Elena Beretta | New York University Abu Dhabi (original) (raw)

Papers by Elena Beretta

Abstract. In order to reconstruct small changes in the interface of an elastic inclusion from mod... more Abstract. In order to reconstruct small changes in the interface of an elastic inclusion from modal measurements, we rigorously derive an asymptotic for-mula which is in some sense dual to the leading-order term in the asymptotic expansion of the perturbations in the eigenvalues due to interface changes of the inclusion. Based on this (dual) formula we propose an algorithm to reconstruct the interface perturbation. We also consider an optimal way of representing the interface change and the reconstruction problem using in-complete data. A discussion on resolution is included. Proposed algorithms are implemented numerically to show their viability. 1.

SIAM Journal on Imaging Sciences

In this paper, we derive new asymptotic expansions for the solutions of higher order elliptic equ... more In this paper, we derive new asymptotic expansions for the solutions of higher order elliptic equations in the presence of small inclusions. As a byproduct, we derive a topological derivative based algorithm for the reconstruction of piecewise smooth functions. This algorithm can be used for edge detection in imaging, topological optimization, and inverse problems, such as quantitative photoacoustic tomography, for which we demonstrate the effectiveness of our asymptotic expansion method numerically.

In this paper we deal with the problem of determining perfectly insulating regions (cavities) fro... more In this paper we deal with the problem of determining perfectly insulating regions (cavities) from boundary measurements in a nonlinear elliptic equation arising from cardiac electrophysiology. With minimal regularity assumptions on the cavities, we first show well-posedness of the direct problem and then prove uniqueness of the inverse problem. Finally, we propose a new reconstruction algorithm by means of a phase-field approach rigorously justified via Γ-convergence.

arXiv: Analysis of PDEs, 2017

In this work we tackle the reconstruction of discontinuous coefficients in a semilinear elliptic ... more In this work we tackle the reconstruction of discontinuous coefficients in a semilinear elliptic equation from the knowledge of the solution on the boundary of the domain, an inverse problem motivated by biological application in cardiac electrophysiology. We formulate a constraint minimization problem involving a quadratic mismatch functional enhanced with a regularization term which penalizes the perimeter of the inclusion to be identified. We introduce a phase-field relaxation of the problem, replacing the perimeter term with a Ginzburg-Landau-type energy. We prove the Gamma\GammaGamma-convergence of the relaxed functional to the original one (which implies the convergence of the minimizers), we compute the optimality conditions of the phase-field problem and define a reconstruction algorithm based on the use of the Fr\`echet derivative of the functional. After introducing a discrete version of the problem we implement an iterative algorithm and prove convergence properties. Several nume...

arXiv: Analysis of PDEs, 2011

We derive asymptotic expansions for the displacement at the boundary of a smooth, elastic body in... more We derive asymptotic expansions for the displacement at the boundary of a smooth, elastic body in the presence of small inhomogeneities. Both the body and the inclusions are allowed to be anisotropic. This work extends prior work of CapdeBoscq and Vogelius ({\em Math. Modelling Num. Anal.} 37, 2003) for the conductivity case. In particular, we obtain an asymptotic expansion of the difference between the displacements at the boundary with and without inclusions, under Neumann boundary conditions, to first order in the measure of the inclusions. We impose no geometric conditions on the inclusions, which need only be measurable sets. The first-order correction contains an elastic moment tensor MM\MMMM that encodes the effect of the inclusions. In the case of thin, strip-like, planar inhomogeneities we obtain a formula for MM\MMMM only in terms of the elasticity tensors, which we assume strongly convex, their inverses, and a frame on the curve that supports the inclusion. We prove uniquenes...

Abstract. We study the inverse boundary value problem for the Helmholtz equation using the Dirich... more Abstract. We study the inverse boundary value problem for the Helmholtz equation using the Dirichlet-toNeumann map at selected frequencies as the data. We develop an explicit iterative reconstruction of the wavespeed using a multi-level nonlinear projected steepest descent iterative scheme in Banach spaces. We consider wavespeeds containing conormal singularities. A conditional Lipschitz estimate for the inverse problem holds for wavespeeds of the form of a linear combination of piecewise constant functions with an underlying domain partitioning, and gives a framework in which the scheme converges. The stability constant grows exponentially as the number of subdomains in the domain partitioning increases. To mitigate this growth of the stability constant, we introduce a hierarchy of compressive approximations of the solution to the inverse problem with piecewise constant functions. We establish an upper bound of the stability constant, which constrains the compression rate of the so...

We study a two-dimensional model of elastic dislocations motivated by applications in geophysics.... more We study a two-dimensional model of elastic dislocations motivated by applications in geophysics. In this model, the displacement satis es the system of linear elasticity with mixed displacement-traction homogeneous boundary conditions in the complement of an open curve in a bounded planar domain, and has a speci ed jump, the slip, across the curve, while the traction is continuous there. The sti ness tensor is allowed to be anisotropic and inhomogeneous. We prove well-posedness of the direct problem in a variational setting, assuming the coe cients are Lipschitz continuous. Using unique continuation arguments, we then establish uniqueness in the inverse problem of determining the dislocation curve and the slip from a single measurement of the displacement on an open patch of the traction-free part of the boundary. Uniqueness holds when the elasticity operators admits a suitable decomposition and the curve satis es additional geometric assumptions. This work extends the results in A...

arXiv: Numerical Analysis, 2014

We study the inverse boundary value problem for the Helmholtz equation using the Dirichlet-to-Neu... more We study the inverse boundary value problem for the Helmholtz equation using the Dirichlet-to-Neumann map at selected frequency as the data. We develop an explicit reconstruction of the wavespeed using a multi-level nonlinear projected steepest descent iterative scheme in Banach spaces. We consider wavespeeds containing discontinuities. A conditional Lipschitz stability estimate for the inverse problem holds for wavespeeds of the form of a linear combination of piecewise constant functions with an underlying domain partitioning, and gives a framework in which the scheme converges. The stability constant grows exponentially as the number of subdomains in the domain partitioning increases. To mitigate this growth of the stability constant, we introduce hierarchical compressive approximations of the solution to the inverse problem with piecewise constant functions. We establish an optimal bound of the stability constant, which leads to a condition on the compression rate pertaining to ...

We consider a model for elastic dislocations in geophysics. We model a portion of the Earth's... more We consider a model for elastic dislocations in geophysics. We model a portion of the Earth's crust as a bounded, inhomogeneous elastic body with a buried fault surface, along which slip occurs. We prove well-posedness of the resulting mixed-boundary-value-transmission problem, assuming only bounded elastic moduli. We establish uniqueness in the inverse problem of determining the fault surface and the slip from a unique measurement of the displacement on an open patch at the surface, assuming in addition that the Earth's crust is an isotropic, layered medium with Lame coefficients piecewise Lipschitz on a known partition and that the fault surface is a graph with respect to an arbitrary coordinate system. These results substantially extend those of the authors in {Arch. Ration. Mech. Anal.} {\bf 263} (2020), n. 1, 71--111.

In this paper we provide a representation formula for boundary voltage perturbations caused by in... more In this paper we provide a representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction in a simplified {\em monodomain model} describing the electric activity of the heart. We derive such a result in the case of a nonlinear problem. Our long-term goal is the solution of the inverse problem related to the detection of regions affected by heart ischemic disease, whose position and size are unknown. We model the presence of ischemic regions in the form of small inhomogeneities. This leads to the study of a boundary value problem for a semilinear elliptic equation. We first analyze the well-posedness of the problem establishing some key energy estimates. These allow us to derive rigorously an asymptotic formula of the boundary potential perturbation due to the presence of the inhomogeneities, following an approach similar to the one introduced by Capdeboscq and Vogelius in \cite{capvoge} in the case of the linear condu...

The reconstruction in quantitative coupled physics imaging often requires that the solutions of c... more The reconstruction in quantitative coupled physics imaging often requires that the solutions of certain PDEs satisfy some non-zero constraints, such as the absence of critical points or nodal points. After a brief review of several methods used to construct such solutions, I will focus on a recent approach that combines the Runge approximation and the Whitney embedding

In this paper we develop theoretical analysis and numerical reconstruction techniques for the sol... more In this paper we develop theoretical analysis and numerical reconstruction techniques for the solution of an inverse boundary value problem dealing with the nonlinear, time-dependent monodomain equation, which models the evolution of the electric potential in the myocardial tissue. The goal is the detection of a small inhomogeneity omegavarepsilon\omega_\varepsilonomegavarepsilon (where the coefficients of the equation are altered) located inside a domain Omega\OmegaOmega starting from observations of the potential on the boundary partialOmega\partial \OmegapartialOmega. Such a problem is related to the detection of myocardial ischemic regions, characterized by severely reduced blood perfusion and consequent lack of electric conductivity. In the first part of the paper we provide an asymptotic formula for electric potential perturbations caused by internal conductivity inhomogeneities of low volume fraction, extending the results published in [7] to the case of three-dimensional, parabolic problems. In the second part we implement a recons...

Using a distributed representation formula of the Gateaux derivative of the Dirichlet to Neumann ... more Using a distributed representation formula of the Gateaux derivative of the Dirichlet to Neumann map with respect to movements of a polygonal conductivity inclusion, [11], we extend the results obtained in [8] proving global Lipschitz stability for the determination of a polygonal conductivity inclusion embedded in a layered medium from knowledge of the Dirichlet to Neumann map.

We derive asymptotic expansions for the displacement at the boundary of a smooth, elastic body in... more We derive asymptotic expansions for the displacement at the boundary of a smooth, elastic body in the presence of small inhomogeneities. Both the body and the inclusions are allowed to be anisotropic. This work extends prior work of CapdeBoscq and Vogelius (Math. Modelling Num. Anal. 37, 2003) for the conductivity case. In particular, we obtain an asymptotic expansion of the difference between the displacements at the boundary with and without inclusions, under Neumann boundary conditions, to first order in the measure of the inclusions. We impose no geometric conditions on the inclusions, which need only be measurable sets. The first-order correction contains a moment or polarization tensor M that encodes the effect of the inclusions. In the case of thin, strip-like, planar inhomogeneities we obtain a formula for M only in terms of the elasticity tensors, which we assume strongly convex, their inverses, and a frame on the curve that supports the inclusion. We prove uniqueness of M ...

Journal of Mathematical Economics

Nonlinearity

In this paper we consider the monodomain model of cardiac electrophysiology. After an analysis of... more In this paper we consider the monodomain model of cardiac electrophysiology. After an analysis of the well-posedness of the model we determine an asymptotic expansion of the perturbed potential due to the presence of small conductivity inhomogeneities (modelling small ischemic regions in the cardiac tissue) and use it to detect the anomalies from partial boundary measurements. This is done by determining the topological gradient of a suitable boundary misfit functional. The robustness of the algorithm is confirmed by several numerical experiments.

Mathematics in Engineering

We consider an inverse problem regarding the detection of small conductivity inhomogeneities in a... more We consider an inverse problem regarding the detection of small conductivity inhomogeneities in a boundary value problem for a semilinear elliptic equation. For such a problem, that is related to cardiac electrophysiology, an asymptotic expansion for the boundary potential due to the presence of small conductivity inhomogeneities was established in [4]. Starting from this we derive Lipschitz continuous dependence estimates for the corresponding inverse problem.

Applicable Analysis

We derive Lipschitz stability estimates for the Hausdorff distance of polygonal conductivity incl... more We derive Lipschitz stability estimates for the Hausdorff distance of polygonal conductivity inclusions in terms of the Dirichlet-to-Neumann map.

Abstract. In order to reconstruct small changes in the interface of an elastic inclusion from mod... more Abstract. In order to reconstruct small changes in the interface of an elastic inclusion from modal measurements, we rigorously derive an asymptotic for-mula which is in some sense dual to the leading-order term in the asymptotic expansion of the perturbations in the eigenvalues due to interface changes of the inclusion. Based on this (dual) formula we propose an algorithm to reconstruct the interface perturbation. We also consider an optimal way of representing the interface change and the reconstruction problem using in-complete data. A discussion on resolution is included. Proposed algorithms are implemented numerically to show their viability. 1.

SIAM Journal on Imaging Sciences

In this paper, we derive new asymptotic expansions for the solutions of higher order elliptic equ... more In this paper, we derive new asymptotic expansions for the solutions of higher order elliptic equations in the presence of small inclusions. As a byproduct, we derive a topological derivative based algorithm for the reconstruction of piecewise smooth functions. This algorithm can be used for edge detection in imaging, topological optimization, and inverse problems, such as quantitative photoacoustic tomography, for which we demonstrate the effectiveness of our asymptotic expansion method numerically.

In this paper we deal with the problem of determining perfectly insulating regions (cavities) fro... more In this paper we deal with the problem of determining perfectly insulating regions (cavities) from boundary measurements in a nonlinear elliptic equation arising from cardiac electrophysiology. With minimal regularity assumptions on the cavities, we first show well-posedness of the direct problem and then prove uniqueness of the inverse problem. Finally, we propose a new reconstruction algorithm by means of a phase-field approach rigorously justified via Γ-convergence.

arXiv: Analysis of PDEs, 2017

In this work we tackle the reconstruction of discontinuous coefficients in a semilinear elliptic ... more In this work we tackle the reconstruction of discontinuous coefficients in a semilinear elliptic equation from the knowledge of the solution on the boundary of the domain, an inverse problem motivated by biological application in cardiac electrophysiology. We formulate a constraint minimization problem involving a quadratic mismatch functional enhanced with a regularization term which penalizes the perimeter of the inclusion to be identified. We introduce a phase-field relaxation of the problem, replacing the perimeter term with a Ginzburg-Landau-type energy. We prove the Gamma\GammaGamma-convergence of the relaxed functional to the original one (which implies the convergence of the minimizers), we compute the optimality conditions of the phase-field problem and define a reconstruction algorithm based on the use of the Fr\`echet derivative of the functional. After introducing a discrete version of the problem we implement an iterative algorithm and prove convergence properties. Several nume...

arXiv: Analysis of PDEs, 2011

We derive asymptotic expansions for the displacement at the boundary of a smooth, elastic body in... more We derive asymptotic expansions for the displacement at the boundary of a smooth, elastic body in the presence of small inhomogeneities. Both the body and the inclusions are allowed to be anisotropic. This work extends prior work of CapdeBoscq and Vogelius ({\em Math. Modelling Num. Anal.} 37, 2003) for the conductivity case. In particular, we obtain an asymptotic expansion of the difference between the displacements at the boundary with and without inclusions, under Neumann boundary conditions, to first order in the measure of the inclusions. We impose no geometric conditions on the inclusions, which need only be measurable sets. The first-order correction contains an elastic moment tensor MM\MMMM that encodes the effect of the inclusions. In the case of thin, strip-like, planar inhomogeneities we obtain a formula for MM\MMMM only in terms of the elasticity tensors, which we assume strongly convex, their inverses, and a frame on the curve that supports the inclusion. We prove uniquenes...

Abstract. We study the inverse boundary value problem for the Helmholtz equation using the Dirich... more Abstract. We study the inverse boundary value problem for the Helmholtz equation using the Dirichlet-toNeumann map at selected frequencies as the data. We develop an explicit iterative reconstruction of the wavespeed using a multi-level nonlinear projected steepest descent iterative scheme in Banach spaces. We consider wavespeeds containing conormal singularities. A conditional Lipschitz estimate for the inverse problem holds for wavespeeds of the form of a linear combination of piecewise constant functions with an underlying domain partitioning, and gives a framework in which the scheme converges. The stability constant grows exponentially as the number of subdomains in the domain partitioning increases. To mitigate this growth of the stability constant, we introduce a hierarchy of compressive approximations of the solution to the inverse problem with piecewise constant functions. We establish an upper bound of the stability constant, which constrains the compression rate of the so...

We study a two-dimensional model of elastic dislocations motivated by applications in geophysics.... more We study a two-dimensional model of elastic dislocations motivated by applications in geophysics. In this model, the displacement satis es the system of linear elasticity with mixed displacement-traction homogeneous boundary conditions in the complement of an open curve in a bounded planar domain, and has a speci ed jump, the slip, across the curve, while the traction is continuous there. The sti ness tensor is allowed to be anisotropic and inhomogeneous. We prove well-posedness of the direct problem in a variational setting, assuming the coe cients are Lipschitz continuous. Using unique continuation arguments, we then establish uniqueness in the inverse problem of determining the dislocation curve and the slip from a single measurement of the displacement on an open patch of the traction-free part of the boundary. Uniqueness holds when the elasticity operators admits a suitable decomposition and the curve satis es additional geometric assumptions. This work extends the results in A...

arXiv: Numerical Analysis, 2014

We study the inverse boundary value problem for the Helmholtz equation using the Dirichlet-to-Neu... more We study the inverse boundary value problem for the Helmholtz equation using the Dirichlet-to-Neumann map at selected frequency as the data. We develop an explicit reconstruction of the wavespeed using a multi-level nonlinear projected steepest descent iterative scheme in Banach spaces. We consider wavespeeds containing discontinuities. A conditional Lipschitz stability estimate for the inverse problem holds for wavespeeds of the form of a linear combination of piecewise constant functions with an underlying domain partitioning, and gives a framework in which the scheme converges. The stability constant grows exponentially as the number of subdomains in the domain partitioning increases. To mitigate this growth of the stability constant, we introduce hierarchical compressive approximations of the solution to the inverse problem with piecewise constant functions. We establish an optimal bound of the stability constant, which leads to a condition on the compression rate pertaining to ...

We consider a model for elastic dislocations in geophysics. We model a portion of the Earth's... more We consider a model for elastic dislocations in geophysics. We model a portion of the Earth's crust as a bounded, inhomogeneous elastic body with a buried fault surface, along which slip occurs. We prove well-posedness of the resulting mixed-boundary-value-transmission problem, assuming only bounded elastic moduli. We establish uniqueness in the inverse problem of determining the fault surface and the slip from a unique measurement of the displacement on an open patch at the surface, assuming in addition that the Earth's crust is an isotropic, layered medium with Lame coefficients piecewise Lipschitz on a known partition and that the fault surface is a graph with respect to an arbitrary coordinate system. These results substantially extend those of the authors in {Arch. Ration. Mech. Anal.} {\bf 263} (2020), n. 1, 71--111.

In this paper we provide a representation formula for boundary voltage perturbations caused by in... more In this paper we provide a representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction in a simplified {\em monodomain model} describing the electric activity of the heart. We derive such a result in the case of a nonlinear problem. Our long-term goal is the solution of the inverse problem related to the detection of regions affected by heart ischemic disease, whose position and size are unknown. We model the presence of ischemic regions in the form of small inhomogeneities. This leads to the study of a boundary value problem for a semilinear elliptic equation. We first analyze the well-posedness of the problem establishing some key energy estimates. These allow us to derive rigorously an asymptotic formula of the boundary potential perturbation due to the presence of the inhomogeneities, following an approach similar to the one introduced by Capdeboscq and Vogelius in \cite{capvoge} in the case of the linear condu...

The reconstruction in quantitative coupled physics imaging often requires that the solutions of c... more The reconstruction in quantitative coupled physics imaging often requires that the solutions of certain PDEs satisfy some non-zero constraints, such as the absence of critical points or nodal points. After a brief review of several methods used to construct such solutions, I will focus on a recent approach that combines the Runge approximation and the Whitney embedding

In this paper we develop theoretical analysis and numerical reconstruction techniques for the sol... more In this paper we develop theoretical analysis and numerical reconstruction techniques for the solution of an inverse boundary value problem dealing with the nonlinear, time-dependent monodomain equation, which models the evolution of the electric potential in the myocardial tissue. The goal is the detection of a small inhomogeneity omegavarepsilon\omega_\varepsilonomegavarepsilon (where the coefficients of the equation are altered) located inside a domain Omega\OmegaOmega starting from observations of the potential on the boundary partialOmega\partial \OmegapartialOmega. Such a problem is related to the detection of myocardial ischemic regions, characterized by severely reduced blood perfusion and consequent lack of electric conductivity. In the first part of the paper we provide an asymptotic formula for electric potential perturbations caused by internal conductivity inhomogeneities of low volume fraction, extending the results published in [7] to the case of three-dimensional, parabolic problems. In the second part we implement a recons...

Using a distributed representation formula of the Gateaux derivative of the Dirichlet to Neumann ... more Using a distributed representation formula of the Gateaux derivative of the Dirichlet to Neumann map with respect to movements of a polygonal conductivity inclusion, [11], we extend the results obtained in [8] proving global Lipschitz stability for the determination of a polygonal conductivity inclusion embedded in a layered medium from knowledge of the Dirichlet to Neumann map.

We derive asymptotic expansions for the displacement at the boundary of a smooth, elastic body in... more We derive asymptotic expansions for the displacement at the boundary of a smooth, elastic body in the presence of small inhomogeneities. Both the body and the inclusions are allowed to be anisotropic. This work extends prior work of CapdeBoscq and Vogelius (Math. Modelling Num. Anal. 37, 2003) for the conductivity case. In particular, we obtain an asymptotic expansion of the difference between the displacements at the boundary with and without inclusions, under Neumann boundary conditions, to first order in the measure of the inclusions. We impose no geometric conditions on the inclusions, which need only be measurable sets. The first-order correction contains a moment or polarization tensor M that encodes the effect of the inclusions. In the case of thin, strip-like, planar inhomogeneities we obtain a formula for M only in terms of the elasticity tensors, which we assume strongly convex, their inverses, and a frame on the curve that supports the inclusion. We prove uniqueness of M ...

Journal of Mathematical Economics

Nonlinearity

In this paper we consider the monodomain model of cardiac electrophysiology. After an analysis of... more In this paper we consider the monodomain model of cardiac electrophysiology. After an analysis of the well-posedness of the model we determine an asymptotic expansion of the perturbed potential due to the presence of small conductivity inhomogeneities (modelling small ischemic regions in the cardiac tissue) and use it to detect the anomalies from partial boundary measurements. This is done by determining the topological gradient of a suitable boundary misfit functional. The robustness of the algorithm is confirmed by several numerical experiments.

Mathematics in Engineering

We consider an inverse problem regarding the detection of small conductivity inhomogeneities in a... more We consider an inverse problem regarding the detection of small conductivity inhomogeneities in a boundary value problem for a semilinear elliptic equation. For such a problem, that is related to cardiac electrophysiology, an asymptotic expansion for the boundary potential due to the presence of small conductivity inhomogeneities was established in [4]. Starting from this we derive Lipschitz continuous dependence estimates for the corresponding inverse problem.

Applicable Analysis

We derive Lipschitz stability estimates for the Hausdorff distance of polygonal conductivity incl... more We derive Lipschitz stability estimates for the Hausdorff distance of polygonal conductivity inclusions in terms of the Dirichlet-to-Neumann map.