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The paper is devoted to a problem of acquiring elastic properties of a composite material from the vibration testing data with a simplified experimental acquisition scheme. The specimen is considered to abide by the linear elasticity laws and subject to viscoelastic damping. The boundary value problem for transverse movement of such a specimen in the frequency domain is formulated and solved with finite-element method. The correction method is suggested for the finite element matrices to account for the mass of the accelerometer. The problem of acquiring the elastic parameters is then formulated as a nonlinear least-square optimization problem. The usage of the automatic differentiation technique for stable and efficient computation of the gradient and hessian allows to use well-studied first and second order local optimization methods. We also explore the possibility of generating initial guesses for local minimization by heuristic global methods. The results of the numerical experiments on simulated data are analyzed in order to provide insights for the following real life experiments. [ABSTRACT FROM AUTHOR]

A fast numerical algorithm for solving the Cauchy problem for elliptic equations with variable coefficients in standard calculation domains (rectangles, circles, or rings) is proposed. The algorithm is designed to calculate the heat flux at the inaccessible boundary. It is based on the separation of variables method. This approach employs a finite difference approximation and allows obtaining a solution to a discrete problem in arithmetic operations of the order of N ⁢ ln ⁡ N , where 푁 is the number of grid points. As a rule, iterative procedures are needed to solve the Cauchy problem for elliptic equations. The currently available direct algorithms for solving the Cauchy problem have been developed only for (Laplace, Helmholtz) operators with constant coefficients and for use of analytical solutions for problems with such operators. A novel feature of the results of the present paper is that the direct algorithm can be used for an elliptic operator with variable coefficients (of a special form). It is important that in this case no analytical solution to the problem can be obtained. The algorithm significantly increases the range of problems that can be solved. It can be used to create devices for determining in real time heat fluxes on the parts of inhomogeneous constructions that cannot be measured. For example, to determine the heat flux on the inner radius of a pipe made of different materials. [ABSTRACT FROM AUTHOR]

To estimate the depth errors in a subsurface model obtained from the inversion of seismic data, the stationary‐phase approximation in a two‐dimensional constant‐velocity model with a dipped reflector is applied to migration with a time‐shift extension. This produces two asymptotic solutions: one is a straight line, and the other is a curve. If the velocity differs from the true one, a closed‐form expression of the depth error follows from the depth and apparent dip of the reflector as well as the position of the amplitude peak at a non‐zero time shift, where the two solutions meet and the extended migration image focuses. The results are compared to finite‐frequency results from a finite‐difference code. A two‐dimensional synthetic example with a salt diapir illustrates how depth errors can be estimated in an inhomogeneous model after inverting the seismic data for the velocity model. [ABSTRACT FROM AUTHOR]

In this paper, we present the coisotropic embedding theorem as a tool to provide a solution for the inverse problem of the calculus of variations for a particular class of implicit differential equations, namely the equations of motion of free Electrodynamics. [ABSTRACT FROM AUTHOR]

We consider the problem of state estimation from a few linear measurements, where the state to recover is an element of the manifold M of solutions of a parameter-dependent equation. The state is estimated using prior knowledge on M coming from model order reduction. Variational approaches based on linear approximation of M , such as PBDW, yield a recovery error limited by the Kolmogorov width of M . To overcome this issue, piecewise-affine approximations of M have also been considered, that consist in using a library of linear spaces among which one is selected by minimizing some distance to M . In this paper, we propose a state estimation method relying on dictionary-based model reduction, where space is selected from a library generated by a dictionary of snapshots, using a distance to the manifold. The selection is performed among a set of candidate spaces obtained from a set of ℓ 1 -regularized least-squares problems. Then, in the framework of parameter-dependent operator equations (or PDEs) with affine parametrizations, we provide an efficient offline-online decomposition based on randomized linear algebra, that ensures efficient and stable computations while preserving theoretical guarantees. [ABSTRACT FROM AUTHOR]

This paper is concerned with the backward problem of recovering initial value for a homogeneous time-space fractional pseudo parabolic differential equation. Since the problem is ill-posed, a version of modified quasi boundary value method is used as the method of regularization for obtaining stable approximations. Error analysis and parameter choice strategies are done for both a priori and a posteriori cases. It is shown that, the order of the convergence rate can exceed 2 3. Hence, the obtained rate improves upon the previously known rates of order 2 3 for the a priori case, and 1 2 for the a posteriori case for the considered problem. [ABSTRACT FROM AUTHOR]

A multiple-input and multiple-output planar actuator is proposed, which utilizes mechanically driven stator magnet arrays to levitate a permanent magnet mover. A state of levitation and actuation is obtained by mechanically altering the orientation of the stator magnets to control the forces and torques on the mover. A challenge for the design and control of the actuator is inverting the relationship between the force and stator magnet rotation angles, as there is no closed-form analytical solution. In this study, a feed-forward neural network is applied to model the forward relation between stator magnet angle input and a force and torque output to reduce the forward computation time for the design process and for error estimation in real-time applications. Additionally, the neural network is considered for inverting the solution for a motion profile sampled at 1000 Hz. The developed forward model is able to calculate the forces and torques on the mover a factor 10 faster than the equivalent charge or Fourier model with an absolute error of 3 mN and 0.1 mNm for the forces and torques, respectively, and a feed-forward neural network is able to accurately learn an inverse solution for small motion profiles. [ABSTRACT FROM AUTHOR]

Objective. We introduce a robust image reconstruction algorithm named residual-guided Golub–Kahan iterative reconstruction technique (RGIRT) designed for sparse-view computed tomography (CT), which aims at high-fidelity image reconstruction from a limited number of projection views. Approach. RGIRT utilizes an inner-outer dual iteration framework, with a flexible least square QR (FLSQR) algorithm implemented in the inner iteration and a restarted iterative scheme applied in the outer iteration. The inner FLSQR employs a flexible Golub–Kahan bidiagonalization method to reduce the size of the inverse problem, and a weighted generalized cross-validation method to adaptively estimate the regularization hyper-parameter. The inner iteration efficiently yields the intermediate reconstruction result, while the outer iteration minimizes the residual and refines the solution by using the result obtained from the inner iteration. Main results. The reconstruction performance of RGIRT is evaluated and compared to other reference methods (FBPConvNet, SART-TV, and FLSQR) using projection data from both numerical phantoms and real experimental Micro-CT data. The experimental findings, from testing various numbers of projection views and different noise levels, underscore the robustness of RGIRT. Meanwhile, theoretical analysis confirms the convergence of residual for our approach. Significance. We propose a robust iterative reconstruction algorithm for x-ray CT scans with sparse views, thereby shortening scanning time and mitigating excessive ionizing radiation exposure to small animals. [ABSTRACT FROM AUTHOR]

In this paper, two-dimensional (2D) heat equations on disjoint rectangles are considered. The solutions are connected by interface Robin's-type internal conditions. The problem has external Dirichlet boundary conditions that, in the forward (direct) formulation, are given functions. In the inverse problem formulation, the Dirichlet conditions are unknown functions, and the aim is to be reconstructed upon integral observations. Well-posedness both for direct and inverse problems is established. Using the given 2D integrals of the unknown solution on each of the domains and the specific interface boundary conditions, we reduce the 2D inverse problem to a forward heat 1D one. The resulting 1D problem is solved using the explicit Saul'yev finite difference method. Numerical test examples are discussed to illustrate the efficiency of the approach. [ABSTRACT FROM AUTHOR]

A diffusion equation involving integral convolution in time variable with arbitrary kernel and nonlocal boundary conditions is considered. The existence and uniqueness results for two inverse problems of determining source terms (space- and time-dependent sources) along with diffusion concentration from appropriate over-specified conditions are presented. A bi-orthogonal system of functions is used to have series representation of the solutions of the inverse problems. Several special cases such as standard diffusion, multi-term diffusion, and tempered diffusion equations are discussed, and some examples are provided. [ABSTRACT FROM AUTHOR]

Pseudo-parabolic equations are commonly used as mathematical models in mechanics, biology, and physics to address various applied problems. One particular application involves describing moisture transfer dynamics in subsoil layers using pseudo-parabolic equations. This manuscript examines the inverse problem (IP) of identifying the moisture transfer function, along with the time-varying potential and source control terms, in a linear pseudo-parabolic equation from known moisture moments. We prove the existence of a unique solution to the inverse problem for sufficiently small times by employing the contraction principle. The inverse problem is reformulated as a nonlinear least-squares minimization problem, with the unknowns subjected to simple bounds. To guarantee stability, the Tikhonov regularization technique is utilized. For the numerical discretization, we develop the Crank-Nicolson finite-difference method to solve the direct problem. To solve the nonlinear least-squares minimization problem, we utilize the built-in subroutine lsqnonlin from the MATLAB optimization toolbox. We present and thoroughly discuss numerical outcomes for a benchmark test example, providing insights into the performance and effectiveness of the proposed methodology. [ABSTRACT FROM AUTHOR]

Full waveform inversion stands at the forefront of seismic imaging technologies, pivotal in retrieving high‐resolution subsurface velocity models. Its application is especially profound when imaging complex geologies such as salt bodies, which are regions notoriously challenging, yet essential given their hydrocarbon potential. However, with the power of full waveform inversion comes the intrinsic challenge of estimating the associated uncertainties. Such uncertainties are crucial in understanding the reliability of subsurface models, particularly in terrains like subsalt regions. Addressing this, we advocate for a nuanced approach employing the Stein variational gradient descent algorithm. Through a judicious use of a limited number of velocity model particles and the integration of random field‐based perturbations, our methodology provides a local representation of the uncertainties inherent in full waveform inversion. Our evaluations, based on the Marmousi model, showcase the robustness of the proposed technique. Yet, it is our exploration into salt‐intensive terrains, leveraging data from the Sigsbee 2A synthetic model and the Gulf of Mexico, that emphasizes the method's versatility. Findings indicate pronounced uncertainties along salt boundaries and in the deeper subsalt sediments, contrasting the minimal uncertainties in non‐salt terrains. However, anomalies like salt canyons present unique challenges, potentially due to the interplay of multi‐scattering effects. Emphasizing the scalability and cost‐effectiveness of this approach, we highlight its potential for large‐scale industrial applications in full waveform inversion, while also underscoring the necessity for prudence when integrating these uncertainty insights into subsequent seismic‐driven geological and reservoir modelling. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for a one-dimensional hyperbolic equation whose lower-order coefficient and right-hand side oscillate in time with a high frequency and the amplitude of the lower-order coefficient is small. Under study is the reconstruction of the cofactors of these rapidly oscillating functions independent of the space variable from a partial asymptotics of a solution at some point of the space. The classical theory of inverse problems examines the numerous problems of determining unknown sources, and coefficients without rapid oscillations for various evolutionary equations, where the exact solution to the direct problem appears in the additional overdetermination condition. Equations with rapidly oscillating data are often encountered in modeling the physical, chemical, and other processes that occur in media subjected to high-frequency electromagnetic, acoustic, vibrational, and others fields, which demonstrates the topicality of perturbation theory problems on the reconstruction of unknown functions in high-frequency equations. We give some nonclassical algorithm for solving such problems that lies at the junction of asymptotic methods and inverse problems. In this case the overdetermination condition involves a partial asymptotics of solution of a certain length rather than the exact solution. [ABSTRACT FROM AUTHOR]

We solve the simultaneous recovery of thermal conductivity and a high-frequency coefficient of a source in a one-dimensional initial-boundary value problem for the heat equation with Dirichlet boundary conditions and an inhomogeneous initial condition from some information on the partial asymptotics of a solution. We show that the coefficients can be restored from some data on the asymptotics of a solution, which is constructed and justified. This article was inspired by Denisov's research on a variety of inverse problems without accounting for high-frequency oscillations. Also, we continue the research by Levenshtam and his students which firstly addressed the inverse problems for parabolic equations with high-frequency coefficients and developed the relevant methodology. In contrast to the previous research of the case that only the source function or its factors are unknown, we assume that the thermal conductivity and the factor of a source function are unknown simultaneously. [ABSTRACT FROM AUTHOR]

We consider a hyperbolic equation with variable leading part and nonlinearity in the lower-order term. The coefficients of the equation are smooth functions constant beyond some compact domain in the three-dimensional space. A plane wave with direction falls to the heterogeneity from the exterior of this domain. A solution to the corresponding Cauchy problem for the original equation is measured at boundary points of the domain for a time interval including the moment of arrival of the wave at these points. The unit vector is assumed to be a parameter of the problem and can run through all possible values sequentially. We study the inverse problem of determining the coefficient of the nonlinearity on using this information about solutions. We describe the structure of a solution to the direct problem and demonstrate that the inverse problem reduces to an integral geometry problem. The latter problem consists of constructing the desired function on using given integrals of the product of this function and a weight function. The integrals are taken along the geodesic lines of the Riemannian metric associated with the leading part of the differential equation. We analyze this new problem and find some stability estimate for its solution, which yields a stability estimate for solutions to the inverse problem. [ABSTRACT FROM AUTHOR]

In the realm of high-tech materials and energy applications, accurately measuring the transient heat flow at media boundaries and the internal thermal conductivity of materials in harsh heat exchange environments poses a significant challenge when using conventional direct measurement methods. Consequently, the study of photothermal parameter reconstruction in translucent media, which relies on indirect measurement techniques, has crucial practical value. Current research on reconstructing photothermal properties within participating media typically focuses on single-objective or time-invariant properties. There is a pressing need to develop effective methods for the simultaneous reconstruction of time-varying thermal flow fields and internal thermal conductivity at the boundaries of participating media. This paper introduces a computational model based on the numerical simulation theory of internal heat transfer systems in participating media, stochastic particle swarm optimization algorithms, and Kalman filter technology. The model aims to enable the simultaneous reconstruction of various thermal parameters within the target medium. Our results demonstrate that under varying levels of measurement noise, the inversion results for different target parameters exhibit slight oscillations around the true values, leading to a reduction in reconstruction accuracy. However, overall, the model demonstrates robustness and accuracy in ideal conditions, validating its effectiveness. [ABSTRACT FROM AUTHOR]

MIT (magnetic induction tomography) image reconstruction from data acquired with a single, small inductive sensor has unique requirements not found in other imaging modalities. During the course of scanning over a target, measured inductive loss decreases rapidly with distance from the target boundary. Since inductive loss exists even at infinite separation due to losses internal to the sensor, all other measurements made in the vicinity of the target require subtraction of the infinite-separation loss. This is accomplished naturally by treating infinite-separation loss as an unknown. Furthermore, since contributions to inductive loss decline with greater depth into a conductive target, regularization penalties must be decreased with depth. A pair of squared L2 penalty norms are combined to form a 2-term Sobolev norm, including a zero-order penalty that penalizes solution departures from a default solution and a first-order penalty that promotes smoothness. While constraining the solution to be non-negative and bounded from above, the algorithm is used to perform image reconstruction on scan data obtained over a 4.3 cm thick phantom consisting of bone-like features embedded in agarose gel, with the latter having a nominal conductivity of 1.4 S/m. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we study an inverse problem for a linear third-order pseudoparabolic equation. The investigated inverse problem consists of determining a space-dependent coefficient of the right-hand side of a pseudoparabolic equation. As an additional information a final overdetermination condition is considered. Under the suitable conditions on the data of the problem, the unique solvability of the considered inverse problem is established. The stability of solutions is also proved. The established results are also true for inverse problems for parabolic equations, which could be obtained as a regularization of the studied pseudoparabolic equation. In addition, the pseudoparabolic problem is discretized using the cubic B-spline functions and recast as a nonlinear least-squares minimization of the Tikhonov regularization function. Numerically, this is effectively solved using the MATLAB subroutine lsqnonlin. Both exact and noisy data are inverted. Numerical results for three benchmark test examples are presented and discussed. Moreover, the von Neumann stability analysis is also discussed. [ABSTRACT FROM AUTHOR]

Probabilistic methods for geophysical inverse problems allow the use of arbitrarily complex prior information in principle. Geostatistical techniques, such as multiple-point statistics (MPS), for describing spatial correlation models and higher-order statistics have been proposed to achieve this inversion task, in which stochastic algorithms such as Markov chain Monte Carlo (McMC) are incorporated. However, stochastic sampling and optimization often require a large number of iterations, and thus geostatistical sampling of the prior model can become computationally demanding. To overcome this challenge, a deep learning model, namely conditional generative adversarial networks (CGANs), is proposed, which allows one to perform a random walk to sample the complex prior distribution. CGANs simulate conditional realizations conditioned to the available hard conditioning data, that is, direct measurements, while preserving the geometrical structure of the model parameters of interest and replicating the sequential Gibbs sampling algorithm. Despite the need for a training step, for a large number of simulations, CGANs are more efficient than traditional geostatistical simulation algorithms such as single normal equation simulation (SNESIM). The proposed methodology is used as part of the extended Metropolis algorithm to predict the distributions of categorical facies in two examples, a dune environment in the Gobi Desert and a channel system in an idealized subsurface reservoir, from indirect observational data such as acoustic impedance. The inversion results are compared to the extended Metropolis algorithm using standard MPS sampling. [ABSTRACT FROM AUTHOR]

First‐arrival travel‐time tomography is frequently used in ocean‐bottom seismometer surveys for estimating subsurface velocity. However, due to a lack of seismometer stations or ray sampling, the tomographic images' spatial resolution and quality are typically low. Inspired by the multiple imaging of ocean‐bottom seismometer data, in this study, I developed a mirror tomography method to incorporate the travel times of refraction multiples in first‐arrival travel‐time tomography. Specifically, the travel times of refraction multiples were treated as virtual first‐arrival travel times from the mirror positions of the stations. This technique enhances ray coverage and stabilizes the inversion. I confirmed that the travel times of refraction multiples were consistent with the first‐arrival travel times calculated using numerical modelling at the mirror position of the station. Synthetic examples showed that the mirror tomography scheme may enhance ray coverage and model resolution. Mirror tomography may compensate for the uneven distribution of travel‐time picks caused by the loss of the ocean‐bottom seismometer. [ABSTRACT FROM AUTHOR]

In this work we present a preliminary study on the use of a multi-frequency topological derivative based method for the detection of flaws in welding joints of steel plates. Simulated measured data at a set of receivers obtained after producing oscillatory excitations at different frequencies at some emitters will be processed. The performance of the method will be illustrated in a gallery of numerical experiments covering highly demanding situations that include very reduced numbers of emitters and receivers, highly noisy data, and several flaws of different sizes. [ABSTRACT FROM AUTHOR]

A nonnegative integer p is realizable by a graph-theoretical invariant I if there exists a graph G such that I (G) = p . The inverse problem for I consists of finding all nonnegative integers p realizable by I. In this paper, we consider and solve the inverse problem for the Mostar index, a recently introduced graph-theoretical invariant which attracted a lot of attention in recent years in both the mathematical and the chemical community. We show that a nonnegative integer is realizable by the Mostar index if and only if it is not equal to one. Besides presenting the complete solution to the problem, we also present some empirical observations and outline several open problems and possible directions for further research. [ABSTRACT FROM AUTHOR]

In a lot of engineering applications, one has to recover the temperature of a heat body having a surface that cannot be measured directly. This raises an inverse problem of determining the temperature on the surface of a body using interior measurements, which is called the sideways problem. Many papers investigated the problem (with or without fractional derivatives) by using the Cauchy data u(x0,t),ux(x0,t)$$ u\left({x}_0,t\right),{u}_x\left({x}_0,t\right) $$ measured at one interior point x=x0∈(0,L)$$ x={x}_0\in \left(0,L\right) $$ or using an interior data u(x0,t)$$ u\left({x}_0,t\right) $$ and the assumption limx→∞u(x,t)=0$$ {\lim}_{x\to \infty }u\left(x,t\right)=0 $$. However, the flux ux(x0,t)$$ {u}_x\left({x}_0,t\right) $$ is not easy to measure, and the temperature assumption at infinity is inappropriate for a bounded body. Hence, in the present paper, we consider a fractional sideways problem, in which the interior measurements at two interior point, namely, x=1$$ x=1 $$ and x=2$$ x=2 $$, are given by continuous data with deterministic noises and by discrete data contaminated with random noises. We show that the problem is severely ill‐posed and further constructs an a posteriori optimal truncation regularization method for the deterministic data, and we also construct a nonparametric regularization for the discrete random data. Numerical examples show that the proposed method works well. [ABSTRACT FROM AUTHOR]

AbstractThe Kennedy and O’Hagan (KOH) calibration framework uses coupled Gaussian processes (GPs) to meta-model an expensive simulator (first GP), tune its “knobs” (calibration inputs) to best match observations from a real physical/field experiment and correct for any modeling bias (second GP) when predicting under new field conditions (design inputs). There are well-established methods for placement of design inputs for data-efficient planning of a simulation campaign in isolation, that is, without field data: space-filling, or via criterion like minimum integrated mean-squared prediction error (IMSPE). Analogues within the coupled GP KOH framework are mostly absent from the literature. Here we derive a closed form IMSPE criterion for sequentially acquiring new simulator data for KOH. We illustrate how acquisitions space-fill in design space, but concentrate in calibration space. Closed form IMSPE precipitates a closed-form gradient for efficient numerical optimization. We demonstrate that our KOH-IMSPE strategy leads to a more efficient simulation campaign on benchmark problems, and conclude with a showcase on an application to equilibrium concentrations of rare earth elements for a liquid–liquid extraction reaction. [ABSTRACT FROM AUTHOR]

Predicting the water abundance of coal‐bearing strata is crucial for ensuring mining safety. However, owing to the dispersion and attenuation characteristics caused by pore fluid flow, it is difficult to estimate the water abundance of coal seam roof aquifers using seismic data. To overcome this challenge, we provide the relationship between the frequency‐dependent seismic wave velocity and water saturation based on the Chapman fracture model and the mixing fluid model. We propose a seismic dispersion attribute technique that can use dispersion information as an indicator of water abundance. Numerical experiment results show that the water saturation of the sandstone aquifer is positively correlated with the dispersion attribute. The results of low‐frequency rock physical experiments are roughly consistent with those predicted by the model for the given parameters. Using seismic dispersion attribute inversion and the frequency slice wavelet transform method, we predicted the water abundance of sandstone in the coal seam roof of the Yuwang coal mine in Yunnan Province, China. The predicted sandstone water abundance was compatible with the actual water‐rich scenario observed in well logs and downhole drilling in the study area. Therefore, the method proposed herein has the potential to quantitatively determine the water abundances of sandstone aquifers in coal seam roofs. [ABSTRACT FROM AUTHOR]

The reflection waveform inversion is a powerful technique to build a large‐scale velocity model of the subsurface by fitting the reflected recorded seismic waves. The reflection waveform inversion is designed based on the pillar concept of model and data‐scale separation. Therefore, its success is related to the ability of its forward modelling engine to separate reflected events distinctly from other propagation modes (diving waves, multiples, etc.). However, the standard Born modelling based on the two‐way wave equation may generate internal multiples in case of an insufficient smooth background model. These internal multiples may lead to a distorted sensitivity kernel, which adds more non‐linearity to the inverse problem. In addition, simulating the wave equation using two‐way propagators is still an overburden step of the algorithm especially in large three‐dimensional real surveys. In this proposal, we introduce an alternative to the two‐way wave equation by using a one‐way approach for the reflection waveform inversion. The Born modelling based on one‐way propagators significantly reduces the computational cost and I think it should be allows to relax the smooth background velocity model assumption by restricting the forward modelling to primary reflected waves. After a brief theoretical description of the one‐way waveform inversion, we present an application of the algorithm on the real marine dataset to review its promises and pitfalls. Our approach produces an acceptable large‐scale velocity model whose accuracy is confirmed by the migrated image and the offset gathers. [ABSTRACT FROM AUTHOR]

We consider an inverse problem for diffusion equation involving fractional Laplacian operator in space and Hilfer fractional derivatives in time with Dirichlet zero boundary conditions. The inverse problem is to recover time-dependent source term and diffusion concentration with an integral type over-determination condition. We discuss the analytical solution of the inverse problem and prove the existence and uniqueness of the analytical solution. Some special cases and examples for the considered inverse problem are provided. [ABSTRACT FROM AUTHOR]

With the growing demand of mineral consumption, the management of the mining waste is crucial. Cemented paste backfill (CPB) is one of the techniques developed by the mining industry to fill the voids generated by the excavation of underground spaces. The CPB process is the subject of various studies aimed at optimizing its implementation in the field. In this article, we focus on the modelling of the backfill phase where it has been shown in Vigneaux et al. (Cem. Concr. Res. 164:107038, 2023) that a viscoplastic lubrication model can be used to describe CPB experiments. The aim here is to propose an accelerated method for performing the parameters' estimation of the properties of the paste (typically its rheological properties), with an inverse problem procedure based on observed height profiles of the paste. The inversion procedure is based on a metamodel built from an initial partial differential equation model, thanks to a polynomial chaos expansion coupled with a principal component analysis. [ABSTRACT FROM AUTHOR]

The reconstruction of the spatial complex conductivity σ + j ω ε 0 ε r from complex valued impedance measurements forms the inverse problem of complex electrical impedance tomography or complex electrical capacitance tomography. Regularized Gauß-Newton schemes have been proposed for their solution. However, the necessary computation of the Jacobian is known to be computationally expensive, as standard techniques such as adjoint field methods require additional simulations. In this work, we show a more efficient way to computationally access the Jacobian matrix. In particular, the presented techniques do not require additional simulations, making the use of the Jacobian, free of additional computational costs. [ABSTRACT FROM AUTHOR]

We propose an inverse problem of determining the mechanical and variable-order parameters of the Euler–Bernoulli beam on viscoelastic foundation. For this goal, we develop a fully-discrete Hermite finite element scheme for this model and analyze the corresponding error estimates. The Levenberg–Marquardt method is then applied to determine the multiple parameters. Extensive numerical experiments are performed under practical settings to demonstrate the behavior of the proposed model and the efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we address stability results in determining the time-dependent scalar and vector potentials appearing in the convection-diffusion equation from the knowledge of the Cauchy data set. We prove Hölder-type stability estimates. The key tool used in this work is the geometric optics solution. [ABSTRACT FROM AUTHOR]

In this paper, the problem we investigate is to simultaneously identify the source term and initial value of the time fractional diffusion equation. This problem is ill-posed, i.e., the solution (if exists) does not depend on the measurable data. We give the conditional stability result under the a-priori bound assumption for the exact solution. The modified Tikhonov regularization method is used to solve this problem, and under the a-priori and the a-posteriori selection rule for the regularization parameter, the convergence error estimations for this method are obtained. Finally, numerical example is given to prove the effectiveness of this regularization method. [ABSTRACT FROM AUTHOR]

This paper considers the inverse problem of determining the time‐dependent coefficient in the time‐fractional diffusion‐wave equation. In this case, an initial boundary value problem was set for the fractional diffusion‐wave equation, and an additional condition was given for the inverse problem of determining the coefficient from this equation. First of all, it was considered the initial boundary value problem. By the Fourier method, this problem is reduced to equivalent integral equations. Then, using the Mittag‐Leffler function and the generalized singular Gronwall inequality, we get a priori estimate for solution via unknown coefficient which we will need to study of the inverse problem. The inverse problem is reduced to the equivalent integral of equation of Volterra type. The principle of contracted mapping is used to solve this equation. Local existence and global uniqueness results are proved. The stability estimate is also obtained. [ABSTRACT FROM AUTHOR]

We consider four common defects that appear in an infinite single-layer graphene lattice, i.e. single vacancy V 1 (5 − 9) , Stone-Wales SW (55 − 77) , and double vacancies V 2 (5 − 8 − 5) and V 2 (555 − 777). Investigating the honeycomb pattern of the graphene, we make use of the concept of shortest closed paths (periodic orbits) in the underlying topological structure. Using properties of the shortest closed paths of odd length we present and prove mathematically an algorithm that classifies which one of these defects occurs. [ABSTRACT FROM AUTHOR]

The main aims of this paper are to investigate the existence and uniqueness results for the solution of an inverse source problem for a multidimensional, semilinear, backward parabolic equation, subject to Dirichlet boundary conditions, and to propose an iterative difference scheme for the numerical solution of the problem. The unique solvability of the difference scheme is established, as well. In the foundation of the theoretical results, some tools and facts from the operator theory, contraction principle and the Banach fixed-point theorem are applied. Furthermore, a comprehensive numerical analysis and several illuminating visualizations are carried out by employing the iterative difference schemes proposed on some test problems. [ABSTRACT FROM AUTHOR]

We present a new version of the FILDSIM code (Galdon-Quíroga et al 2018 Plasma Phys. Control. Fusion 60 105005), which significantly refines the modelling of the fast-ion loss detector (FILD) signal. We demonstrate that the FILD weight functions computed using this new version of FILDSIM are more accurate relative to synthetic benchmarks than those computed using the previous version. Thus, the new version enables higher-quality velocity-space sensitivity modelling and reconstructions. We validate the improvements on experimental data from discharge #75620 at TCV. Additionally, we present a novel approach for characterizing FILDs through a gross FILD measurement and a gross weight function based on the calculations from the new version of FILDSIM. We use them to characterize the TCV FILD. [ABSTRACT FROM AUTHOR]

This paper is devoted to some inverse problems of recovering the nonlinearity for the Jordan–Moore–Gibson–Thompson equation, which is a third order nonlinear acoustic equation. This equation arises, for example, from the wave propagation in viscous thermally relaxing fluids. The well-posedness of the nonlinear equation is obtained with the small initial and boundary data. By the second order linearization to the nonlinear equation, and construction of complex geometric optics solutions for the linearized equation, the uniqueness of recovering the nonlinearity is derived. [ABSTRACT FROM AUTHOR]

Kinetic process models are widely applied in science and engineering, including atmospheric, physiological and technical chemistry, reactor design, or process optimization. These models rely on numerous kinetic parameters such as reaction rate, diffusion or partitioning coefficients. Determining these properties by experiments can be challenging, especially for multiphase systems, and researchers often face the task of intuitively selecting experimental conditions to obtain insightful results. We developed a numerical compass (NC) method that integrates computational models, global optimization, ensemble methods, and machine learning to identify experimental conditions with the greatest potential to constrain model parameters. The approach is based on the quantification of model output variance in an ensemble of solutions that agree with experimental data. The utility of the NC method is demonstrated for the parameters of a multi-layer model describing the heterogeneous ozonolysis of oleic acid aerosols. We show how neural network surrogate models of the multiphase chemical reaction system can be used to accelerate the application of the NC for a comprehensive mapping and analysis of experimental conditions. The NC can also be applied for uncertainty quantification of quantitative structure–activity relationship (QSAR) models. We show that the uncertainty calculated for molecules that are used to extend training data correlates with the reduction of QSAR model error. The code is openly available as the Julia package KineticCompass. [ABSTRACT FROM AUTHOR]

We consider and combine two methods to solve the inverse problem of locating dipolar sources in a restricted domain from measurements of the Cauchy data on a part of the boundary for the Laplace equation. We begin by extending the boundary measurements, which are accessible only on a part of the boundary by solving an extremal bounded problem in an annulus. The inverse problem of source identification is then solved using the reciprocity gap concept applied to the completed data. In order to evaluate the effectiveness of source detection with regard to the expansion of the data, some numerical tests are provided. [ABSTRACT FROM AUTHOR]

We consider the identification of spatially distributed parameters under H¹ regularization. Solving the associated minimization problem by Gauss-Newton iteration results in linearized problems to be solved in each step that can be cast as boundary value problems involving a low-rank modification of the Laplacian. Using an algebraic multigrid as a fast Laplace solver, the Sherman-Morrison-Woodbury formula can be employed to construct a preconditioner for these linear problems which exhibits excellent scaling w.r.t. the relevant problem parameters. We first develop this approach in the functional setting, thus obtaining a consistent methodology for selecting boundary conditions that arise from the H¹ regularization. We then construct a method for solving the discrete linear systems based on combining any fast Poisson solver with the Woodbury formula. The efficacy of this method is then demonstrated with scaling experiments. These are carried out for a common nonlinear parameter identification problem arising in electrical resistivity tomography. [ABSTRACT FROM AUTHOR]

The Cauchy problem for the heat equation with zero right-hand side is considered. The initial function is assumed to belong to the space of tempered distributions. The problem of determining the support of the initial function from solution values at some fixed time is studied. Necessary and sufficient conditions for the support to lie in a given convex compact set are obtained. These conditions are formulated in terms of the solution's decay rate at infinity. A sharp constant in the exponential for the Landis–Oleinik conjecture on the nonexistence of fast decaying solutions is found. [ABSTRACT FROM AUTHOR]

Fast estimation is a critical feature of the proposed modeling approach as a "forward" fast solver that has a key role in solving the "inverse" problem involved in crack sizing, especially for real‐time applications. To this end, two different inversion methods have been proposed to estimate the crack depth. The first one uses an interpolation model of the 3D direct model simulation data. The second approach is based on artificial neural networks (ANN). In this article, the rotating uniform eddy current (RUEC) probe is used to detect the normal magnetic component Bz which is defined as the characteristic signal for reconstructing the crack length and depth concurrently. Using the first approach, The Bz characteristic 3D surface is presented, and this can be modeled by a fitted polynomial interpolation equation. Thus, the crack depth can be inverted using this equation referring to the experimental or simulated Bz signal and the crack length. The ANN is outlined to determine the crack depth based on the simulated Bz signature. The move from the first inversion method to the second was flexible and useful, in which the interpolation model is used as a defect signal generator for fast building of a large and efficient database for ANN training. Both of the proposed methods proved the objectivity and accuracy of the inversion results and offered more robust engineering support for automated NDT, reducing production time and increasing productivity. [ABSTRACT FROM AUTHOR]

This study investigated the potential of machine-learning-based stroke image reconstruction in capacitively coupled electrical impedance tomography. The quality of brain images reconstructed using the adversarial neural network (cGAN) was examined. The big data required for supervised network training were generated using a two-dimensional numerical simulation. The phantom of an axial cross-section of the head without and with impact lesions was an average of a three-centimeter-thick layer corresponding to the height of the sensing electrodes. Stroke was modeled using regions with characteristic electrical parameters for tissues with reduced perfusion. The head phantom included skin, skull bone, white matter, gray matter, and cerebrospinal fluid. The coupling capacitance was taken into account in the 16-electrode capacitive sensor model. A dedicated ECTsim toolkit for Matlab was used to solve the forward problem and simulate measurements. A conditional generative adversarial network (cGAN) was trained using a numerically generated dataset containing samples corresponding to healthy patients and patients affected by either hemorrhagic or ischemic stroke. The validation showed that the quality of images obtained using supervised learning and cGAN was promising. It is possible to visually distinguish when the image corresponds to the patient affected by stroke, and changes caused by hemorrhagic stroke are the most visible. The continuation of work towards image reconstruction for measurements of physical phantoms is justified. [ABSTRACT FROM AUTHOR]

Metamaterial-based coded aperture imaging (MCAI) is a forward-looking radar imaging technique based on wavefront modulation. The scattering coefficients of the target can resolve as an ill-posed inverse problem. Data-based deep-learning methods provide an efficient, but expensive, way for target reconstruction. To address the difficulty in collecting paired training data, an untrained deep radar-echo-prior-based MCAI (DMCAI) optimization model is proposed. DMCAI combines the MCAI model with a modified U-Net for predicting radar echo. A joint loss function based on deep-radar echo prior and total variation is utilized to optimize network weights through back-propagation. A target reconstruction strategy by alternatively using the imaginary and real part of the radar echo signal (STAIR) is proposed to solve the DMCAI. It makes the target reconstruction task turn into an estimation from an input image by the U-Net. Then, the optimized weights serve as a parametrization that bridges the input image and the target. The simulation and experimental results demonstrate the effectiveness of the proposed approach under different SNRs or compression measurements. [ABSTRACT FROM AUTHOR]

Effective identification of dynamic loads on vehicle systems is crucial for evaluating the structural integrity of railway vehicles. The inverse problem method has been proposed for dynamic load identification to determine the impact load on the carbody. Due to the sparse nature of the impact load, the combination of the l1-norm and TwIST (two-step iterative threshold shrinkage method) methods is used to reconstruct the impact load, taking into account the influence of sensor placement on the identification results. The MOHBA (Multi-objective Optimisation Honey Badger Algorithm) was proposed to simultaneously select the regularisation and iteration parameters. This algorithm aims to improve the accuracy of subsequence load reconstruction by considering the influence of noise levels when selecting the measured data. The impact load experiment was conducted on the scale carbody, and the single-point impact load and multi-point impact load matched well with the actual load. Satisfactory load identification results can still be achieved when the noise level is within 15%. When arranging measurement points on the boundary of the vehicle body, satisfactory load identification results can also be achieved. [ABSTRACT FROM AUTHOR]

In many subsurface studies, facies and petrophysical properties are two important reservoir parameters that are closely correlated. They are routinely used in well interpretation, hydrocarbon reserve calculation and production profile prediction. These two parameters have commonly been determined in two separate tasks because of their mathematical differences (facies are discrete, and petrophysical properties are continuous). However, this is incorrect because facies and petrophysical properties are often strongly correlated. Therefore, we propose a new joint inversion method of facies and petrophysical properties based on a bi‐level optimization model. In the bi‐level optimization model, the upper‐level problem is the petrophysical property inversion while the lower‐level problem can identify the facies and add a facies‐related constraint for the upper‐level optimization. We also develop a new genetic algorithm for the discrete‐continuous inversion problem based on the bi‐level optimization model because the inversion problem usually has multiple local solutions. In addition, rock physics and statistics are combined in the inversion process. A rock physics model is used to establish the basic relationships between the petrophysical and elastic parameters, and the statistical approach is used to describe the intrinsic connection among the multiple reservoir parameters based on well log data. The numerical experiments indicate that the traditional separate prediction method and the new joint inversion method can quickly obtain more accurate results. In the application examples of real data, the inversion results can be matched to the well log data within the limits of the input data resolution, which further verifies the reliability and application potential of this new method. [ABSTRACT FROM AUTHOR]

Let f (t 1 , ... , t n) be a nondegenerate integral quadratic form. We analyze the asymptotic behavior of the function D f (X) , the number of integers of absolute value up to X represented by f. When f is isotropic or n is at least 3, we show that there is a δ (f) ∈ Q ∩ (0 , 1) such that D f (X) ∼ δ (f) X and call δ (f) the density of f. We consider the inverse problem of which densities arise. Our main technical tool is a Near Hasse Principle: a quadratic form may fail to represent infinitely many integers that it locally represents, but this set of exceptions has density 0 within the set of locally represented integers. [ABSTRACT FROM AUTHOR]

This study is the multidimensional inverse problem for a viscoelasticity equation. This inverse problem aims to reconstruct the displacement and convolutional kernel of the integral term simultaneously, from the measurement described as the overdetermination condition. The main result is the theorem of global unique solvability of the inverse problem in the class of functions continuously in the time variable t and analytic in the space variable. The methods of scales of Banach spaces of real analytic functions of variable and weight norms in the class of continuous functions are applied. [ABSTRACT FROM AUTHOR]

Background: Active infrared thermography is proved to be viable and attractive for non-destructive evaluation of interfacial defects like delaminations in a coating-substrate system. But it is a challenging task to accurately quantify small and deeply buried defects from thermal images, due to the inevitable effects of lateral heat diffusion and measurement noise. Objective: The aim of this work is to estimate the size and pattern of defects at the interface of a two-layer system with high accuracy and high reliability based on step heating thermography. Methods: To characterize the effect of defect on the heat flow, a virtual heat flux is assumed at the interface, which is reconstructed from measured surface temperature by solving a three-dimensional inverse problem. The inverse solution is obtained using the Green's function and regularization techniques, and then used for estimating the defect pattern by threshold segmentation. An improvement on computational efficiency is achieved by an iteratively substitution of nodal temperature. Results: Simulations with synthetic data generated by a finite element model validate the feasibility of this approach. Results obtained from experiments for an Aluminum oxide/steel system show the robustness of this approach, when temperatures are contaminated with measurement noise. Both the performance on estimation of various defect shapes and the effects of regularization are discussed. Conclusion: This study show that the present approach brings an improvement in accuracy and reliability for the estimation of size and pattern of defects with various diameter-to-depth ratios, in comparison with conventional techniques. [ABSTRACT FROM AUTHOR]

Measuring the size distribution and temperature of high-temperature dispersed particles, particularly in-flame soot, holds paramount importance across various industries. Laser-induced incandescence (LII) stands out as a potent non-contact diagnostic technology for in-flame soot, although its effectiveness is hindered by uncertainties associated with pre-determined thermal properties. To tackle this challenge, our study proposes a multi-parameter inversion strategy—simultaneous inversion of particle size distribution, thermal accommodation coefficient, and initial temperature of in-flame soot aggregates using time-resolved LII signals. Analyzing the responses of different heat transfer sub-models to temperature rise demonstrates the necessity of incorporating sublimation and thermionic emission for accurately reproducing LII signals of high-temperature dispersed particles. Consequently, we selected a particular LII model for the multi-parameter inversion strategy. Our research reveals that LII-based particle sizing is sensitive to biases in the initial temperature of particles (equivalent to the flame temperature), underscoring the need for the proposed multi-parameter inversion strategy. Numerical results obtained at two typical flame temperatures, 1100 K and 1700 K, illustrate that selecting an appropriate laser fluence enables the simultaneous inversion of particle size distribution, thermal accommodation coefficient, and initial particle temperatures of soot aggregates with high accuracy and confidence using the LII technique. [ABSTRACT FROM AUTHOR]