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A reduced basis warm-start iterative solver for the parameterized linear systems

Shijin Hou, Yanlai Chen, Yinhua Xia

TL;DR

This work introduces a Reduced Basis Warm-Start (RBWS) strategy that leverages the L1ROC reduced basis method to generate a high-fidelity initial guess for parametrized linear systems arising from discretized PDEs, followed by refinement with a PCG solver. By comparing RBWS against zero-initialization and RB-preconditioned iterations, the study demonstrates substantial reductions in iteration counts and online solve time, particularly when using a multigrid preconditioner (RBI-MGCG). It also analyzes fundamental limitations of using Reduced Basis Methods as preconditioners at machine-precision accuracy, showing that the RB space may need to grow with iteration, which can negate online gains (as shown for RBI-MSRBCG). The numerical results on two 3D diffusion problems establish the practical utility of RBWS for real-time or many-query contexts, while highlighting cost trade-offs between offline training and online performance.

Abstract

This paper proposes and tests the first-ever reduced basis warm-start iterative method for the parametrized linear systems, exemplified by those discretizing the parametric partial differential equations. Traditional iterative methods are usually used to obtain the high-fidelity solutions of these linear systems. However, they typically come with a significant computational cost which becomes challenging if not entirely untenable when the parametrized systems need to be solved a large number of times (e.g. corresponding to different parameter values or time steps). Classical techniques for mitigating this cost mainly include acceleration approaches such as preconditioning. This paper advocates for the generation of an initial prediction with controllable fidelity as an alternative approach to achieve the same goal. The proposed reduced basis warm-start iterative method leverages the mathematically rigorous and efficient reduced basis method to generate a high-quality initial guess thereby decreasing the number of iterative steps. Via comparison with the iterative method initialized with a zero solution and the RBM preconditioned and initialized iterative method tested on two 3D steady-state diffusion equations, we establish the efficacy of the proposed reduced basis warm-start approach.

A reduced basis warm-start iterative solver for the parameterized linear systems

TL;DR

This work introduces a Reduced Basis Warm-Start (RBWS) strategy that leverages the L1ROC reduced basis method to generate a high-fidelity initial guess for parametrized linear systems arising from discretized PDEs, followed by refinement with a PCG solver. By comparing RBWS against zero-initialization and RB-preconditioned iterations, the study demonstrates substantial reductions in iteration counts and online solve time, particularly when using a multigrid preconditioner (RBI-MGCG). It also analyzes fundamental limitations of using Reduced Basis Methods as preconditioners at machine-precision accuracy, showing that the RB space may need to grow with iteration, which can negate online gains (as shown for RBI-MSRBCG). The numerical results on two 3D diffusion problems establish the practical utility of RBWS for real-time or many-query contexts, while highlighting cost trade-offs between offline training and online performance.

Abstract

This paper proposes and tests the first-ever reduced basis warm-start iterative method for the parametrized linear systems, exemplified by those discretizing the parametric partial differential equations. Traditional iterative methods are usually used to obtain the high-fidelity solutions of these linear systems. However, they typically come with a significant computational cost which becomes challenging if not entirely untenable when the parametrized systems need to be solved a large number of times (e.g. corresponding to different parameter values or time steps). Classical techniques for mitigating this cost mainly include acceleration approaches such as preconditioning. This paper advocates for the generation of an initial prediction with controllable fidelity as an alternative approach to achieve the same goal. The proposed reduced basis warm-start iterative method leverages the mathematically rigorous and efficient reduced basis method to generate a high-quality initial guess thereby decreasing the number of iterative steps. Via comparison with the iterative method initialized with a zero solution and the RBM preconditioned and initialized iterative method tested on two 3D steady-state diffusion equations, we establish the efficacy of the proposed reduced basis warm-start approach.
Paper Structure (16 sections, 3 theorems, 42 equations, 3 figures, 3 tables, 5 algorithms)

This paper contains 16 sections, 3 theorems, 42 equations, 3 figures, 3 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Preconditioned conjugate gradient method
  4. Multigrid preconditioner
  5. Multispace reduced basis preconditioner
  6. Reduced basis warm-start iterative solvers
  7. L1-based reduced over-collocation method
  8. Main algorithm
  9. MSRB Update: Factors toward its low efficiency.
  10. Numerical tests
  11. Example 1
  12. Example 2
  13. Results on convergence and efficiency of RBWS
  14. Numerical confirmation of the efficiency degradation of the RB preconditioner
  15. Conclusion
  16. ...and 1 more sections

Key Result

Lemma 3.1

Given that the linear system is well-conditioned and the Kolmogorov n-width of the error manifold decreases at a polynomial or exponential rate uniformly with respect to $k$, it follows that the dimension $n$ of the reduced basis manifold must increase as the iteration index $k$ increases.

Figures (3)

  • Figure 1: Left: The convergence result with the iteration. Right: The total computational cost as the number of online solvers increases. (Top: Example 1, Bottom: Example 2.)
  • Figure 2: The convergence of the relative residual $r_N$ with the RB dimension $N$. (Left: Example 1, Right: Example 2.)
  • Figure 3: Decay of the eigenvalues for the residual snapshots computed by the RBI-MSRBCG method. (Left: Example 1, Right: Example 2.)

Theorems & Definitions (5)

  • Lemma 3.1
  • proof
  • Remark 3.1
  • Theorem 1
  • Corollary 1