Three-dimensional Inverse Problem Research Articles

An optimal transport approach for 3D electrical impedance tomography

Abstract This work solves the three-dimensional inverse boundary value problem with the quadratic Wasserstein distance (W 2), which originates from the optimal transportation (OT) theory. The computation of the W 2 distance on the manifold surface is boiled down to solving the generalized Monge–Ampère equation, whose solution is directly related to the gradient of the W 2 distance. An efficient first-order method based on iteratively solving Poisson’s equation is introduced to solve the fully nonlinear elliptic equation. Combining with the adjoint-state technique, the optimization framework based on the W 2 distance is developed to solve the three-dimensional electrical impedance tomography problem. The proposed method is especially suitable for severely ill-posed and highly nonlinear inverse problems. Numerical experiments demonstrate that our method improves the stability and outperforms the traditional regularization methods.

New Estimates of the Potential Schrödinger Equation

The Schrödinger equation, a fundamental construct in quantum mechanics, plays a pivotal role in understanding the properties and behaviors of quantum systems across various scientific fields, including but not limited to physics, economics, and geophysics. This research paper delves into the exploration of novel invariants associated with the Schrödinger equation, uncovering previously unrecognized symmetries and properties that open new pathways for theoretical and applied scientific exploration. The investigation reveals that the energy of bound states remains invariant under transformations of the coordinate system, highlighting a universal symmetry embedded within the equation. This discovery, coupled with an analysis of the variation in scattering amplitudes’ phase with different coordinate systems, not only enriches the theoretical framework of quantum mechanics but also has significant practical implications across various domains such as fluid dynamics, material science, and medical imaging. Furthermore, by applying the principles of the Poincaré-Riemann-Hilbert boundary-value problem, the study offers a novel approach to estimate potentials within the Schrödinger equation, advancing the three-dimensional inverse problem of quantum scattering theory. This methodological innovation allows for a more profound understanding of the unitary scattering operator, facilitating enhanced modeling of complex systems and phenomena. The implications of these findings extend beyond theoretical physics, offering transformative insights and applications in areas ranging from fluid dynamics, where it aids in refining models for the Navier-Stokes equations, to seismic exploration, tomography, and ultrasound imaging, where it enhances phase selection and interpretation of measurement results. In essence, this research not only contributes to a deeper understanding of the Schrödinger equation’s fundamental properties but also paves the way for interdisciplinary advancements, demonstrating the profound impact of theoretical discoveries on practical applications in diverse scientific and technological domains.

Solution of the Vector Three-Dimensional Inverse Problem on an Inhomogeneous Dielectric Hemisphere Using a Two-Step Method

This work is devoted to the development and implementation of a two-step method for solving the vector three-dimensional inverse diffraction problem on an inhomogeneous dielectric scatterer having the form of a hemisphere characterized by piecewise constant permittivity. The original boundary value problem for Maxwell’s equations is reduced to a system of integro-differential equations. An integral formulation of the vector inverse diffraction problem is proposed and the uniqueness of the solution of the first-kind integro-differential equation in special function classes is established. A two-step method for solving the vector inverse diffraction problem on the hemisphere is developed. Unlike traditional approaches, the two-step method for solving the inverse problem is non-iterative and does not require knowledge of the exact initial approximation. Consequently, there are no issues related to the convergence of the numerical method. The results of calculations of approximate solutions to the inverse problem are presented. It is shown that the two-step method is an efficient approach to solving vector problems in near-field tomography.

Fast Algorithm for Solving Some Three-Dimensional Inverse Problems of Magnetometry

Fast Algorithm for Solving Some Three-Dimensional Inverse Problems of Magnetometry

A fast Bayesian parallel solution framework for large-scale parameter estimation of 3D inverse heat transfer problems

The inverse heat transfer problems (IHTP) have a wide range of applications in the engineering field. Bayesian methods using Markov Chain Monte Carlo (MCMC) have long been considered as a robust and effective method for solving inverse problems. However, the discretization of the problem domain by the spatio-temporal Galerkin skill, i.e., the finite element interpolation also includes the time dimension, making the scale of the unknown parameters extremely difficult for Bayesian calculations. In this paper, a fast Bayesian parallel sampling (FBPS) framework is proposed for large-scale parameter estimation of benchmark three-dimensional inverse heat transfer problems (3D-IHTP). The FBPS we developed achieves a parameter computation scale of 105 magnitude within minutes, through dimensionality reduction of the space-time dependent problem domain. The Hamiltonian Monte Carlo (HMC) sampler, which is proven to be more efficient for high-dimensional parameter estimation, is employed. Through several simulation tests of IHTP, it was confirmed that the solving efficiency of FBPS surpasses that of the traditional Bayesian strategy significantly. Finally, FBPS is successfully developed to estimate the unknown heat flux on the chip heat sink and pack interface effectively, given some simulated high resolution measurement data. The reliability and efficiency show that FBPS has the potential to support efficient prediction techniques for a class of IHTPs in engineering applications.

Structural Tectonic Scheme Creation Based on Seismic-Gravity Models and Isostasy Usage: Ural Case

Process of Earth’s density models creation leads to the solution of direct and inverse gravimetry problems. The inverse problem of gravimetry is a classic example of an ill-posed problem: in the common statement, its solution is not unique and unstably depends on input data. Therefore, it is necessary to determine solutions belonging substantial sets of correctness, choosing reasonable models of an initial approximation. In this paper the application of complex interpretation methods of seismic and gravitational data for the creation of three-dimensional models of crust and the upper mantle are presented. Original algorithms and programs were developed for implementation of these methods. They contain solution of non-linear (structural) inverse problem and the solution of the linear three-dimensional inverse problem taking note of the side sources. Coefficients of the “density-velocity” correlation formulas for a number of geo-traverses were defined. Also, we suggest a technique of tectonic maps construction, which is based on the lithostatic pressure calculation. Its idea can be applied to both two- and three-dimensional cases. In the 2D case we show the way to split the mantle to blocks with vertical boundaries. If lithostatic compensation hypothesis is adopted, the method also allows one to calculate density value for each block. Such separation of the mantle helps to diminish discrepancy between model and observed fields. In 3D case we suggest a method, which can be used to construct tectonic structure maps with information about approximate depth and height of each tectonic block.

"Fast" algorithm for solving some three-dimensional inverse problems of magnetometry

Рассматриваются типичные трехмерные обратные задачи магниторазведки, а именно: определения векторной плотности магнитных диполей в исследуемой области земной коры по измеряемым на поверхности компонентам вектора (и/или тензора градиента) магнитной индукции. Эти задачи, являясь, как правило, некорректно поставленными, могут быть решены стандартными методами регуляризации. Однако при этом для такого решения на достаточно подробных сетках за время порядка нескольких минут требуются значительные вычислительные ресурсы (вычислительные кластеры, суперкомпьютеры и т.д.). В статье предлагается новый «быстрый» регуляризующий алгоритм решения таких трехмерных задач, позволяющий получать их приближенное решение на персональном компьютере средней производительности за десятки секунд или несколько минут. Кроме того, используемый подход позволяет за сравнимое время вычислить и апостериорную оценку точности найденного решения, и это дает возможность оценить качество решения при интерпретации результатов. Алгоритмы решения обратной задачи и апостериорной оценки точности решения апробируются при решении модельных обратных задач и используются при обработке экспериментальных данных.

On the features of inhomogeneous residual stress identification using the digital speckle interferometry and the hole-drilling method

A phenomenological approach to the actual problem of determining the inhomogeneous residual stress-strain state in the components of high-tech engineering systems at the stages of their design and operation is presented. The approach is based on physical and mechanical methods of measuring displacements. Current physical models describe the physical regularities of the residual states attributed to changes in the structure by the interaction of defects and dislocations in the field of micro- and meso-stresses. At the same time, there are the problems of the transition to the macrolevel, the construction of multilevel models, and the conversion of these models in engineering practice. In the framework of phenomenological approaches, in the general case, the solution of this problem requires the solution of three-dimensional inverse problems of thermoelasticity. A well-known mechanical method for determining a uniform field of residual elastic stresses recommended by ASTM E837 is described. The method proposed earlier by one of the authors for determining an inhomogeneous (in the plane) field of residual elastic stresses is discussed. A method of the three-dimensional inhomogeneous residual elastic stress-strain state determination based on the experimental determination of the displacement vector components by the method of step-by-step point hole-drilling and data of digital speckle interferometry and digital image correlation is developed. The constitutive relations for the components of the displacement vector are written in the form of Volterra integral operators. The basic operator functions are the functions of four variables, i.e., the coordinates of the cylindrical system (r, θ, z) associated with the hole, and the hole depth h. A method for verification of the basic functions is presented. The problem is reduced to the determination of three displacement functions of three variables: hole radius r, h, and z. Numerical simulation of basic functions is carried out. The obtained results are consistent with the known experimental data and calculated values of the deformation on the surface depending on the depth of the hole according to the ASTM E837 Standard.

Physics-informed UNets for discovering hidden elasticity in heterogeneous materials

Soft biological tissues often have complex mechanical properties due to variation in structural components. In this paper, we develop a novel UNet-based neural network model for inversion in elasticity (El-UNet) to infer the spatial distributions of mechanical parameters from strain maps as input images, normal stress boundary conditions, and domain physics information. We show superior performance – both in terms of accuracy and computational cost – by El-UNet compared to fully-connected physics-informed neural networks in estimating unknown parameters and stress distributions for isotropic linear elasticity. We characterize different variations of El-UNet and propose a self-adaptive spatial loss weighting approach. To validate our inversion models, we performed various finite-element simulations of isotropic domains with heterogenous distributions of material parameters to generate synthetic data. El-UNet is faster and more accurate than the fully-connected physics-informed implementation in resolving the distribution of unknown fields. Among the tested models, the self-adaptive spatially weighted models had the most accurate reconstructions in equal computation times. The learned spatial weighting distribution visibly corresponded to regions that the unweighted models were resolving inaccurately. Our work demonstrates a computationally efficient inversion algorithm for elasticity imaging using convolutional neural networks and presents a potential fast framework for three-dimensional inverse elasticity problems that have proven unachievable through previously proposed methods.

A novel method for simultaneous determination of thermophysical properties and boundary conditions of phase change problems based on element differential method

Thermophysical properties and boundary conditions for phase change thermal management systems are challenging to be accurately determined, due to the phase change heat transfer phenomenon and complex working conditions. In this work, the element differential method (EDM) and a gradient-based method are combined to simultaneously predict thermal conductivity, mass specific heat, and boundary heat flux in two-dimensional (2D) and three-dimensional (3D) inverse phase change problems, for the first time. The multi-parameter identification for the 3D physical model with phase change is more general than the previous attempts. Moreover, the effective heat capacity method is employed to deal with phase change problems, to improve efficiency. The sensitivity coefficient is accurately determined by the complex-variable-differentiation method (CVDM) in the multi-parameter prediction. Finally, the effect of measurement points, measurement errors, and initial guessed values on the multi-parameter identification are investigated. This study demonstrates that the present method has good accuracy, efficiency, stability, and robustness in dealing with transient nonlinear inverse problems during phase change process.

An experimental inverse problem in determining spatially varying surface heat flux using measured temperature readings

An experimental investigation to estimate the unknown spatially dependent surface heat flux of a flat plate based on the surface temperature readings taken by a thermography camera and the conjugate gradient method (CGM) is examined in this study. In the present three-dimensional inverse problem, the functional form of the heat flux is considered to be initially unknown; thus, the functional form of the heat flux is classified as the category of function estimation. In addition, air velocity is specified as the working condition, not the heat transfer coefficient; therefore, it is defined as an inverse heat conduction-convection conjugated problem (IHCCCP). The experimental study for estimating the unknown spatially dependent surface heat flux has not been examined previously in the IHCCCP. The estimations of the inverse solutions are conducted experimentally with two designed plate heaters and various inlet air velocities and plate thicknesses, and the total heat losses are found to be approximately 5% of the original design heating power. The experimental results indicated that for a uniform heating design (Design A), the estimated heat fluxes are always very accurate if the heat losses are deducted from the design power. For the nonuniform heating design (Design B), due to the existence of two discontinuities in the heating power, the estimated relative errors for heat fluxes are found to be between 16% and 18% using 6 to 8 iterations, reflecting the nature of the inverse problem, i.e., a small measurement error will amplify the error of the estimates; in addition, the estimated heat flux becomes less accurate as the plate thickness increases. Finally, numerical simulations with errorless measurements are performed, and the results reveal that the estimated errors for heat fluxes can be decreased to approximately 8% with 150 iterations for Design B.

Parametric level-set inverse problems with stochastic background estimation

We study parametric shape reconstruction inverse problems in which the object of interest is embedded in a heterogeneous background medium that is known only approximately. We model the background medium as a Gaussian random field and pose shape reconstruction as a stochastic programming problem in which we seek to minimize the expected value, with respect to the background field, of a stochastic objective function. We develop a computationally efficient algorithm based on the sample average approximation that reduces the effect of uncertainty in the background medium on shape recovery. We demonstrate that by using accelerated stochastic gradient descent, we can apply our method to large-scale problems. The capabilities of our method are demonstrated on a simple two-dimensional model problem and in a more demanding application to a three-dimensional inverse conductivity problem in geophysical imaging.

“Fast” Solution of the Three-Dimensional Inverse Problem of Quasi-Static Elastography with the Help of the Small Parameter Method

“Fast” Solution of the Three-Dimensional Inverse Problem of Quasi-Static Elastography with the Help of the Small Parameter Method

Weighted Radial Basis Collocation Method for the Nonlinear Inverse Helmholtz Problems

In this paper, a meshfree weighted radial basis collocation method associated with the Newton’s iteration method is introduced to solve the nonlinear inverse Helmholtz problems for identifying the parameter. All the measurement data can be included in the least-squares solution, which can avoid the iteration calculations for comparing the solutions with part of the measurement data in the Galerkin-based methods. Appropriate weights are imposed on the boundary conditions and measurement conditions to balance the errors, which leads to the high accuracy and optimal convergence for solving the inverse problems. Moreover, it is quite easy to extend the solution process of the one-dimensional inverse problem to high-dimensional inverse problem. Nonlinear numerical examples include one-, two- and three-dimensional inverse Helmholtz problems of constant and varying parameter identification in regular and irregular domains and show the high accuracy and exponential convergence of the presented method.

3D inversion of 2D electrical resistivity data for geotechnical analysis

Geophysical methods are effective tools for geotechnical analysis. In particular, the two-dimensional electrical resistivity method is widely applied in the determination of broken, cracked, and karst cave structures in many countries around the world. The information given by this method plays an important role in preventing and mitigating the risk of geological hazards caused by geological structures. Geotechnical analysis is also an important step in civil construction. However, the interpretation of two-dimensional problems with individual sections still has certain limitations, such as the delineation of the spatial distribution of interesting objects in the entire survey area. We propose a new data processing procedure for the two-dimensional electrical resistivity methods based on rearranging the data as the input of three-dimensional inversion. If the distribution of survey lines is dense enough, we can use two-dimensional data to solve the three-dimensional inverse problems. This inversion will give us a diagram of resistivity distribution in three-dimensional space, which is intuitive and high detail in identifying interesting objects. In this study, the authors have solved the three-dimensional inverse problems from the two-dimensional electrical resistivity survey carried out in the Hoa Lac area, west of Hanoi city. The result shows that the resistivity distribution in three-dimensional space not only increases the intuition but also the accuracy in reflecting the spatial distribution of hidden underground objects. This also defines the localization and determination of fracture zones in the study area as more reliable than explained in each section according to the results of solving two-dimensional inverse problems. The effectiveness of this new method has been proven when compared with drilling results in the studied area.

Fast Solution Algorithm for a Three-Dimensional Inverse Multifrequency Problem of Scalar Acoustics with Data in a Cylindrical Domain

A new algorithm for the stable solution of a three-dimensional scalar inverse problem of acoustic sounding of an inhomogeneous medium in a cylindrical region is proposed. The data of the problem is the complex amplitude of the wave field, measured outside the region of acoustic inhomogeneities in a cylindrical layer. Using the Fourier transform and Fourier series, the inverse problem is reduced to solving a set of one-dimensional Fredholm integral equations of the first kind, to the subsequent calculation of the complex amplitude of the wave field in the region of inhomogeneity, and then to finding the required sound velocity field in this region. The algorithm allows solving the inverse problem on a personal computer of average performance for sufficiently fine three-dimensional grids in tens of seconds. A numerical study of the accuracy of the proposed algorithm for solving model inverse problems at various frequencies is carried out, and the issues of stability of the algorithm with respect to data perturbations are investigated.

Applying three-dimensional inversion to the frequency-domain response converted from transient electromagnetic data for a rectangular fixed loop

The computationally expensive cost and complexity are practical challenges in solving the three-dimensional (3-D) inverse problem in the time domain, and many real-world surveys are still inverted in one dimension (1-D). We present a new way to invert transient electromagnetic (TEM) data from a rectangular fixed loop to image subsurface electrical conductivity, which overcomes this limitation and permits efficient implementation of the inversion process on a personal computer. A TEM signal was transformed to the real and imaginary components of a frequency-domain (FD) electromagnetic (EM) response. The transformation was performed by a regularization inversion method. The recovered FD EM response was used as the data for the 3-D inversion. The forward problem and sensitivity were solved by using a staggered-grid finite-difference technique in the FD, and the background Green's function was carried out using the virtual interface method. The non-linear conjugate gradient method (NLCG) was employed to solve the 3-D inverse problem. A synthetic example revealed the basic functionality of the approach, and the FD inversion of a TEM field dataset acquired in the northeastern Qinshui Basin, Shanxi Province, China demonstrated a real-world application. Our inversion approach is general and suitable for different-type sources, such as a vertical magnetic dipole or horizontal electric dipole.

A Nonlinear Inverse Design Problem for a Pipe Type Heat Exchanger Equipped with Internal Z-Shape Lateral Fins and Ribs

A non-linear three-dimensional inverse shape design problem was investigated for a pipe type heat exchanger to estimate the design variables of continuous lateral ribs on internal Z-shape lateral fins for maximum thermal performance factor η. The design variables were considered as the positions, heights, and number of ribs while the physical properties of air were considered as a polynomial function of temperature; this makes the problem non-linear. The direct problem was solved using software package CFD-ACE+, and the Levenberg–Marquardt method (LMM) was utilized as the optimization tool because it has been proven to be a powerful algorithm for solving inverse problems. Z-shape lateral fins were found to be the best thermal performance among Z-shape, S-shape, and V-shape lateral fins. The objective of this study was to include continuous lateral ribs to Z-shape lateral fins to further improve η. Firstly, the numerical solutions of direct problem were solved using both polynomial and constant air properties and then compared with the corrected solutions to verify the necessity for using polynomial air properties. Then, four design cases, A, B, C and D, based on various design variables were conducted numerically, and the resultant η values were computed and compared. The results revealed that considering continuous lateral ribs on the surface of Z-shape lateral fins can indeed improve η value at the design working condition Re = 5000. η values of designs A, B and C were approximately 13% higher than that for Z-shape lateral fins, however, when the rib numbers were increased, i.e., design D, the value of η became only 11.5 % higher. This implies that more ribs will not guarantee higher η value.

An Improved Subspace-Regularized DBIM-MLGFIM Method for Three-Dimensional Inverse Scattering Problems

The repetitive computation of the forward problem in distorted Born iterative method (DBIM)-type inversion method is time-consuming and memory-costly, especially in dealing with 3-D inverse scattering problem. An improved subspace-regularized DBIM is proposed for the efficient solution of the 3-D inverse problem, which introduces the multilevel Green’s function interpolation method (MLGFIM) to accelerate the computation of the forward problem. In order to stabilize the inversion, an improved subspace-regularized DBIM is used with one new subspace-regularized induced current term in the cost function in virtue of truncated singular value decomposition (TSVD). To further reduce the computation burden, the incident field induced with updated inhomogeneous background medium is used approximately as the total field in the cost function, which saves one forward solver calculation in each iteration. Besides, the resource-intensive multiplication of inhomogeneous Green’s function can be avoided, which further improves the efficiency. Three numerical and two Fresnel experimental examples are given to demonstrate both the efficiency of the forward solver and the accuracy of the inversion algorithm.

Solution of the three-dimensional inverse elastography problem for parametric classes of inclusions

We study the three-dimensional inverse problem of elastography, that is finding the Young's modulus of a biological tissue from known values of its vertical displacements. In this way, one can find inclusions with Young's modulus several times higher than its known background value. Such inclusions are interpreted as tumours. A quasistatic statement of the problem is used in which the fragment of the tissue is considered as a linearly elastic body. It is assumed that the geometry of inclusions is specified parametrically, and the Young's modulus inside and outside the inclusions is constant. The task is reduced to finding the number of inclusions, parameters defining their shape and the Young's modulus inside the inclusions. To solve the problem, a special algorithm is proposed and justified. The results of numerical experiments on solving a three-dimensional model problems are presented. A comparison is made of the solutions to the inverse problem in the three-dimensional domain and two-dimensional inverse problems in selected cross-sections of this domain. It is found that 2D inverse problems do not always allow one to find a true 3D solution. For one of solved inverse problems, a-posteriori error estimates for Young modulus and geometric parameters of inclusions are obtained.