Hessian‐based model reduction: large‐scale inversion and prediction

Lieberman, Chad; Fidkowski, Krzysztof J.; Willcox, Karen; Waanders, Bart van Bloemen

doi:10.1002/fld.3650

Cited by 23 publications

(19 citation statements)

References 25 publications

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“…As we will illustrate numerically in Section 5.1, these approximations can be viewed as natural limiting cases of our approach. They are also closely related to previous efforts in dimensionality reduction that propose only Hessian-based [55] or prior-based [61] reductions. In contrast with these previous efforts, here we will consider versions of Hessian-and prior-based reductions that do not discard prior information in the remaining directions.…”

Section: 41mentioning

confidence: 73%

Optimal Low-rank Approximations of Bayesian Linear Inverse Problems

Spantini¹,

Solonen²,

Cui³

et al. 2015

SIAM J. Sci. Comput.

117

223

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Abstract. In the Bayesian approach to inverse problems, data are often informative, relative to the prior, only on a low-dimensional subspace of the parameter space. Significant computational savings can be achieved by using this subspace to characterize and approximate the posterior distribution of the parameters. We first investigate approximation of the posterior covariance matrix as a low-rank update of the prior covariance matrix. We prove optimality of a particular update, based on the leading eigendirections of the matrix pencil defined by the Hessian of the negative log-likelihood and the prior precision, for a broad class of loss functions. This class includes the Förstner metric for symmetric positive definite matrices, as well as the Kullback-Leibler divergence and the Hellinger distance between the associated distributions. We also propose two fast approximations of the posterior mean and prove their optimality with respect to a weighted Bayes risk under squared-error loss. These approximations are deployed in an offline-online manner, where a more costly but data-independent offline calculation is followed by fast online evaluations. As a result, these approximations are particularly useful when repeated posterior mean evaluations are required for multiple data sets. We demonstrate our theoretical results with several numerical examples, including high-dimensional X-ray tomography and an inverse heat conduction problem. In both of these examples, the intrinsic low-dimensional structure of the inference problem can be exploited while producing results that are essentially indistinguishable from solutions computed in the full space.Key words. inverse problems, Bayesian inference, low-rank approximation, covariance approximation, Förstner-Moonen metric, posterior mean approximation, Bayes risk, optimality 1. Introduction. In the Bayesian approach to inverse problems, the parameters of interest are treated as random variables, endowed with a prior probability distribution that encodes information available before any data are observed. Observations are modeled by their joint probability distribution conditioned on the parameters of interest, which defines the likelihood function and incorporates the forward model and a stochastic description of measurement or model errors. The prior and likelihood then combine to yield a probability distribution for the parameters conditioned on the observations, i.e., the posterior distribution. While this formulation is quite general, essential features of inverse problems bring additional structure to the Bayesian update. The prior distribution often encodes some kind of smoothness or correlation among the inversion parameters; observations typically are finite, few in number, and corrupted by noise; and the observations are indirect, related to the inversion parameters by the action of a forward operator that destroys some information. A key consequence of these features is that the data may be informative, relative to the prior, only on a low-dimensional subspace o...

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Section: 41mentioning

confidence: 73%

Optimal Low-rank Approximations of Bayesian Linear Inverse Problems

Spantini¹,

Solonen²,

Cui³

et al. 2015

SIAM J. Sci. Comput.

117

223

View full text Add to dashboard Cite

show abstract

“…• In [154], the authors consider a convection-diffusion model of contaminant transport on a complex three-dimensional domain, parameterized by the initial distribution of contaminant concentration. A POD model (section 3.3) with structure-exploiting parametric sampling reduces the dimension of the problem from more than one million states to 800 states, with no loss of accuracy in the parameter inversion estimates or in the subsequent predictions of future contaminant evolution.…”

Section: Parametric Model Reduction In Actionmentioning

confidence: 99%

“…In these cases, if one treats the initial condition as the parameter set, then the parameter dimension d becomes as large as the system dimension n leading to a high-dimensional parameter space. In some cases this high dimension can be efficiently sampled by exploiting system structure; see, e.g., [27,154] for recent work that considers initial condition parameters for problems that have linear state dependence.…”

Section: 7mentioning

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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“…Such techniques have been successfully applied in domains ranging from computational fluid mechanics [1,2,3], aeroelasticity [4,5] and computational haemodynamics [6] to circuit simulation [7], finance [8] and acoustics [9]. The low cost associated with the solution of reduced order models (ROMs) has in turn allowed their use to accelerate real-time analysis [5,10], PDE-constrained optimization [6,11,12,13,14,15] and uncertainty quantification [16,17] problems. In all these cases -and more generally whenever interested to solve parametrized PDEs many times, for several parametric instances -a suitable offline/online stratagem becomes mandatory to gain a strong computational speedup.…”

Section: Introductionmentioning

confidence: 99%

Efficient model reduction of parametrized systems by matrix discrete empirical interpolation

Negri

Manzoni

Amsallem

2015

Journal of Computational Physics

129

106

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In this work, we apply a Matrix version of the so-called Discrete Empirical Interpolation (MDEIM) for the efficient reduction of nonaffine parametrized systems arising from the discretization of linear partial differential equations. Dealing with affinely parametrized operators is crucial in order to enhance the online solution of reduced-order models (ROMs). However, in many cases such an affine decomposition is not readily available, and must be recovered through (often) intrusive procedures, such as the empirical interpolation method (EIM) and its discrete variant DEIM. In this paper we show that MDEIM represents a very efficient approach to deal with complex physical and geometrical parametrizations in a non-intrusive, efficient and purely algebraic way. We propose different strategies to combine MDEIM with a state approximation resulting either from a reduced basis greedy approach or Proper Orthogonal Decomposition. A posteriori error estimates accounting for the MDEIM error are also developed in the case of parametrized elliptic and parabolic equations. Finally, the capability of MDEIM to generate accurate and efficient ROMs is demonstrated on the solution of two computationally-intensive classes of problems occurring in engineering contexts, namely PDE-constrained shape optimization and parametrized coupled problems.

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Hessian‐based model reduction: large‐scale inversion and prediction

Cited by 23 publications

References 25 publications

Optimal Low-rank Approximations of Bayesian Linear Inverse Problems

Optimal Low-rank Approximations of Bayesian Linear Inverse Problems

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

Efficient model reduction of parametrized systems by matrix discrete empirical interpolation

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