Model Reduction for Large-Scale Systems with High-Dimensional Parametric Input Space

By Tan Bui-Thanh, Karen E. Willcox, Omar Ghattas. SIAM Journal on Scientific Computing, 2008

418 citations recorded by the source corpus.

A model-constrained adaptive sampling methodology is proposed for the reduction of large-scale systems with high-dimensional parametric input spaces. Our model reduction method uses a reduced basis approach, which requires the computation of high-fidelity solutions at a number of sample points throughout the parametric input space. A key challenge that must be addressed in the optimization, control, and probabilistic settings is the need for the reduced models to capture variation over this parametric input space, which, for many applications, will be of high dimension. We pose the task of determining appropriate sample points as a PDE-constrained optimization problem, which is implemented using an efficient adaptive algorithm that scales well to systems with a large number of parameters. The methodology is demonstrated using examples with parametric input spaces of dimension 11 and 21, which describe thermal analysis and design of a heat conduction fin, and compared with statistically based sampling methods. For these examples, the model-constrained adaptive sampling leads to reduced models that, for a given basis size, have error several orders of magnitude smaller than that obtained using the other methods.

Subfields: Computational Theory and Mathematics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e76bf8884fe09c3f7e0ff29f648c8d7a0b0a8347

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Citations in the corpus

Application of POD and DEIM on dimension reduction of non-linear miscible viscous fingering in porous media (2011, methodology): “It is possible to apply more efficient algorithms for selecting snapshots such as those proposed in [ 38 – 40 ].”
A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems (2015, positive): “For problems with more than a handful of parameters, the greedy sampling approach can be formulated as a sequence of adaptive model-constrained optimization problems [61].”
Sparse-grid, reduced-basis Bayesian inversion (2015, background): “The problem of efficient computational treatment of high-dimensional, multiparametric forward maps in computational Bayesian inversion in Partial Differential Equations (PDEs for short) attracted considerable attention recently; we mention only [6, 7, 5, 44].”
Physically based modeling in catchment hydrology at 50: Survey and outlook (2015, methodology): “Recent research in this field has been spurred by advances in proper orthogonal decomposition and reduced basis finite element methods [Kunisch and Volkwein, 2001; Vermeulen et al., 2004; Grepl and Patera, 2005; Bui-Thanh et al., 2008; Quarteroni et al., 2011; Manzoni, 2014].”
Survey of multifidelity methods in uncertainty propagation, inference, and optimization (2018, background): “Many sampling strategies for selecting the inputs are based on greedy approaches, see [203, 181, 33, 94] and [21] for detailed discussions.”
Interpolation among reduced‐order matrices to obtain parameterized models for design, optimization and probabilistic analysis (2009, methodology): “In some cases, the problem structure admits such a decomposition of the system matrices; otherwise, it can be obtained from the low-order terms of a Taylor series expansion [14, 16, 17].”
Constraint reduction procedures for reduced‐order subsurface flow models based on POD–TPWL (2015, methodology): “A Petrov–Galerkin procedure was also used in [32] for linear model reduction, and recently in [25] for POD–TPWL compositional reservoir simulation.”
Bilevel approaches for learning of variational imaging models (2015, methodology): “In the context of data driven learning approaches we mention statistically grounded methods for optimal model design [42] and marginalization [13, 53], adaptive and multiscale regularization [76, 35, 39], learning in the context of sparse coding and dictionary learning [64, 57, 56, 85, 66],…”
Nonintrusive reduced‐order modeling of parametrized time‐dependent partial differential equations (2013, methodology): “It would be of interest to develop greedy versions of the prop osed approach to improve computational efficiency further (see [10] for a detailed expos iti n of the ROM-greedy algorithm for stationary parametrized problems, and [27, 28] for more gen eral considerations on greedy approaches).”
Surrogate and reduced-order modeling: a comparison of approaches for large-scale statistical inverse problems [Chapter 7] (2010, methodology): “Other choices forΨ are also possible, such as one that minimizes a least-squares weighted-residual (Bui-Thanh et al. 2008; Maday et al. 2002; Rovas 2003; Rozza and Veroy 2006), or one that includes output information through use of adjoint solutions (Lall et al. 2002; Willcox and Peraire 2002).”
The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows (2012, methodology): “that greedy sampling methods tailored for model reduction [38, 6, 39, 40, 33] could alternatively be used to determine Dtrain.”