Surrogate and Reduced‐Order Modeling: A Comparison of Approaches for Large‐Scale Statistical Inverse Problems

Frangos, Michalis; Marzouk, Youssef; Willcox, Karen; Waanders, Bart van Bloemen

doi:10.1002/9780470685853.ch7

Cited by 110 publications

(118 citation statements)

References 76 publications

(71 reference statements)

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“…For example, [32] shows that in a two-dimensional driven cavity flow example for a viscoelastic material, a projection-based reduced model with 60 degrees of freedom performs significantly better than a coarser discretization with 14,803 degrees of freedom. The paper [100] compares projection-based reduced models to stochastic spectral approximations in a statistical inverse problem setting and concludes that, for an elliptic problem with low parameter dimension, the reduced model requires fewer offline simulations to achieve a desired level of accuracy, while the polynomial chaos-based surrogate is cheaper to evaluate in the online phase. In [74] parametric reduced models are compared with Kriging models for a thermal fin design problem and for prediction of contaminant transport.…”

Section: Parametric Model Reduction In Actionmentioning

confidence: 99%

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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Abstract. Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for which the equations governing the system behavior depend on a set of parameters. Examples include parameterized partial differential equations and large-scale systems of parameterized ordinary differential equations. The goal of parametric model reduction is to generate low-cost but accurate models that characterize system response for different values of the parameters. This paper surveys state-of-the-art methods in projection-based parametric model reduction, describing the different approaches within each class of methods for handling parametric variation and providing a comparative discussion that lends insights to potential advantages and disadvantages in applying each of the methods. We highlight the important role played by parametric model reduction in design, control, optimization, and uncertainty quantification-settings that require repeated model evaluations over different parameter values.

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Section: Parametric Model Reduction In Actionmentioning

confidence: 99%

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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“…in order to compute sample statistics such as expectations, variances, and higher moments). Here we do not treat in detail statistical inverse problems and the various approaches that have been proposed in what is a vast field of applied statistics, but limit ourselves to mention that (i) reduction techniques prove to be mandatory also within a statistical approach (as detailed in [20,22,39]) and that (ii) the reduced basis framework is suitable also for uncertainty quantification [26] and more general probabilistic problems [8,46]. In this paper we identify two approaches to deal with uncertainty.…”

Section: A Statistical Framework For Inverse Problemsmentioning

confidence: 99%

“…Offline-Online decomposition. Under the affinity assumption (19)- (20), RB operators can be written isolating the parametric contribution as follows:…”

Section: Thanks To the (Considerably) Reduced Dimension O(n )mentioning

confidence: 99%

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A reduced computational and geometrical framework for inverse problems in hemodynamics

Lassila

Manzoni

Quarteroni³

et al. 2013

Numer Methods Biomed Eng

103

104

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SUMMARYThe solution of inverse problems in cardiovascular mathematics is computationally expensive. In this paper we apply a domain parametrization technique to reduce both the geometrical and computational complexity of the forward problem, and replace the finite element solution of the incompressible NavierStokes equations by a computationally less expensive reduced basis approximation. This greatly reduces the cost of simulating the forward problem. We then consider the solution of inverse problems in both the deterministic sense, by solving a least-squares problem, and in the statistical sense, by using a Bayesian framework for quantifying uncertainty. Two inverse problems arising in haemodynamics modelling are considered: (i) a simplified fluid-structure interaction model problem in a portion of a stenosed artery for quantifying the risk of atherosclerosis by identifying the material parameters of the arterial wall based on pressure measurements; (ii) a simplified femoral bypass graft model for robust shape design under uncertain residual flow in the main arterial branch identified from pressure measurements.

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“…1 Interesting examples of intrusive techniques exploit multiple spatial resolutions of the forward model (Higdon et al, 2003;Christen and Fox, 2005;Efendiev et al, 2006), models with tunable accuracy (Korattikara et al, 2013;Bal et al, 2013), or projection-based reduced order models (Frangos et al, 2010;Lieberman et al, 2010;Cui et al, 2014). or simulated from, respectively (Andrieu and Roberts, 2009;Marin et al, 2011). …”

Section: Introductionmentioning

confidence: 99%

Accelerating Asymptotically Exact MCMC for Computationally Intensive Models via Local Approximations

Conrad

Marzouk

Pillai

et al. 2016

Journal of the American Statistical Association

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We construct a new framework for accelerating Markov chain Monte Carlo in posterior sampling problems where standard methods are limited by the computational cost of the likelihood, or of numerical models embedded therein. Our approach introduces local approximations of these models into the Metropolis-Hastings kernel, borrowing ideas from deterministic approximation theory, optimization, and experimental design. Previous efforts at integrating approximate models into inference typically sacrifice either the sampler's exactness or efficiency; our work seeks to address these limitations by exploiting useful convergence characteristics of local approximations. We prove the ergodicity of our approximate Markov chain, showing that it samples asymptotically from the exact posterior distribution of interest. We describe variations of the algorithm that employ either local polynomial approximations or local Gaussian process regressors. Our theoretical results reinforce the key observation underlying this paper: when the likelihood has some local regularity, the number of model evaluations per MCMC step can be greatly reduced without biasing the Monte Carlo average. Numerical experiments demonstrate multiple order-of-magnitude reductions in the number of forward model evaluations used in representative ODE and PDE inference problems, with both synthetic and real data.

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Surrogate and Reduced‐Order Modeling: A Comparison of Approaches for Large‐Scale Statistical Inverse Problems

Cited by 110 publications

References 76 publications

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

A reduced computational and geometrical framework for inverse problems in hemodynamics

Accelerating Asymptotically Exact MCMC for Computationally Intensive Models via Local Approximations

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