We analyze the inverse spectral problem on the half line associated with elastic surface waves. Here, we extend the treatment of Love waves [5] to Rayleigh waves. Under certain conditions, and assuming that the Poisson ratio is constant, we establish uniqueness and present a reconstruction scheme for the S-wave speed with multiple wells from the semiclassical spectrum of these waves.

This work investigates the elasticity imaging inverse problem of tumor identification in a fully incompressible medium through a family of inverse problems in a nearly incompressible medium. We develop an inversion framework for saddle point problems that goes far beyond the elasticity imaging inverse problem and applies to a wide variety of inverse problems. We introduce a family of convex optimization

We present new joint reconstruction and regularization techniques inspired by ideas in microlocal analysis and lambda tomography, for the simultaneous reconstruction of the attenuation coefficient and electron density from x-ray transmission (i.e., x-ray CT) and backscattered data (assumed to be primarily Compton scattered). To demonstrate our theory and reconstruction methods, we consider the ‘parallel

The iteratively regularized Gauss–Newton method (IRGNM) is a prominent method for solving nonlinear inverse problems. Based on a modified discrepancy principle, in this paper we propose for the IRGNM in Banach spaces a heuristic rule which is purely data driven and requires no information on the noise level. Under the tangential cone condition on the forward operator and the variational source conditions

In seismic waveform inversion, the reconstruction of the subsurface properties is usually carried out using approximative wave propagation models to ensure computational efficiency. The viscoelastic nature of the subsurface is often unaccounted for, and two popular approximations—the acoustic and linearized Born inversion—are widely used. This leads to reconstruction errors since the approximations

This paper is concerned with the mathematical analysis of experimental methods for the estimation of the power of an uncorrelated, extended aeroacoustic source from measurements of correlations of pressure fluctuations. We formulate a continuous, infinite dimensional model describing these experimental techniques based on the convected Helmholtz equation in ##IMG## [http://ej.iop.org/images/0266-5

We shall study in this paper the convergence rates of the Tikhonov regularized solutions for the recovery of the radiativities in elliptic and parabolic systems in general dimensional spaces. The conditional stability estimates are first derived. Due to the difficulty of the verification of the existing source conditions or nonlinearity conditions of the inverse radiativity problems in high dimensional

In this paper, we consider the minimization of a Tikhonov functional with an ℓ 1 penalty for solving linear inverse problems with sparsity constraints. One of the many approaches used to solve this problem uses the Nemskii operator to transform the Tikhonov functional into one with an ℓ 2 penalty term but a nonlinear operator. The transformed problem can then be analyzed and minimized using standard

The ensemble Kalman filter method can be used as an iterative particle numerical scheme for state dynamics estimation and control-to-observable identification problems. In applications it may be required to enforce the solution to satisfy equality constraints on the control space. In this work we deal with this problem from a constrained optimization point of view, deriving corresponding optimality

We consider the variational regularization for inverse problems in a general form. Based on the discrepancy principle, we propose a heuristic parameter choice rule for choosing the regularization parameter which does not require the information on the noise level and is therefore purely data driven. Under variational source conditions, we obtain a posteriori error estimates. According to the Bakushinskii

In this paper we revisit the discrepancy principle for Landweber iteration for solving linear as well as nonlinear inverse problems in Banach spaces and prove a new convergence result which requires neither the Gâteaux differentiability of the forward operator nor the reflexivity of the image space. Therefore, we expand the applied range of the discrepancy principle for Landweber iteration to cover

This paper analyzes inverse scattering for the one-dimensional Helmholtz equation in the case where the wave speed is piecewise constant. Scattering data recorded for an arbitrarily small interval of frequencies is shown to determine the wave speed uniquely, and a direct reconstruction algorithm is presented. The algorithm is exact provided data is recorded for a sufficiently wide range of frequencies

We consider the problem of the recovery of a Robin coefficient on a part γ ⊂ ∂Ω of the boundary of a bounded domain Ω from the principal eigenvalue and the boundary values of the normal derivative of the principal eigenfunction of the Laplace operator with Dirichlet boundary condition on ∂Ω\ γ . We prove the uniqueness, as well as local Lipschitz stability of the inverse problem. Moreover, we present

We study an inverse problem for light sheet fluorescence microscopy (LSFM), where the density of fluorescent molecules needs to be reconstructed. Our first step is to present a mathematical model to describe the measurements obtained by an optic camera during an LSFM experiment. Two meaningful stages are considered: excitation and fluorescence. We propose a paraxial model to describe the excitation

We introduce a fractional magnetic operator involving a magnetic potential and an electric potential. We formulate an inverse problem for the fractional magnetic operator. We determine the electric potential from the exterior partial measurements of the associated Dirichlet-to-Neumann map by using Runge approximation property.

We present a method for computing A-optimal sensor placements for infinite-dimensional Bayesian linear inverse problems governed by PDEs with irreducible model uncertainties. Here, irreducible uncertainties refers to uncertainties in the model that exist in addition to the parameters in the inverse problem, and that cannot be reduced through observations. Specifically, given a statistical distribution

This paper addresses the numerical simulation of three-dimensional time-dependent inverse source problems of acoustic waves. The reconstructions of both multiple stationary point sources and a moving point source are considered. The modified method of fundamental solutions (MMFS), which expands the solution utilizing the time convolution of Green’s function and the signal function, is proposed to solve

We analyze the inverse spectral problem on the half line associated with elastic surface waves. Here, we focus on Love waves. Under certain generic conditions, we establish uniqueness and present a reconstruction scheme for the S -wavespeed with multiple wells from the semiclassical spectrum of these waves.

In this paper, we establish two Carleman estimates for a stochastic degenerate parabolic equation. The first one is for the backward stochastic degenerate parabolic equation with singular weight function. Combining this Carleman estimate and an approximate argument, we prove the null controllability of the forward stochastic degenerate parabolic equation with the gradient term. The second one is for

This work deals with the inverse thermoacoustic tomography (TAT) problem. It is a biomedical, multi-wave imaging technique, based on the photoacoustic effect (generation of sound from light) that was discovered in 1880 by Alexander Graham Bell. The inverse problem we are concerned in is the inverse source problem of recovering small absorbers in a bounded domain ##IMG## [http://ej.iop.org/images/0

This paper considers the inverse problem of recovering state-dependent source terms in a reaction–diffusion system from overposed data consisting of the values of the state variables either at a fixed finite time (census-type data) or a time trace of their values at a fixed point on the boundary of the spatial domain. We show both uniqueness results and the convergence of an iteration scheme designed

This paper is concerned with inverse acoustic scattering problem of inferring the position and shape of a sound-soft obstacle from phaseless far-field data. We propose the Bayesian approach to recover sound-soft disks, line cracks and kite-shaped obstacles through properly chosen incoming waves in two dimensions. Given the Gaussian prior measure, the well-posedness of the posterior measure in the Bayesian

We consider the two scale asymptotic expansion for a transmission problem modeling scattering by a bounded inhomogeneity with a periodic coefficient in the lower order term of the Helmholtz equation. The squared index of refraction is assumed to be a periodic function of the fast variable, specified over the unit cell with characteristic size ϵ . Since the convergence of the boundary correctors to

Recovering a function or high-dimensional parameter vector from indirect measurements is a central task in various scientific areas. Several methods for solving such inverse problems are well developed and well understood. Recently, novel algorithms using deep learning and neural networks for inverse problems appeared. While still in their infancy, these techniques show astonishing performance for

In this work, we propose a new criterion for choosing the regularization parameter in Tikhonov regularization when the noise is white Gaussian. The criterion minimizes a lower bound of the predictive risk, when both data norm and noise variance are known, and the parameter choice involves minimizing a function whose solution depends only on the signal-to-noise ratio. Moreover, when neither noise variance

The inverse scattering method is performed to the complex modified Korteweg–de Vries equation with zero boundary condition by developing an appropriate Riemann–Hilbert problem (RHP). We solve the RHP when reflection coefficient has multiple higher-order poles, and display the formulae of bound-state (BS) solitons and multiple BS solitons which correspond to one higher-order pole and multiple higher-order

We consider the problem of reconstructing the optical properties of a highly-scattering medium from coherent acousto-optic measurements. A method to solve the problem is proposed that is based on an inverse problem with internal data for a system of radiative transport equations.

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We consider the problem of estimating the density of buyers and vendors in a nonlinear parabolic price formation model using measurements of the price and the transaction rate. Our approach is based on a work by Puel (Puel J-P 2002 C. R. Acad. Sci., Paris 335 (2) 161–166), and results in an optimal control problem. We analyze this problems and provide stability estimates for the controls as well as

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In this paper, we consider the Navier–Stokes equation with memory term in a three-dimensional bounded domain. The equation is the so-called Oldroyd fluid model equation, which can describe the stress relaxation as well as the retardation of deformation due to the memory term. For this equation we considered the inverse problem for recovering the kernel of memory term in this model equation from the

Let ##IMG## [http://ej.iop.org/images/0266-5611/36/6/064002/ipab7d2cieqn1.gif] {$\mathcal{G}$} be a compact group and let ##IMG## [http://ej.iop.org/images/0266-5611/36/6/064002/ipab7d2cieqn2.gif] {${f}_{ij}\in C\left(\mathcal{G}\right)$} . We define the non-unique games (NUG) problem as finding ##IMG## [http://ej.iop.org/images/0266-5611/36/6/064002/ipab7d2cieqn3.gif] {${g}_{1},\dots ,\;{g}_{n}\in

In this work we consider a generalized bilevel optimization framework for solving inverse problems. We introduce fractional Laplacian as a regularizer to improve the reconstruction quality, and compare it with the total variation regularization. We emphasize that the key advantage of using fractional Laplacian as a regularizer is that it leads to a linear operator, as opposed to the total variation

In this paper, one dimensional conformable fractional Dirac-type integro differential system is considered. The asymptotic formulae for the solutions, eigenvalues and nodal points are obtained. We investigate the inverse nodal problem and give an effective procedure for solving the inverse nodal problem with respect to given a dense subset of nodal points.

This paper focuses on the data completion problem, which is well known to be an ill-posed inverse problem. We propose a dual regularization strategy without regularization parameter, based on the minimization of a functional which, instead of acting on the space of solutions, acts on the space of data. We prove the well-posedness of the minimization problem and the convergence of our regularized solution

Multi-energy computed tomography (ME-CT) is a medical imaging modality aiming to reconstruct the spatial density of materials from the attenuation properties of probing x-rays. For each line in two- or three-dimensional space, ME-CT measurements may be written as a nonlinear mapping from the integrals of the unknown densities of a finite number of materials along said line to an equal or larger number

We consider an inverse scattering problem for the Schrödinger operator in two dimensions. The aim of this work is to discuss some first numerical results on Saito’s formula. Saito’s formula is an explicit integral formula, which at the high-frequency limit gives a uniqueness result for the inverse scattering problem. The numeric approach is quite straight-forward: we take a large enough fixed wave

An analogue of the total variation prior for the normal vector field along the boundary of piecewise flat shapes in 3D is introduced. A major class of examples are triangulated surfaces as they occur for instance in finite element computations. The analysis of the functional is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere

In this paper we consider a final value problem for a diffusion equation with time-space fractional differentiation on a bounded domain D of ##IMG## [http://ej.iop.org/images/0266-5611/36/5/055011/ipab730dieqn1.gif] {${\mathbb{R}}^{k}$} , k ≥ 1, which includes the fractional power ##IMG## [http://ej.iop.org/images/0266-5611/36/5/055011/ipab730dieqn2.gif] {${\mathcal{L}}^{\beta }$} , 0 < β ≤ 1, of a

In this paper we propose a new approach for tomographic reconstruction with spatially varying regularization parameter. Our work is based on the SA-TV image restoration model proposed by Dong et al (2011 J. Math. Imag. Vis. 40 82–104) where an automated parameter selection rule for spatially varying parameters has been proposed. Their parameter selection rule, however, only applies if measured imaging

In this paper, we study ℓ 1 − αℓ 2 (0 < α ⩽ 1) minimization methods for signal and image reconstruction with impulsive noise removal. The data fitting term is based on ℓ 1 fidelity between the reconstruction output and the observational data, and the regularization term is based on ℓ 1 − αℓ 2 nonconvex minimization of the reconstruction output or its total variation. Theoretically, we show that under

We present a new technique, based on semivariogram methodology, for obtaining point estimates for use in prior modeling for solving Bayesian inverse problems. This method requires a connection between Gaussian processes with covariance operators defined by the Matérn covariance function and Gaussian processes with precision (inverse-covariance) operators defined by the Green’s functions of a class

An analogue of the total variation prior for the normal vector field along the boundary of smooth shapes in 3D is introduced. The analysis of the total variation of the normal vector field is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere manifold. It is shown that spheres are stationary points when the total variation

We consider an inverse time-dependent source problem governed by a distributed time-fractional diffusion equation using interior measurement data. Such a problem arises in some ultra-slow diffusion phenomena in many applied areas. Based on the regularity result of the solution to the direct problem, we establish the solvability of this inverse problem as well as the conditional stability in suitable

This work characterizes, analytically and numerically, two major effects of the quadratic Wasserstein ( W 2 ) distance as the measure of data discrepancy in computational solutions of inverse problems. First, we show, in the infinite-dimensional setup, that the W 2 distance has a smoothing effect on the inversion process, making it robust against high-frequency noise in the data but leading to a reduced

We discuss the determination of the Lamé parameters of an elastic material by the means of boundary measurements. We will combine previous results of Eskin–Ralston and Isakov to prove inverse results in the case of bounded domains with partial data. Moreover, we generalise these results to domains with cylindrical ends.

In this note we consider the stability of posterior measures occurring in Bayesian inference w.r.t. perturbations of the prior measure and the log-likelihood function. This extends the well-posedness analysis of Bayesian inverse problems. In particular, we prove a general local Lipschitz continuous dependence of the posterior on the prior and the log-likelihood w.r.t. various common distances of probability

In this paper we propose a new class of iterative regularization methods for solving ill-posed linear operator equations. The prototype of these iterative regularization methods is in the form of second order evolution equation with a linear vanishing damping term, which can be viewed not only as an extension of the asymptotical regularization, but also as a continuous analog of the Nesterov’s acceleration

Phase retrieval with prior information can be cast as a nonsmooth and nonconvex optimization problem. To decouple the signal and measurement variables, we introduce an auxiliary variable and reformulate it as an optimization with an equality constraint. We then solve the reformulated problem by graph projection splitting (GPS), where the two proximity subproblems and the graph projection step can be

One-bit compressed sensing aims to recover unknown sparse signals from extremely quantized linear measurements which just capture their signs. In this paper, we propose a nonconvex ℓ p (0 < p < 1) minimization model for one-bit compressed sensing problem and define the set of ℓ p effectively s -sparse signals which contains genuinely s -sparse signals. Utilizing properties of covering number, we show

Layer stripping is a method for solving inverse boundary value problems for elliptic PDEs, originally proposed in the literature for solving the Calderón problem of electrical impedance tomography (EIT), where the data consist of the Neumann-to-Dirichlet operator on the boundary. Defining a tangent–normal coordinate system near the boundary, the data are extended to a family of boundary operators on

We generate data-driven reduced order models (ROMs) for inversion of the one and two dimensional Schrödinger equation in the spectral domain given boundary data at a few frequencies. The ROM is the Galerkin projection of the Schrödinger operator onto the space spanned by solutions at these sample frequencies. The ROM matrix is in general full, and not good for extracting the potential. However, using

In many modern imaging applications the desire to reconstruct high resolution images, coupled with the abundance of data from acquisition using ultra-fast detectors, have led to new challenges in image reconstruction. A main challenge is that the resulting linear inverse problems are massive. The size of the forward model matrix exceeds the storage capabilities of computer memory, or the observational

Tikhonov regularization is a popular approach to obtain a meaningful solution for ill-conditioned linear least squares problems. A relatively simple way of choosing a good regularization parameter is given by Morozov’s discrepancy principle. However, most approaches require the solution of the Tikhonov problem for many different values of the regularization parameter, which is computationally demanding

We consider the inverse problem of recovering the spherically symmetric sound speed, density and attenuation in the Sun from the observations of the acoustic field randomly excited by turbulent convection. We show that observations at two heights above the photosphere and at two frequencies above the acoustic cutoff frequency uniquely determine the solar parameters. We also present numerical simulations

We propose and study a data completion algorithm for recovering missing data from the knowledge of Cauchy data on parts of the same boundary. The algorithm is based on surface representation of the solution and is presented for the Helmholtz equation. This work is an extension of the data completion algorithm proposed by the two last authors where the case of data available of a closed boundary was

We consider the reconstruction of a compactly supported source term in the constant-coefficient Helmholtz equation in R 3 , from the measurement of the outgoing solution at a source-enclosing sphere. The measurement is taken at a finite number of frequencies. We explicitly characterize certain finite-dimensional spaces of sources that can be stably reconstructed from such measurements. The characterization

Spectral computed tomography (CT) reconstructs the same scanned object from projections of multiple narrow energy windows, and it can be used for material identification and decomposition. However, the multi-energy projection dataset has a lower signal-noise-ratio (SNR), resulting in poor reconstructed image quality. To address this thorny problem, we develop a spectral CT reconstruction method, namely

Magnetic resonance fingerprinting (MRF) is a technique for quantitative estimation of spin- relaxation parameters from magnetic-resonance data. Most current MRF approaches assume that only one tissue is present in each voxel, which neglects intravoxel structure, and may lead to artifacts in the recovered parameter maps at boundaries between tissues. In this work, we propose a multicompartment MRF model

Broken ray transforms (BRTs) are typically considered to be reciprocal, meaning that the transform is independent of the direction in which a photon travels along a given broken ray. However, if the photon can change its energy (or be absorbed and re-radiated at a different frequency) at the vertex of the ray, then reciprocity is lost. In optics, non-reciprocal BRTs are applicable to imaging problems