Adaptive POD–Krylov Enrichment

• The paper introduces an adaptive method that dynamically augments reduced subspaces using Krylov vectors to efficiently solve large-scale, time-dependent, or nonlinear systems.
• It leverages singular value decomposition and residual-based criteria to optimally select and truncate POD modes, ensuring a balance between computational cost and simulation precision.
• The approach integrates hyperreduction and a three-stage augmented Krylov-CG algorithm to maintain accuracy in the presence of evolving topologies and strong nonlinearities.

Adaptive POD–Krylov Enrichment refers to a class of model-reduction techniques designed to accelerate the solution of large-scale parametric or time-dependent problems, especially in computational mechanics, by adaptively integrating Proper Orthogonal Decomposition (POD) with Krylov subspace methods. The central objective is to construct and adapt low-dimensional, goal-oriented subspaces that facilitate efficient iterative solution of sequences of linear or nonlinear systems—typically with slowly varying system matrices—while controlling both computational cost and accuracy (Carlberg et al., 2015, Kerfriden et al., 2011). These methods offer a principled framework to optimally balance dimensionality reduction and online enrichment, crucial for scenarios where strong nonlinearity or topology evolution renders static reduced bases insufficient.

1. Theoretical Foundations

POD–Krylov enrichment originates from two key algorithmic streams. First, Proper Orthogonal Decomposition (POD) generates a reduced basis by extracting dominant directions from a set of high-fidelity solution "snapshots," minimizing projection error in a given metric; truncation is governed by an energy criterion, often retaining the smallest number of modes to capture a prescribed fraction of system dynamics or outputs. Second, Krylov subspace methods, particularly Conjugate Gradient (CG) for symmetric-positive-definite (SPD) systems, form the workhorse of large-scale linear and nonlinear solvers. Krylov-subspace recycling (or augmentation) addresses scenarios where multiple related systems must be solved: it reuses information from past Krylov iterates to expedite subsequent solutions (Carlberg et al., 2015).

The adaptive enrichment paradigm bridges these: it adaptively constructs a goal-oriented POD basis using accumulated Krylov vectors as snapshots, employs a systematic metric selection to align with desired output or error norms, and continuously augments the basis as dictated by online deficiency criteria (such as residual stagnation or detected model inadequacy) (Kerfriden et al., 2011).

2. Adaptive POD Basis Construction and Truncation

Given a collection of candidate vectors $S_j \in \mathbb{R}^{n\times s}$, a weight vector $w \in \mathbb{R}^s$, and a (pseudo)metric $\Theta \succeq 0$, the goal-oriented POD subspace $\mathcal{Y}_y$ of dimension $y$ is constructed to minimize the weighted sum of projection errors: $$\mathcal{Y}_y = \arg\min_{\dim(\mathcal{U})=y}\;\sum_{i=1}^s \|s_i - P_{\mathcal{U}}s_i\|_{\Theta}^2,$$ where $\|v\|_{\Theta}^2 = v^T\Theta v$.

A weighted snapshot matrix $$\widehat{S} = \Theta^{1/2}S_j \operatorname{diag}(w)$$ undergoes a thin singular value decomposition (SVD) $\widehat{S}=U\Sigma V^T$. The POD basis is given as

$$\Phi_y = S_j\operatorname{diag}(w)V_{:,1:y}\Sigma_{1:y,1:y}^{-1},$$

with columns that are $\Theta$-orthonormal and span the optimal subspace. The dimension $y$ is set to satisfy a prescribed energy retention threshold: $$\frac{\sum_{i=1}^y \sigma_i^2}{\sum_{i=1}^s \sigma_i^2} \geq \eta,$$ where $\eta \in (0,1]$.

The method aligns truncation with solver objectives by custom selection of:

• Snapshots: All available Krylov basis vectors from prior solves.
• Metric $\Theta$: $A_{j-1}$ for SPD matrix energy-norm or $C^TC$ for output-oriented goals.
• Weights $w$: Ideally computed from exact solutions, but in practice approximated using previous-solve or recursively blended (RBF-type) formulations. This yields near-optimal subspaces for future inexact solves even when $A_j$ and $b_j$ vary (Carlberg et al., 2015).

Enrichment is triggered when the Krylov vector pool exceeds a storage threshold $m$; then, POD is applied, truncating to $y$ modes to tightly control augmenting subspace dimension.

3. Three-Stage Augmented Krylov-CG Algorithms

After POD-based augmentation, solution of each new system $A_jx_j = b_j$ proceeds via a three-stage strategy:

1. Stage 1: Direct Galerkin projection onto a small core subspace $V_1$, solving $V_1^T A_j V_1 y_1 = V_1^T(b_j-A_jx^{(0)})$ for $y_1$.
2. Stage 2: Iterative CG in the full augmenting subspace $V$, targeting increments in $\operatorname{range}(V)$. This solves $(V^T A_j V)\delta = V^T(b_j - A_j x^{(1)})$. Due to enforced $A_j$-orthogonality, the reduced system remains well-conditioned.
3. Stage 3: Full-space augmented CG, orthogonalizing new search directions with respect to $[V_1, V]$ and running preconditioned CG (PCG) to the required tolerance $\tau_j$.

This strategy exploits both the optimal ordering of POD vectors (principal components are addressed first) and the stability resulting from careful basis orthonormalization (Carlberg et al., 2015). All required reduced-system solves leverage re-used Cholesky factorizations.

4. Goal-Oriented Adaptivity and Online Basis Enrichment

For problems with evolving topology or strong nonlinearity, fixed POD bases rapidly lose representativity. Adaptive POD–Krylov enrichment diagnoses model inadequacy via dual-residual monitoring:

• Reduced residual $r_r$: The norm of the reduced model’s residual.
• Full residual $r_f$: The norm of the full system residual, checked only when the reduced residual is small.

If $r_r \leq \text{New,R}$ but $r_f > \text{New}$, the algorithm triggers an online basis enrichment step by splitting the Newton-Krylov update into a reduced-subspace solve and a PCG correction in the $K_t$-orthogonal complement. Each new Krylov vector appended (properly $K$-orthogonalized) systematically extends the reduced basis. This approach, denoted "C-POD" in the literature, ensures maintained accuracy at a sharply reduced cost—especially when coupled with hyperreduction strategies for problems with localized nonlinearities or damage (Kerfriden et al., 2011).

Basis size growth is controlled via periodic SVD-based compression when the size exceeds prescribed maxima, always retaining leading POD and enriched modes.

5. Integration with Hyperreduction and Implementation

Hyperreduction further accelerates computations by projecting the Galerkin test onto a selective Petrov–Galerkin sampling (matrix $E$), limiting evaluation of internal forces $F_\text{int}$ and assembly of the tangent stiffness matrix $K_t$ to a controlled integration domain. The full-space residual and stiffness are recomputed only in localized patches requiring enrichment, substantially reducing the cost per enrichment (Kerfriden et al., 2011).

Algorithmic integration is as follows:

• Offline: Collect snapshots and determine an initial POD basis by truncated SVD.
• Online (time step loop): Use the reduced model with hyperreduction. Upon Newton convergence deficiency, switch to online Krylov enrichment, append the new vector, and recompute reduced operators.
• Projection and Correction: PCG corrections are done in the $K_t$-orthogonal space, and correction vectors are appended to the reduced basis modularly.

6. Conditioning, Optimal Ordering, and Theoretical Guarantees

If $V$ is constructed such that $V^T A_j V = I$, then for small changes in $A_j$

$$\kappa(V^T A_j V) \leq 1 + \sum_k \|A_k - A_{k-1}\|\|V_k\|^2$$

where $\kappa(\cdot)$ is the condition number. POD-derived modes are optimally ordered: the first $y$ capture maximum projection energy, so even a very small subspace core can drastically accelerate convergence.

Error in computed weights compared to the ideal is bounded by

$$\|w^{\text{ideal}} - w^{\text{prev}}\| \leq \frac{1}{\sigma_{\min}(K_{j-1})} (\|A_j - A_{j-1}\|\|r_j\| + \|r_j - r_{j-1}\|)$$

and subspace distance between different metrics/weights is controlled by eigengap and perturbations, ensuring stable augmentation (Carlberg et al., 2015).

7. Numerical Performance and Applications

In solid-mechanics benchmarks such as the “pancake” and “I-beam” test cases (size > 27,000 DOF), the adaptive POD–Krylov enrichment outperforms both fixed POD and pure recycling schemes. Key empirical results include:

• Comparable or improved wall-times over full-memory ("no-truncation") recycling, with much smaller augmenting subspaces.
• Significant reduction in preconditioner applications compared to unrecycled FOM.
• Output-oriented POD subspaces (with $C^TC$ metrics) significantly improve quantities of interest, especially at moderate tolerances (Carlberg et al., 2015).
• In highly nonlinear situations or damage-driven evolution, adaptive enrichment maintains validation error <1% at $2\times$ the cost of basic POD, as opposed to $20$–$30\times$ for full Newton-Krylov (Kerfriden et al., 2011).

The adaptive POD–Krylov approach, particularly when integrated with Petrov–Galerkin hyperreduction, represents a systematic and flexible bridge between classical MOR and Newton–Krylov solvers, providing accurate, efficient solutions with robust online error control.