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Adaptive $h$-refinement for reduced-order models

Kevin Carlberg

arXiv:1404.0442v3math.NA

TL;DR

Reduced-order models can be inaccurate when online phenomena are absent from offline training, motivating a posteriori refinement without new full-order solves. The paper splits basis vectors according to an offline clustering tree and selects refinements with dual-weighted residuals; experiments show accurate capture of previously untrained phenomena, while refinement adds evaluation overhead and the method omits hyper-reduction.

  • Problem

    ROM accuracy is not guaranteed outside training data, limiting their use for applications requiring reliable predictions of complex online phenomena.

  • Method

    The method enriches ROMs online by splitting selected basis vectors with disjoint support, using an offline k-means tree and dual-weighted residual indicators to target output error.

  • Results

    Numerical examples on the inviscid Burgers equation show accurate prediction of phenomena absent from the training data used to construct the reduced basis.

  • Takeaways & Limitations

    A completely refined basis is equivalent to the full-order model, so h-adaptivity can reach any prescribed tolerance and capture previously unobserved behavior.

  • Takeaways & Limitations

    The study omits hyper-reduction for general nonlinear or non-affine residuals, and adaptive evaluation is roughly twice as slow as a similarly sized non-adaptive ROM.

Abstract

from arXiv · show

This work presents a method to adaptively refine reduced-order models \emph{a posteriori} without requiring additional full-order-model solves. The technique is analogous to mesh-adaptive $h$-refinement: it enriches the reduced-basis space online by `splitting' a given basis vector into several vectors with disjoint support. The splitting scheme is defined by a tree structure constructed offline via recursive $k$-means clustering of the state variables using snapshot data. The method identifies the vectors to split online using a dual-weighted-residual approach that aims to reduce error in an output quantity of interest. The resulting method generates a hierarchy of subspaces online without requiring large-scale operations or full-order-model solves. Further, it enables the reduced-order model to satisfy \emph{any prescribed error tolerance} regardless of its original fidelity, as a completely refined reduced-order model is mathematically equivalent to the original full-order model. Experiments on a parameterized inviscid Burgers equation highlight the ability of the method to capture phenomena (e.g., moving shocks) not contained in the span of the original reduced basis.

1. Introduction

Reduced-order models make repeated high-fidelity simulations tractable, but their accuracy can fail when online phenomena are absent from training data. The proposed method adaptively refines the ROM online through basis splitting, avoiding additional full-order solves while targeting output error.

  • Motivation: ROMs reduce expensive high-fidelity simulations by constructing a low-dimensional basis offline and projecting full-order equations during inexpensive online evaluations.The approach targets many-query and real-time settings.
  • Accuracy challenge: Online accuracy is not guaranteed because phenomena absent from offline training data are missing from ROM predictions.A priori guarantees generally apply only to finite training points.
  • Accuracy challenge: Missing error control limits ROM use in optimization, uncertainty quantification, and highly nonlinear dynamical systems requiring efficient refinement.The paper motivates refinement for phenomena that training may not fully capture.
  • Related approaches: Existing a posteriori improvements either add full-order solutions to the basis or incur large-scale operations, while a priori adaptive methods build separate offline ROMs.These approaches do not provide the proposed solve-free online refinement mechanism.
  • Contribution: The method splits selected basis vectors into disjoint-support vectors using an offline k-means tree and selects refinements online with dual-weighted residuals.It generates a hierarchy of subspaces without large-scale operations or high-fidelity solves.
  • Contribution: A completely refined basis is mathematically equivalent to the full-order model, allowing the ROM to meet any prescribed online error tolerance under modest conditions.This property provides the method’s stated failsafe capability.

2. Problem formulation

The paper formulates parameterized full-order systems and their reduced projections, then defines adaptive refinement as a goal-oriented process that enlarges the reduced space through basis splitting. The refinement hierarchy uses clustered state-variable supports and error indicators to target output accuracy.

  • 2.1. Full-order model: The full-order formulation represents states x_k, parameters μ, residual operators, iterations, and scalar outputs z_k of interest.These quantities define the parameterized input–output problem addressed by model reduction.
  • 2.1. Full-order model: For large state dimension n, solving parameterized full-order systems and evaluating outputs is prohibitively expensive in many-query and real-time applications.The formulation covers discretized PDEs, ODEs, and parameterized linear systems.
  • 2.2. Reduced-order model: ROMs construct a low-dimensional trial basis V and test basis W offline, then seek online solutions in x̄ + range(V) while enforcing residual orthogonality to range(W).The reduced coordinates determine the approximate state within the trial subspace.
  • 2.2. Reduced-order model: General nonlinear or non-affine residuals may require hyper-reduction for n-independent online costs, but this work omits such approximations and leaves them for future work.The stated scope is therefore the non-hyper-reduced setting.
  • 2.2. Reduced-order model: When W = A_n(x; μ)V, Petrov–Galerkin projection is equivalent to Galerkin projection on the modified residual r_k := A_n(x; μ)^T r̃_k.The relation includes Galerkin, balanced-truncation, least-squares Petrov–Galerkin, and stability-oriented choices of A_n.
  • 2.3. Objective: adaptive refinement: Adaptive refinement detects insufficient accuracy, constructs a higher-dimensional basis containing the original range, recomputes the solution, and repeats until the desired output accuracy is reached.The objective is explicitly goal-oriented toward reducing errors in z_k.
  • 2.3. Objective: adaptive refinement: The proposed h-refinement splits selected basis vectors into multiple disjoint-support vectors, with a tree from k-means clustering defining the refinement mechanism.The tree maps coarse basis vectors to their fine children.

3. Refinement mechanism

The method refines a reduced basis online by splitting basis vectors according to an offline tree, producing nested subspaces with provable convergence properties. Under stated support conditions, complete refinement recovers the full-order model, while clustering-based trees can preserve low-dimensional representations.

  • 3.1. Tree data structure: Basis vectors are split into child vectors with disjoint support defined by a tree, while preserving the original nonzero values.Each split replaces one vector with as many child vectors as the node has children.
  • 3.1. Tree data structure: The tree requires complete root support, disjoint child supports whose union equals the parent, and assignment of every element to a leaf node.These conditions underpin the method’s hierarchical-subspace and full-order-model equivalence properties.
  • 3.2. Algebraic refinement: The refinement hierarchy uses exact prolongation from coarse to fine coordinates, unlike typical mesh refinement, whose prolongation is often interpolatory.The fine basis contains at least as many vectors as the coarse basis, and restriction can be defined using a pseudoinverse.
  • 3.3. Properties: The method generates hierarchical subspaces and, for a priori convergent ROMs, guarantees monotonic solution convergence under refinement.The convergence rate is unknown, so stringent tolerances may not preserve low dimensionality, although experiments report accurate responses with low-dimensional refined bases.
  • 3.3. Properties: If the original reduced basis covers every state element, complete splitting yields a reduced-order model equivalent to the full-order model.The completely split basis can be transformed into the identity basis under the theorem’s support condition.

4. Dual-weighted residual error indicators

The method estimates output-relevant refinement error using a dual-weighted-residual approach with adjoint solves, while approximating fine-basis quantities from coarse solutions to avoid expensive full fine-basis solves.

  • 4. Dual-weighted residual error indicators: The error-indicator methodology adapts duality-based error control to reduced-order models through goal-oriented dual-weighted residuals and adjoint solves.The approach is intended for online refinement at the iteration level.
  • 4. Dual-weighted residual error indicators: The method approximates the fine solution's output and residual by first-order expansions about the coarse solution, then solves for the state error.
  • 4. Dual-weighted residual error indicators: A prolongated coarse adjoint replaces a q-dimensional fine-basis adjoint solve, yielding a cheaply computable error estimate.This approximation avoids solves associated with the fine basis V h.
  • 4. Dual-weighted residual error indicators: The method differs from prior reduced-order dual-weighted residual work because it supports adaptive refinement and basis hierarchies without full-order dual solves.

5. Adaptive h-refinement algorithm

The online algorithm repeatedly solves the ROM, refines the basis when the full-order residual exceeds tolerance, and resets it periodically to control growth and amortize refinement work.

  • 5. Adaptive h-refinement algorithm: The algorithm refines until the full-order residual meets its prescribed tolerance, and complete basis splitting makes the ROM mathematically equivalent to the full-order model.Thus, the refinement process can reach any tolerance supported by the full-order residual criterion.
  • 5. Adaptive h-refinement algorithm: The online workflow first computes a ROM solution, refines when the full-order residual exceeds ϵ, and repeats the solve-refine cycle until convergence.The ROM solve itself satisfies the separate tolerance ϵROM.
  • 5. Adaptive h-refinement algorithm: The error-estimation stage solves a coarse adjoint, constructs the fine basis and adjoint representation, and computes the fine error-estimate vector.
  • 5. Adaptive h-refinement algorithm: Periodic basis resets prevent monotonic basis growth and amortize refinement work across later time steps when the solution changes little.The pre-refinement solution can also initialize an iterative ROM solve.
  • 5. Adaptive h-refinement algorithm: Refinement marks parent vectors with above-average child error contributions, splits them according to the tree, and updates the reduced basis and tree nodes.A more sophisticated variant groups children so each group contributes roughly the same fraction of the parent's error.

6. Numerical experiments: parameterized inviscid Burgers’ equation

Experiments on parameterized inviscid Burgers’ equation show that online h-refinement improves reduced-order accuracy for previously unseen shock locations, while accuracy and cost depend on refinement settings and tolerances.

  • 6.1. Fixed inputs: h-adaptive refinement reduced error substantially: with average basis dimension p̄ = 44.3, it outperformed non-adaptive ROMs of dimensions p = 45 and p = 150, whose errors were 43.9% and 8.5%.The best-performing adaptive configuration used p(0) = 10 and nreset = 50.
  • 6.1. Fixed inputs: The unrefined ROM remained highly inaccurate because training snapshots omitted shock locations beyond approximately x = 60, whereas h-adaptivity captured these previously unobserved phenomena.
  • 6.1. Fixed inputs: Smaller initial bases and more frequent resets reduced average basis size but increased refinement calls, making intermediate parameter values fastest online.
  • 6.1. Fixed inputs: Adaptive evaluation with p̄ = 44.3 took roughly twice as long as a non-adaptive ROM with p = 45 because of adaptation overhead.The authors expect this overhead to become relatively smaller for larger problem sizes.
  • 6.1. Fixed inputs: Tighter full-order-model tolerances improved the ROM solution but increased computational cost through larger average bases and more refinement calls per time step.
  • 6.2. Input variation: In the varying-inputs case, more informative snapshots improved the non-adaptive ROM, but h-adaptivity was still required to reduce errors below 1%.

7. Conclusions

The work presents an adaptive h-refinement method that improves reduced-order models a posteriori without large-scale operations, and demonstrates accurate prediction of phenomena absent from training data. It also identifies extensions involving complexity reduction, coarsening, alternative refinement, stability, and extrapolation.

  • The method improves ROM solutions a posteriori without incurring the large-scale operations required by existing approaches.
  • Numerical examples on the inviscid Burgers equation show accurate prediction of phenomena absent from the training data.
  • Future work includes integrating complexity reduction and developing more sophisticated adaptive coarsening mechanisms.
  • Further directions include adaptive p-refinement, alternative tree construction, and assessing the refinement method’s effect on ROM stability.
  • Richardson extrapolation could improve output approximation but requires knowledge of the ROM convergence rate as basis vectors are added.

Appendix A. Refinement algorithm with multiple trees

The multiple-tree refinement mechanism groups a flagged vector’s children according to their contributions to the parent error, reducing over-refinement at the cost of additional bookkeeping. The algorithm updates both the reduced basis and the associated tree structures.

  • Children of a flagged parent are divided into groups contributing roughly equal fractions α of the parent’s total error.
  • Grouping children avoids over-refinement when a parent has many children, but increases the required bookkeeping.
  • The algorithm maintains tree structures while separating children, updating bases and trees, and removing redundant basis vectors.
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