Solution Of Inverse Problem Research Articles

Coherence-enhancing anisotropic diffusion filter for 3D high-resolution reconstruction of P-wave velocity and density using full-waveform inversion: Application to a North Sea ocean bottom cable data set

Regularization is a central topic in the study of the solutions of ill-posed inverse problems. High-resolution seismic imaging using full-waveform inversion (FWI) belongs to this category of problems. Regularization through anisotropic diffusion, a technique that emerged in the field of image processing, is an interesting alternative to conventional regularization strategies. Exploiting the structural information of a given image, it has the capability to smooth this image along its main structures. The main difficulty is how to design the anisotropic diffusion operator. The concept of coherence enhancing proposed in 2D is extended in 3D and applied so as to filter and enhance the structural coherence of the model updates within an FWI algorithm. The benefits of this strategy are investigated on a 2D synthetic experiment before considering the multiparameter inversion of a 3D field data set from the North Sea up to 10 Hz. From this data, the vertical velocity and the density are simultaneously reconstructed. Compared with a conventional nonstationary Gaussian regularization strategy, the models obtained using the coherence-enhancing anisotropic diffusion strategy indicate an enhanced resolution, especially for the density model. The high-resolution reflectivity image computed from the impedance volume clearly illustrates the benefit this filtering approach can deliver in terms of structural interpretation.

Gauss-Newton inversion with node-based basis functions: Application to imaging of seabed minerals in an area with rough bathymetry

The Gauss-Newton method has good convergence properties when used for the solution of seismic and electromagnetic inversion problems. One main issue is the high numerical cost. The numerical cost can be reduced if the optimization domain can be decoupled from the simulation domain such that the number of optimization parameters is much smaller than the number of grid nodes required for accurate simulation results. Overparameterization can be avoided. The decoupling can be achieved in a rigorous manner with the use of node-based basis functions. We provide a generic derivation of the method that is easily specialized for seismic and electromagnetic problems. The transformations between the optimization domain and the simulation domain are most effective if both domains can be described by rectilinear grids. A variable seabed depth causes difficulty. We introduce a transform from the true bathymetry to a flat seabed that solves this problem. The method is validated by application to synthetic and real electromagnetic data sets. The real data are acquired at the slow spreading Mohns Ridge located east of Greenland and southwest of Svalbard. We provide a discussion on the interpretation of these data for an inverse scheme using the transverse isotropy with a vertical symmetry axis approximation. We offer some insights on how to interpret inversion results in the case of exploration for marine minerals. The interpretation differs from a hydrocarbon exploration setting owing to the presence of vertical conductors due to the formation water circulation and vertical resistors due to volcanic intrusions.

Hybrid Deep Learning and Sensitivity Operator-Based Algorithm for Identification of Localized Emission Sources

Hybrid approaches combining machine learning with traditional inverse problem solution methods represent a promising direction for the further development of inverse modeling algorithms. The paper proposes an approach to emission source identification from measurement data for advection–diffusion–reaction models. The approach combines general-type source identification and post-processing refinement: first, emission source identification by measurement data is carried out by a sensitivity operator-based algorithm, and then refinement is done by incorporating a priori information about unknown sources. A general-type distributed emission source identified at the first stage is transformed into a localized source consisting of multiple point-wise sources. The second, refinement stage consists of two steps: point-wise source localization and emission rate estimation. Emission source localization is carried out using deep learning with convolutional neural networks. Training samples are generated using a sensitivity operator obtained at the source identification stage. The algorithm was tested in regional remote sensing emission source identification scenarios for the Lake Baikal region and was able to refine the emission source reconstruction results. Hence, the aggregates used in traditional inverse problem solution algorithms can be successfully applied within machine learning frameworks to produce hybrid algorithms.

Convolution kernel determination problem in the third order Moore–Gibson–Thompson equation

This article is concerned with the study of the inverse problem of determining the difference kernel in a Volterra type integral term function in the third-order Moore–Gibson–Thompson (MGT) equation. First, the initial-boundary value problem is reduced to an equivalent problem. Using the Fourier spectral method, the equivalent problem is reduced to a system of integral equations. The existence and uniqueness of the solution to the integral equations are proved. The obtained solution to the integral equations of Volterra-type is also the unique solution to the equivalent problem. Based on the equivalence of the problems, the theorem of the existence and uniqueness of the classical solutions of the original inverse problem is proved.

Tomographic Inverse Problems: Mathematical Challenges and Novel Applications

This workshop brought together researchers working on mathematical problems related to tomography, with a particular emphasis on novel applications and associated mathematical challenges. Examples of respective issues represented in the workshop were tomographic imaging with Compton cameras or coupled-physics imaging, resolution and aliasing, vector and tensor field tomography, diffraction tomography, magnetic particle tomography, and limited data, all of which are motivated by the many modern applications. These topics were complemented by novel algorithmic strategies in the solution of tomographic inverse problems, such as stochastic methods and machine learning techniques. Bringing together mathematical and scientific researchers working on these different mathematical problems created a fruitful interchange with novel ideas and strong impact for the future of the field.

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.

Study on the influence of internal modulation parameters on the temperature calculation performance of matrix-searching CCD pyrometer

Matrix-searching CCD temperature measurement technology can integrate the complex signal modulation process of CCD because of its forward solution of inverse problem. But in the existing research, there is still a lack of discussion on the temperature calculation ability of this technology in the internal modulation process. Therefore, the research on the influence of temperature calculation performance of matrix-searching CCD pyrometer is carried out. Based on the three-channel temperature equation and the radiation acquisition theory of the detector, the evaluation model of effective measurement range and temperature resolution is defined and constructed. Combined with the calibration experiment, the effects of gain and gamma internal modulation parameters on the effective temperature measurement range and temperature resolution are discussed. Through the analysis of the simulation results, the use and effects of the two internal modulation parameters are obtained, which not only improves the theoretical system of the matrix-seeking temperature measurement method, but also provides theoretical guidance for the parameter selection of the method under different conditions.

Photoacoustic image reconstruction with anobjective function using TGV and ESTGV asaregularization functional.

Photoacoustic tomographic imaging is a non-invasive medical diagnostic technology for visualizing biological tissue. However, the inverse problem and noise in photoacoustic signals often cause blurred images. Existing regularization methods struggle with staircasing artifacts and edge preservation. To overcome this, an objective function incorporating total generalized variation (TGV) is proposed. However, it failed with high-density Gaussian noise. To address this, an extended version called edge-guided second-order TGV (ESTGV) is introduced. For sparsification, wavelet transform and discrete cosine transform are introduced, while the fast-composite-splitting algorithm is employed for the inverse problem solution. Experimental validation demonstrates the potential of these approaches.

X-ray-based overlay metrology using reciprocal space slicing analysis

Overlay serves as the pivotal performance indicator for lithography tools, and its prompt and precise measurement significantly underpins the process yield control. At present, diffraction-based overlay metrology employing optical wavelengths encounters constraints in terms of measurement sensitivity. When transitioning to x-ray wavelengths, the critical-dimension small-angle x-ray scattering (CDSAXS) method for nanostructure characterization necessitates reciprocal space mapping (RSM) and inverse problem solving, resulting in substantial throughput constraints. In this work, we propose an x-ray-based overlay metrology using reciprocal space slicing analysis (RSS), yielding high-precision overlay measurement at one single angle of incidence (AOI). Moreover, we examine the robustness of the proposed method against errors stemming from overlay target grating fabrication and measurement processes, substantiating its efficacy as a novel x-ray-based overlay metrology and unveiling the potential application of x-ray-based techniques within the realm of integrated circuit metrology.

Application of machine learning regression models to inverse eigenvalue problems

In this work, we study the numerical solution of inverse eigenvalue problems from a machine learning perspective. Two different problems are considered: the inverse Sturm-Liouville eigenvalue problem for symmetric potentials and the inverse transmission eigenvalue problem for spherically symmetric refractive indices. Firstly, we solve the corresponding direct problems to produce the required eigenvalues datasets in order to train the machine learning algorithms. Next, we consider several examples of inverse problems and compare the performance of each model to predict the unknown potentials and refractive indices respectively, from a given small set of the lowest eigenvalues. The supervised regression models we use are k-Nearest Neighbours, Random Forests and Multi-Layer Perceptron. Our experiments show that these machine learning methods, under appropriate tuning on their parameters, can numerically solve the examined inverse eigenvalue problems.

Distributed Deep Learning Optimization of Heat Equation Inverse Problem Solvers

Distributed Deep Learning Optimization of Heat Equation Inverse Problem Solvers

DRIP: deep regularizers for inverse problems

In this paper we consider inverse problems that are mathematically ill-posed. That is, given some (noisy) data, there is more than one solution that approximately fits the data. In recent years, deep neural techniques that find the most appropriate solution, in the sense that it contains a-priori information, were developed. However, they suffer from several shortcomings. First, most techniques cannot guarantee that the solution fits the data at inference. Second, while the derivation of the techniques is inspired by the existence of a valid scalar regularization function, such techniques do not in practice rely on such a function, and therefore veer away from classical variational techniques. In this work we introduce a new family of neural regularizers for the solution of inverse problems. These regularizers are based on a variational formulation and are guaranteed to fit the data. We demonstrate their use on a number of highly ill-posed problems, from image deblurring to limited angle tomography.

Reconstruction of time-dependent right-hand side in parabolic equations on disjoint domains

We study inverse problems of reconstructing the time-dependent right-hand side from point observation in a one dimensional parabolic equation on disjoint intervals. These problems are ill-possed, i.e. very slight errors in the additional input may cause relatively significant errors in the output of the left and right internal right-hand side. In this work, we construct computational algorithms, using the loaded equation method. First, we perform a decomposition with respect to the unknown source of the inverse problem solutions. Then the inverse problems are reduced to a loaded parabolic equation problems. The well-posedness of the inverse problems is studied on the base of loaded equation ones. The numerical performance of the approach is realized by finite difference schemes, solved with decomposition algorithms. Computational experiments show the efficiency of the method.

Hydrodynamics of transient processes in a well with an electric submersible pump

The relevance. Associated with the problem of increase in efficiency of algorithms for monitoring and controlling the dynamic operation modes of wells equipped with an electric submersible pump with adjustable supply. The result summarizes and develops the previously published solution on the design of a dynamic well model, through the transition from linearized correlations to quadratic relationships between states at key points of a lift. These points properly reflect the observed forms of behavior in transient and equilibrium operating modes at full start-stop control ranges. The main aim. Updated description of a comprehensive quadratic barometric model of the «reservoir–lift–electric pump–well head» type well, focused on the tasks of automatic control and regulation in real time. Objects. Well with frequency-regulated submersible pump. Methods. Material balance, percolation, hydrostatics, quadratic kinetics of friction losses, numerical modeling of differential equations, finite-dimensional description of the complex hydrodynamics of the well, taking into account the potential and kinetic head losses in the lift. Results: (1) Updated mathematical description of the hydrodynamics of the well with an electric submersible pump creates the basis for: more reliable solution of inverse problems of parametric model support according to the data of field control, refined assessment of the adjustment potential, predicting the dynamics of a possible system output beyond the boundaries of functional stability, technologies of numerical and analytical design of optimal solutions when selecting the parameters of arrangement and laws of operational regulation, synthesis of dynamic observers of an extended state vector synchronously operating with the dynamics of controlling well-head and deep pressure. (2) The results of the computational analysis demonstrate a complex behavior that does not match the graph of the inflow and pump supply, when changing the frequency and wellhead pressure. This explains the observed effect of the multi-tempoity of transient processes during starts and stops and possible instability of the supply with a sharp decrease in frequency. (3) Estimates of discrepancy between the solutions for a quadratic model and a linearized prototype indicate an increase in errors of linearized analysis with strong deviations of operating modes from the equilibrium-calculated states. The refinements in dynamically disturbed operating modes delivered by the updated model are important for improving the reliability of operational control systems and parametric estimation for data control mode states.

An integrated 3D digital model of stratigraphy, petrophysics and karstified fracture network for the Cristal Cave, NE-Brazil

Digital Outcrop Models (DOMs) are virtual representations of geological features. Although DOMs are widely used tools in geosciences, their integration with other datasets remains relatively underexplored. We combined a DOM, derived from a photogrammetric survey of a carbonate sequence, with lithostratigraphic, petrophysical (porosity, permeability and uniaxial compressive strength), fracture distribution, and karst dissolution information to compose a single integrated three-dimensional digital model. The study site is one of the entrances of the Cristal Cave (São Francisco Craton, Northeastern Brazil), which has been used as a structural and diagenetic outcrop analog for the Brazilian pre-salt carbonate reservoirs. Data from fracture distributions, measured on the exposed surfaces of the cave, were used to build a Discrete Fracture Network, based on the solution of the stereology inverse problem. Fracture apertures were then modified to generate different scenarios of karstification, thus composing Discrete Fracture and Karst Networks. This integrative approach brought relevant insights into the cave development due to dissolution along fracture clusters. Our methodology offers better geological data handling to build static models to be used in a fluid flow modeling environment, contributing to bridge the gap between geophysics/geology and engineering approaches.

Guided acoustic waves in thin epitaxial films: Experiment and inverse problem solution for NiTi

Despite the fundamental and technological importance of the elastic constants, a suitable method for their full characterization in epitaxial films is missing. Here we show that transient grating spectroscopy (TGS) with highly k-vector-selective generation and detection of acoustic waves is capable of determination of all independent elastic coefficients of an epitaxial thin film grown on a single-crystalline substrate. This experimental setup enables detection of various types of guided acoustic waves and evaluation of the directional dependence of their speeds of propagation. For the studied model system, which is a 3μm thin epitaxial film of the NiTi shape memory alloy on an MgO substrate, the TGS angular maps include Rayleigh-type surface acoustic waves as well as Sezawa-type and Love-type modes, delivering rich information on the elastic response of the film under different straining modes. The resulting inverse problem, which means the calculation of the elastic constants from the TGS maps, is subsequently solved using the Ritz–Rayleigh numerical method. Using this approach, tetragonal elastic constants of the NiTi film and their changes with the austenite→martensite phase transition are analyzed.

Development of a method for the on-line spectral characterization of Bremsstrahlung photons from an electron accelerator

The use of irradiators based on electron accelerator technologies coupled with an electron–photon conversion target requires an accurate and continuous characterization of the Bremsstrahlung spectrum produced. The indirect methods developed so far are either unsuitable for the constraints of on-line measurement or they call for excessively invasive devices. We present here a method for on-line reconstruction of the photon spectrum from the measurement of the energy deposited in a scintillator irradiated on its edge, using a simple, compact and robust device.The approach to reconstructing the Bremsstrahlung spectrum from the measurement of its energy deposited in a scintillator, translates mathematically into the solution of a linear inverse problem. The classical resolution of the problem is disrupted by its very nature and by the presence of noise in the experimental measurement of the deposited energy in the scintillator, but it is possible to apply methods compatible with the on-line constraint. In this paper, a method is implemented on the basis of Tikhonov regularization and, with the use of an effective tool, allows a multi-parametric study to be carried out in order to meet the on-line measurement constraint. In particular, the aim is to optimize the choice of scintillator and the properties of the acquisition system to be used.

Adaptive Spectral Inversion for inverse medium problems

A nonlinear optimization method is proposed for the solution of inverse medium problems with spatially varying properties. To avoid the prohibitively large number of unknown control variables resulting from standard grid-based representations, the misfit is instead minimized in a small subspace spanned by the first few eigenfunctions of a judicious elliptic operator, which itself depends on the previous iteration. By repeatedly adapting both the dimension and the basis of the search space, regularization is inherently incorporated at each iteration without the need for extra Tikhonov penalization. Convergence is proved under an angle condition, which is included into the resulting Adaptive Spectral Inversion (ASI) algorithm. The ASI approach compares favorably to standard grid-based inversion using L 2-Tikhonov regularization when applied to an elliptic inverse problem. The improved accuracy resulting from the newly included angle condition is further demonstrated via numerical experiments from time-dependent inverse scattering problems.

Correction of the optimal regulator based on solving the inverse problem of optimal stabilization with vector control

Objective. The problem of designing controllers that implement a given programmed movement of a controlled object and the problem of determining the movement of a dynamic system are two main problems in classical control theory. This article discusses the solution of direct and inverse optimal stabilization problems. The state vector is assumed to be completely available for measurement.Method. Based on the optimality ratio linking the weight coefficients of the quadratic quality functional and the optimal gain matrix, which closes the control object, it is proposed to use a numerical method for determining the functional matrices. Mathematical models of autonomous fully controlled objects were used for the study, the formation of which was carried out randomly, in particular, according to the normal distribution law.Result. The initial stage of the solution is associated with modal synthesis, the result of which is a proportional regulator that provides stabilization of the control object by the location of the poles of the synthesized system. The next step is to determine the weighting coefficients of the functional by numerically solving the optimality ratio. The final stage is the solution of the direct optimal stabilization problem, which is based on the Lagrange variational problem. As a result, the optimal regulator is calculated, which, when switched on in a closed system instead of a modal one, reduces the duration of the transient process.Conclusion. The proposed approach of the authors allows minimizing to a certain extent the transients of the adjusted control system.

Convexification for the viscocity solution for a coefficient inverse problem for the radiative transfer equation

A coefficient inverse problem for the radiative transport equation is considered. The globally convergent numerical method, the so-called convexification, is developed. For the first time, the viscosity solution is considered for a boundary value problem for the resulting system of two coupled partial differential equations. A Lipschitz stability estimate is proved for this boundary value problem using a Carleman estimate for the Laplace operator. Next, the global convergence analysis is provided via that Carleman estimate. Results of numerical experiments demonstrate a high computational efficiency of this approach.