Reduced Krylov Basis Methods for Parametric Partial Differential Equations

TL;DR

The paper introduces Reduced Krylov Basis Methods (RKBMs) for parametric PDEs, constructing the reduced basis in a Krylov subspace rather than from solution snapshots. By solving a high-fidelity problem once with a preconditioner $B\approx A(\mu_0)^{-1}$, the offline basis $\mathcal{H}_m$ spans the Krylov subspace $\mathcal{K}_m$ and enables fast online evaluations of reduced solutions $\hat{u}_\mu$ for many parameters. The work provides convergence analyses for both SPD (RCGBM) and nonsymmetric/indefinite (RKBM$_1$, RKBM$_2$) cases, including invariant Krylov-subspace properties and GMRES/BiCG-type bounds tied to Kolmogorov $n$-width, and it extends to multiple-parameter settings via multi-space variants (mRKBM, mRCGBM). Numerical experiments across H(curl) Maxwell, convection-diffusion, and elasticity verify rapid error decay with modest Krylov-space dimensions and demonstrate substantial online efficiency gains. The approach offers a snapshot-free, user-friendly alternative to greedy or POD-based RBMs with strong theoretical grounding and practical impact for large-scale parametric PDEs.

Abstract

This work is on a user-friendly reduced basis method for solving a family of parametric PDEs by preconditioned Krylov subspace methods including the conjugate gradient method, generalized minimum residual method, and bi-conjugate gradient method. The proposed methods use a preconditioned Krylov subspace method for a high-fidelity discretization of one parameter instance to generate orthogonal basis vectors of the reduced basis subspace. Then large-scale discrete parameter-dependent problems are approximately solved in the low-dimensional Krylov subspace. As shown in the theory and experiments, only a small number of Krylov subspace iterations are needed to simultaneously generate approximate solutions of a family of high-fidelity and large-scale systems in the reduced basis subspace. This reduces the computational cost dramatically because (1) to construct the reduced basis vectors, we only solve one large-scale problem in the high-fidelity level; and (2) the family of large-scale problems restricted to the reduced basis subspace have much smaller sizes.

Figures (6)

• Figure 1: Piecewise constant diffusion coefficient $\alpha_\mu=\sum_{i=1}^4\nu_i\mathbbm{1}_{\Omega_i}$.
• Figure 1: $\log_{10}(\|\bm{u}_{\tilde{\mu}_i}-\hat{\bm{u}}_{\tilde{\mu}_i}\|_{H({\rm curl})}/\|\bm{u}_{\tilde{\mu}_i}\|_{H({\rm curl})})$ of ${\rm RCGBM}(\mathbb{B},\mathcal{P},(1,1),(1,2),m)$ for $\tilde{\mu}_i\in\mathcal{P}$, where $\tilde{\mu}_1=(1,1), \ldots, \tilde{\mu}_{25}=(3,3)$ are ordered in lexicographic way, $ne=1872064$.
• Figure 2: $\log_{10}(|u_{\tilde{\mu}_i}-\hat{u}_{\tilde{\mu}_i}|_{H^1(\Omega)}/|u_{\tilde{\mu}_i}|_{H^1(\Omega)})$ of ${\rm RKBM}_1(\mathbb{B},\mathbb{M},\mathcal{P},(1,\frac{\pi}{2}),(1,0),m)$ for $\tilde{\mu}_i\in\mathcal{P}$, where $\tilde{\mu}_1=(0.4,0), \ldots, \tilde{\mu}_{25}=(2,2\pi)$ are ordered in lexicographic way, $nv=263169$.
• Figure 3: $\log_{10}(|u_{\tilde{\mu}_i}-\hat{u}_{\tilde{\mu}_i}|_{H^1(\Omega)}/|u_{\tilde{\mu}_i}|_{H^1(\Omega)})$ of ${\rm RKBM}_2(\mathbb{B},\mathcal{P},(1,\frac{\pi}{2}),(1,0),m)$ for $\tilde{\mu}_i\in\mathcal{P}$, where $\tilde{\mu}_1=(0.4,0), \ldots, \tilde{\mu}_{25}=(2,2\pi)$ are ordered in lexicographic way, $nv=263169$.
• Figure 4: $\log_{10}(|\!|\!|u_{\tilde{\mu}_i}-\hat{u}_{\tilde{\mu}_i}|\!|\!|/|\!|\!|u_{\tilde{\mu}_i}|\!|\!|)$ of ${\rm mRCGBM}(\mathbb{B},\mathcal{P},\mu_0,\mu_1,24)$, ${\rm mRCGBM}(\mathbb{B}, \mathcal{P}, \mu_0,\mu_1,\mu_2, 12)$, ${\rm mRCGBM}(\mathbb{B},\mathcal{P},\mu_0,\mu_1,\mu_2,\mu_3, 4)$, where $\tilde{\mu}_i \in \mathcal{P}$, $nv = 274625$.

Theorems & Definitions (11)

• Remark 2.1
• Lemma 2.2
• Proof 1
• Theorem 2.3
• Proof 2
• Corollary 2.4
• Proof 3
• Theorem 3.1
• Proof 4
• Theorem 3.2