Stochastic Spectral Embedding

Marelli, Stefano; Wagner, Paul-Remo; Lataniotis, Christos; Sudret, Bruno

doi:10.1615/int.j.uncertaintyquantification.2020034395

Cited by 29 publications

(49 citation statements)

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“…Stochastic spectral embedding (SSE) is a multi-level approach to surrogate modeling originally proposed in Marelli et al (2020). It attempts to approximate a given square-integrable…”

Section: Stochastic Spectral Embeddingmentioning

confidence: 99%

“…The original algorithm for computing an SSE was presented in Marelli et al (2020). It recursively partitions the input domain D X and constructs truncated expansions of the residual.…”

Section: Modifications To the Original Algorithmmentioning

confidence: 99%

“…In practice, the residual expansions R ,p S (X ,p ) are computed using a fixed experimental design X and corresponding model evaluations L(X ). The algorithm then only requires the specification of a partitioning strategy and a termination criterion, as detailed in Marelli et al (2020).…”

Section: Modifications To the Original Algorithmmentioning

confidence: 99%

“…Stochastic spectral embedding (SSE) is a metamodelling technique suitable for approximating functions with complex localized features recently developed in Marelli et al (2020).…”

Section: Introductionmentioning

confidence: 99%

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Bayesian model inversion using stochastic spectral embedding

Wagner,

Marelli,

Sudret

2020

Preprint

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In this paper we propose a new sampling-free approach to solve Bayesian model inversion problems that extends the recently introduced spectral likelihood expansions (SLE) method.The latter solves the inverse problem by expanding the likelihood function onto a global polynomial basis orthogonal w.r.t. the prior distribution. This gives rise to analytical expressions for key statistics of the Bayesian posterior distribution, such as evidence, posterior moments and posterior marginals by simple post-processing of the expansion coefficients.It is well known that in most practically relevant scenarios, likelihood functions have close to compact support, which causes the global SLE approach to fail due to the high polynomial degree required for an accurate spectral representation. To solve this problem, we herein replace the global polynomial expansion from SLE with a recently proposed method for local spectral expansion refinement called stochastic spectral embedding (SSE). This surrogatemodeling method was developed for functions with high local complexity. To increase the efficiency of SSE, we enhance it with an adaptive sample enrichment scheme. We show that SSE works well for likelihood approximations and retains the relevant spectral properties of SLE, thus preserving analytical expressions of posterior statistics.To assess the performance of our approach, we include three case studies ranging from low to high dimensional model inversion problems that showcase the superiority of the SSE approach compared to SLE and present the approach as a promising alternative to existing inversion frameworks.

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“…Stochastic spectral embedding (SSE) is a multi-level approach to surrogate modeling originally proposed in Marelli et al (2020). It attempts to approximate a given square-integrable…”

Section: Stochastic Spectral Embeddingmentioning

confidence: 99%

“…The original algorithm for computing an SSE was presented in Marelli et al (2020). It recursively partitions the input domain D X and constructs truncated expansions of the residual.…”

Section: Modifications To the Original Algorithmmentioning

confidence: 99%

Section: Modifications To the Original Algorithmmentioning

confidence: 99%

“…Stochastic spectral embedding (SSE) is a metamodelling technique suitable for approximating functions with complex localized features recently developed in Marelli et al (2020).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Bayesian model inversion using stochastic spectral embedding

Wagner,

Marelli,

Sudret

2020

Preprint

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show abstract

“…Various classes of surrogate models, such as those based on Gaussian process models or kriging (Sacks et al, 1989;Rasmussen, 2003) and polynomial chaos expansions (PCEs) (Xiu and Karniadakis, 2002;Blatman and Sudret, 2011), can be employed in Bayesian inverse problems (Nagel, 2019;Higdon et al, 2015;Marzouk and Xiu, 2009;Marzouk et al, 2007;Wagner et al, 2020Wagner et al, , 2021a. In this contribution, we rely on regression-based sparse PCE due to their extrapolation capabilities and robustness with respect to noise (Blatman and Sudret, 2011;Lüthen et al, 2021b;Marelli et al, 2021b). Several applications of PCE in the geosciences have been proposed.…”

mentioning

confidence: 99%

Bayesian tomography with prior-knowledge-based parametrization and surrogate modeling

Meles,

Linde,

Marelli

2022

Preprint

Self Cite

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We present a Bayesian tomography framework operating with prior-knowledge-based parametrization that is accelerated by surrogate models. Standard high-fidelity forward solvers (e.g., Finite-Difference Time-Domain schemes) solve wave equations with natural spatial parametrizations based on fine discretization. Similar parametrizations, typically involving tens of thousand of variables, are usually employed to parameterize the subsurface in tomography applications. When the data do not allow to resolve details at such finely parameterized scales, it is often beneficial to instead rely on a priorknowledge-based parametrization defined on a lower dimension domain (or manifold). Due to the increased identifiability in the reduced domain, the concomitant inversion is better constrained and generally faster. We illustrate the potential of a prior-knowledge-based approach by considering ground penetrating radar (GPR) travel-time tomography in a crosshole configuration. An effective parametrization of the input (i.e., the permittivity distributions determining the slowness field) and compression in the output (i.e., the travel-time gathers) spaces are achieved via data-driven principal component decomposition based on random realizations of the prior Gaussian-process model with a truncation determined by the performances of the standard solver on the full and reduced model domains. To accelerate the inversion process, we employ a high-fidelity polynomial chaos expansion (PCE) surrogate model. We investigate the impact of the size of the training set on the performance of the PCE and show that a few hundreds design data sets is sufficient to provide reliable Markov chain Monte Carlo inversion at a fraction of the cost associated with a standard approach involving a fine discretization and physics-based forward solvers. Appropriate uncertainty quantification is achieved by reintroducing the truncated higher-order principle components in the original model space after inversion on the manifold and by adapting a likelihood function that accounts for the fact that the truncated higher-order components are not completely located in the null-space.

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Reliability analysis of structures using probability density evolution method and stochastic spectral embedding surrogate model

Das

Tesfamariam

2023

Earthq Engng Struct Dyn

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This study presents an efficient reliability analysis method using probability density evolution method (PDEM) and stochastic spectral embedding (SSE) based surrogate model. The PDEM is used to estimate the structural response's probability density function (PDF). The PDEM is derived based on the principle of probability conservation where generalized density evolution equations (GDEEs) are decoupled from the physical system. The GDEEs are solved using finite difference method coupled with total variation diminishing, in which a set of representative points of random parameters are generated using the generalized F-discrepancy scheme. To obtain satisfactory accuracy of the numerical solution, representative points are needed, which becomes computationally expensive for complex structures. To reduce the computation burden, the SSE is used, which approximates the original response surface. The SSE is a class of supervised machine learning algorithm where it is trained by few observations and enables output prediction as spectral representation. This is achieved by minimizing the residual using domain decomposition technique. To illustrate the proposed SSE-based PDEM, three numerical examples are investigated, including the reliability analysis of four-branch problem and shear building frame subjected to ground acceleration, and the reliability-based design optimization of a moment-resisting frame coupled with the nonlinear energy sink with negative stiffness and sliding friction. Numerical results show that the proposed SSE-based PDEM can estimate failure probability using a very small number of representative points without compromising accuracy compared with Monte Carlo simulation, which leads to a reduction in computational costs.

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Stochastic Spectral Embedding

Cited by 29 publications

References 0 publications

Bayesian model inversion using stochastic spectral embedding

Bayesian model inversion using stochastic spectral embedding

Bayesian tomography with prior-knowledge-based parametrization and surrogate modeling

Reliability analysis of structures using probability density evolution method and stochastic spectral embedding surrogate model

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