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Level-set physics-informed neural networks for domain inverse problems of gravimetry

T0 review · 3 major / 4 minor · reviewed 2026-07-12 · grok-4.5

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Pith's one-line read Level-set PINNs recover unknown mass domains from surface gravity by treating the interface as a continuous network whose zero level set is the boundary.

desk verdict Solid engineering of level-set PINNs for a well-posed subclass of inverse gravimetry; the interface-aware backprop and its O(τ) analysis are the real additions. read the letter →

arxiv 2607.03772 v1 pith:WEPEKUKS submitted 2026-07-04 math.NA cs.NA

Jingnan Yao , Wenbin Li , Jianliang Qian This is my paper · ORCID
classification math.NAcs.NA MSC 65N2149Q1068T0786A22
keywords level-setmethodphysics-informedneuralnetworksdomaininverseproblemgravimetryinterface-awareback-propagationsigned-distancefunction
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Gravity inversion is classically ill-posed: many different underground mass distributions can produce the same surface measurements. Restricting the unknown to a domain D of known density contrast restores uniqueness under mild geometric conditions. The paper shows that this domain inverse problem can be solved by physics-informed neural networks once the discontinuous characteristic function of D is replaced by a continuous level-set function whose zero contour is the interface. The key technical device is an interface-aware back-propagation rule that widens the numerical support of the interface derivative during training, preventing vanishing or exploding gradients while still keeping the recovered density sharp. Adaptive collocation near the interface and carefully chosen network architectures (ReLU for the level-set, tanh for the potential) complete the scheme. Synthetic two- and three-dimensional tests recover circles, parallelograms, spheres and mixed bodies whose predicted gravity matches the data, demonstrating that the method works in practice.

What carries the argument

Interface-aware back-propagation: the numerical Heaviside is kept thin (parameter τ) in the forward pass, but its derivative is replaced by a thicker kernel (˜τ > τ) in the backward pass, so that a broader set of collocation points can drive interface motion without destroying sharpness.

What would settle it

A synthetic experiment in which the true density contrast is deliberately misspecified, or the true domain violates the star-shaped/convexity hypotheses, and the recovered zero level-set fails to match the true boundary while the data residual remains small.

Watch

Extended reading notes

Core claim

A continuous neural-network level-set function, trained by a physics-informed loss whose interface derivative is deliberately widened during back-propagation, recovers the unknown domain of a volume-mass anomaly from surface gravity measurements while preserving a sharp interface.

Load-bearing premise

The density contrast is known in advance and non-negative, and the unknown domain satisfies one of the geometric constraints that make the domain inverse problem unique.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes level-set physics-informed neural networks for the domain inverse problem of gravimetry. This problem restores uniqueness (Theorem 2.1) to the severely ill-posed inverse gravimetry problem when the density contrast f is known a priori and non-negative and the anomalous domain D satisfies geometric constraints (star-shaped, convex in one direction, etc.). The unknown domain is encoded as the zero level-set of a continuous neural network ϕ_ω, yielding the density ρ_ω = f H_τ(ϕ_ω). A composite PINN loss (PDE residual, boundary gravity data, far-field condition, eikonal and length regularizers, parameter regularization) is minimized by Adam. To mitigate exploding/vanishing gradients induced by the sharp interface, an interface-aware back-propagation redefines the Heaviside derivative as δ_˜τ with ˜τ > τ while keeping a small τ in the forward pass; Lemma 3.1 and Proposition 3.1 bound the resulting loss discrepancy by O(τ). Adaptive collocation near the interface and architecture selection (ReLU for ϕ_ω, tanh for the potential U_θ) justified by W^{1,∞} approximation theory are also introduced. Synthetic 2-D and 3-D recoveries, together with standard-PINN baselines that fail, are presented as evidence of effectiveness.

Significance. If the claims hold, the work supplies a mesh-free neural method for a well-posed subclass of inverse gravimetry and, more broadly, for domain/interface reconstruction problems. The interface-aware gradient redefinition together with its rigorous O(τ) error analysis (Proposition 3.1) is a concrete technical contribution that can transfer to other PINN-based free-boundary or multiphase problems. Explicit approximation-capacity bounds for the chosen ReLU and tanh architectures (Corollaries 3.4–3.6) and clear ablation against naïve PINNs that ignore the level-set structure are additional strengths. The method stays strictly inside the uniqueness class of Theorem 2.1 and does not over-claim.

major comments (3)
  1. [Section 3.2 and Table 1] The loss weights λ1–λ5, interface thicknesses τ/˜τ and refinement width C_refine are chosen purely heuristically (Table 1) and held fixed for every example. No sensitivity study or adaptive-weighting experiment is supplied. Because λ1 = λ2 = 20 dominate the data terms, the reported geometric recoveries could be fragile; this is load-bearing for the claim that the method is effective.
  2. [Proposition 3.1] The O(τ(c−1))+O(τ³) bound that justifies the interface-aware strategy assumes that ϕ_ω is a signed-distance function near ∂D and that collocation points are uniformly distributed. In the actual algorithm the eikonal penalty is only soft (λ3 = 1) and adaptive refinement concentrates points, so the hypothesis of Proposition 3.1 is not guaranteed along the optimization trajectory. A numerical verification of |˜L_pde − L_pde| would close the gap.
  3. [Section 3.5.3] With the arbitrary constant c0 = 1 the guaranteed approximation error for the tanh network U_θ is ε ≈ 0.27 (Section 3.5.3). While the authors remark that practice outperforms the bound, an error of this magnitude relative to typical potential values leaves the theoretical capacity argument for the Laplacian-sensitive potential incomplete.
minor comments (4)
  1. [Equation (3.18)] The surrogate derivative is written H′_τ(s) := δ_˜τ(s) (Eq. 3.18), overwriting the true derivative. A distinct symbol (e.g., δ^surr) would avoid notational confusion.
  2. [Section 4] All examples employ noise-free synthetic data generated from the exact integral equation with perfectly known f. A brief noise-robustness or f-perturbation test would better support the practical claim, even though the theoretical setting of Theorem 2.1 is respected.
  3. [Section 4.1] Pre-training of both networks (Section 4.1) is presented as essential because of initialization sensitivity, yet no ablation without pre-training is shown.
  4. [Section 1] The discussion of related PINN-interface papers (Section 1) correctly notes that they treat forward problems; a one-sentence explicit contrast of the inverse setting would sharpen the novelty claim.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity; the interface-aware strategy and PINN loss are independently derived and analyzed, with only a non-load-bearing self-citation of prior uniqueness results.

  1. uniqueness imported from authors [Section 2, Theorem 2.1 and surrounding text]
    "Theorem 2.1. ... Given the modulus of gravity acceleration, |∇U|, on Σ0, and given f≥0 in Ω, the domain D can be uniquely determined if one of the following constraints is satisfied: ... Theorem 2.1 implies that the domain inverse problem of gravimetry admits uniqueness. ... As noted in [1], the domain inverse problem is a correctness class of the inverse gravimetry; we shall therefore study neural-network approaches for the domain inverse problems."

    Uniqueness is cited from a cluster that includes the authors' own prior papers ([18] Li-Qian 2021 and related works). While the citation is not load-bearing for the new PINN algorithm or the error analysis of Proposition 3.1, it is the sole justification offered for restricting the inverse problem to the domain class; the step is therefore a mild instance of uniqueness imported from the authors rather than an external, independently verified fact used only for context.

full rationale

The paper's core technical contributions (level-set representation of the domain inverse problem inside a PINN residual, the interface-aware redefinition H'_τ(s) := δ_˜τ(s) with ˜τ > τ, the O(τ) error bound of Proposition 3.1 under the signed-distance assumption, adaptive collocation refinement, and the ReLU/tanh architecture choices justified by W^{1,∞} approximation theory) are derived from first principles of the level-set method and automatic differentiation; none of them reduce by construction to fitted inputs or to the uniqueness statement. Theorem 2.1 (the correctness class) is imported from the literature, including prior work by two of the present authors, but it is used only as motivation for restricting attention to the domain inverse problem; the subsequent algorithmic development and the synthetic recovery experiments do not rely on that uniqueness proof as a hidden premise. No parameters are fitted to data and then re-presented as predictions, and the numerical examples remain inside the stated geometric hypotheses. The single minor self-citation therefore does not elevate the circularity score above 1.

Assumptions & free parameters 4 free parameters · 3 assumptions · 1 invented entities

The central claim rests on classical uniqueness theorems for the domain inverse problem, standard Sobolev approximation results for neural networks, and a handful of hand-chosen numerical hyper-parameters; no new physical entities are postulated.

free parameters (4)
  • loss weights λ1…λ5
    Chosen heuristically so that each loss term remains of comparable magnitude during training; values listed in Table 1.
  • interface thickness τ and ˜τ
    τ = 10^{-2}, ˜τ = 2τ fixed by hand to keep the interface sharp while widening the support of the derivative.
  • refinement band C_refine
    Set to 0.18; controls how many extra collocation points are added near the current zero level-set.
  • network depth/width (L=6, width=300)
    Selected after theoretical upper-bound calculations; not optimized by cross-validation.
assumptions (3)
  • domain assumption Uniqueness of the domain inverse problem under the geometric constraints of Theorem 2.1 (star-shaped, convex in one direction, etc.) when f ≥ 0 is known.
    Invoked in Section 2 to justify restricting attention to the domain inverse problem; taken from Isakov and subsequent level-set papers.
  • standard math Signed-distance functions belong to W^{1,∞} and gravity potentials of L^∞ densities belong to W^{1,∞}.
    Propositions 3.2–3.3; used to select ReLU/tanh architectures via known approximation theorems.
  • standard math Yarotsky’s and Zhang–Lu–Zhao’s approximation rates for ReLU and tanh networks on W^{n,∞} balls.
    Theorems 3.2–3.3 and Corollaries; supply the depth/width bounds used to justify the chosen architectures.
invented entities (1)
  • interface-aware back-propagation (H'_τ := δ_˜τ with ˜τ > τ)
    purpose: Widen the support of the interface derivative during training while keeping the forward density sharp.
    Defined in Section 3.3; justified by the error estimate of Proposition 3.1 but has no independent experimental existence outside the paper’s own numerics.

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Cite this review

Pith. "Pith review of Level-set physics-informed neural networks for domain inverse problems of gravimetry." pith.science (2026). https://pith.science/paper/WEPEKUKS

@misc{pith2026260703772,
  author       = {Pith},
  title        = {Pith review of: Level-set physics-informed neural networks for domain inverse problems of gravimetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WEPEKUKS}},
  note         = {Machine review of arXiv:2607.03772}
}
read the original abstract

We propose level-set physics-informed neural networks (PINNs) for domain inverse problems of gravimetry. The domain inverse problem establishes a correctness class for ill-posed inverse gravimetry, which we solve within the PINNs framework. Directly representing the domain inverse problem via neural networks is problematic due to the discontinuous nature of interfaces. We consider a level-set formulation where the neural network represents a continuous level-set function, and its zero level-set depicts sharp interfaces. To overcome the challenges of exploding and vanishing gradients caused by sharp interfaces during training, we propose an interface-aware backpropagation strategy. By redefining the derivative associated with interface evolution, this strategy enables a broader support region to drive the evolution process. Detailed analysis is provided to justify the efficacy of this strategy. Additionally, we introduce a simple procedure for adaptive refinement of collocation points near interfaces. The selection of network architecture is investigated by studying the solution spaces and the approximation properties of neural networks. Finally, extensive 2D and 3D numerical examples demonstrate the effectiveness of the proposed method.

Figures

Figures reproduced from arXiv: 2607.03772 by the authors.

Figure 1
Figure 1. Initialize ϕω(x). (a) The output of the pre-trained ϕω(x) in our 2D ex￾periment, where the red curve indicates its zero level set. (b) The output of the pre-trained ϕω(x) in our 3D experiment, where we plot its zero-level-set isosurface. 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 z 1.5 2.0 2.5 3.0 3.5 4.0 4.5 (a) 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 y 0.5 0.6 0.7 0.8 0.9 (b) [PITH_FULL_IMAGE:fig… view at source ↗
Figure 2
Figure 2. Initialize Uθ(x). (a) The output of the pre-trained Uθ(x) in 2D experiment. (b) The output of the pre-trained Uθ(x) in 3D experiment, where we only display it at z = 0. 4.2. 2D example. 4.2.1. Example 1: A circle and a parallelogram [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 5
Figure 5. The inverted solution successfully recovers the shape of the density model. [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figures from the paper (15 more)
Figure 3
Figure 3. Figure 3: Example 1: A circle and a parallelogram. Left: True density. Right: [PITH_FULL_IMAGE:figures/full_fig_p018_3.png]
Figure 4
Figure 4. Figure 4: Example 1: Gravity-data fitting. The first row shows the fitting for [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: Example 1: Convergence history. The values of the individual loss terms [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Baseline Comparison for Example 1: Standard PINNs. Left: True density. [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: Baseline Comparison for Example 1: Standard PINNs. The performance of [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: Baseline Comparison for Example 1: Standard PINNs. Convergence history. [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 9
Figure 9. Figure 9: Comparison for Example 1: Standard PINNs with a sigmoid activation for [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]
Figure 10
Figure 10. Figure 10: Comparison for Example 1: Standard PINNs with a sigmoid activation for [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]
Figure 11
Figure 11. Figure 11: Comparison for Example 1: Standard PINNs with a sigmoid activation for [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: Example 2: Four spherical anomalies. (a) True and inverted density [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: Example 2: Gravity-data fitting on the top measurement surface ( [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]
Figure 14
Figure 14. Figure 14: Example 3: Anomalies with mixed geometries. (a) True and inverted [PITH_FULL_IMAGE:figures/full_fig_p024_14.png]
Figure 15
Figure 15. Figure 15: Example 3: Gravity-data fitting on the top measurement surface ( [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]
Figure 16
Figure 16. Figure 16: Example 4: Anomalies with depth-dependent density contrast. (a) True [PITH_FULL_IMAGE:figures/full_fig_p025_16.png]
Figure 17
Figure 17. Figure 17: Example 4: Gravity-data fitting on the top measurement surface ( [PITH_FULL_IMAGE:figures/full_fig_p026_17.png]

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Reference graph

Works this paper leans on

300 extracted references

  1. [1]

    Journal of Computational Physics , volume=

    A multiple transferable neural network method with domain decomposition for elliptic interface problems , author=. Journal of Computational Physics , volume=. 2025 , publisher=

    2025

  2. [2]

    arXiv preprint arXiv:2512.10123 , year=

    A Model-Guided Neural Network Method for the Inverse Scattering Problem , author=. arXiv preprint arXiv:2512.10123 , year=

    arXiv

  3. [3]

    2026 , publisher =

    Li, Wenbin and Deng, Xingyu , title =. 2026 , publisher =

    2026

  4. [4]

    Communications in Computational Physics , volume=

    An efficient neural-network and finite-difference hybrid method for elliptic interface problems with applications , author=. Communications in Computational Physics , volume=

  5. [5]

    Computer Methods in Applied Mechanics and Engineering , volume=

    A shallow physics-informed neural network for solving partial differential equations on static and evolving surfaces , author=. Computer Methods in Applied Mechanics and Engineering , volume=. 2024 , publisher=

    2024

  6. [6]

    Journal of Computational Physics , volume=

    A discontinuity capturing shallow neural network for elliptic interface problems , author=. Journal of Computational Physics , volume=. 2022 , publisher=

    2022

  7. [7]

    Annals of Applied Mathematics , volume=

    A discontinuity and cusp capturing PINN for Stokes interface problems with discontinuous viscosity and singular forces , author=. Annals of Applied Mathematics , volume=

  8. [8]

    Journal of Computational Physics , volume=

    A cusp-capturing PINN for elliptic interface problems , author=. Journal of Computational Physics , volume=. 2023 , publisher=

    2023

Show all 300 references
  1. [9]

    IMA Journal of Numerical Analysis , pages=

    Point source identification using singularity-enriched neural networks , author=. IMA Journal of Numerical Analysis , pages=. 2026 , publisher=

    2026

  2. [10]

    , author=

    Solving elliptic optimal control problems via neural networks and optimality system. , author=. Advances in Computational Mathematics , volume=. 2025 , publisher=

    2025

  3. [11]

    Journal of Scientific Computing , volume=

    Scientific machine learning through physics--informed neural networks: Where we are and what's next , author=. Journal of Scientific Computing , volume=. 2022 , publisher=

    2022

  4. [12]

    Inverse Problems , volume=

    Deep learning methods for partial differential equations and related parameter identification problems , author=. Inverse Problems , volume=. 2023 , publisher=

    2023

  5. [13]

    Journal of Computational physics , volume=

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations , author=. Journal of Computational physics , volume=. 2019 , publisher=

    2019

  6. [14]

    Geophysics , volume=

    Constructing piecewise-constant models in multidimensional minimum-structure inversions , author=. Geophysics , volume=. 2008 , publisher=

    2008

  7. [15]

    Inverse problems , volume=

    Numerical tools for analysis and solution of Fredholm integral equations of the first kind , author=. Inverse problems , volume=

  8. [16]

    Gravity field of the Moon from the Gravity Recovery and Interior Laboratory (

    Zuber, Maria T and Smith, David E and Watkins, Michael M and Asmar, Sami W and Konopliv, Alexander S and Lemoine, Frank G and Melosh, H Jay and Neumann, Gregory A and Phillips, Roger J and Solomon, Sean C and others , journal=. Gravity field of the Moon from the Gravity Recove...

    2013

  9. [17]

    Journal of Geophysical Research: Planets , volume=

    Potential anomalies on a sphere: Applications to the thickness of the lunar crust , author=. Journal of Geophysical Research: Planets , volume=. 1998 , publisher=

    1998

  10. [18]

    2007 , publisher=

    Table of integrals, series, and products, Seventh Edition , author=. 2007 , publisher=

    2007

  11. [19]

    Journal of Machine Learning Research , volume=

    Deep network approximation: Beyond relu to diverse activation functions , author=. Journal of Machine Learning Research , volume=

  12. [20]

    Error bounds for approximations with deep

    Yarotsky, Dmitry , journal=. Error bounds for approximations with deep. 2017 , publisher=

    2017

  13. [21]

    arXiv preprint arXiv:2508.05141 , year=

    Deep Neural Networks with General Activations: Super-Convergence in Sobolev Norms , author=. arXiv preprint arXiv:2508.05141 , year=

  14. [22]

    Corless, Robert M and Gonnet, Gaston H and Hare, David EG and Jeffrey, David J and Knuth, Donald E , journal=. On the. 1996 , publisher=

    1996

  15. [23]

    Inverse Problems , volume=

    A level-set structural approach for multi-physics joint inversion using full-waveform and gravity data , author=. Inverse Problems , volume=. 2025 , publisher=

    2025

  16. [24]

    International journal of computer vision , volume=

    A multiphase level set framework for image segmentation using the Mumford and Shah model , author=. International journal of computer vision , volume=. 2002 , publisher=

    2002

  17. [25]

    Journal of Scientific Computing , volume=

    A level set method for the Dirichlet k-partition problem , author=. Journal of Scientific Computing , volume=. 2021 , publisher=

    2021

  18. [26]

    Nonlinear analysis: theory, methods & applications , volume=

    Extending viscosity solutions to eikonal equations with discontinuous spatial dependence , author=. Nonlinear analysis: theory, methods & applications , volume=. 2000 , publisher=

    2000

  19. [27]

    International Journal of Computer Vision , volume=

    Numerical shape-from-shading for discontinuous photographic images , author=. International Journal of Computer Vision , volume=. 2001 , publisher=

    2001

  20. [28]

    Medical physics , volume=

    First-arrival traveltime sound speed inversion with a priori information , author=. Medical physics , volume=. 2014 , publisher=

    2014

  21. [29]

    IEEE Transactions on sonics and ultrasonics , volume=

    Reconstruction of ultrasound propagation speed distributions in soft tissue: time-of-flight tomography , author=. IEEE Transactions on sonics and ultrasonics , volume=. 1977 , publisher=

    1977

  22. [30]

    Geophysical Journal International , volume=

    Applications of seismic travel-time tomography , author=. Geophysical Journal International , volume=. 1987 , publisher=

    1987

  23. [31]

    and Gustavsson, K

    Zahedi, S. and Gustavsson, K. and Kreiss, G. , date-added =. A conservative level set method for contact line dynamics , volume =. J. Comput. Phys. , pages =

  24. [32]

    , date-added =

    Spelt, P.D.M. , date-added =. A level-set approach for simulations of flows with multiple moving contact lines with hysteresis , volume =. J. Comput. Phys. , pages =

  25. [33]

    and Xu, S

    Zhao, Q. and Xu, S. and Ren, W. , date-added =. A level set method for simulation of moving contact lines in three dimensions , volume =. Commun. Comput. Phys. , number =

  26. [34]

    and van Rees, W.M

    Gabbard, J. and van Rees, W.M. , date-added =. A high-order finite difference method for moving immersed domain boundaries and material interfaces , volume =. J. Comput. Phys. , number =

  27. [35]

    Elliptic Problems in Nonsmooth Domains , year =

    Grisvard, Pierre , publisher =. Elliptic Problems in Nonsmooth Domains , year =

  28. [36]

    A comparative study of structural similarity and regularization for joint inverse problems governed by PDEs , volume =

    Crestel, Benjamin and Stadler, Georg and Ghattas, Omar , journal =. A comparative study of structural similarity and regularization for joint inverse problems governed by PDEs , volume =

  29. [37]

    Joint inversion approaches for geophysical electromagnetic and elastic full-waveform data , volume =

    Abubakar, Aria and Gao, G and Habashy, Tarek M and Liu, J , journal =. Joint inversion approaches for geophysical electromagnetic and elastic full-waveform data , volume =

  30. [38]

    Joint two-dimensional DC resistivity and seismic travel time inversion with cross-gradients constraints , volume =

    Gallardo, Luis A and Meju, Max A , journal =. Joint two-dimensional DC resistivity and seismic travel time inversion with cross-gradients constraints , volume =

  31. [39]

    Joint inversion: a structural approach , volume =

    Haber, E and Oldenburg, D , journal =. Joint inversion: a structural approach , volume =

  32. [40]

    Joint inversion of refraction and gravity data for the three-dimensional topography of a sediment--basement interface , volume =

    Afnimar and Koketsu, Kazuki and Nakagawa, Koichi , journal =. Joint inversion of refraction and gravity data for the three-dimensional topography of a sediment--basement interface , volume =

  33. [41]

    Integrated gravity and wide-angle seismic inversion for two-dimensional crustal modelling , volume =

    Nielsen, Lars and Jacobsen, Bo Holm , journal =. Integrated gravity and wide-angle seismic inversion for two-dimensional crustal modelling , volume =

  34. [42]

    A global integration platform for optimizing cooperative modeling and simultaneous joint inversion of multi-domain geophysical data , volume =

    Dell'Aversana, Paolo and Bernasconi, Giancarlo and Chiappa, Fabio and others , journal =. A global integration platform for optimizing cooperative modeling and simultaneous joint inversion of multi-domain geophysical data , volume =

  35. [43]

    An optimal transport approach for seismic tomography: Application to 3D full waveform inversion , volume =

    M. An optimal transport approach for seismic tomography: Application to 3D full waveform inversion , volume =. Inverse Problems , number =

  36. [44]

    Total variation regularization for seismic waveform inversion using an adaptive primal dual hybrid gradient method , volume =

    Yong, Peng and Liao, Wenyuan and Huang, Jianping and Li, Zhenchun , journal =. Total variation regularization for seismic waveform inversion using an adaptive primal dual hybrid gradient method , volume =

  37. [45]

    Efficient 1.5 D full waveform inversion in the Laplace-Fourier domain , volume =

    Kontakis, Apostolos and Rovetta, Diego and Colombo, Daniele and Sandoval-Curiel, Ernesto , journal =. Efficient 1.5 D full waveform inversion in the Laplace-Fourier domain , volume =

  38. [46]

    An overview of full-waveform inversion in exploration geophysics , volume =

    Virieux, Jean and Operto, St. An overview of full-waveform inversion in exploration geophysics , volume =. Geophysics , number =

  39. [47]

    FWIGAN: Full-waveform inversion via a physics-informed generative adversarial network , volume =

    Yang, Fangshu and Ma, Jianwei , journal =. FWIGAN: Full-waveform inversion via a physics-informed generative adversarial network , volume =

  40. [48]

    Notes on perfectly matched layers (

    Johnson, Steven G , journal =. Notes on perfectly matched layers (

  41. [49]

    Deepwave , url =

    Richardson, Alan , doi =. Deepwave , url =

  42. [50]

    Learning on the correctness class for domain inverse problems of gravimetry , volume =

    Chen, Yihang and Li, Wenbin , journal =. Learning on the correctness class for domain inverse problems of gravimetry , volume =

  43. [51]

    Adam: A method for stochastic optimization , year =

    Kingma, Diederik P and Ba, Jimmy , journal =. Adam: A method for stochastic optimization , year =

  44. [52]

    A stochastic gradient descent approach with partitioned-truncated singular value decomposition for large-scale inverse problems of magnetic modulus data , volume =

    Li, Wenbin and Wang, Kangzhi and Fan, Tingting , journal =. A stochastic gradient descent approach with partitioned-truncated singular value decomposition for large-scale inverse problems of magnetic modulus data , volume =

  45. [53]

    Inverse theory and applications in geophysics , year =

    Zhdanov, Michael S , publisher =. Inverse theory and applications in geophysics , year =

  46. [54]

    U-net: Convolutional networks for biomedical image segmentation , year =

    Ronneberger, Olaf and Fischer, Philipp and Brox, Thomas , booktitle =. U-net: Convolutional networks for biomedical image segmentation , year =. doi:10.1007/978-3-319-24574-4_28 , fbooktitle =

    doi

  47. [55]

    Bianchi, Davide and Lai, Guanghao and Li, Wenbin , journal =

  48. [56]

    A Multitask Deep Learning for Simultaneous Denoising and Inversion of 3-

    Zhang, Lianzhi and Zhang, Guibin and Fan, Zhenyu and Ma, Jianwei , journal =. A Multitask Deep Learning for Simultaneous Denoising and Inversion of 3-

  49. [57]

    Deep learning for 3-

    Zhang, Lianzhi and Zhang, Guibin and Liu, Yao and Fan, Zhenyu , journal =. Deep learning for 3-

  50. [58]

    Yang, Qianguo and Hu, Xiangyun and Liu, Shuang and Jie, Qu and Wang, Huaijiang and Chen, Qiuhua , journal =. 3-

  51. [59]

    Zhou, Xinyi and Chen, Zhaoxi and Lv, Yandong and Wang, Shuai , journal =. 3-

  52. [60]

    Deep learning 3

    Huang, Rui and Liu, Shuang and Qi, Rui and Zhang, Yujie , journal =. Deep learning 3

  53. [61]

    Ge, Rong and Huang, Furong and Jin, Chi and Yuan, Yang , booktitle =

  54. [62]

    An alternative view: When does

    Kleinberg, Robert and Li, Yuanzhi and Yuan, Yang , booktitle =. An alternative view: When does

  55. [63]

    Barzilai-Borwein step size for stochastic gradient descent , year =

    Tan, Conghui and Ma, Shiqian and Dai, Yu-Hong and Qian, Yuqiu , booktitle =. Barzilai-Borwein step size for stochastic gradient descent , year =

  56. [64]

    Curiously fast convergence of some stochastic gradient descent algorithms , volume =

    Bottou, L. Curiously fast convergence of some stochastic gradient descent algorithms , volume =. Proceedings of the symposium on learning and data science, Paris , pages =

  57. [65]

    Stochastic gradient descent tricks , year =

    Bottou, L. Stochastic gradient descent tricks , year =. Neural networks: Tricks of the trade , pages =

  58. [66]

    Efficient mini-batch training for stochastic optimization , year =

    Li, Mu and Zhang, Tong and Chen, Yuqiang and Smola, Alexander J , booktitle =. Efficient mini-batch training for stochastic optimization , year =

  59. [67]

    A level-set algorithm for the inverse problem of full magnetic gradient tensor data , volume =

    Li, Wenbin and Qian, Jianliang , journal =. A level-set algorithm for the inverse problem of full magnetic gradient tensor data , volume =

  60. [68]

    Computers & Mathematics with Applications , volume=

    Newton-type Gauss--Seidel Lax--Friedrichs high-order fast sweeping methods for solving generalized eikonal equations at large-scale discretization , author=. Computers & Mathematics with Applications , volume=. 2020 , publisher=

    2020

  61. [69]

    Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods , volume =

    Li, Wenbin and Qian, Jianliang , journal =. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods , volume =

  62. [70]

    Kantorovich-Rubinstein metric based level-set methods for inverting modulus of gravity-force data , volume =

    Li, Wenbin and Qian, Jianliang , journal =. Kantorovich-Rubinstein metric based level-set methods for inverting modulus of gravity-force data , volume =

  63. [71]

    How good is

    Safran, Itay and Shamir, Ohad , booktitle =. How good is

  64. [72]

    A stochastic approximation method , year =

    Robbins, Herbert and Monro, Sutton , journal =. A stochastic approximation method , year =

  65. [73]

    Robust stochastic approximation approach to stochastic programming , volume =

    Nemirovski, Arkadi and Juditsky, Anatoli and Lan, Guanghui and Shapiro, Alexander , journal =. Robust stochastic approximation approach to stochastic programming , volume =

  66. [74]

    Optimization for simulation: Theory vs

    Fu, Michael C , journal =. Optimization for simulation: Theory vs. practice , volume =

  67. [75]

    Optimization methods for large-scale machine learning , volume =

    Bottou, L. Optimization methods for large-scale machine learning , volume =. SIAM Review , number =

  68. [76]

    Imaging cargo containers using gravity gradiometry , volume =

    Kirkendall, Barry and Li, Yaoguo and Oldenburg, Douglas , journal =. Imaging cargo containers using gravity gradiometry , volume =

  69. [77]

    Optimal approximations by piecewise smooth functions and associated variational problems , volume =

    Mumford, David and Shah, Jayant , journal =. Optimal approximations by piecewise smooth functions and associated variational problems , volume =

  70. [78]

    A local level-set method for 3D inversion of gravity-gradient data , volume =

    Lu, Wangtao and Qian, Jianliang , journal =. A local level-set method for 3D inversion of gravity-gradient data , volume =

  71. [79]

    Kantorovich-Rubinstein misfit for inverting gravity-gradient data by the level-set method , volume =

    Huang, Guanghui and Zhang, Xinming and Qian, Jianliang , journal =. Kantorovich-Rubinstein misfit for inverting gravity-gradient data by the level-set method , volume =

  72. [80]

    Analysis of Regularized Kantorovich--Rubinstein Metric and Its Application to Inverse Gravity Problems , volume =

    Huang, Guanghui and Qian, Jianliang , journal =. Analysis of Regularized Kantorovich--Rubinstein Metric and Its Application to Inverse Gravity Problems , volume =

  73. [81]

    Joint inversion of surface and three-component borehole magnetic data , volume =

    Li, Yaoguo and Oldenburg, Douglas W , journal =. Joint inversion of surface and three-component borehole magnetic data , volume =

  74. [82]

    3D inversion of magnetic total gradient data in the presence of remanent magnetization , year =

    Shearer, Sarah and Li, Yaoguo , booktitle =. 3D inversion of magnetic total gradient data in the presence of remanent magnetization , year =

  75. [83]

    Joint inversion of surface and borehole magnetic data: A level-set approach , volume =

    Li, Wenbin and Qian, Jianliang and Li, Yaoguo , journal =. Joint inversion of surface and borehole magnetic data: A level-set approach , volume =

  76. [84]

    Inversion of the magnetic field gradient equation for a magnetic dipole field , year =

    Frahm, Charles P , journal =. Inversion of the magnetic field gradient equation for a magnetic dipole field , year =

  77. [85]

    Method of magnetic source localization using gradient tensor components and rate tensor components , year =

    Wynn, William M , month = jul #. Method of magnetic source localization using gradient tensor components and rate tensor components , year =

  78. [86]

    Inversion of geo-magnetic full-tensor gradiometer data , volume =

    Schneider, M and Stolz, R and Linzen, S and Schiffler, M and Chwala, A and Schulz, M and Dunkel, S and Meyer, H-G , journal =. Inversion of geo-magnetic full-tensor gradiometer data , volume =

  79. [87]

    Detection of buried magnetic objects by a SQUID gradiometer system , volume =

    Meyer, Hans-Georg and Hartung, Konrad and Linzen, Sven and Schneider, Michael and Stolz, Ronny and Fried, Wolfgang and Hauspurg, Sebastian , booktitle =. Detection of buried magnetic objects by a SQUID gradiometer system , volume =

  80. [88]

    Schmidt, Phillip and Clark, David and Leslie, Keith and Bick, Marcel and Tilbrook, David and Foley, Cathy , journal =

  81. [89]

    Peiselt, K and Schmidl, F and Linzen, S and Anton, AS and H. High-. Superconductor Science and Technology , number =

  82. [90]

    Clem, Ted R and Overway, David J and Purpura, John W and Bono, John T and Koch, Roger H and Rozen, James R and Keefe, George A and Willen, Scott and Mohling, Robert A , journal =. High-

  83. [91]

    Advances in sensor development and demonstration of superconducting gradiometers for mobile operation , volume =

    Clem, T R and Froelich, M C and Ovenvay, D J and Purpura, J W and Wiegert, R F and Koch, R H and Lathrop, D K and Rozen, J and Eraker, J H and Schmidt, J M , journal =. Advances in sensor development and demonstration of superconducting gradiometers for mobile operation , volume =

  84. [92]

    Field trials using

    Foley, Catherine P and Leslie, KE and Binks, R and Lewis, Christoper and Murray, Wayne and Sloggett, GJ and Lam, S and Sankrithyan, B and Savvides, N and Katzaros, A and others , journal =. Field trials using

  85. [93]

    Experience with SQUID magnetometers in airborne TEM surveying , volume =

    Lee, James B and Turner, Robert J and Downey, Mark A and Maddever, Arthur and Dart, David L and Foley, Catherine P and Binks, Rex and Lewis, Christopher and Murray, Wayne and Panjkovic, Goran and others , journal =. Experience with SQUID magnetometers in airborne TEM surveying...

  86. [94]

    The magnetic gradient tensor: Its properties and uses in source characterization , volume =

    Schmidt, Phil W and Clark, David A , journal =. The magnetic gradient tensor: Its properties and uses in source characterization , volume =

  87. [95]

    New methods for interpretation of magnetic gradient tensor data , volume =

    Clark, David A , journal =. New methods for interpretation of magnetic gradient tensor data , volume =

  88. [96]

    Full magnetic gradient tensor from triaxial aeromagnetic gradient measurements: Calculation and application , volume =

    Luo, Yao and Wu, Mei-Ping and Wang, Ping and Duan, Shu-Ling and Liu, Hao-Jun and Wang, Jin-Long and An, Zhan-Feng , journal =. Full magnetic gradient tensor from triaxial aeromagnetic gradient measurements: Calculation and application , volume =

  89. [97]

    Estimating source location using normalized magnetic source strength calculated from magnetic gradient tensor data , volume =

    Beiki, Majid and Clark, David A and Austin, James R and Foss, Clive A , journal =. Estimating source location using normalized magnetic source strength calculated from magnetic gradient tensor data , volume =

  90. [98]

    Nabighian, M. N. and Grauch, V.J.S. and Hansen, R. O. and LaFehr, T. R. and Li, Y. and Peirce, J. W. and Phillips, J. D. and Ruder, M. E. , journal =. The historical development of the magnetic method in exploration , volume =

  91. [99]

    Quantum Detection Meets Archaeology--Magnetic Prospection with SQUIDs, Highly Sensitive and Fast , year =

    Linzen, Sven and Schultze, Volkmar and Chwala, Andreas and Sch. Quantum Detection Meets Archaeology--Magnetic Prospection with SQUIDs, Highly Sensitive and Fast , year =. New Technologies for Archaeology , pages =

  92. [100]

    Zhdanov, Michael S and. 3. SEG Technical Program Expanded Abstracts 2012 , pages =

    2012

  93. [101]

    Magnetic mapping for the detection and characterization of

    Munschy, Marc and Boulanger, Daniel and Ulrich, Patrice and Bouiflane, Mustapha , journal =. Magnetic mapping for the detection and characterization of

  94. [102]

    , publisher =

    Datta, B.N. , publisher =. Numerical linear algebra and applications , year =

  95. [103]

    A fast sweeping method for static convex

    Qian, Jianliang and Zhang, Yong-Tao and Zhao, Hong-Kai , journal =. A fast sweeping method for static convex

  96. [104]

    , journal =

    Sethian, J.A. , journal =. A fast marching level set method for monotonically advancing fronts , volume =

  97. [105]

    Well-to-well seismic measurements , volume =

    Bois, P and La Porte, M and Lavergne, M and Thomas, G , journal =. Well-to-well seismic measurements , volume =

  98. [106]

    and Coulouvrat, F

    Loubeau, A. and Coulouvrat, F. , journal =. Effects of meteorological variability on sonic boom propagation from hypersonic aircraft , volume =

  99. [107]

    Hjelle,. A. Mathematical Geosciences , number =

  100. [108]

    Plessix, R. -E. , date-added =. A review of the adjoint-state method for computing the gradient of a functional with geophysical applications , volume =

  101. [109]

    J. J. Duistermaat , date-added =. Fourier integral operators , year =

  102. [110]

    and Li, B

    Qian, J. and Li, B. , date-added =. In preparation , pages =

  103. [111]

    and Li, B

    Qian, J. and Li, B. , date-added =. Parallel high-order fast sweeping methods on. In preparation , pages =

  104. [112]

    and Keller, J

    Karal Jr., F.C. and Keller, J. B. , date-added =. Elastic wave propagation in homogeneous and inhomogeneous media , volume =. J. Acoust. Soc. Am. , number =

  105. [113]

    Lu and J

    W. Lu and J. Qian and R. Burridge , date-added =

  106. [114]

    Qian and R

    J. Qian and R. Burridge , date-added =. Ray theory for isotropic elasticity near a point source

  107. [115]

    Qian and R

    J. Qian and R. Burridge , date-added =. Uniform expansion near a point source for a self-adjoint scalar wave equation , year =

  108. [116]

    and Shilov, G.E

    Gelfand, I.M. and Shilov, G.E. , date-added =. Generalized Functions , volume =

  109. [117]

    A multiple level-set method for 3

    Li, Wenbin and Lu, Wangtao and Qian, Jianliang and Li, Yaoguo , date-added =. A multiple level-set method for 3

  110. [118]

    Zworski , date-added =

    M. Zworski , date-added =. Semiclassical Analysis , year =

  111. [119]

    Li and W

    W. Li and W. Lu and J. Qian and Y. Li , booktitle =. A multiple level set method for three-dimensional inversion of magnetic data , year =

  112. [120]

    Lu and J

    W. Lu and J. Qian and R. Burridge , booktitle =. Babich-expansion based Fast

  113. [121]

    Fang and J

    J. Fang and J. Qian and L. Zepeda-Nunez and H. Zhao , date-added =. A hybrid approach to solve the high-frequency. J. Comp. Phys. , pages =

  114. [122]

    Glowinski and H

    R. Glowinski and H. Liu and S. Leung and J. Qian , date-added =. A Finite Element/Operator-Splitting Method for the Numerical Solution of the Two Dimensional Elliptic

  115. [123]

    Lu and R

    W. Lu and R. Burridge and J. Qian , date-added =. EXTENDING

  116. [124]

    Glowinski and S

    R. Glowinski and S. Leung and J. Qian , date-added =. A SIMPLE EXPLICIT OPERATOR-SPLITTING METHOD FOR EFFECTIVE HAMILTONIANS , volume =. SIAM J. Sci. Comput. , pages =

  117. [125]

    and Lu, Y

    Lu, W. and Lu, Y. and Qian, J. , date-added =. SIAM J. Applied Math. , pages =

  118. [126]

    Kwan, W. and S. Leung, S. and Wang, X-P and Qian, J. , date-added =. A Fast Huygens Sweeping Method for Capturing Paraxial Multi-color Optical Self-focusing in Nematic Liquid Crystals , volume =. Journal of Computational Physics , pages =

  119. [127]

    and Lam, C

    Chung, E. and Lam, C. and Qian, J. , date-added =. A ray-based. Journal of Computational Physics , pages =

  120. [128]

    and Sheppard, M

    Glowinski, R and Shiau, L. and Sheppard, M. , date-added =. Numerical Methods for a Class of Nonlinear Integral-differential Equations , volume =. Calcolo , number =

  121. [129]

    and Camilli, F

    Cacace, S. and Camilli, F. , date-added =. A generalized

  122. [130]

    and Chambolle, A

    Cacace, S. and Chambolle, A. and Monneau, R. , date-added =. A posteriori error estimates for the effective. Numer. Math. , pages =

  123. [131]

    and Lions, P.-L

    Lasry, J.-M. and Lions, P.-L. , date-added =. Mean field games , volume =. Jpn. J. Math. , pages =

  124. [132]

    and Gomes, D

    Camilli, F. and Gomes, D. and Capuzzo Dolcetta, I. , date-added =. Error estimates for the approximation of the effective. Appl. Math. Optim. , pages =

  125. [133]

    and Tran, H

    Qian, J. and Tran, H. and Yu, Y. , date-added =. Math. Ann. , pages =

  126. [134]

    Lions, P. L. and Souganidis, P.E. , date-added =. Correctors for the homogenization of. Commun. Pure Appl. Math. , pages =

  127. [135]

    , date-added =

    Glowinski, R. , date-added =. Finite element methods for incompressible viscous flow, In

  128. [136]

    , date-added =

    Xin, J. , date-added =. An introduction to fronts in random media, In

  129. [137]

    and Takei, R

    Oberman, A. and Takei, R. and Vladimirsky, A. , date-added =. Homogenization of metric. SIAM Multiscale Model. Simul. , number =

  130. [138]

    and Jensen, R

    Barron, E.N. and Jensen, R. , date-added =. Semicontinuous Viscosity Solutions For. Comm. Partial Differential Equations , pages =

  131. [139]

    , date-added =

    Evans, L.C. , date-added =. Some new. Calc. Var. Partial Differential Equations , number =

  132. [140]

    and Rorro, M

    Falcone, M. and Rorro, M. , booktitle =. Optimization techniques for the computation of the effective

  133. [141]

    and Rorro, M

    Falcone, M. and Rorro, M. , date-added =. On a variational approximation of the effective. Numerical Mathematics and Advanced Applications , pages =

  134. [142]

    and Camilli, F

    Achdou, Y. and Camilli, F. and Dolcetta, I.C. , date-added =. Homogrenization of. Math. Models and Methods in Appl. Sci. , number =

  135. [143]

    and Rorro, M

    Falcone, M. and Rorro, M. , date-added =. A parallel semi-. Communications to SIMAI Congress , pages =

  136. [144]

    , date-added =

    Rorro, M. , date-added =. An approximation scheme for the effective. Applied Numerical Mathematics , pages =

  137. [145]

    and Yu, Y

    Luo, S. and Yu, Y. and Zhao, H. , date-added =. A new approximation for effective. SIAM Multiscale Model. Simul. , number =

  138. [146]

    and Shu, C.-W

    Gottlieb, S. and Shu, C.-W. , date-added =. Total Variation Diminishing. Mathematics of Computation , pages =

  139. [147]

    and Oberman, A

    Gomes, D.A. and Oberman, A. , date-added =. Computing the effective. SIAM J. Control and Optimization , number =

  140. [148]

    and Kozlov, V.V

    Arnold, V.I. and Kozlov, V.V. and Neishtadt, A.I. , date-added =. Mathematical Aspects of Classical and Celestial Mechanics , year =

  141. [149]

    Friedlander, F. G. , date-added =. Private communication to the third author , year =

  142. [150]

    and Qian, J

    Fang, J. and Qian, J. and Zepeda-Nunez, L. and Zhao, H.-K , date-added =. Learning Dominant Wave Directions For Plane Wave Methods For High-Frequency

  143. [151]

    Glowinski and S

    R. Glowinski and S. Y. Leung and H. Liu and J. Qian , date-added =. On the numerical solution of an obstacle problem for the

  144. [152]

    Qian and S

    J. Qian and S. Leung and B. Xu , date-added =. An improved full waveform inversion method based on the Kantorovich-Rubinstein TV norm , year =

  145. [153]

    Qian and W

    J. Qian and W. Lu and X. Zhang , date-added =. A fast high-resolution level-set method based on the

  146. [154]

    L. C. Evans , booktitle =. Partial differential equations and

  147. [155]

    Santambrogio , date-added =

    F. Santambrogio , date-added =. Optimal Transport for Applied Mathematics , year =

  148. [156]

    Villani , date-added =

    C. Villani , date-added =. Topics in Optimal Transportation , volume =

  149. [157]

    J. D. Benamou and B. D. Froese and A. M. Oberman , date-added =. Numerical solution of the optimal transportation problem using the Monge-Ampere equation , volume =. J. Comput. Phys. , pages =

  150. [158]

    Engquist and B

    B. Engquist and B. D. Froese , date-added =. Application of the. Commun. Math. Sci. , number =

  151. [159]

    J. D. Benamou and Y. Brenier , date-added =. Mixed. Journal of Optimization Theory and Applications , number =

  152. [160]

    Benamou and Y

    J.-D. Benamou and Y. Brenier , date-added =. A computational fluid mechanics solution to the. Numer. Math. , pages =

  153. [161]

    Brenier , date-added =

    Y. Brenier , date-added =. Polar factorization and monotone rearrangement of vector-valued functions , volume =

  154. [162]

    Lellmann and D

    J. Lellmann and D. Lorenz and C. Schonlieb and T. Valkonen , date-added =. Imaging with

  155. [163]

    Metivier and R

    L. Metivier and R. Brossier and Q. Merigot and E. Oudet and J. Virieux , date-added =. An optimal transport approach for seismic tomography: application to 3

  156. [164]

    Metivier and R

    L. Metivier and R. Brossier and Q. Merigot and E. Oudet and J. Virieux , date-added =. Measuring the misfit between seismograms using an optimal transport distance: application to full waveform inversion , volume =

  157. [165]

    Metivier and A

    L. Metivier and A. Allain and R. Brossier and Q. Merigot and E. Oudet and J. Virieux , date-added =. Optimal transport for mitigating cycle skipping in full-waveform inversion: a graph-space transform approach , volume =

  158. [166]

    Cuturi and G

    M. Cuturi and G. Peyre , date-added =. A smoothed dual approach for variational

  159. [167]

    Wang and J

    X. Wang and J. Qian and H. Wang , date-added =. Efficient angle-domain common-imaging gathers using Cauchy condition based polarization vectors , volume =

  160. [168]

    Lax , date-added =

    P. Lax , date-added =. Asymptotic solutions of oscillatory initial value problems , volume =. Duke Math. J. , pages =

  161. [169]

    and Qian, J

    Lu, W. and Qian, J. and Burridge, R. , date-added =. Babich-like ansatz for three-dimensional point-source. Multiscale Model. Simul. , number =

  162. [170]

    and Qian, J

    Lu, W. and Qian, J. and Burridge, R. , date-added =. Babich's Expansion and the Fast. J. Comput. Phys. , pages =

  163. [171]

    and Leung, S

    Glowinski, R. and Leung, S. and Qian, J , date-added =. Operator-Splitting Based Fast Sweeping Methods for Isotropic Wave Propagation in a Moving Fluid , volume =. SIAM J. Sci. Comput. , pages =

  164. [172]

    and Qian, J

    Li, W. and Qian, J. , date-added =. Joint inversion of gravity and traveltime data using a level-set based structural parameterization , volume =

  165. [173]

    and Lu, W

    Li, W. and Lu, W. and Qian, J. and Li, Y. , date-added =. A multiple level set method for three-dimensional inversion of magnetic data , volume =

  166. [174]

    and Kong, J

    Tsang, L. and Kong, J. A. and Ding, K-H , publisher =. Scattering of

  167. [175]

    and Huganess, S

    Toflove, A. and Huganess, S. C. , publisher =. Computational Electrodynamics: The Finite Difference Time Domain Method, Second Editions , year =

  168. [176]

    and Lu, Y

    Lu, W. and Lu, Y. Y. , date-modified =. Efficient boundary integral equation method for photonic crystal fibers , volume =. Journal of Lightwave Technology , number =

  169. [177]

    , journal =

    Alkhalifah, T. , journal =. Efficient traveltime compression for

  170. [178]

    Boyd, J. P. , publisher =. Chebyshev and

  171. [179]

    , publisher =

    Hadamard, J. , publisher =. Lectures on Cauchy's Problem in Linear Partial Differential Equations , year =

  172. [180]

    Chew, W. C. , publisher =. Waves and fields in inhomogeneous media , year =

  173. [181]

    and Hilbert, D

    Courant, R. and Hilbert, D. , publisher =. Methods of Mathematical Physics, Volume II , year =

  174. [182]

    Jensen, F. B. and Kuperman, W. A. and Porter, M. B. and Schmidt, H. , publisher =. Computational Ocean Acoustics , year =

  175. [183]

    Brekhovskikh, L. M. and Lysanov, Y. P. , publisher =. Fundamentals of Ocean Acoustics , year =

  176. [184]

    and Plemmons, R

    Berman, A. and Plemmons, R. , publisher =. Nonnegative matrices in the mathematical sciences , year =

  177. [185]

    and Levinson, N

    Coddington, E. and Levinson, N. , publisher =. Theory of ordinary differential equations , year =

  178. [186]

    , publisher =

    LeVeque, R. , publisher =. Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems , year =

  179. [187]

    Pierce, A. D. , publisher =

  180. [188]

    and Blanc, E

    Le Pichon, A. and Blanc, E. and Hauchecorne, A. , publisher =

  181. [189]

    and Hecht, B

    Novotny, L. and Hecht, B. , publisher =

  182. [190]

    Kornhauser, E. T. , journal = jasa, pages =

  183. [191]

    and Coulouvrat, F

    Blumrich, R. and Coulouvrat, F. and Heimann, D. , journal = jasa, pages =

  184. [192]

    and Xu, B

    Qian, J. and Xu, B. and Leung, S. and Anastasio, M. A. , journal =

  185. [193]

    and Osher, S

    Darbon, J. and Osher, S. , journal =

  186. [194]

    and Harris, J

    Zhu, T. and Harris, J. M. and Biondi, B. , journal =

  187. [195]

    and Harris, J

    Zhu, T. and Harris, J. M. , journal =

  188. [196]

    Qian, Jianliang and Stefanov, Plamen and Uhlmann, Gunther and Zhao, Hongkai , journal =

  189. [197]

    and Anastasio, Mark A

    Huang, Chao and Wang, Kun and Nie, Liming and Wang, Lihong V. and Anastasio, Mark A. , journal =

  190. [198]

    and Biondi, Biondo , journal =

    Zhu, Tieyuan and Harris, Jerry M. and Biondi, Biondo , journal =

  191. [199]

    , journal =

    Zhu, Tieyuan and Harris, Jerry M. , journal =

  192. [200]

    and Zhang, M

    Xiong, T. and Zhang, M. and Zhang, Y. T. and Shu, C.-W. , journal = jsc, pages =. Fast sweeping fifth order WENO scheme for static Hamilton-Jacobi equations with accurate boundary treatment , volume =

  193. [201]

    and Zhdanov, M

    Cai, H. and Zhdanov, M. , journal =. Application of

  194. [202]

    Leveille, J. P. and Jones, I. F. and Zhou, Z. Z. and Wang, B. and Liu, F. , journal =. Subsalt imaging for exploration, production, and development: A review , volume =

  195. [203]

    and Miller, D

    Farmer, P. and Miller, D. and Pleprzak, A. and Rutledge, J. and Woods, R. , journal =. Exploring the subsalt , volume =

  196. [204]

    Bain, J. E. and Weyand, J. and Weber, M. , note =. Resolving complex salt features using gravity and magnetics , year =

  197. [205]

    Krahenbuhl, R. A. and Li, Y. , date-modified =. Inversion of gravity data using a binary formulation , volume =. Geophysical Journal International , number =

  198. [206]

    Bain, J. E. and Horscroft, T. R. and Weyand, J. and Saad, A. H. and D. N. Bulling , journal =

  199. [207]

    Gibson, R. I. and Millegan, P. S. , publisher =. Geologic applications of gravity and magnetics: case histories , year =

  200. [208]

    Barbosa, V. C. F. and Silva, J. B. C. and Medeiros, W. E. , journal =. Gravity inversion of a discontinuous relief stabilized by weighted smoothness constraints on depth , volume =

  201. [209]

    and Li, Y

    Cheng, D. and Li, Y. and Larner, K. , booktitle =. Inversion of gravity data for base salt , year =

  202. [210]

    , booktitle =

    Cuturi, M. , booktitle =

  203. [211]

    Silva Dias, F. J. S. and Barbosa, V. C. F. and Silva, J. B. C. , booktitle =. Adaptive learning gravity inversion for

  204. [212]

    Silva Dias, F. J. S. and Barbosa, V. C. F. and Silva, J. B. C. , journal = geo, pages =. Adaptive learning

  205. [213]

    and Hall, S

    Ennen, C. and Hall, S. , booktitle =. Structural mapping of the

  206. [214]

    and Annecchione, M

    Hatch, D. and Annecchione, M. , booktitle =. Gravity gradient interpretation of salt bodies in nil-zone regimes , year =

  207. [215]

    , journal =

    Li, Y. , journal =. 3-

  208. [216]

    Krahenbuhl, R. A. and Li, Y. , journal =. Inversion of gravity data using a binary formulation , volume =

  209. [217]

    and Oldenburg, D

    Li, Y. and Oldenburg, D. W. , journal =. 3-

  210. [218]

    Oldenburg, D. W. and Li, Y. and Farquharson, C. G. and Kowalczyk, P. and Aravanis, T. and King, A. and Zhang, P. and Watts, A. , journal =. Applications of geophysical inversions in mineral exploration , volume =

  211. [219]

    Aminzadeh, Fred , journal =

  212. [220]

    , journal =

    Santosa, F. , journal =. A level-set approach for inverse problems involving obstacles , volume =

  213. [221]

    and Lesselier, D

    Litman, A. and Lesselier, D. and Santosa, F. , journal =. Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set , volume =

  214. [222]

    , journal =

    Burger, M. , journal =. A level set method for inverse problems , volume =

  215. [223]

    and Solna, K

    Hou, S. and Solna, K. and Zhao, H. , journal =. Imaging of location and geometry for extended targets using the response matrix , volume =

  216. [224]

    and Lesselier, D

    Dorn, O. and Lesselier, D. , journal =. Level set methods for inverse scattering , volume =

  217. [225]

    Miled, M. B. H. and Miller, E. L. , journal =. A projection-based level-set approach to enhance conductivity anomaly reconstruction in electrical resistance tomography , volume =

  218. [226]

    and Ascher, U

    van den Doel, K. and Ascher, U. M. and Leitao, A. , journal =. Multiple level sets for piecewise constant surface reconstruction in highly ill-posed problems , volume =

  219. [227]

    and Herty, M

    Papadopoulos, D. and Herty, M. and Rath, V. and Behr, M. , journal =. Identification of uncertainties in the shape of geophysical objects with level sets and the adjoint method , volume =

  220. [228]

    and Farquharson, C

    Zheglova, P. and Farquharson, C. G. and Hurich, C. A. , journal =. 2-D reconstruction of boundaries with level set inversion of traveltimes , volume =

  221. [229]

    A fast local level set adjoint state method for first arrival transmission traveltime tomography with discontinuous slowness , volume =

    Li, Wenbin and Leung, Shingyu , journal =. A fast local level set adjoint state method for first arrival transmission traveltime tomography with discontinuous slowness , volume =

  222. [230]

    A level-set adjoint-state method for crosswell transmission-reflection traveltime tomography , volume =

    Li, Wenbin and Leung, Shingyu and Qian, Jianliang , journal =. A level-set adjoint-state method for crosswell transmission-reflection traveltime tomography , volume =

  223. [231]

    A level set method for imaging salt structures using gravity data , volume =

    Li, Wenbin and Lu, Wangtao and Qian, Jianliang , date-modified =. A level set method for imaging salt structures using gravity data , volume =. Geophysics , number =

  224. [232]

    and Rector, J

    Zhang, L. and Rector, J. W. and Hoversten, G. M. , journal = gji, pages =. Eikonal solver in the celerity domain , volume =

  225. [233]

    Jorgensen, G. J. and Kisabeth, J. L. , booktitle =. Joint

  226. [234]

    and Qian, J

    Lu, W. and Qian, J. , journal =. A local level set method for three-dimensional inversion of gravity gradient data , volume =

  227. [235]

    Bracewell, R. N. , isbn =. The Fourier Transform and Its Applications , year =

  228. [236]

    Zhao, H. K. and Chan, T. and Merriman, B. and Osher, S. , journal =. A variational level set approach to multiphase motion , volume =

  229. [237]

    and Hecht, B

    Novotny, L. and Hecht, B. , date-modified =. Principles of nano-optics , year =

  230. [238]

    , publisher =

    Zhdanov, M. , publisher =. Geophysical inverse theory and regularization problems , volume =

  231. [239]

    Bracewell, R. N. , isbn =. The Fourier Transform and Its Applications , url =. 2000 , bdsk-url-1 =

    2000

  232. [240]

    Leli\`evre , school =

    Peter G. Leli\`evre , school =

  233. [241]

    and Zhao, H.-K

    Engquist, B. and Zhao, H.-K. , journal =. An approximate separability of

  234. [242]

    and Lafitte, O

    Benamou, J.-D. and Lafitte, O. and Sentis, R. and Solliec, I. , journal = jcam, pages =. A geometrical optics-based numerical method for high frequency electromagnetic fields computation near fold caustics--

  235. [243]

    Popov, M. M. and Psencik, I. , journal =. Computation of ray amplitudes in inhomogeneous media with curved interfaces , volume =

  236. [244]

    , journal =

    Psencik, I. , journal =. Ray amplitudes of compressional, shear and converted seismic body waves in

  237. [245]

    and Oldenburg, D

    Li, Y. and Oldenburg, D. , journal =

  238. [246]

    and Oldenburg, D

    Leli\`evre, P. and Oldenburg, D. , journal =. Magnetic forward modelling and inversion for high susceptibility , volume =

  239. [247]

    and Leitao, A

    DeCezaro, A. and Leitao, A. and Tai, X-C , journal =. On multiple level-set regularization methods for inverse problems , volume =

  240. [248]

    , journal =

    Tai, X-C and Chan, T. , journal =. A survey on multiple level set methods with applications for identifying piecewise constant functions , volume =

  241. [249]

    Chung, E. T. and Tai, X-C and Chan, T. , journal = jcp, pages =. Electrical impedance tomography using level set representation and total variational regularization , volume =

  242. [250]

    Bhattacharyya, B. K. , journal =. A generalized multibody model for inversion of magnetic anomalies , volume =

  243. [251]

    and Zhdanov, M

    Portniaguine, O. and Zhdanov, M. S. , journal = geo, number =

  244. [252]

    and Hansen, R

    Wang, X. and Hansen, R. O. , journal = geo, number =. Inversion for magnetic anomalies of arbitrary three-dimensional bodies , volume =

  245. [253]

    and Leung, S

    Glowinski, R. and Leung, S. and Qian, J. , journal = siims, pages =. A penalization-regularization-operator splitting method for eikonal-based traveltime tomography , volume =

  246. [254]

    and Baskar, S

    Dahiya, D. and Baskar, S. and Coulouvrat, F. , journal = sisc, pages =. Characteristic Fast Marching Method for Monotonically Propagating Fronts in a Moving Medium , volume =

  247. [255]

    V. M. Babich , journal =

  248. [256]

    and Lai, R

    Ozolins, V. and Lai, R. and Caflisch, R. and Osher, S. , journal = pnas, pages =

  249. [257]

    , journal = sinum, pages =

    Strang, G. , journal = sinum, pages =

  250. [258]

    , journal =

    Strang, G. , journal =. Accurate partial difference methods

  251. [259]

    Avila, G. S. and Keller, J. B. , journal = cpam, pages =

  252. [260]

    and Stoffa, P

    Hu, C. and Stoffa, P. , journal = geo, pages =. Slowness-driven

  253. [261]

    and Lailly, P

    Ehinger, A. and Lailly, P. and Marfurt, K. , journal = geo, pages =

  254. [262]

    and Nichols, D

    Audebert, F. and Nichols, D. and Rekdal, T. and Biondi, B. and Lumley, D. and Urdaneta, H. , journal = geo, pages =. Imaging complex geology structure with single-arrival

  255. [263]

    and Fomel, S

    Li, S. and Fomel, S. , journal = geo, pages =

  256. [264]

    and Luo, S

    Qian, J. and Luo, S. and Burridge, R. , journal = geo, pages =. Fast

  257. [265]

    and Kress, R

    Colton, D. and Kress, R. , publisher =

  258. [266]

    , publisher =

    Glowinski, R. , publisher =

  259. [267]

    , edition =

    Glowinski, R. , edition =

  260. [268]

    , publisher =

    Nedlec, J. , publisher =

  261. [269]

    and Le Tallec, P

    Glowinski, R. and Le Tallec, P. , publisher =

  262. [270]

    and Dean, E

    Glowinski, R. and Dean, E. J. and Guidoboni, G. and Juarez, L. H. and Pan, T.-W. , journal =. Applications of operator-splitting methods to the direct numerical simulation of particulate and free-surface flows and to the numerical solution of the two-dimensional elliptic

  263. [271]

    Taylor, M. E. , booktitle =. Finsler structures and wave propagation , year =

  264. [272]

    , booktitle =

    Glowinski, R. , booktitle =. Viscous flow simulations by finite element methods and related numerical techniques , year =

  265. [273]

    , booktitle =

    Glowinski, R. , booktitle =. Splitting methods for the numerical solution of the incompressible

  266. [274]

    and Glowinski, R

    Dacorogna, B. and Glowinski, R. and Kuznetzov, Y. and Pan, T. W. , booktitle =. On a conjugate gradient/

  267. [275]

    and Solna, K

    Hou, S. and Solna, K. and Zhao, H.-K. , journal = jcp, pages =

  268. [276]

    and Hairer, M

    Hairer, E. and Hairer, M. , booktitle =

  269. [277]

    and Qian, J

    Lu, W. and Qian, J. , journal = geo, pages =. A Local Level Set Method for Three-dimensional Inversion of Gravity Gradiometry Data , volume =

  270. [278]

    and Bellefleur, G

    Huang, J.-W. and Bellefleur, G. , journal = gji, pages =. Joint transmission and reflection traveltime tomography using the fast sweeping method and the adjoint-state technique , volume =

  271. [279]

    and Lu, Y

    Lu, W. and Lu, Y. Y. , journal =. Efficient boundary integral equation method for photonic crystal fibers , volume =

  272. [280]

    and Li, Y

    Krahenbuhl, R. and Li, Y. , journal =. Inversion of gravity data using a binary formulation , volume =

  273. [281]

    and Symes, W

    Sei, A. and Symes, W. W. , booktitle =. Gradient calculation of the traveltime cost function without ray tracing , year =

  274. [282]

    and Talwani, M

    Condi, F. and Talwani, M. , booktitle =. Resolution and efficient inversion of gravity gradiometry , year =

  275. [283]

    and Amundsen, L

    Zhou, H. and Amundsen, L. and Zhang, G. , booktitle =. Fundamental Issues in Full Waveform Inversion , year =

  276. [284]

    and Kisabeth, J

    Jorgensen, G. and Kisabeth, J. , booktitle =. Joint 3

  277. [285]

    , booktitle =

    Li, Y. , booktitle =. Processing gravity gradiometer data using an equivalent source technique , year =

  278. [286]

    , booktitle =

    Li, Y. , booktitle =

  279. [287]

    and Jorgensen, G

    Routh, P. and Jorgensen, G. and Kisabeth, J. , booktitle =. Base of the salt imaging using gravity and tensor gravity data , year =

  280. [288]

    , booktitle =

    Pica, A. , booktitle =. Fast and accurate finite-difference solutions of the

  281. [289]

    and Ellis, R

    Zhdanov, M. and Ellis, R. and Mukherjee, S. , journal =. Three dimensional regularized focusing inversion of gravity gradient tensor component data , volume =

  282. [290]

    and Florio, G

    Fedi, M. and Florio, G. , booktitle =

  283. [291]

    , journal =

    Fedi, M. , journal =

  284. [292]

    and Liu, X

    Zhdanov, M. and Liu, X. and Wilson, G. and Wan, L. , journal =. Potential field migration for rapid imaging of gravity gradiometry data , volume =

  285. [293]

    An improved fast local level set method for three-dimensional inverse gravimetry , volume =

    Lu, Wangtao and Leung, Shingyu and Qian, Jianliang , journal =. An improved fast local level set method for three-dimensional inverse gravimetry , volume =

  286. [294]

    and Kim, S

    Koh, K. and Kim, S. and Boyd, S. , journal =. l1 \_ ls:

  287. [295]

    Slotnick, M. M. , journal =. Curvature of equipotential surfaces , volume =

  288. [296]

    and Pilkington, M

    Fedi, M. and Pilkington, M. , journal =. Understanding imaging methods for potential field data , volume =

  289. [297]

    , journal =

    Pilkington, M. , journal =. Analysis of gravity gradiometer inverse problems using optimal design measures , volume =

  290. [298]

    , journal =

    Li, X. , journal =. Efficient 3D gravity and magnetic modeling , year =

  291. [299]

    Lee, J. B. , journal =

  292. [300]

    and Pears, G

    Fullagar, P. and Pears, G. , booktitle =. From gravity gradient to density gradient , year =

Pith tools

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