[Paper Review] Reduced-basis method for the iterative solution of parametrized symmetric positive-definite linear systems

Ngoc Cuong Nguyen, Yanlai Chen|arXiv (Cornell University)|Apr 17, 2018

Advanced Numerical Methods in Computational Mathematics24 references3 citations

TL;DR

This paper introduces a novel reduced basis (RB) method for efficiently solving parametrized symmetric positive-definite (SPD) linear systems by using a greedy algorithm to construct a low-dimensional subspace and applying it as a standalone iterative solver or as a preconditioner in the conjugate gradient (CG) method. The approach achieves convergence rates comparable to multigrid-preconditioned CG but with significantly lower computational cost per iteration, making it viable for large-scale problems where multigrid is impractical due to memory or timing constraints.

ABSTRACT

We present a class of reduced basis (RB) methods for the iterative solution of parametrized symmetric positive-definite (SPD) linear systems. The essential ingredients are a Galerkin projection of the underlying parametrized system onto a reduced basis space to obtain a reduced system; an adaptive greedy algorithm to efficiently determine sampling parameters and associated basis vectors; an offline-online computational procedure and a multi-fidelity approach to decouple the construction and application phases of the reduced basis method; and solution procedures to employ the reduced basis approximation as a {\em stand-alone iterative solver} or as a {\em preconditioner} in the conjugate gradient method. We present numerical examples to demonstrate the performance of the proposed methods in comparison with multigrid methods. Numerical results show that, when applied to solve linear systems resulting from discretizing the Poisson's equations, the speed of convergence of our methods matches or surpasses that of the multigrid-preconditioned conjugate gradient method, while their computational cost per iteration is significantly smaller providing a feasible alternative when the multigrid approach is out of reach due to timing or memory constraints for large systems. Moreover, numerical results verify that this new class of reduced basis methods, when applied as a stand-alone solver or as a preconditioner, is capable of achieving the accuracy at the level of the {\em truth approximation} which is far beyond the RB level.

Motivation & Objective

• To develop a reduced basis (RB) method tailored for iterative solution of parametrized symmetric positive-definite (SPD) linear systems arising in parametric PDEs.
• To address the computational bottleneck of repeatedly solving such systems over many parameter values in design, optimization, or uncertainty quantification.
• To decouple offline RB construction from online solution application via an offline-online computational procedure.
• To enable the RB approximation to function as both a standalone iterative solver and an effective preconditioner for CG.
• To demonstrate that the method maintains high accuracy, matching the truth approximation level, even beyond the RB space's intrinsic resolution.

Proposed method

• The method employs a Galerkin projection of the parametrized system onto a reduced basis (RB) space to form a smaller, reduced linear system.
• An adaptive greedy algorithm selects sampling parameters and associated solution snapshots to efficiently construct the RB space, ensuring rapid error decay.
• The approach implements an offline-online decomposition: offline, the RB space and associated operators are constructed; online, only the reduced system is solved.
• The RB approximation is used either as a standalone iterative solver (RBI) or as a preconditioner in the conjugate gradient (RBCG) method.
• A multi-fidelity approach is employed to decouple the construction and application phases, enhancing computational efficiency.
• A posteriori error estimators certify the accuracy of the RB solution relative to the high-fidelity truth solution.

Experimental results

Research questions

• RQ1 Can a reduced basis method be effectively adapted to serve as an iterative solver or preconditioner for parametrized SPD linear systems?
• RQ2 How does the convergence rate of the RB-based iterative method compare to classical multigrid-preconditioned CG in terms of iteration count?
• RQ3 What is the computational cost per iteration of the RB-based schemes compared to multigrid-preconditioned CG, and how does this affect overall efficiency?
• RQ4 To what extent does increasing the RB dimension improve convergence speed and robustness across different parameter values?
• RQ5 Can the RB-based method maintain high accuracy—on par with the truth approximation—despite using a low-dimensional subspace?

Key findings

• For Poisson’s equation discretizations, the RB-based methods match or exceed the convergence speed of multigrid-preconditioned CG, achieving similar or better iteration counts.
• The computational cost per iteration of the RB schemes is significantly lower than that of multigrid-preconditioned CG, leading to orders-of-magnitude faster total solve times.
• Even with a one-dimensional RB space, the method reduces the number of iterations by half compared to standard CG, and with a five-dimensional RB, the iteration count drops to one-tenth.
• When applied as a standalone solver (RBI) or as a preconditioner (RBCG), the method achieves accuracy at the level of the truth approximation, far exceeding the RB space's nominal resolution.
• On large systems with over 2 million degrees of freedom, the RBCG method remains efficient and scalable, while multigrid becomes impractical due to memory and complexity constraints from multiple restriction/prolongation operators.
• The convergence performance of the RB-based schemes improves monotonically with increasing RB dimension, as shown by the average number of steps decreasing across parameter samples.