Exploiting Sparsity in Solving PDE-Constrained Inverse Problems: Application to Subsurface Flow Model Calibration

Golmohammadi, Azarang; Khaninezhad, M-Reza M.; Jafarpour, Behnam

doi:10.1007/978-1-4939-8636-1_12

Cited by 2 publications

(1 citation statement)

References 81 publications

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“…The inverse problem can be defined as finding the functions u and y that minimize the discrepancy with respect to the observed data while satisfying the governing equations and boundary conditions [2,3], that is, min (…”

Section: Introductionmentioning

confidence: 99%

Physics-Informed Machine Learning with Conditional Karhunen-Loève Expansions

Tartakovsky

Barajas-Solano

2019

Preprint

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We present a new physics-informed machine learning approach for the inversion of PDE models with heterogeneous parameters. In our approach, the space-dependent partially-observed parameters and states are approximated via Karhunen-Loève expansions (KLEs). Each of these KLEs is then conditioned on their corresponding measurements, resulting in low-dimensional models of the parameters and states that resolve observed data. Finally, the coefficients of the KLEs are estimated by minimizing the norm of the residual of the PDE model evaluated at a finite set of points in the computational domain, ensuring that the reconstructed parameters and states are consistent with both the observations and the PDE model to an arbitrary level of accuracy.In our approach, KLEs are constructed using the eigendecomposition of covariance models of spatial variability. For the model parameters, we employ a parameterized covariance model calibrated on parameter observations; for the model states, the covariance is estimated from a number of forward simulations of the PDE model corresponding to realizations of the parameters drawn from their KLE. We apply the proposed approach to identifying heterogeneous log-diffusion coefficients in diffusion equations from spatially sparse measurements of the log-diffusion coefficient and the solution of the diffusion equation. We find that the proposed approach compares favorably against state-of-the-art point estimates such as maximum a posteriori estimation and physics-informed neural networks.

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Section: Introductionmentioning

confidence: 99%

Physics-Informed Machine Learning with Conditional Karhunen-Loève Expansions

Tartakovsky

Barajas-Solano

2019

Preprint

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Weighted sparsity regularization for source identification for elliptic PDEs

Elvetun,

Nielsen

2023

Journal of Inverse and Ill-Posed Problems

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This investigation is motivated by PDE-constrained optimization problems arising in connection with electrocardiograms (ECGs) and electroencephalography (EEG). Standard sparsity regularization does not necessarily produce adequate results for these applications because only boundary data/observations are available for the identification of the unknown source, which may be interior. We therefore study a weighted ℓ 1 \ell^{1} -regularization technique for solving inverse problems when the forward operator has a significant null space. In particular, we prove that a sparse source, regardless of whether it is interior or located at the boundary, can be exactly recovered with this weighting procedure as the regularization parameter 𝛼 tends to zero. Our analysis is supported by numerical experiments for cases with one and several local sources. The theory is developed in terms of Euclidean spaces, and our results can therefore be applied to many problems.

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Exploiting Sparsity in Solving PDE-Constrained Inverse Problems: Application to Subsurface Flow Model Calibration

Cited by 2 publications

References 81 publications

Physics-Informed Machine Learning with Conditional Karhunen-Loève Expansions

Physics-Informed Machine Learning with Conditional Karhunen-Loève Expansions

Weighted sparsity regularization for source identification for elliptic PDEs

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