi-minimal residual (WQMR) strategy for the QMR_SYM method to solve large-scale complex symmetric shifted linear systems arising in electronic structure simulations. By introducing an optimized weight matrix, the method—named shifted QMR_SYM(B)—significantly reduces computational cost in updating approximate solutions, achieving performance competitive with the shifted COCG method while eliminating the need to select a seed system.

ABSTRACT

We consider the solution of complex symmetric shifted linear systems. Such systems arise in large-scale electronic structure simulations and there is a strong need for the fast solution of the systems. With the aim of solving the systems efficiently, we consider a special case of the QMR method for non-Hermitian shifted linear systems and propose its weighted quasi-minimal residual approach. A numerical algorithm, referred to as shifted QMR\_SYM($B$), is given by the choice of a particularly cost-effective weight. Numerical examples are presented to show the performance of the shifted QMR\_SYM($B$) method.

Motivation & Objective

• Address the high computational cost of updating approximate solutions in the shifted QMR_SYM method for large numbers of shifted linear systems.
• Overcome the limitation of the shifted COCG method, which requires careful seed system selection to avoid convergence failure.
• Leverage the shift-invariance property of Krylov subspaces to reuse basis vectors across multiple shifted systems.
• Develop a cost-effective weighting strategy that minimizes residual updates without sacrificing convergence behavior.
• Provide a robust, efficient alternative to existing Krylov methods for complex symmetric shifted systems in electronic structure calculations.

Proposed method

• Specialize the QMR method for non-Hermitian shifted systems to the complex symmetric case, resulting in the shifted QMR_SYM method.
• Identify that updating approximate solutions is the most time-consuming operation in shifted QMR_SYM when the number of shifts is large.
• Propose a weighted quasi-minimal residual (WQMR) approach to reduce the cost of solution updates by introducing a specific weight matrix.
• Choose a cost-effective weight matrix B that enables real arithmetic operations in residual and solution updates, reducing floating-point operations.
• Integrate the WQMR strategy into the QMR_SYM framework to form the new algorithm, shifted QMR_SYM(B), which maintains convergence properties.
• Ensure the method does not require seed system selection, unlike the shifted COCG method, by exploiting the shift-invariance of Krylov subspaces.

Experimental results

Research questions

• RQ1 What is the dominant computational bottleneck in the shifted QMR_SYM method when solving a large number of shifted linear systems?
• RQ2 Can a weighted quasi-minimal residual strategy reduce the cost of updating approximate solutions without compromising convergence?
• RQ3 How does the performance of the proposed shifted QMR_SYM(B) method compare to the shifted COCG method in terms of computational time and robustness?
• RQ4 Does the new method eliminate the need for seed system selection, a key limitation of the shifted COCG method?
• RQ5 Can the proposed method achieve efficiency comparable to the shifted COCG method while being applicable to a broader class of complex symmetric shifted systems?

Key findings

• The shifted QMR_SYM method requires significantly more computational cost per iteration than the shifted COCG method due to complex arithmetic in solution updates.
• Updating approximate solutions becomes the most time-consuming part of the shifted QMR_SYM method when the number of shifted systems exceeds a threshold (e.g., m > 200).
• The proposed shifted QMR_SYM(B) method reduces computational cost to levels comparable to the shifted COCG method by using a real-weighted strategy.
• For m = 1501, shifted QMR_SYM(B) achieves a computational time ratio of approximately 1.0 relative to shifted COCG, indicating competitive performance.
• The method maintains robustness without requiring seed system selection, unlike the shifted COCG method.
• Numerical experiments on large electronic structure problems confirm that shifted QMR_SYM(B) is a viable and efficient alternative for systems with many shifts.