Convergence analysis of surrogate-based methods for Bayesian inverse problems

Yan, Liang; Zhang, Yuanxiang

doi:10.1088/1361-6420/aa9417

Cited by 26 publications

(15 citation statements)

References 32 publications

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“…the prior weighted L 2 -norm, then the approximate posterior converges to the exact posterior with at least the same rate [30,41]. This result has been improved recently in [46] where it was showed that the convergence rate of the posterior approximation is at least twice as large as the convergence rate of the surrogate, for general priors. However, constructing an accurate surrogate over the entire support of the prior might not be feasible and is in fact often unnecessary.…”

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confidence: 82%

Multilevel Adaptive Sparse Leja Approximations for Bayesian Inverse Problems

Farcaș¹,

Latz²,

Ullmann³

et al. 2020

SIAM J. Sci. Comput.

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Deterministic interpolation and quadrature methods are often unsuitable to address Bayesian inverse problems depending on computationally expensive forward mathematical models. While interpolation may give precise posterior approximations, deterministic quadrature is usually unable to efficiently investigate an informative and thus concentrated likelihood. This leads to a large number of required expensive evaluations of the mathematical model. To overcome these challenges, we formulate and test a multilevel adaptive sparse Leja algorithm. At each level, adaptive sparse grid interpolation and quadrature are used to approximate the posterior and perform all quadrature operations, respectively. Specifically, our algorithm uses coarse discretizations of the underlying mathematical model to investigate the parameter space and to identify areas of high posterior probability. Adaptive sparse grid algorithms are then used to place points in these areas, and ignore other areas of small posterior probability. The points are weighted Leja points. As the model discretization is coarse, the construction of the sparse grid is computationally efficient. On this sparse grid, the posterior measure can be approximated accurately with few expensive, fine model discretizations. The efficiency of the algorithm can be enhanced further by exploiting more than two discretization levels. We apply the proposed multilevel adaptive sparse Leja algorithm in numerical experiments involving elliptic inverse problems in 2D and 3D space, in which we compare it with Markov chain Monte Carlo sampling and a standard multilevel approximation.

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mentioning

confidence: 82%

Multilevel Adaptive Sparse Leja Approximations for Bayesian Inverse Problems

Farcaș¹,

Latz²,

Ullmann³

et al. 2020

SIAM J. Sci. Comput.

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“…It also introduces some subtle theoretical and practical difficulties. From a theoretical standpoint, since the surrogate effectively changes every time it is updated (i.e., the forward model changes), standard convergence analyses and proofs of ergodicity and detailed balance found in the literature (e.g., Yan & Zhang, 2017) cannot be easily adopted. Given the numerical character of this work, here we do not attempt such analyses and/or proofs but acknowledge that a general description and validation of the method will require an in-depth study of these properties; we reserve this for a future study.…”

Section: Some Final Remarksmentioning

confidence: 99%

Fast Stokes Flow Simulations for Geophysical‐Geodynamic Inverse Problems and Sensitivity Analyses Based On Reduced Order Modeling

et al. 2020

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Markov chain Monte Carlo (MCMC) methods have become standard in Bayesian inference and multi‐observable inversions in almost every discipline of the Earth sciences. In the case of geodynamic and/or coupled geophysical‐geodynamic inverse problems, however, the computational cost associated with the solution of large‐scale 3‐D Stokes forward problems has rendered probabilistic formulations impractical. Here we present a novel and extremely efficient method to produce ultrafast solutions of the 3‐D Stokes problem for MCMC simulations. Our approach combines the individual benefits of Reduced Basis techniques, goal‐oriented error formulations, and MCMC algorithms to produce an accurate and computationally efficient surrogate for the forward problem. Importantly, the surrogate adapts itself during the MCMC simulation according to the history of the chain and the goals of the inversion. This maximizes the efficiency of the forward problem and removes the need for preinversion off‐line computations to build a surrogate. We demonstrate the benefits and limitations of the method with several numerical examples and show that in all cases the computational cost is of the order of <1% compared to a traditional MCMC approach. The method is general enough to be applied to a range of problems, including uncertainty quantification/propagation, adjoint‐based geodynamic inversions, sensitivity analyses in mantle convection problems, and in the creating surrogate models for complex forward problems (e.g., heat transfer, seismic tomography, and magnetotellurics).

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“…In addition, the error and convergence analysis for surrogates in Bayesian inversion is far from complete (see e.g. [59,64] for recent studies), and requires further work.…”

Section: Surrogatesmentioning

confidence: 99%

Bayesian Parameter Identification in Cahn--Hilliard Models for Biological Growth

Kahle¹,

Lam²,

Latz³

et al. 2019

SIAM/ASA J. Uncertainty Quantification

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We consider the inverse problem of parameter estimation in a diffuse interface model for tumour growth. The model consists of a fourth-order Cahn-Hilliard system and contains three phenomenological parameters: the tumour proliferation rate, the nutrient consumption rate, and the chemotactic sensitivity. We study the inverse problem within the Bayesian framework and construct the likelihood and noise for two typical observation settings. One setting involves an infinite-dimensional data space where we observe the full tumour. In the second setting we observe only the tumour volume, hence the data space is finite-dimensional. We show the well-posedness of the posterior measure for both settings, building upon and improving the analytical results in [C. Kahle and K.F. Lam, Appl. Math. Optim. (2018)]. A numerical example involving synthetic data is presented in which the posterior measure is numerically approximated by the sequential Monte Carlo approach with tempering. are able to ignore apoptosis (programmed cell death) signals, remain elusive to attacks from the immune system, and, most dangerously, have the ability to induce the growth of new blood vessels towards itself (angiogenesis). This leads to the spreading of cancer to other parts of the body, and the formation of secondary tumours (metastasis).The study of tumour growth can be roughly divided according to the physical and chemical phenomena occuring at three scales [47]: the tissue scale which is commonly observed in experiments involving movement of cells (such as metastasis and growth into the extracellular matrix) and nutrient diffusion; the cellular scale consisting of activities and interactions between individual cells such as mitosis and the activation of receptors; and sub-cellular scale where genetic mutations and DNA degradation occur. We focus on the tissue-scaled phenomena, as they are the first to be detected in a routine diagnosis, and can be described fairly well with help of continuum models consisting of differential equations.Since the seminal work in [11] and [27] where simple mathematical models for tumour growth are employed, there has been an explosion in the number of models proposed for modelling the multiscale nature of cancer, see for instance [19,21,47] and the references cited therein. The diversity of model variants reflects the difficulties when we try to identify key biological phenomena that are responsible for experimental observations.As metastasis is an important hallmark of cancer, we restrict our attention to continuum models that can capture such events. Continuum models often rely on a mathematical description to distinguish tumour tissue from healthy host tissues. To be able to capture metastasis the models have to allow for some form of topological change of the separation layers between the tumour and the host tissues. The classical description represents the separation layers as idealised hypersurfaces, known also as the sharp interface approach. In this case complicated boundary conditions have to be imposed t...

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Convergence analysis of surrogate-based methods for Bayesian inverse problems

Cited by 26 publications

References 32 publications

Multilevel Adaptive Sparse Leja Approximations for Bayesian Inverse Problems

Multilevel Adaptive Sparse Leja Approximations for Bayesian Inverse Problems

Fast Stokes Flow Simulations for Geophysical‐Geodynamic Inverse Problems and Sensitivity Analyses Based On Reduced Order Modeling

Bayesian Parameter Identification in Cahn--Hilliard Models for Biological Growth

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