Partial Boundary Measurements Research Articles

Stability estimates for an inverse problem for Schrödinger operators at high frequencies from arbitrary partial boundary measurements

In this paper, we study the partial data inverse boundary value problem for the Schrödinger operator at a high frequency in a bounded domain with smooth boundary in , . Assuming that the potential is known in a neighborhood of the boundary, we obtain the logarithmic stability when both Dirichlet data and Neumann data are taken on arbitrary open subsets of the boundary where the two sets can be disjointed. Our results also show that the logarithmic stability can be improved to the one of Hölder type in the high frequency regime. To achieve those goals, we used a method by combining the CGO solution, Runge approximation and Carleman estimate.

Recovery of multiple parameters in subdiffusion from one lateral boundary measurement

This work is concerned with numerically recovering multiple parameters simultaneously in the subdiffusion model from one single lateral measurement on a part of the boundary, while in an incompletely known medium. We prove that the boundary measurement corresponding to a fairly general boundary excitation uniquely determines the order of the fractional derivative and the polygonal support of the diffusion coefficient, without knowing either the initial condition or the source. The uniqueness analysis further inspires the development of a robust numerical algorithm for recovering the fractional order and diffusion coefficient. The proposed algorithm combines small-time asymptotic expansion, analytic continuation of the solution and the level set method. We present extensive numerical experiments to illustrate the feasibility of the simultaneous recovery. In addition, we discuss the uniqueness of recovering general diffusion and potential coefficients from one single partial boundary measurement, when the boundary excitation is more specialized.

Numerical Reconstruction of Radiative Sources from Partial Boundary Measurements

We consider an inverse source problem in the stationary radiative transport through an absorbing and scattering medium in two dimensions. Using the angularly resolved radiation measured on an arc of the boundary, we propose a numerical algorithm to recover the source in the convex hull of this arc. The method involves an unstable step of inverting a bounded operator whose range is not closed. We show that the continuity constant of the discretized inverse grows at most linearly with the discretization step, thus stabilizing the problem. Numerical examples presented show the effectiveness of the proposed method.

A numerical algorithm for inverse problem from partial boundary measurement arising from mean field game problem

In this work, we consider a novel inverse problem in mean-field games (MFGs). We aim to recover the MFG model parameters that govern the underlying interactions among the population based on a limited set of noisy partial observations of the population dynamics under the limited aperture. Due to its severe ill-posedness, obtaining a good quality reconstruction is very difficult. Nonetheless, it is vital to recover the model parameters stably and efficiently in order to uncover the underlying causes for population dynamics for practical needs. Our work focuses on the simultaneous recovery of running cost and interaction energy in the MFG equations from a finite number of boundary measurements of population profile and boundary movement. To achieve this goal, we formalize the inverse problem as a constrained optimization problem of a least squares residual functional under suitable norms. We then develop a fast and robust operator splitting algorithm to solve the optimization using techniques including harmonic extensions, three-operator splitting scheme, and primal-dual hybrid gradient method. Numerical experiments illustrate the effectiveness and robustness of the algorithm.

A noniterative reconstruction method for solving a time-fractional inverse source problem from partial boundary measurements

In this paper, a noniterative method for solving an inverse source problem governed by the two-dimensional time-fractional diffusion equation is proposed. The basic idea consists in reconstructing the geometrical support of the unknown source from partial boundary measurements of the associated potential. A Kohn–Vogelius type shape functional is considered together with a regularization term penalizing the relative perimeter of the unknown set of anomalies. Identifiability result is derived and uniqueness of a minimizer is ensured. The shape functional measuring the misfit between the solutions of two auxiliary problems containing information about the boundary measurements is minimized with respect to a finite number of ball-shaped trial anomalies by using the topological derivative method. In particular, the second-order topological gradient is exploited to devise an efficient and fast noniterative reconstruction algorithm. Finally, some numerical experiments are presented, showing different features of the proposed approach in reconstructing multiple anomalies of varying shapes and sizes by taking noisy data into account.

Simplified reconstruction of layered materials in EIT

This short note considerably simplifies a reconstruction method by the author (Garde, 2020), for reconstructing piecewise constant layered conductivities (PCLC) from partial boundary measurements in electrical impedance tomography. Theory from monotonicity-based reconstruction of extreme inclusions eliminates most of the bookkeeping related to multiple components of each layer, and also simplifies the involved test operators. Moreover, the method no longer requires a priori lower and upper bounds to the unknown conductivity values.

Solving an inverse acousto-elastic scattering problems by combining full-waveform redatuming and time reversal

The aim of this paper is to propose a new method to solve the inverse scattering problem in a non homogeneous elastic medium, from data recorded in an acoustic medium. We aim to determine the Young modulus of elastic unknown “inclusions”, i.e. not observable solid objects, located in the elastic medium, from partial aperture boundary measurements. This approach combines a time-reversal approach and redatuming - a technique that consists in virtually moving the sensors from the original acquisition location to an arbitrary position - which allows to reduce the size of the inspected medium. Thanks to this strategy, we can recover the properties of the scatterer, this technique being also relatively insensitive to noise in the data.

Determining a potential of the parabolic equation from partial boundary measurements * *This work was supported by NSFC (61671009).

We concern the operator ∂ t − Δ + q with the Dirichlet boundary conditions on a bounded domain in , n ⩾ 3, and show that the knowledge of the Dirichlet-to-Neumann map measured on a very small subset of the boundary determines the potential uniquely.

A stable non-iterative reconstruction algorithm for the acoustic inverse boundary value problem

<p style='text-indent:20px;'>We present a non-iterative algorithm to reconstruct the isotropic acoustic wave speed from the measurement of the Neumann-to-Dirichlet map. The algorithm is designed based on the boundary control method and involves only computations that are stable. We prove the convergence of the algorithm and present its numerical implementation. The effectiveness of the algorithm is validated on both constant speed and variable speed, with full and partial boundary measurement as well as different levels of noise.</p>

On Calderón’s inverse inclusion problem with smooth shapes by a single partial boundary measurement

We consider Calderón’s inverse inclusion problem of recovering the shape of an anomalous inhomogeneity embedded in a homogeneous conductivity by the associated electric boundary measurements. It is a longstanding problem whether one can establish the unique recovery result by a single boundary measurement. In several existing works, it is shown that corner singularities can help to resolve the problem within polygonal or polyhedral shapes. In this paper, under a generic technical condition, we show that the corner singularity can be relaxed to be a certain high-curvature condition and derive novel unique recovery results within smooth shapes.

Stable recovery of a non-compactly supported coefficient of a Schrödinger equation on an infinite waveguide

<p style='text-indent:20px;'>We study the stability issue for the inverse problem of determining a coefficient appearing in a Schrödinger equation defined on an infinite cylindrical waveguide. More precisely, we prove the stable recovery of some general class of non-compactly and non periodic coefficients appearing in an unbounded cylindrical domain. We consider both results of stability from full and partial boundary measurements associated with the so called Dirichlet-to-Neumann map.</p>

On the determination of ischemic regions in the monodomain model of cardiac electrophysiology from boundary measurements**In loving memory of Maria Assunta Pozio.

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.

Time reversal for elastic scatterer location from acoustic recording

The aim of this paper is to study the feasibility of time-reversal methods in a non homogeneous elastic medium, from data recorded in an acoustic medium. We aim to determine, from partial aperture boundary measurements, the presence and some physical properties of elastic unknown “inclusions”, i.e. not observable solid objects, located in the elastic medium. We first derive a variational formulation of the acousto-elastic problem, from which one constructs a time-dependent finite element method to solve the forward, and then, the time reversed problem. Several criteria, derived from the reverse time migration framework, are then proposed to construct images of the inclusions, and to determine their locations. The dependence of the approach to several parameters (aperture, number of sources, etc.) is also investigated. In particular, it is shown that one can differentiate between a benign and malignant close inclusions. This technique is fairly insensitive to noise in the data.

Stable determination of polygonal inclusions in Calderón’s problem by a single partial boundary measurement

We are concerned with the Calderón problem of determining the unknown conductivity of a body from the associated boundary measurement. We establish a logarithmic type stability estimate in terms of the Hausdorff distance in determining the support of a convex polygonal inclusion by a single partial boundary measurement. We also derive the uniqueness result in a more general scenario where the conductivities are piecewise constants supported in a nested polygonal geometry. Our methods in establishing the stability and uniqueness results have a significant technical initiative and a strong potential to apply to other inverse boundary value problems.

On the Kohn–Vogelius formulation for solving an inverse source problem

An inverse source problem related to the Poisson equation is the main concern of this work. Specifically, we deal with the reconstruction of a mass distribution in a geometrical domain from a partial boundary measurement of the associated potential. The considered problem is motivated by various applications such as the identification of geological anomalies underneath the Earth's surface. The proposed approach is based on the Kohn–Vogelius formulation and the topological derivative method. An explicit second-order sensitivity related to circular shaped anomalies is calculated for different examples of the Kohn–Vogelius type functional. Then, the optimal location and size of the unknown support of the mass distribution are characterized as the solution to a minimization problem. The resulting reconstruction procedure is non-iterative and robust with respect to noisy data. Finally, we produce numerical results from four different examples of the Kohn–Vogelius type functional. The results first demonstrate the method and then compare the robustness of each functional in solving the inverse source problem.

On the identification of the heat conductivity distribution from partial dynamic boundary measurements

ABSTRACT This work deals with an inverse boundary value problem arising from the equation of heat conduction. We reconstruct small perturbations of the (isotropic) heat conductivity distribution from partial (on accessible part of the boundary) dynamic boundary measurements and for finite interval in time. By constructing of appropriate test functions, using a control method, we provide a rigorous derivation of the inverse Fourier transform of the perturbations in the diffusion coefficient as the leading order of an appropriate averaging of the partial dynamic boundary measurements.

Performance Output Tracking for Multidimensional Heat Equation Subject to Unmatched Disturbance and Noncollocated Control

This paper investigates performance output tracking for a boundary controlled multidimensional heat equation. It is assumed that the so-called total disturbance (which is composed of internal possibly nonlinear uncertainty and external disturbance) the equation is subject to is on one part of the boundary, and that the control is applied on the rest of the boundary. Using only partial boundary measurement, we first propose an extended state observer to estimate both system state and the total disturbance. This allows us to design a servomechanism and then an output feedback controller.Under the condition that both the reference signal and the disturbance vanish or belong to spaces H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> (0, ∞; L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> (Γ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> )) and L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (0, ∞; L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (Γ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> )), respectively. We show that the over-all control strategy achieves three objectives on the system performance: first, exponentially output tracking for arbitrary given reference signal; second, uniformly boundedness of all internal signals; and, third, the internal asymptotic stability of the closed-loop system. In addition, the control strategy turns out to be robust to the measurement noise. We provide numerical experiments to illustrate the effectiveness of the proposed control strategy.

A noniterative reconstruction method for the inverse potential problem with partial boundary measurements

In this paper, a noniterative reconstruction method for solving the inverse potential problem is proposed. The forward problem is governed by a modified Helmholtz equation. The inverse problem consists in the reconstruction of a set of anomalies embedded into a geometrical domain from partial or total boundary measurements of the associated potential. Since the inverse problem is written in the form of an ill‐posed boundary value problem, the idea is to rewrite it as a topology optimization problem. In particular, a shape functional measuring the misfit between the solution obtained from the model and the data taken from the boundary measurements is minimized with respect to a set of ball‐shaped anomalies by using the concept of topological derivatives. It means that the shape functional is expanded asymptotically and then truncated up to the desired order term. The resulting truncated expansion is trivially minimized with respect to the parameters under consideration that leads to a noniterative second order reconstruction algorithm. As a result, the reconstruction process becomes very robust with respect to the noisy data and independent of any initial guess. Finally, some numerical experiments are presented showing the capability of the proposed method in reconstructing multiple anomalies of different sizes and shapes by taking into account complete or partial boundary measurements.

An inverse dipole source problem in inhomogeneous media: Application to the EEG source localization in neonates

Abstract In this paper we present some aspects of an inverse source problem in an elliptic equation with varying coefficients, using partial Dirichlet boundary measurements. A uniqueness result is established for dipolar sources including their number. Additionally, assuming the number of dipoles known, a stability result is obtained and an efficient numerical identification method is developed. Finally, numerical experiments illustrate the effectiveness of the approach and a discussion is given on electroencephalography (EEG) in neonates.

Carleman estimate for the Schrödinger equation and application to magnetic inverse problems

Abstract In this paper, we prove Lipschitz stable determination of the complex-valued electric potential and the direction of the magnetic field appearing in the dynamic Schrodinger equation with static coefficients, by finitely many partial boundary measurements of the solution. This is by means of the Bukhgeim–Klibanov method, based on an appropriate Carleman estimate. Since the time symmetrization of the static magnetic Schrodinger equation around t = 0 is not possible, we preliminarily establish a Carleman inequality specifically designed for this problem.