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A computational framework for infinite-dimensional Bayesian inverse problems. Part I: The linearized case, with application to global seismic inversion

arXiv 1308.1313 ↗First posted 6 Aug 2013math.NA cs.NA math.OCcited by 32

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Abstract (arXiv)

We present a computational framework for estimating the uncertainty in the numerical solution of linearized infinite-dimensional statistical inverse problems. We adopt the Bayesian inference formulation: given observational data and their uncertainty, the governing forward problem and its uncertainty, and a prior probability distribution describing uncertainty in the parameter field, find the posterior probability distribution over the parameter field. The prior must be chosen appropriately in order to guarantee well-posedness of the infinite-dimensional inverse problem and facilitate computation of the posterior. Furthermore, straightforward discretizations may not lead to convergent approximations of the infinite-dimensional problem. And finally, solution of the discretized inverse problem via explicit construction of the covariance matrix is prohibitive due to the need to solve the forward problem as many times as there are parameters. Our computational framework builds on the infinite-dimensional formulation proposed by Stuart (A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), pp. 451-559), and incorporates a number of components aimed at ensuring a convergent discretization of the underlying infinite-dimensional inverse problem. The framework additionally incorporates algorithms for manipulating the prior, constructing a low rank approximation of the data-informed component of the posterior covariance operator, and exploring the posterior that together ensure scalability of the entire framework to very high parameter dimensions. We demonstrate this computational framework on the Bayesian solution of an inverse problem in 3D global seismic wave propagation with hundreds of thousands of parameters.

Key facts. Cited by 32 papers in the map (data through 7 Oct 2026). First posted on arXiv 6 Aug 2013.

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Citing papers per year: 2021: 5; 2022: 7; 2023: 8; 2024: 4; 2025: 4; 2026: 4.

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  1. 2022Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learningcited by 43
  2. 2021Projected Wasserstein gradient descent for high-dimensional Bayesian inferencecited by 24
  3. 2022Differentiable Programming for Earth System Modelingcited by 21
  4. 2022Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problemscited by 21
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  7. 2022Bayesian model calibration for block copolymer self-assembly: Likelihood-free inference and expected information gain computation via measure transportcited by 11
  8. 2023A Hybrid Deep Neural Operator/Finite Element Method for Ice-Sheet Modelingcited by 11
  9. 2023Adaptive operator learning for infinite-dimensional Bayesian inverse problemscited by 11
  10. 2023HomPINNs: homotopy physics-informed neural networks for solving the inverse problems of nonlinear differential equations with multiple solutionscited by 8
  11. 2023Solving High-Dimensional Inverse Problems with Auxiliary Uncertainty via Operator Learning with Limited Datacited by 4
  12. 2021Bayesian neural network priors for edge-preserving inversioncited by 2
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  4. 2026Dual-space posterior sampling for Bayesian inference in constrained inverse problemscited by 2
  5. 2023An autoencoder compression approach for accelerating large-scale inverse problemscited by 1
  6. 2023Robust A-Optimal Experimental Design for Bayesian Inverse Problemscited by 1
  7. 2023Stochastic PDE representation of random fields for large-scale Gaussian process regression and statistical finite element analysiscited by 1
  8. 2024Discovering intrinsic multi-compartment pharmacometric models using Physics Informed Neural Networkscited by 1
  9. 2024Probabilistic Approach to Black-Box Binary Optimization with Budget Constraints: Application to Sensor Placementcited by 1
  10. 2025Bayesian Inference for PDE-based Inverse Problems using the Optimization of a Discrete Losscited by 1
  11. 2021Stochastic Learning Approach to Binary Optimization for Optimal Design of Experimentscited by 0
  12. 2022Further analysis of multilevel Stein variational gradient descent with an application to the Bayesian inference of glacier ice modelscited by 0
  13. 2022Posterior sampling with CNN-based, Plug-and-Play regularization with applications to Post-Stack Seismic Inversioncited by 0
  14. 2023Object based Bayesian full-waveform inversion for shear elastographycited by 0
  15. 2024Survey of Deep Learning and Physics-Based Approaches in Computational Wave Imagingcited by 0
  16. 2025Enhanced uncertainty quantification variational autoencoders for the solution of Bayesian inverse problemscited by 0
  17. 2025Learning Generalizable Neural Operators for Inverse Problemscited by 0
  18. 2026Variance-Reduced Diffusion Sampling via Target Score Identitycited by 0
  19. 2026Hierarchical Inference and Closure Learning via Adaptive Surrogates for ODEs and PDEscited by 0
  20. 2026ShaplEIG: Bayesian Experimental Design for Shapley Value Estimationcited by 0