Inverse Boundary Value Problem For The Helmholtz Equation: Quantitative Conditional Lipschitz Stability Estimates

Beretta, Elena; Hoop, Maarten V. de; Faucher, Florian; Scherzer, Otmar

doi:10.1137/15m1043856

Cited by 30 publications

(41 citation statements)

References 23 publications

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“…In the language of regularization theory, the transition to finitely sampled B-spline objects corresponds to imposing a (very strong) source condition. Similarly as proven in [1,4] for other severely ill-posed problems, such a "finite-resolution" source condition enables Lipschitz-stability estimates for image-reconstruction from truncated Fresneldata. This is seen by combining the quasi-band-limitation results from §5.2 with the leakage estimates from §4.3: Withη f ∆ as defined in Theorem 4.4, the constant is given by…”

Section: Stability Estimatesmentioning

confidence: 76%

Locality estimates for Fresnel-wave-propagation and stability of x-ray phase contrast imaging with finite detectors

Maretzke

2018

Inverse Problems

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Coherent wave-propagation in the near-field Fresnel-regime is the underlying contrast-mechanism to (propagation-based) X-ray phase contrast imaging (XPCI), an emerging lensless technique that enables 2D-and 3D-imaging of biological soft tissues and other light-element samples down to nanometer-resolutions. Mathematically, propagation is described by the Fresnel-propagator, a convolution with an arbitrarily non-local kernel. As real-world detectors may only capture a finite field-of-view, this non-locality implies that the recorded diffraction-patterns are necessarily incomplete. This raises the question of stability of image-reconstruction from the truncated data -even if the complex-valued wave-field, and not just its modulus, could be measured. Contrary to the latter restriction of the acquisition, known as the phase-problem, the finite-detector-problem has not received much attention in literature. The present work therefore analyzes locality of Fresnelpropagation in order to establish stability of XPCI with finite detectors. Image-reconstruction is shown to be severely ill-posed in this setting -even without a phase-problem. However, quantitative estimates of the leaked wave-field reveal that Lipschitz-stability holds down to a sharp resolution limit that depends on the detector-size and varies within the field-of-view. The smallest resolvable lengthscale is found to be ≈ 1/f times the detector's aspect length, wheref is the Fresnel number associated with the latter scale. The stability results are extended to phaseless imaging in the linear contrast-transfer-function regime.(1) Although we will refer to "real-valued" signals throughout the work, note that all the results obtained for such trivially extend to signals given by real-functions multiplied by a global complex phase.

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Section: Stability Estimatesmentioning

confidence: 76%

Locality estimates for Fresnel-wave-propagation and stability of x-ray phase contrast imaging with finite detectors

Maretzke

2018

Inverse Problems

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“…Note that we always just invert the negative Laplacian D and not the parameter dependent operator D c or D a as it would be required in a reduced formulation (1). For the inverse source problem (18), the cost functional is already quadratic, so just one Newton step is required and coinides with the original regularized minimization problem (18).…”

Section: Gauss-newton Sqp Methodsmentioning

confidence: 99%

“…However, this requires to work in nonreflexive spaces both in parameter and in data space, which makes an analysis challenging. 6.6901e-06 6.6154e-06 6.6154e-06 3.2887e-05 3.2532e-05 3.2744e-05 errspot 1 2.4023e-06 0 0 6.7780e-07 0 0 errspot 2 7.3665e-06 0 0 0.0462 0 0 errspot 3 1.0890e-05 0 0 J(x δ k ,u δ k ) J(x 0 ,u 0 ) 6.2155e-06 6.0569e-06 8.0316e-07 1.6318e-04 1.6286e-04 1.6286e-04 errspot 1 6.9611e-10 0 0 1.2184e-10 0 0 errspot 2 1.7502e-06 0 0 0.7183 0.7145 0.7145 errspot 3 1.5392…”

Section: Conclusion and Remarksmentioning

confidence: 99%

Regularization of inverse problems via box constrained minimization

Hungerländer¹,

Kaltenbacher²,

Rendl³

2020

Inverse Problems &Amp; Imaging

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In the present paper we consider minimization based formulations of inverse problems (x, Φ) ∈ argmin {J (x, Φ; y) : (x, Φ) ∈ M ad (y)} for the specific but highly relevant case that the admissible set M δ ad (y δ ) is defined by pointwise bounds, which is the case, e.g., if L ∞ constraints on the parameter are imposed in the sense of Ivanov regularization, and the L ∞ noise level in the observations is prescribed in the sense of Morozov regularization. As application examples for this setting we consider three coefficient identification problems in elliptic boundary value problems.Discretization of (x, Φ) with piecewise constant and piecewise linear finite elements, respectively, leads to finite dimensional nonlinear box constrained minimization problems that can numerically be solved via Gauss-Newton type SQP methods. In our computational experiments we revisit the suggested application examples. In order to speed up the computations and obtain exact numerical solutions we use recently developed active set methods for solving strictly convex quadratic programs with box constraints as subroutines within our Gauss-Newton type SQP approach.

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“…Controlling the number of unknowns influences the resolution of the outcome, but also the stability and convergence of the procedure. The use of piecewise constant coefficients appears natural for numerical applications, and is also motivated by stability results (Alessandrini & Vessella, 2005;Beretta et al, 2016). However, such a decomposition can lead to an artificial 'block' representation (cf.…”

Section: Introductionmentioning

confidence: 99%

“…However, such a decomposition can lead to an artificial 'block' representation (cf. Beretta et al (2016); Faucher (2017)) which would not be appropriate in terms of resolution. For this reason, a piecewise linear model representation is explored by Alessandrini et al (2018Alessandrini et al ( , 2019, still motivated by the stability properties.…”

Section: Introductionmentioning

confidence: 99%

Eigenvector models for solving the seismic inverse problem for the Helmholtz equation

Faucher

Scherzer

Barucq³

2020

Geophysical Journal International

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We study the seismic inverse problem for the recovery of subsurface properties in acoustic media. In order to reduce the ill-posedness of the problem, the heterogeneous wave speed parameter is represented using a limited number of coefficients associated with a basis of eigenvectors of a diffusion equation, following the regularization by discretization approach. We compare several choices for the diffusion coefficient in the partial differential equations, which are extracted from the field of image processing. We first investigate their efficiency for image decomposition (accuracy of the representation with respect to the number of variables and denoising). Next, we implement the method in the quantitative reconstruction procedure for seismic imaging, following the Full Waveform Inversion method, where the difficulty resides in that the basis is defined from an initial model where none of the actual structures is known. In particular, we demonstrate that the method is efficient for the challenging reconstruction of media with salt-domes. We employ the method in two and three-dimensional experiments, and show that the eigenvector representation compensates for the lack of low-frequency information, it eventually serves us to extract guidelines for the implementation of the method.

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Inverse Boundary Value Problem For The Helmholtz Equation: Quantitative Conditional Lipschitz Stability Estimates

Cited by 30 publications

References 23 publications

Locality estimates for Fresnel-wave-propagation and stability of x-ray phase contrast imaging with finite detectors

Locality estimates for Fresnel-wave-propagation and stability of x-ray phase contrast imaging with finite detectors

Regularization of inverse problems via box constrained minimization

Eigenvector models for solving the seismic inverse problem for the Helmholtz equation

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