11th Applied Inverse Problems Conference
September 4 - 8, 2023 | Göttingen, Germany
Conference Agenda
Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).
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Session Overview | |
| Location: VG3.104 |
| Date: Monday, 04/Sept/2023 | |
| 1:30pm - 3:30pm | MS29 1: Eigenvalues in inverse scattering Location: VG3.104 Session Chair: Martin Halla Session Chair: Peter Monk |
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Interior transmission eigenvalue trajectories 1Forschungszentrum Jülich GmbH, Germany; 2Karlsruhe Institute of Technology, Germany
Complex-valued eigenvalue trajectories parametrized by a constant index of refraction are investigated for the interior transmission problem. Several properties are derived for the unit disk such as that the only intersection points with the real axis are Dirichlet eigenvalues of the Laplacian. For general sufficiently smooth scatterers in two dimensions the only trajectorial limit points are shown to be Dirichlet eigenvalues of the Laplacian as the refractive index tends to infinity. Additionally, numerical results for several scatterers are presented which give rise to an underlying one-to-one correspondence between these two eigenvalue families which is finally stated as a conjecture.
A new family of modified interior transmission eigenvalues for a fluid-solid interaction 1University of Delaware; 2University of Oviedo
We study a new family of modified interior transmission eigenvalues for the interaction of a bounded elastic body (the target) embedded in an unbounded compressible inviscid fluid (the acoustic medium). This problem is modelled with the elastodynamic and acoustic equations in the time-harmonic regime, and the interaction of the two media is represented through the dynamic and kinematic boundary conditions; these are are two transmission conditions posed on the wet boundary that represent the equilibrium of forces, and the equality of the normal displacements of the solid and the fluid, respectively.
For such a model problem, we propose a new family of modified interior transmission eigenvalues (mITP eigenvalues), which depends on a tunable parameter $\gamma$ that can help increase the sensitivity of the eigenvalues to changes in the scatterer. We analyze the distribution of the mITP eigenvalues on the complex plane, in particular we show that they are real valued, and that they either fill the whole real line or define a discrete subset with no finite accumulation point. We also justify theoretically that they can be approximated from measurements of the far field pattern corresponding to incident plane waves by solving a collection of modified far field equations.
Furthermore, for a suitable choice of the parameter $\gamma$, our theory is more complete: it includes a proof of the discreteness of the mTIP eigenvalues, an upper bound for them, and a physical interpretation of the largest of them via a Courant min-max principle.
We finally provide numerical results for synthetic data to give an insight of the expected perfomance of the mITP eigenvalues if used as target signatures in applications.
[1] F. Cakoni, D. Colton, S. Meng, P. Monk. Stekloff eigenvalues in inverse scattering, SIAM J. Appl. Math. 76(4): 1737-1763, 2016.
[2] S. Cogar, D. Colton, S. Meng, P. Monk. Modified transmission eigenvalues in inverse scattering theory, Inverse Probl. 33(12): 125002, 2017.
[3] M. Levitin, P. Monk, V. Selgas,.Impedance eigenvalues in linear elasticity, SIAM J. on Appl. Math. 81(6), 2021.
[4] P. Monk, V. Selgas. Modified transmission eigenvalues for inverse scattering in a fluid-solid interaction problem, Research in the Mathematical Sciences 9(3), 2022.
Computation of transmission eigenvalues in singular configurations using a corner perfectly matched layer 1POEMS (CNRS-ENSTA Paris-INRIA), Institut Polytechnique de Paris, Palaiseau, France; 2IDEFIX (EDF-ENSTA Paris-INRIA), Institut Polytechnique de Paris, Palaiseau, France
In scattering, transmission eigenvalues are complex wavenumbers at which there exists an incident field that produces a vanishingly-small scattered far field. These eigenvalues solve the interior transmission eigenvalue problem (ITEP), which is a non-selfadjoint eigenvalue problem formulated on the support of the scatterer. In this work, we consider the discretization of the ITEP in two-dimensional cases where the difference between the parameters of the scatterer and that of the background medium changes sign at some point $O$ on the boundary of the scatterer. This sign change implies the existence of strongly-oscillating singularities localized around $O$, which prevent $H^{1}$-conforming finite element discretizations from approximating transmission eigenvalues, even when the corresponding modes are in $H^{1}$. In this talk we will demonstrate how transmission eigenvalues can be approximated by solving a modified ITEP; the modification consists in applying a suitable perfectly matched layer in a neighborhood of $O$, whose job is intuitively to tame strongly-oscillating singularities without inducing spurious reflections.
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| 4:00pm - 6:00pm | MS29 2: Eigenvalues in inverse scattering Location: VG3.104 Session Chair: Martin Halla Session Chair: Peter Monk |
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A duality between scattering poles and transmission eigenvalues in scattering theory Rutgers University, United States of America
Spectral properties of operators associated with scattering phenomena carry essential information about the scattering media. The theory of scattering resonances is a rich and beautiful part of scattering theory and, although the notion of resonances is intrinsically dynamical, an elegant mathematical formulation comes from considering them as the poles of the meromorphic extension of the scattering operator. The scattering poles exist and they are complex with negative imaginary part. They capture physical information by identifying the rate of oscillations with the real part of a pole and the rate of decay with its imaginary part. At a scattering pole, there is a non-zero scattered field in the absence of the incident field. On the flip side of this characterization of the scattering poles one could ask if there are frequencies for which there exists an incident field that doesn’t scatterer by the scattering object. The answer to this question for scattering by inhomogeneous media leads to the introduction of transmission eigenvalues.
In this talk we discuss a conceptually unified approach for characterizing and determining scattering poles and transmission eigenvalues for the scattering problem for inhomogenous media. Our approach explores a duality stemming from interchanging the roles of incident and scattered fields in our analysis. Both sets are related to the kernel of the relative scattering operator mapping incident fields to scattered fields, corresponding to the exterior scattering problem for the transmission eigenvalues, and the interior scattering problem for scattering poles.
Prolate eigensystem and its application in Born inverse scattering Chinese Academy of Sciences, China, People's Republic of
This talk is concerned with the generalized prolate spheroidal wave functions/eigenvalues (in short prolate eigensystem) and their application in two dimensional Born inverse medium scattering problems. The prolate eigenfunctions are eigenfunctions of a Fourier integral operator; they remarkably extend analytically to the whole space, are doubly orthogonal, and are complete in the class of band-limited functions. We first establish a Picard criterion for reconstructing the contrast using the prolate eigensystem, where the reconstruction formula can also be understood in the viewpoint of data processing and analytic extrapolation. Another salient feature associated with the generalized prolate spheroidal wave functions is that the prolate basis for a disk is also a basis for a Sturm-Liouville differential operator. With the help of Sturm-Liouville theory, we estimate the $L^2$ approximation error for a spectral cutoff approximation of $H^s$ functions, $0<s\le1$. This yields a spectral cutoff regularization strategy for noisy data and an explicit stability estimate for contrast in $H^s$ ($0<s\le1$) in the full aperture case. In the limited-aperture and multi-frequency cases, we also obtain spectral cutoff regularization strategies for noisy data and stability estimates for a class of contrast. Numerical examples are currently being investigated and a few preliminary examples are provided to illustrate the application of prolate eigensystem in inverse scattering problems.
Scattering from corners and other singularities LUT University, Finland
I will present a summary of my and my collaborators' work on fixed wavenumber scattering from corners and other geometric shapes of interest from the past 10 years. Our early work showed that in potential scattering, corners produce patterns in the far-field which cannot be cancelled by any other structure nearby or far away. This led to interesting finds such as unique shape determination of polyhedral or pixelated scattering potentials by the far-field made by any single incident wave. It also led to the study of how geometry of the domain affects the distribution of energy of the transmission eigenfunctions. Complete understanding is still away, and different geometrical configurations are being studied. In this talk I present shortly past results and also newer results related to general conical singularities and scattering screens.
The inverse spectral problem for a spherically symmetric refractive index using modified transmission eigenvalues National Technical University of Athens
In recent years, the classic transmission eigenvalue problem has risen in importance in inverse scattering theory. In this work, we discuss the introduction of a modification that corresponds to an artificial metamaterial background [1] and pose the inverse problem for determining a spherically symmetric refractive index from these modified eigenvalues. We show that uniqueness can be established under some assumptions for the magnitude of a fixed wavenumber and the unknown refractive index [2].
[1] D. Gintides, N. Pallikarakis, K. Stratouras. On the modified transmission eigenvalue problem with an artificial metamaterial background, Res. Math. Sci. 8, 2021.
[2] D. Gintides, N. Pallikarakis, K. Stratouras. Uniqueness of a spherically symmetric
refractive index from modified transmission eigenvalues, Inverse Problems 38, 2022.
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| Date: Tuesday, 05/Sept/2023 | |
| 1:30pm - 3:30pm | MS29 3: Eigenvalues in inverse scattering Location: VG3.104 Session Chair: Martin Halla Session Chair: Peter Monk |
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A new family of nearly singular interior transmission eigenvalues for inverse scattering Georg-August Universität Göttingen, Germany
I propose a new family of nearly singular interior transmission eigenvalue problems for inverse scattering. For a known support of the inhomogeneity the eigenvalues allow to identify e.g. the piece-wise constant values of the refractive index. For an unknown support of the inhomogeneity the eigenvalues allow to construct an indicator function, I present an analysis for ideal settings and numerical examples for general cases.
Interior transmission problems related to imaging periodic layers 1INRIA, France; 2ENIT, Tunisia
The extension of sampling methods to the imaging of locally perturbed periodic layers [1] requires the analysis of interior transmission problems in unbounded waveguides. The resulting problem is no longer of Fredholm type and its study necessitates additional tools to those classically used for the case of bounded domains. We shall present a proof of well posdeness in the case of absorbing background using Floquet-Bloch transform and finite dimensional approximation with respect to the Floquet-Bloch variable. The analysis of absorption free problem and related transmission eigenvalues is an open problem that we shall also briefly discuss. We plan to additionally present the so-called differential sampling method where some specific single Floquet-Bloch variable transmission eigenvalue problems shows up. These problems had been adressed in the case of periodically distributed defects in [2].
[1] Y. Boukari, H. Haddar, N. Jenhani. Analysis of sampling methods for imaging a periodic layer and its defects, Inverse Problems, 2023.
[2] F. Cakoni, H. Haddar, T.-P. Nguyen. New interior transmission problem applied to a single Floquet-Bloch mode imaging of local perturbations in periodic media, Inverse Problems, 2018.
Transparent scatterers and transmission eigenvalues of infinite multiplicity 1CNRS, École polytechnique, France; 2Steklov Mathematical Institute, Russia
We give a short review of old and recent results on scatterers with transmission eigenvalues of infinite multiplicity,
including transparent scatterers. Our examples include potentials from the Schwartz class and multipoint potentials of Bethe - Peierls type.
[1] P.G. Grinevich, R.G. Novikov. Russian Mathematical Surveys 77:1021-1028, 2022. https://doi.org/10.4213/rm10080e
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| 4:00pm - 6:00pm | MS42: Inverse Problems with Anisotropy Location: VG3.104 Session Chair: Kim Knudsen |
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A density property for tensor products of gradients of harmonic functions National Yang Ming Chiao Tung University, Taiwan
In this talk I will present a recent result showing that linear combinations of tensor products of $k$ gradients of harmonic functions, with $k$ at least three, are dense in $C(\overline{\Omega})$, for any bounded domain $\Omega$ in dimension 3 or higher. This kind of density result has applications to inverse problems for elliptic quasilinear equations/systems in divergence form, where the nonlinear part of the "conductivity'' is anisotropic. The talk will be based on two papers written in collaboration with A. Feizmohammadi.
Reconstructing anisotropic conductivities on manifolds University of Jyväskylä, Finland
We study the problem of recovering an electrical anisotropic conductivity from interior power density measurements on a two-dimensional Riemannian manifold. This problem arises in Acousto-Electric Tomography and is motivated by the geometric Calderón problem of recovering the metric from the Dirichlet-to-Neumann map. In contrast to the geometric Calderón problem, we consider a conductive Riemannian manifold and treat the conductivity and metric separately. Assuming that the metric is known, for two-dimensional Riemannian manifolds with genus zero, we highlight in this talk that under certain assumptions on the power density data it is possible to recover the conductivity uniquely and constructively from the data. We illustrate our findings with a numerical experiment and comment on how added noise on the manifold affects the reconstructed conductivity.
Imaging anisotropic conductivities from current densities The Chinese University of Hong Kong
In this talk, we discuss a reconstruction algorithm for imaging an anisotropic conductivity tensor in a second-order elliptic PDE with a nonzero Dirichlet boundary condition from internal current densities. The approach is based on a regularized output least-squares formulation with the standard $L^2$ penalty, which is then discretized by the standard Galerkin finite element method. We discuss the analysis of the discretized problem, especially the convergence of the discrete approximations with respect to the mesh size, using the discrete counterpart of $H$-convergence, when the discretization is uniform or adaptive. We present extensive two-dimensional numerical examples to show the efficiency of the proposed method.
Stability and reconstruction for anisotropic inverse problems. University of Limerick, Ireland
In this talk we investigate the issues of stability and reconstruction in inverse problems in the presence of anisotropy. As is well-known, there is a fundamental obstruction to the unique determination of the anisotropic conductivity of materials. Such obstruction is based on the observation that any deffeomorphism of a domain $\Omega$ that keeps its boundary $\partial\Omega$ fixed, changes the conductivity in $\Omega$ by keeping the boundary measurements unchanged. In this talk we will investigate how to circumvent this obstruction and restore well-posedness in the problem.
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| Date: Wednesday, 06/Sept/2023 | |
| 9:00am - 11:00am | MS53: Uniqueness and stability in inverse problems for partial differential equations Location: VG3.104 Session Chair: Sonia Foschiatti Session Chair: Elisa Francini Session Chair: Eva Sincich |
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Stability for the inverse problem of the determination on an inclusion in a Schrödinger type equation using Cauchy data. Università degli Studi di Trieste, Italy
We consider the stability issue for a broad class of inverse problems described by second-order elliptic equations with anisotropic and scalar coefficients that are finite-dimensional. This class of problems encompasses the well-studied conductivity equation, the Helmholtz equation and the Schrödinger equation. The applications of this study range from medicine, for example EIT, where the coefficient to be reconstructed is the conductivity, to the reconstruction of the wave-speed in a medium. It is well known that these inverse problems are ill-posed.
In this talk we prove a logarithmic stability estimate for the inverse problem that regards the determination of an inclusion in terms of local Cauchy data, since the Dirichlet to Neumann map that can encode the data at the boundary is not always available. This talk is based on a joint work with Eva Sincich.
Refined instability estimates for two inverse problems National Taiwan University, Taiwan
Many inverse problems are known to be ill-posed. The ill-posedness can be manifested by an instability estimate of exponential type, first derived by Mandache. Inspired by Mandache's idea, in this talk, I would like to refinements of the instability estimates for two inverse problems, including the inverse inclusion problem and the inverse scattering problem. The aim is to derive explicitly the dependence of the instability estimates on key parameters.
The first topic of this talk is to show how the instability depends
on the depth of the hidden inclusion and the conductivity of the background
medium. The second topic is to justify the optimality of increasing
stability in determining the near-field of a radiating solution of the Helmholtz
equation from the far-field pattern.
Stability estimates for the inverse fractional conductivity problem University of Cambridge, United Kingdom
We study the stability of an inverse problem for the fractional conductivity equation on bounded smooth domains. We obtain a logarithmic stability estimate for the inverse problem under suitable a priori bounds on the globally defined conductivities. The argument has three main ingredients: 1. the logarithmic stability of the related inverse problem for the fractional Schrödinger equation by Rüland and Salo; 2. the Lipschitz stability of the exterior determination problem; 3. utilizing and identifying nonlocal analogies of Alessandrini's work on the stability of the classical Calderón problem. The main contribution of the article is the resolution of the technical difficulties related to the last mentioned step. Furthermore, we show the optimality of the logarithmic stability estimates, following the earlier works by Mandache on the instability of the inverse conductivity problem, and by Rüland and Salo on the analogous problem for the fractional Schrödinger equation.
Uniqueness and stability for anisotropic inverse problems. University of Limerick, Ireland
In this talk we investigate the issue of uniqueness and stability for certain inverse problems which forward problem is modelled by a second order elliptic partial differential equation. As is well known, there is a fundamental obstruction to uniquely determine physical properties of anisotropic materials from boundary maps/measurements. In fact, any diffeomorphism of the domain under investigation, that keeps the domain's boundary fixed, changes its material's properties without changing its boundary measurements. In this talk we will provide some positive answers to the issues of uniqueness and stability of certain type of anisotropy in terms to the correspondent boundary measurements.
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| Date: Thursday, 07/Sept/2023 | |
| 1:30pm - 3:30pm | MS20 1: Recent advances in inverse problems for elliptic and hyperbolic equations Location: VG3.104 Session Chair: Ru-Yu Lai |
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Determining a nonlinear hyperbolic system with unknown sources and nonlinearity National Yang Ming Chiao Tung University, Taiwan
This talk is devoted to some inverse boundary problems associated with a time-dependent semilinear hyperbolic equation, where both nonlinearity and sources (including initial displacement and initial velocity) are unknown. It is shown in several
generic scenarios that one can uniquely determine the nonlinearity and/or the sources by using passive or active boundary observations. In order to exploit the nonlinearity and the sources simultaneously, we develop a new technique, which combines the observability for linear wave equations and an approximation property with higher order linearization for the semilinear hyperbolic equation
Uniqueness in an inverse problem of fractional elasticity University of Bonn, Germany
We study an inverse problem for fractional elasticity. In analogy to the classical problem of linear elasticity, we consider the unique recovery of the Lamé parameters associated to a linear, isotropic fractional elasticity operator from fractional Dirichlet-to-Neumann data. In our analysis we make use of a fractional matrix Schrödinger equation via a generalization of the so-called Liouville reduction, a technique classically used in the study of the scalar conductivity equation. We conclude that unique recovery is possible if the Lamé parameters agree and are constant in the exterior, and their Poisson ratios agree everywhere. Our study is motivated by the significant recent activity in the field of nonlocal elasticity.
This is a joint work with Prof. Maarte de Hoop and Prof. Mikko Salo.
Calderon problem for elliptic systems via complex ray transform University of Zurich, Switzerland
Let $(M, g)$ be a Riemannian manifold embedded (up to a conformal factor) into the product $\mathbb{R}^2 \times (M_0, g_0)$, let $A$ be a skew-Hermitian matrix of $1$-forms and let $Q$ be a matrix potential. In this talk, I will explain how to simultaneously recover the pair $(A, Q)$, up to gauge-equivalence, from the associated Dirichlet-to-Neumann map of the Schroedinger operator $d_A^*d_A + Q := (d + A)^* (d + A) + Q$. Techniques involve constructing complex geometric optics (CGO) solutions and analysing a complex ray transform that arises. This improves on the previously known results.
Asymptotics Applied to Small Volume Inverse Shape Problems Purdue University, United States of America
We consider two inverse shape problems coming from diffuse optical tomography and inverse scattering. For both problems, we assume that there are small volume subregions that we wish to recover using the measured Cauchy data. We will derive an asymptotic expansion involving their respective fields. Using the asymptotic expansion, we derive a MUSIC-type algorithm for the Reciprocity Gap Functional, which we prove can recover the subregion(s) with a finite amount of Cauchy data. Numerical examples will be presented for both problems in two dimensions in the unit circle.
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| 4:00pm - 6:00pm | MS20 2: Recent advances in inverse problems for elliptic and hyperbolic equations Location: VG3.104 Session Chair: Ru-Yu Lai |
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Fixed angle inverse scattering for velocity University of Delaware, United States of America
An inhomogeneous acoustic medium is probed by a plane wave and the resultant time dependent wave is measured on the boundary of a ball enclosing the inhomogeneous part of the medium. We describe our partial results about the recovery of the velocity of the medium from the boundary measurement. This is a formally determined inverse problem for the wave equation, consisting of the recovery of a non-constant coefficient of the principal part of the operator from the boundary measurement.
Inverse Problems for Some Nonlinear Schrodinger Equations 浙江大学, China, People's Republic of
In this talk, I will demonstrate the higher order linearization approach to solve several inverse boundary value problems for nonlinear PDEs, modeling for example nonlinear optics, including nonlinear magnetic Schrodinger equation and its fractional version. Considering partial data problems, the problem will be reduced to solving for the coefficient functions from their integrals against multiple linear solutions that vanish on part of the boundary. We will focus our discussion on choices of linear solutions and some microlocal anlaysis tools and ideas in proving injectivity of the coefficient function from its integral transforms such as the FBI transform.
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| Date: Friday, 08/Sept/2023 | |
| 1:30pm - 3:30pm | MS31 1: Inverse Problems in Elastic Media Location: VG3.104 Session Chair: Andrea Aspri Session Chair: Ekaterina Sherina |
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Hybrid Inverse Problems for Nonlinear Elasticity 1Leibniz University of Hannover, Germany; 2Inria, Chile
We consider the Saint-Venant model in 2 dimensions for nonlinear elasticity. Under the hypothesis the fluid is incompressible, we recover the displaced field and the Lame parameter $\mu$ from power density measurements. A stability estimate is shown to hold for small displacement fields, under some natural hypotheses on the direction of the displacement. The techniques introduced show the difficulties of using hybrid imaging techniques for non-linear inverse problems.
Quantitative reconstruction of viscoelastic media with attenuation model uncertainty. 1Makutu, Inria Bordeaux, France; 2Faculty of Mathematics, University of Vienna, Austria
We consider the inverse wave problem of reconstructing the properties of a viscoelastic medium. The data acquisition corresponds to probing waves that are sent and measured outside of the sample of interest, in the configuration of non-intrusive inversion. In media with attenuation, waves lose energy when propagating through the domain. Attenuation is a frequency-dependent phenomenon with several models existing [1], each leading to different models of wave equations. Therefore, in addition to adding unknown coefficients to the inverse problem, the attenuation law characterizing a medium is typically unknown prior to the reconstruction, hence further increasing the ill-posedness.
In this work, we consider time-harmonic waves which are convenient to unify the different models of attenuation using complex-valued parameters. We illustrate the difference in wave propagation depending on the attenuation law and carry out the reconstruction with attenuation model uncertainty [2]. That is, we perform the reconstruction procedure with different attenuation models used for the (synthetic) data generation and for the reconstruction. In this way, we show the robustness of the reconstruction method. Furthermore, we investigate a configuration with reflecting boundary surrounding the sample. To handle the resulting multiple reflections, we introduce a strategy of reconstruction with a progression of complex frequencies. We illustrate with experiments of ultrasound imaging.
[1] J. M. Carcione. Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media, third ed., Elsevier, 2015.
[2] F. Faucher, O. Scherzer. Quantitative inverse problem in visco-acoustic media under attenuation model uncertainty, Journal of Computational Physics 472: 111685, 2023. https://doi.org/10.1016/j.jcp.2022.111685
An Intensity-based Inversion Method for Quasi-Static Optical Coherence Elastography 1University of Vienna, Austria; 2Medical University of Vienna, Austria; 3Johann Radon Institute Linz, Austria
We consider optical coherence elastography, which is an emerging research field but still lacking precision and reproducibility. Elastography as an imaging modality aims at mapping of the biomechanical properties of a given sample. This problem is widely used in Medicine, in particular for the non-invasive identification of malignant formations inside the human skin or tissue biopsies during surgeries. In term of diagnostics accuracy, one is interested in quantitative values mapped on top of the visualisation of the sample rather then only qualitative images.
In this work, we discuss a general intensity-based approach to the inverse problem of quasi-static elastography, under any deformation model. From a pair of tomographic scans obtained by an imaging modality of choice, e.g. as X-ray, ultrasound, magnetic resonance, optical imaging or other, we aim to recover one or a set of unknown material parameters describing the sample. This approach has been briefly introduced in [1], under the name of intensity-based inversion method, and applied for recovery of the Young's modulus of a set of samples imaged with Optical Coherence Tomography. Here, we mainly focus on investigating the intensity-based inversion approach in the Inverse Problems framework. Furthermore, we illustrate the performance of the inversion method on twelve silicone elastomer phantoms with inclusions of varying size and stiffness.
[1] L. Krainz, E. Sherina, S. Hubmer, M. Liu, W. Drexler, O. Scherzer. Quantitative Optical Coherence Elastography: A Novel Intensity-Based Inversion Method Versus Strain-Based Reconstructions. IEEE J. Sel. Topics Quantum Electron. 29(4): 1-16, 2023. DOI: 10.1109/JSTQE.2022.3225108.
On the identification of cavities and inclusions in linear elasticity with a phase-field approach 1New York University Abu Dhabi, United Arab Emirates; 2Università degli Studi di Milano, Italy; 3Polytechnic University of Milan, Italy; 4University of Pavia, Italy
I analyze the geometric inverse problem of recovering cavities and inclusions embedded in a linear elastic isotropic medium from boundary displacement measurements.
Starting from a constrained minimization problem involving a boundary quadratic misfit functional with a regularization term penalizing the perimeter of the cavity or inclusion we consider a family of relaxed functionals using a phase-field approach and derive a robust algorithm for the reconstruction of elastic inclusions and cavities modeled as inclusions with a very small elasticity tensor.
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| 4:00pm - 6:00pm | MS31 2: Inverse Problems in Elastic Media Location: VG3.104 Session Chair: Andrea Aspri Session Chair: Ekaterina Sherina |
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An inverse problem for the porous medium equation 1National Yang Ming Chiao Tung University, Taiwan; 2National Institute of Science Education and Research, India; 3Inha University, Korea
The porous medium equation is a degenerate parabolic type quasilinear equation that models, for example, the flow of a gas through a porous medium. In this talk I will present recent results on uniqueness in the inverse boundary value problem for this equation. These are the first such results to be obtained for a degenerate parabolic equation.
Comparison of variational formulations for the direct solution of an inverse problem in linear elasticity 1Boston University, United States of America; 2Rochester Institute of Technology, United States of America
Given one or more observations of a displacement field within a linear elastic, isotropic, incompressible object, we seek to identify the material property distribution within that object. This is a mildly ill-posed inverse problem in linear elasticity. While most common approaches to solving this inverse problem use forward iteration, several variational formulations have been proposed that allow its direct solution. We review five such direct variational formulations for this inverse problem: Least Squares, Adjoint Weighted Equation, Virtual Fields, Inverse Least Squares, Direct Error in Constitutive Eqn. [1, 2, 3, 4, 5]. We briefly review their derivations, their mathematical properties, and their compatibility with Galerkin discretization and numerical solution. We demonstrate these properties through numerical examples.
[1] P. B. Bochev, M. D. Gunzburger. Finite element methods of least-squares type, SIAM Review, 40(4): 789--837, 1998.
[2] P. E. Barbone, C. E. Rivas, I. Harari, U. Albocher, A. A. Oberai, Y. Zhang. Adjoint-weighted variational formulation for the direct solution of inverse problems of general linear elasticity with full interior data, International Journal for Numerical Methods in Engineering 81(13): 1713--1736, 2010.
[3] F. Pierron, M. Grédiac. The Virtual Fields Method: Extracting Constitutive Mechanical
Parameters from Full-field Deformation Measurements, Springer Science & Business Media, 2012.
[4] G. Bal, C. Bellis, S. Imperiale, F. Monard. Reconstruction of constitutive parameters in isotropic linear elasticity from noisy full-field measurements, Inverse Problems 30(12): 125004, 2014.
[5] O. A. Babaniyi, A. A. Oberai, P. E. Barbone. Direct error in constitutive equation formulation for plane stress inverse elasticity problem, Computer Methods in Applied Mechanics and Engineering 314: 3--18, 2017.
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