Data Inversion Problem Research Articles

Inverse boundary value problem for the convection–diffusion equation with local data

We study a local data inverse problem for the time-dependent convection–diffusion equation in a bounded domain where a part of the boundary is treated to be inaccessible. Up on assuming the inaccessible part to be flat, we seek for the unique determination of the time-dependent convection and the density terms from the knowledge of the boundary data measured outside the inaccessible part. In the process, we show that there is a natural gauge in the perturbations, and we prove that this is the only obstruction in the uniqueness result.

Partial data inverse problems for magnetic Schrödinger operators on conformally transversally anisotropic manifolds

We study inverse boundary problems for the magnetic Schrödinger operator with Hölder continuous magnetic potentials and continuous electric potentials on a conformally transversally anisotropic Riemannian manifold of dimension n ⩾ 3 with connected boundary. A global uniqueness result is established for magnetic fields and electric potentials from the partial Cauchy data on the boundary of the manifold provided that the geodesic X-ray transform on the transversal manifold is injective.

Partial Data Inverse Problems for Magnetic Schrödinger Operators with Potentials of Low Regularity

Partial Data Inverse Problems for Magnetic Schrödinger Operators with Potentials of Low Regularity

Local data inverse problem for the polyharmonic operator with anisotropic perturbations

In this article, we study an inverse problem with local data for a linear polyharmonic operator with several lower order tensorial perturbations. We consider our domain to have an inaccessible portion of the boundary where neither the input can be prescribed nor the output can be measured. We prove the unique determination of all the tensorial coefficients of the operator from the knowledge of the Dirichlet and Neumann map on the accessible part of the boundary, under suitable geometric assumptions on the domain.

Deep Neural Network-Oriented Indicator Method for Inverse Scattering Problems Using Partial Data

We consider the inverse scattering problem to reconstruct an obstacle using partial far-field data due to one incident wave. A simple indicator function, which is negative inside the obstacle and positive outside of it, is constructed and then learned using a deep neural network (DNN). The method is easy to implement and effective as demonstrated by numerical examples. Rather than developing sophisticated network structures for the classical inverse operators, we reformulate the inverse problem as a suitable operator such that standard DNNs can learn it well. The idea of the DNN-oriented indicator method can be generalized to treat other partial data inverse problems.

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.

Fractional Calderón problems and Poincaré inequalities on unbounded domains

We generalize many recent uniqueness results on the fractional Calderón problem to cover the cases of all domains with nonempty exterior. The highlight of our work is the characterization of uniqueness and nonuniqueness of partial data inverse problems for the fractional conductivity equation on domains that are bounded in one direction for conductivities supported in the whole Euclidean space and decaying to a constant background conductivity at infinity. We generalize the uniqueness proof for the fractional Calderón problem by Ghosh, Salo and Uhlmann to a general abstract setting in order to use the full strength of their argument. This allows us to observe that there are also uniqueness results for many inverse problems for higher order local perturbations of a lower order fractional Laplacian. We give concrete example models to illustrate these curious situations and prove Poincaré inequalities for the fractional Laplacians of any order on domains that are bounded in one direction. We establish Runge approximation results in these general settings, improve regularity assumptions also in the cases of bounded sets and prove general exterior determination results. Counterexamples to uniqueness in the inverse fractional conductivity problem with partial data are constructed in another companion work.

Generalized statistics: Applications to data inverse problems with outlier-resistance.

The conventional approach to data-driven inversion framework is based on Gaussian statistics that presents serious difficulties, especially in the presence of outliers in the measurements. In this work, we present maximum likelihood estimators associated with generalized Gaussian distributions in the context of Rényi, Tsallis and Kaniadakis statistics. In this regard, we analytically analyze the outlier-resistance of each proposal through the so-called influence function. In this way, we formulate inverse problems by constructing objective functions linked to the maximum likelihood estimators. To demonstrate the robustness of the generalized methodologies, we consider an important geophysical inverse problem with high noisy data with spikes. The results reveal that the best data inversion performance occurs when the entropic index from each generalized statistic is associated with objective functions proportional to the inverse of the error amplitude. We argue that in such a limit the three approaches are resistant to outliers and are also equivalent, which suggests a lower computational cost for the inversion process due to the reduction of numerical simulations to be performed and the fast convergence of the optimization process.

Partial data inverse problems for nonlinear magnetic Schrödinger equations

Partial data inverse problems for nonlinear magnetic Schrödinger equations

A neural network-based approach for bending analysis of strain gradient nanoplates

Purpose of this paper is the presentation of a novel Machine Learning (ML) technique for nanoscopic study of thin nanoplates. The second-order strain gradient theory is used to derive the governing equations and account for size effects. The ML framework is based on Physics-Informed Neural Networks (PINNs), a new concept of Artificial Neural Networks (ANNs) enriched with the mathematical model of the problem. Training of PINNs is performed using a highly efficient learning algorithm, known as Extreme Learning Machine (ELM). Two applications of this ANNs-based method are illustrated: solution of the Partial Differential Equations (PDEs) modeling the flexural response of thin nanoplates (direct problem), and identification of the length scale parameter of the nanoplate mathematical model with the aid of measurement data (inverse problem). Comparison with analytical and Finite Element (FE) solutions demonstrate the accuracy and efficiency of this ML framework as meshfree solver of high-order PDEs. The stability and reliability of the present method are verified through parameter studies on hyperparameters, network architectures, data noise and training initializations. The results presented give evidence of the effectiveness and robustness of this new ML approach for solving both direct and inverse nanoplate problems.

Bayesian Inversion of Time Domain Electromagnetic (TDEM) Synthetic Data using Langevin Monte Carlo (LMC)

The inversion of time domain electromagnetic (TDEM) data is an ill-posed problem. This makes the standard inversion procedure, which is to linearize the related objective function. Then take a deterministic approach to determine a solution that can minimize the said objective function, which has the potential to be trapped in a local minimum. In this study, we solved the problem of TDEM data inversion using the Bayesian framework by generating a sample from the posterior distribution. The posterior distribution contains information related to the uncertainty of the TDEM data and the results of the forward modeling formulation, and prior information about the subsurface parameter model. In conducting the sampling process, we use the Langevin Monte Carlo (LMC) algorithm, one of many gradient-based Markov chain Monte Carlo (MCMC) sampling algorithms. Bayesian inversion was performed on synthetic data generated through the forward modeling of several test models. We used a model with varying thickness and resistivity values for inversion. We also added a prior probability distribution related to the smoothness constraint between resistivity values in adjacent layers allowing sharp and smooth transitions.

Embedded parallel Fourier ptychographic microscopy reconstruction system.

Fourier ptychographic microscopy (FPM) has attracted a wide range of focus for its ability of large space-bandwidth product and quantitative phase imaging. It is a typical computational imaging technique that jointly optimizes imaging hardware and reconstruction algorithms. The data redundancy and inverse problem algorithms are the sources of FPM's excellent performance. But at the same time, this large amount of data processing and complex algorithms also evidently reduce the imaging speed. To accelerate the FPM reconstruction speed, we proposed a fast FPM reconstruction framework consisting of three levels of parallel computation and implemented it with an embedded computing module. In the conventional FPM framework, the sample image is divided into multiple sub-regions to process separately because the illumination angles and defocus distances for different sub-regions may also be different. Our parallel framework first performs digital refocusing and high-resolution reconstruction for each sub-region separately and then stitches the complex sub-regions together to obtain the final high-resolution complex image. The feasibility of the proposed parallel FPM reconstruction framework is verified with different experimental results acquired with the system we built.

An outlier-resistant [formula omitted]-generalized approach for robust physical parameter estimation

In this work we propose a robust methodology to mitigate the undesirable effects caused by outliers to generate reliable physical models. In this way, we formulate the inverse problems theory in the context of Kaniadakis statistical mechanics (or κ-statistics), in which the classical approach is a particular case. In this regard, the errors are assumed to be distributed according to a finite-variance κ-generalized Gaussian distribution. Based on the probabilistic maximum-likelihood method we derive a κ-objective function associated with the finite-variance κ-Gaussian distribution. To demonstrate our proposal’s outlier-resistance, we analyze the robustness properties of the κ-objective function with help of the so-called influence function. In this regard, we discuss the role of the entropic index (κ) associated with the Kaniadakis κ-entropy in the effectiveness in inferring physical parameters by using strongly noisy data. In this way, we consider a classical geophysical data-inverse problem in two realistic circumstances, in which the first one refers to study the sensibility of our proposal to uncertainties in the input parameters, and the second is devoted to the inversion of a seismic data set contaminated by outliers. The results reveal an optimum κ-value at the limit κ→2/3, which is related to the best results.

Uniqueness and stability analysis of final data inverse source problems for evolution equations

This article proposes a unified approach to the issues of uniqueness and Lipschitz stability for the final data inverse source problems of determining the unknown spatial load F ⁢ ( x ) {F(x)} in the evolution equations. The approach is based on integral identities outlined here for the one-dimensional and multidimensional heat equations ρ ⁢ ( x ) ⁢ u t - ( k ⁢ ( x ) ⁢ u x ) x = F ⁢ ( x ) ⁢ G ⁢ ( t ) , ( x , t ) ∈ ( 0 , ℓ ) × ( 0 , T ] , \rho(x)u_{t}-(k(x)u_{x})_{x}=F(x)G(t),\quad(x,t)\in(0,\ell)\times(0,T], and ρ ⁢ ( x ) ⁢ u t - div ⁡ ( k ⁢ ( x ) ⁢ ∇ ⁡ u ) = F ⁢ ( x ) ⁢ G ⁢ ( x , t ) , ( x , t ) ∈ Ω × ( 0 , T ] , Ω ∈ ℝ n , \rho(x)u_{t}-\operatorname{div}(k(x)\nabla u)=F(x)G(x,t),\quad(x,t)\in\Omega% \times(0,T],\,\Omega\in\mathbb{R}^{n}, for the damped wave, and the Euler–Bernoulli beam and Kirchhoff plate equations ρ ⁢ ( x ) ⁢ u t ⁢ t + μ ⁢ ( x ) ⁢ u t - ( r ⁢ ( x ) ⁢ u x ) x = F ⁢ ( x ) ⁢ G ⁢ ( x , t ) , ρ ⁢ ( x ) ⁢ u t ⁢ t + μ ⁢ ( x ) ⁢ u t + ( r ⁢ ( x ) ⁢ u x ⁢ x ) x ⁢ x = F ⁢ ( x ) ⁢ G ⁢ ( t ) , \rho(x)u_{tt}+\mu(x)u_{t}-(r(x)u_{x})_{x}=F(x)G(x,t),\quad\rho(x)u_{tt}+\mu(x)% u_{t}+(r(x)u_{xx})_{xx}=F(x)G(t), for ( x , t ) ∈ ( 0 , ℓ ) × ( 0 , T ] {(x,t)\in(0,\ell)\times(0,T]} , and ρ ⁢ ( x ) ⁢ h ⁢ ( x ) ⁢ u t ⁢ t + μ ⁢ ( x ) ⁢ u t + ( D ⁢ ( x ) ⁢ ( u x 1 , x 1 + ν ⁢ u x 2 , x 2 ) ) x 1 , x 1 + ( D ⁢ ( x ) ⁢ ( u x 2 , x 2 + ν ⁢ u x 1 , x 1 ) ) x 2 , x 2 + 2 ⁢ ( 1 - ν ) ⁢ ( D ⁢ ( x ) ⁢ u x 1 , x 2 ) x 1 , x 2 \displaystyle\rho(x)h(x)u_{tt}+\mu(x)u_{t}+(D(x)(u_{x_{1},x_{1}}+\nu u_{x_{2},% x_{2}}))_{x_{1},x_{1}}+(D(x)(u_{x_{2},x_{2}}+\nu u_{x_{1},x_{1}}))_{x_{2},x_{2% }}+2(1-\nu)(D(x)u_{x_{1},x_{2}})_{x_{1},x_{2}} = F ⁢ ( x ) ⁢ G ⁢ ( t ) , ( x , t ) ∈ Ω T := Ω × ( 0 , T ) , Ω := ( 0 , ℓ 1 ) × ( 0 , ℓ 2 ) , \displaystyle =F(x)G(t),\quad(x,t)\in\Omega_{T}:=\Omega\times(0,T),\,% \Omega:=(0,\ell_{1})\times(0,\ell_{2}), and allows us to prove the uniqueness and stability of the solutions for all considered inverse problems under the same conditions imposed on the load G ⁢ ( t ) {G(t)} or G ⁢ ( x , t ) {G(x,t)} .

Partial data inverse problems for quasilinear conductivity equations

We show that the knowledge of the Dirichlet-to-Neumann maps given on an arbitrary open non-empty portion of the boundary of a smooth domain in $${\mathbb {R}}^n$$ , $$n\ge 2$$ , for classes of semilinear and quasilinear conductivity equations, determines the nonlinear conductivities uniquely. The main ingredient in the proof is a certain $$L^1$$ -density result involving sums of products of gradients of harmonic functions which vanish on a closed proper subset of the boundary.

Runge approximation and stability improvement for a partial data Calderón problem for the acoustic Helmholtz equation

<p style='text-indent:20px;'>In this article, we discuss quantitative Runge approximation properties for the acoustic Helmholtz equation and prove stability improvement results in the high frequency limit for an associated partial data inverse problem modelled on [<xref ref-type="bibr" rid="b3">3</xref>,<xref ref-type="bibr" rid="b35">35</xref>]. The results rely on quantitative unique continuation estimates in suitable function spaces with explicit frequency dependence. We contrast the frequency dependence of interior Runge approximation results from non-convex and convex sets.</p>

High‐frequency stability estimates for a partial data inverse problem

In this article, high‐frequency stability estimates for the determination of the potential in the Schrödinger equation are studied when the boundary measurements are made on slightly more than half the boundary. The estimates reflect the increasing stability property with growing frequency.

Improving Seismic Inversion Robustness via Deformed Jackson Gaussian.

The seismic data inversion from observations contaminated by spurious measures (outliers) remains a significant challenge for the industrial and scientific communities. This difficulty is due to slow processing work to mitigate the influence of the outliers. In this work, we introduce a robust formulation to mitigate the influence of spurious measurements in the seismic inversion process. In this regard, we put forth an outlier-resistant seismic inversion methodology for model estimation based on the deformed Jackson Gaussian distribution. To demonstrate the effectiveness of our proposal, we investigated a classic geophysical data-inverse problem in three different scenarios: (i) in the first one, we analyzed the sensitivity of the seismic inversion to incorrect seismic sources; (ii) in the second one, we considered a dataset polluted by Gaussian errors with different noise intensities; and (iii) in the last one we considered a dataset contaminated by many outliers. The results reveal that the deformed Jackson Gaussian outperforms the classical approach, which is based on the standard Gaussian distribution.

An extended sampling-ensemble Kalman filter approach for partial data inverse elastic problems

Inverse problems are more challenging when only partial data are available in general. In this paper, we propose a two-step approach combining the extended sampling method and the ensemble Kalman filter to reconstruct an elastic rigid obstacle using partial data. In the first step, the approximate location of the unknown obstacle is obtained by the extended sampling method. In the second step, the ensemble Kalman filter is employed to reconstruct the shape. The location obtained in the first step guides the construction of the initial particles of the ensemble Kalman filter, which is critical to the performance of the second step. Both steps are based on the same physical model and use the same scattering data. Numerical examples are shown to demonstrate the effectiveness of the proposed method.

Decentralized and parallel primal and dual accelerated methods for stochastic convex programming problems

Abstract We introduce primal and dual stochastic gradient oracle methods for decentralized convex optimization problems. Both for primal and dual oracles, the proposed methods are optimal in terms of the number of communication steps. However, for all classes of the objective, the optimality in terms of the number of oracle calls per node takes place only up to a logarithmic factor and the notion of smoothness. By using mini-batching technique, we show that the proposed methods with stochastic oracle can be additionally parallelized at each node. The considered algorithms can be applied to many data science problems and inverse problems.