Kevin Carlberg

Adaptive h-refinement for reduced-order models

Our work presents a method to adaptively refine reduced-order models 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. This 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. Furthermore, it enables the reduced-order model to satisfy 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.

Efficient structure-preserving model reduction for nonlinear mechanical systems with application to structural dynamics.

This work proposes a model-reduction methodology that both preserves Lagrangian structure and leads to computationally inexpensive models, even in the presence of high-order nonlinearities. We focus on parameterized simple mechanical systems under Rayleigh damping and external forces, as structural-dynamics models often t this description. The proposed model-reduction methodology directly approximates the quantities that dene the problem’s Lagrangian structure: the Riemannian metric, the potential-energy function, the dissipation function, and the external force. These approximations preserve salient properties (e.g., positive deniteness), behave similarly to the functions they approximate, and ensure computational eciency. Results applied to a simple parameterized trussstructure problem demonstrate the importance of preserving Lagrangian structure and illustrate the method’s ability to generate speedups while maintaining observed stability, in contrast with other model-reduction techniques that do not preserve structure.

Conservative model reduction for finite-volume models

Abstract This work proposes a method for model reduction of finite-volume models that guarantees the resulting reduced-order model is conservative, thereby preserving the structure intrinsic to finite-volume discretizations. The proposed reduced-order models associate with optimization problems characterized by a minimum-residual objective function and nonlinear equality constraints that explicitly enforce conservation over subdomains. Conservative Galerkin projection arises from formulating this optimization problem at the time-continuous level, while conservative least-squares Petrov–Galerkin (LSPG) projection associates with a time-discrete formulation. We equip these approaches with hyper-reduction techniques in the case of nonlinear flux and source terms, and also provide approaches for handling infeasibility. In addition, we perform analyses that include deriving conditions under which conservative Galerkin and conservative LSPG are equivalent, as well as deriving a posteriori error bounds. Numerical experiments performed on a parameterized quasi-1D Euler equation demonstrate the ability of the proposed method to ensure not only global conservation, but also significantly lower state-space errors than nonconservative reduced-order models such as standard Galerkin and LSPG projection.

Corrigendum: Corrigendum to The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows [J. Comput. Phys. 242 (2013) 623-647]

0021-9991/

An On-Line Method for Interpolating Linear Reduced-Order Structural Dynamics Models

see front matter Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jcp.2013.05.022 DOI of original article: http://dx.doi.org/10.1016/j.jcp.2013.02.028 ⇑ Corresponding author. Tel.: +1 925 2946677. E-mail addresses: ktcarlb@sandia.gov (K. Carlberg), cfarhat@stanford.edu (C. Farhat), jcortia@sandia.gov (J. Cortial), amsallem@stan (D. Amsallem). URL: http://sandia.gov/~ktcarlb (K. Carlberg). 1 7011 East Ave., MS 9159, Livermore, CA 94550. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Compan United States Department of Energy under contract DE-AC04-94-AL85000. 2 Durand Building, 496 Lomita Mall, Stanford University, Stanford, CA 94305-3035, United States. Kevin Carlberg a,⇑,1, Charbel Farhat , Julien Cortial , David Amsallem b,2

Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations

A rigorous method for interpolating a set of parameterized linear structural dynamics reduced-order models (ROMs) is presented. By design, this method does not operate on the underlying set of parameterized full-order models. Hence, it is amenable to a real-time and on-line implementation. It is based on mapping appropriately the ROM data onto a tangent space to the manifold of symmetric positive definite matrices, interpolating the mapped data in this space and mapping back the result to the aforementioned manifold. Algorithms for computing the forward and backward mappings are oered for the case where the ROMs are derived from a general Galerkin projection method and the case where they are constructed from modal reduction. The proposed interpolation method is illustrated with applications ranging from the fast dynamic characterization of a parameterized structural model to the fast evaluation of its response to a given input. In all cases, good accuracy is demonstrated at real-time processing speeds.