Forward Solver Research Articles

3D general-measure inversion of crosswell EM data using a direct solver

Abstract We present a three-dimensional (3D) general-measure inversion scheme of crosswell electromagnetic (EM) data in the frequency domain with a direct forward solver. In the forward problem, we discretised the EM Helmholtz equation by the staggered-grid finite difference (SGFD) scheme and solved it using the Intel MKL PARDISO direct solver. By applying a direct solver, we simultaneously solved the multisource forward problems at a given frequency. In the inversion, we integrated a general measure of data misfit and model constraints with linearised least-squares inversion. We reconstructed a model with blocky features by selecting the appropriate measure parameters and model constraints. We used the adjoint equation method to explicitly calculate the Jacobian matrix, which facilitated the determination of an appropriate initial value for the regularisation coefficient in the objective function. We illustrated the inversion scheme using synthetic crosswell EM data with a general measure, the L2 norm, and, specifically, two mixed norms.

Approximation error model (AEM) approach with hybrid methods in the forward-inverse analysis of the transesterification reaction in 3D-microreactors

This work advances the approximation error model approach for the inverse analysis of the biodiesel synthesis using soybean oil and methanol in 3D-microreactors. Two hybrid numerical-analytical approaches of reduced computational cost are considered to offer an approximate forward problem solution for a three-dimensional nonlinear coupled diffusive-convective-reactive model. First, the Generalized Integral Transform Technique (GITT) is applied using approximate non-converged solutions of the 3D model, by adopting low truncation orders in the eigenfunction expansions. Second, the Coupled Integral Equations Approach (CIEA) provides a reduced mathematical model for the average concentrations, which leads to inherently approximate solutions. The AEM approach through the Bayesian framework is illustrated in the simultaneous estimation of kinetic and diffusion coefficients of the transesterification reaction. For this purpose, the fully converged GITT results with higher truncation orders for the 3D partial differential model are employed as reference results to define the approximations errors. The results highlight that either the non-converged solutions via GITT or the reduced model solution obtained via CIEA, when taking into account the model error, are robust and cost-effective alternatives for the inverse analysis of nonlinear convection–diffusion-reaction problems.

Flaw characterization in conductive media based on pulsed Eddy current measurements: A fast non‐iterative inversion approach

This paper proposes a non-iterative procedure for flaw(s) characterization based on Pulsed Eddy Current Testing (PECT) signals analysis. The adopted inversion strategy is based on the use of supervised statistical learning algorithms. A numerical forward solver, based on the Finite Integration Technique (FIT), is used for the generation of the training data (the input-targets couples of the learning algorithm). Predictions are then carried out in almost real-time using a non-linear kernel based regression method, known as kernel ridge regression. It turns out that the direct fit of the regression model to the raw PECT signals may lead to poor prediction accuracy due to the large cardinality of PECT signals. To remedy this problem, an adaptive sampling strategy has been adopted in this work. The performance of the proposed methodology is discussed and compared with solutions proposed in the literature.

Fast Geometric Multigrid Preconditioned Finite- Difference Frequency-Domain Method for Borehole Electromagnetic Sensing

To model the responses of borehole electromagnetic sensing in complicated geological environments, the geometric multigrid preconditioned finite-difference frequency-domain (FDFD) method is examined on its numerical performance as a fast forward modeling solver in this work. The geometric multigrid method is employed as a preconditioner for the biconjugate gradient stabilized method to efficiently solve the linear system resulting from the finite-difference method based on the staggered Yee’s cell. The Gauss–Seidel method is utilized as both presmoother and postsmoother for the multigrid method, and the bilinear approach is employed to construct the restriction and prolongation operators between coarse and fine grids. The F-cycle coarsening scheme is used to balance the convergence and computational efficiency of the multigrid method. Numerical examples of borehole electromagnetic sensing are included to examine the performance of the method, which not only demonstrates its accuracy but also shows its capability as a fast solver for practical application. The geometric multigrid preconditioned FDFD method can work as a good candidate for 3-D modeling-based interpretation and evaluation for borehole electromagnetic sensing.

Inversion of IP-Affected TEM Responses and Its Application in High Polarization Area

The transient electromagnetic (TEM) responses measured in areas with highly polarizable media commonly show induced polarization (IP) effects, which poses difficulties to the TEM data interpretation and inversion with conventional methods. When present, the IP effects break the monotony in TEM decay curves and can even cause sign reversals, and lead to the singularity and non-monotony of inversion method. It is therefore important to take the IP effects into account when processing those TEM data. In this paper, a new inversion method is developed to invert TEM data with IP effects. The Cole-Cole model was introduced to the forward modeling solver so that the IP effects can be accurately calculated. The regularization terms were adapted to the objective function and a global optimization method, the particle swarm optimization algorithm, was used to solve the optimization problem, which weakened the singularity and non-monotony of the inversion greatly. Field data were collected in Wulong gold mine where IP effects were detected. The data were inverted using the method developed in this paper, and the inversion results were able to identify the depth extension of NO. 163 ore-bearing quartz veins and their associated fine-grained diorite veins. The global inversion method in this paper primarily solved the inversion problem of IP-affected TEM responses, and recovered the polarization parameters, which can be used as an aid to the geological interpretation.

Reconstructing the conductivity profile of a graphite block using inductance spectroscopy with data-driven techniques

The radial depth profile of the electrical conductivity of the graphite channels in the UK's advanced gas-cooled reactors (AGRs) can be reconstructed and estimated by solving a non-linear optimisation problem using the mutual inductance spectra of a set of coils. This process is slow, as it requires many iterations of a forward solver. Alternatively, a data-driven approach can be used to provide an initial estimate for the optimisation algorithm, reducing the amount of time it takes to solve the ill-posed inverse problem. Two data-driven approaches are compared: multi-variable polynomial regression (MVPR) and a convolutional neural network (CNN). The training data are generated using a finite element (FE) model and superimposed on a noise floor in the interval [20, 60] dB of the weakest amplitude point in the corresponding spectrum. A total of 5000 simulated datasets are generated for training. The results on smoothed test data show that the two models have a comparable mean percentage error norm of 17.8% for the convolutional neural network and 17.3% for multivariable polynomial regression. A further 500 unsmoothed profiles are tested in order to assess the performance of each algorithm on conductivity distributions where the conductivity of each layer is independent of another. The performance of both algorithms is then assessed on reactor-type test data. The results show that the two data-driven algorithms have a comparable performance when estimating the electrical conductivity depth profile of a typical reactor-type distribution, as well as vast deviations. More generally, it is thought that data-driven approaches for depth profiling of some electromagnetic quantity have the potential to be applied to other ill-posed inverse problems where speed is a priority.

Computational image reconstruction for multi-frequency diffuse optical tomography

Diffuse optical tomography (DOT) is promising to investigate functional and structural information in biological tissue by involving the forward problem and inverse solution. The work within this study develops an image reconstruction algorithm based on frequency-domain DOT using multi-frequency. In the forward problem, numerical simulation performs a diffusion equation to compute the light radiance distribution inside the model geometry with the assumed light intensity and optical properties. While in inverse solution, the absorption and reduced scattering coefficients are reconstructed utilizing light boundary data. To test the algorithm, several simulation cases, such as different contrast ratios between background and inclusion, inclusion size, as well as single and multi-frequency using one to three frequencies each and simultaneously are implemented. For verifying the image reconstruction results of multi-frequency, an objective numerical assessment, namely contrast and size detail is applied with a threshold value to specify the visible inclusion inside the geometry. Reconstructed images indicate that the effectivity of two frequencies in multi-frequency with 100 and 150 MHz has similar results as in single-frequency with 200 MHz. Therefore, the more number of frequencies may not enhance the contrast and image quality.

The design of high-viscosity oil reservoir model based on the inverse problem solution

We propose an approach to the solution of the automatic history-matching problem for constructing models of high-viscosity oil reservoirs. It is based on a special parameterization of the reservoir model with the use of not only the physical but also the geometric parameters and minimization of the functional of residual between observed and computed data of cumulative oil production, bottomhole pressure, oil rate, water and oil cut. In order to obtain physically reasonable values of parameters and reduce the number of iterations when solving the inverse problem, the adaptive regularization is used. We use the finite element method on non-conforming meshes with hexahedral cells to solve the forward problem. The forward problem solution is verified by comparison with the results presented in the 10th SPE Comparative Solution Project. The applicability and effectiveness of the proposed approach are demonstrated on the simulated and real field data obtained on the site of one of the high-viscosity oil fields. Using the simulated data obtained for the complex realistic petroleum reservoir model, we show that the proposed approach allows us to obtain distributions of absolute permeability, porosity, and oil in the reservoir volume as well as to estimate oil reserves and their changes with time. The results of numerical experiments performed using the real field data show that the approach proposed for the history-matching problem solution enables us to make reliable production forecasts. In the forecast period, the deviations of the predicted total oil production from the observed one do not exceed 1.5%.

Parallel solution of 3D forward and inverse problems of airborne electromagnetic survey

The paper is devoted to the comparison of different approaches to parallel solution of 3D forward and inverse problems of airborne electromagnetic (EM) surveys when multidimensional processing of data obtained on large areas is needed. The mathematical models for the solution of 3D forward and inverse problems and different methods of their parallelization are considered. The results of numerical experiments for the airborne EM survey problem, which is typical of prospecting the kimberlite bodies in Eastern Siberia, are presented. It is shown that the combination of different techniques of parallelization in a distributed computational system, taking into account the specifics of the numerical schemes implementation for forward and inverse problems solution, makes it possible to achieve the speedup close to linear.

Simulation of Time-Lapse Resistivity Logging During Two-Phase Well Testing in Petroleum Reservoirs

Well testing methods are widely used in petroleum industry to identify reservoir parameters in-situ, at real reservoir flow conditions. Two-phase tests with water injection extend capabilities of well testing to evaluation of oil-water flow characteristics through the solution of an inverse problem based on well measurements during the test. Near-wellbore water saturation is used as one of the informative measured parameters. However, water saturation is actually not directly measured on wells. Instead, it is interpreted from time-lapse well logging data, with resistivity logging being one of the options. This means that forward and inverse problems are to be extended to incorporate direct simulations of resistivity logging. In this paper we address the forward problem. It is formulated and solved for different types of logging tools, both for open hole and through the casing. A numerical algorithm and software implementation have been developed for repeated solution of an axisymmetric forward problem of resistivity logging – a 2D elliptical partial differential equation for electrical potential. Distributions of formation resistivity at the moments of logging are calculated from distributions of water saturation and salinity. The latter are computed through numerical solution of the forward problem of well testing – a two-phase oil-water flow in the reservoir with changing salinity of the water phase, with full account for real fluid and reservoir properties. The software implementation of the resistivity logging problem has been tested in comparison with the known theoretical solutions. Examples are presented to show how time-lapse changes in resistivity logging data reflect the specifics of the two-phase flow with changing water salinity during the well test.

A Comparison of Time-Domain and Frequency-Domain Microwave Imaging of Experimental Targets

An existing forward-backward time-stepping (FBTS) time-domain quantitative imaging algorithm is augmented with a discontinuous Galerkin method (DGM) forward solver. The resulting DGM-FBTS imaging algorithm is capable of solving the electromagnetic inverse scattering problem using high-order expansions of the fields and unknown target constitutives, decoupling the solution from the unstructured grid. DGM-FBTS provides a time-domain alternative to our previous development of flexible DGM-based frequency-domain imaging codes, and enables us to present a comparison of the performance of time-domain and frequency-domain imaging algorithms for synthetic and experimental targets. For experimental targets, a procedure for obtaining calibrated time-domain data from frequency-domain broadband VNA-collected data is explained and applied to a system with a reduced number of transmitters and receivers to highlight the potential benefits of time-domain methods. Results highlight the potential capabilities of time-domain experimental imaging using the same hardware configuration used to collect broadband frequency-domain measurements and suggest future work on hybrid frequency- and time-domain imaging algorithms and efforts to improve the computational performance of DGM-FBTS.

Particle Size Characterization in Liquid–Solid Dispersion With Aggregates by Broadband Ultrasound Attenuation

Ultrasound attenuation methods for particle size characterization generally assume that the dispersed phases are uniformly dispersed in the continuous phase. However, the aggregation of particles in dispersions affects ultrasound attenuation, which limits the estimation accuracy of the attenuation model. A broadband ultrasound method is proposed, which utilizes a fractal approach to address the acoustic attenuation influence of particle aggregation. First, a chirp-based swept-frequency mode is designed to demodulate the ultrasonic attenuation at a low time cost to ensure the consistency of multifrequency attenuation. Next, considering the effect of particle aggregation on attenuation, the ultrasound attenuation model based on Faran scattering theory is modified by the fractal approach. Then, the inversion of particle size characterization is formulated as an optimization problem of nonlinear and nonconvex functions. An iterative optimization method is proposed to solve the ill-posed inverse problem, in which covariance matrix adaptive evolutionary strategy (CMA-ES) is developed as its inverse solver, and the fractal modified attenuation model is developed as its forward solver. Finally, the experiments were carried out with a mixture of micron-sized polystyrene particles and water. Experimental results indicate that the attenuation predicted by the proposed attenuation model is in good agreement with that measured in liquid–solid dispersion with particle aggregation, and the effectiveness of CMA-ES based on multiple distribution functions is verified for the inversion of particle size characterization.

Predicting Surface Heat Flux on Complex Systems Via Conv-LSTM

Existing algorithms with iterations as the principle for 3D inverse heat conduction problems (IHCPs) are usually time-consuming. With the recent advancements in deep learning techniques, it is possible to apply the neural network to compute IHCPs. In this paper, a new framework based on Convolutional-LSTM is introduced to predict the transient heat flux via measured temperature. The inverse heat conduction models concerned in this work have 3D complex structures with non-linear boundary conditions and thermophysical parameters. In order to reach high precision, a forward solver based on the finite element method is utilized to generate sufficient data for training. The fully trained framework can provide accurate predictions efficiently once the measured temperature and models are acquired. It is believed that the proposed framework offers a new pattern for real-time heat flux inversion.

Solution of 2D MIT Forward Problem by Considering Skin and Proximity Effects in Coils

Previous studies on the forward problem of magnetic induction tomography (MIT) have used simplified Maxwell's equations which assume a constant and position-independent total current density (TCD) inside the coils (ignoring skin effect). Moreover, they assume that TCD is independent of relative position of the coils (ignoring proximity effect). This article presents an improved finite element (FE) modeling for the two-dimensional (2D) forward problem of MIT by incorporating skin and proximity effects in the exciter and sensor coils. Consideration of skin and proximity effects requires the use of a position-dependent TCD in Maxwell's equations inside the coil domain. The FE method implementation of the improved forward method is validated with a simple analytical test problem. To evaluate the performance of the improved method in possible medical and industrial applications with low conductivity regions, a 16 coils 2D MIT system is modeled and the FE method is employed to solve the forward problem using a synthetic phantom. Results show the difference between the real parts of induced voltages obtained from the early and improved method falls into the range which is meaningful in terms of achievable conductivity contrast from the improved one. This discrepancy can generate considerable errors in image reconstruction. By considering skin and proximity effects, the improved method generates lower voltages in the coils suggesting the use of more precise hardware to detect inclusion. Findings show the importance and necessity of using improved method for modeling 2D MIT coils especially at low conductivity applications where measured signals are inherently weak.

Enhanced FEM-Based DBIM Approach for Two-Dimensional Microwave Imaging

The distorted Born iterative method (DBIM) is widely used in microwave imaging to solve the electromagnetic inverse scattering problem iteratively based on the distorted Born approximation. Various methods are used for the forward solver in the DBIM to simulate wave propagation and scattering. In this communication, a new matrix building technique exploiting the basis functions of the finite element method (FEM) is used to create the inverse-problem matrix in a 2-D DBIM algorithm. The proposed approach provides great efficiency as the entries of this matrix are readily calculated from already precomputed matrices in the FEM procedure without the need for interpolation to calculate the field. The FEM-based DBIM is validated with both simulated data and experimental data from a microwave tomography (MWT) system. Results confirm that the proposed FEM implementation can be an efficient and accurate forward solver for this MWT system.

A Hybrid Volume-Surface-Wire Integral Equation for the Anisotropic Forward Problem in Electroencephalography

Solving the electroencephalography (EEG) forward problem is a fundamental step in a wide range of applications including biomedical imaging techniques based on inverse source localization. State-of-the-art electromagnetic solvers resort to a computationally expensive volumetric discretization of the full head to account for its complex and heterogeneous electric profile. The more efficient, popular in biomedical imaging circles, but unfortunately oversimplifying Boundary Element Method (BEM) relies instead on a piecewise-uniform approximation that severely curbs its application in high resolution EEGs. This contribution lifts the standard BEM contraints by treating the local anisotropies with adequate wire and thin volume integral equations that are tailored to specific structures of the fibrous white matter and the inhomogeneous skull. The proposed hybrid integral equation formulation thereby avoids the full volumetric discretization of the head medium and allows for a realistic and efficient BEM-like solution of the anisotropic EEG forward problem. The accuracy and flexibility of the proposed formulation is demonstrated through numerical experiments involving both canonical and realistic MRI-based head models.

Application of the generalized multiscale finite element method in an inverse random source problem

In this work, an inverse random source problem in the fractional diffusion equation with heterogeneous medium is considered. The measurements used are the statistical moments of the realizations of the single point data u(x0,t,ω). We build the representation of the solution u in integral sense, then prove that the unknowns can be bounded by the moments theoretically. For the numerical reconstruction, to handle the highly heterogeneous medium, the generalized multiscale finite element method (GMsFEM) will be employed in the forward problem solver. With the simulated data, we establish an iterative algorithm of regularized Levenberg-Marquardt type, and some numerical results generated from this algorithm are displayed.

Time-dependent lowest term estimation in a 2D bioheat transfer problem with nonlocal and convective boundary conditions

ABSTRACT A solution method for an inverse problem of determining the time-dependent lowest order coefficient of the 2D bioheat Pennes equation with nonlocal boundary conditions and total energy integral overdetermination condition recently appeared in literature is analysed and improved. Improvements include convective boundary condition into the model, the development of an accurate forward solver, accurate determination of total energy, and the proposal of a method for the numerical treatment of the inverse problem from noisy data. In the method, the 2D bioheat Pennes equation is solved by the method of lines based on a highly accurate pseudospectral approach, and sought coefficient values are estimated by the Levenberg–Marquardt method with the discrepancy principle as stopping rule. Numerical experiments are reported to illustrate effectiveness of the proposed method.

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.

Forward and inverse design of kirigami via supervised autoencoder

Machine learning (ML) methods have recently been used as forward solvers to predict the mechanical properties of composite materials. Here, we use a supervised-autoencoder (sAE) to perform inverse design of graphene kirigami, where predicting the ultimate stress or strain under tensile loading is known to be difficult due to nonlinear effects arising from the out-of-plane buckling. Unlike the standard autoencoder, our sAE is able not only to reconstruct cut configurations but also to predict mechanical properties of graphene kirigami and classify the kirigami witheither parallel or orthogonal cuts. By interpolating in the latent space of kirigami structures, the sAE is able to generate novel designs that mix parallel and orthogonal cuts, despite being trained independently on parallel or orthogonal cuts. Our method allows us to both identify novel designs and predict, with reasonable accuracy, their mechanical properties, which is crucial for expanding the search space for materials design.