Forward Solver Research Articles

2D inversion of ultradeep electromagnetic logging measurements for look-ahead applications

Knowing formation ahead of the drill bit can help with hydrocarbon exploration and production, especially for low-angle and vertical wells. The advent of electromagnetic look-ahead technology makes it possible to interrogate formations tens of feet ahead of the drill bit. The formation can be very complex, such as fault and unconformity. However, existing approaches to recover the formation rely primarily on the 1D forward solver, which presumes that the formation is layered and transversely isotropic. For complicated formation, it will give rise to incorrect resistivity distribution. As a consequence, we have investigated and implemented a 2D pixel-based inversion algorithm to interpret the measurements for complex scenarios. We conduct a sensitivity study to illustrate the differences between look-ahead applications and look-around applications. To improve the look-ahead ability, we propose an effective technique to incorporate the prior information. In addition, we develop several examples to demonstrate the performance of the 2D inversion algorithm. It is found that using prior information as a reference model in the objective function enhances inversion performance significantly, and the proposed method can reasonably reconstruct some complex structures.

A Novel Approach for Identifying Hyper-Elastic Material Parameters of Cartilage based on FEM and Neural Networks

Cartilage damage and degeneration may lead to osteoarthritis for both animals and humans. Quantitative studies on the nonlinear hyper-elastic behavior of cartilages are essential to evaluate cartilage tissue deterioration. However, direct identification of the material behavior is not feasible. This paper presents a procedure to characterize the nonlinear mechanical behavior of the cartilage tissue by an inverse method using measurable structural quantities. First, a two-way neural network (NN) is established, which uses the fully trained forward problem neural network instead of the forward problem solver to generate training samples for inverse problem neural network. Moreover, based on the experimental data of the kangaroo shoulder joint, a nonlinear finite element (FE) model is then created to produce a dataset for training the forward network. Furthermore, intensive studies are conducted to examine the performance of our two-way NN method for the prediction of cartilage hyper-elastic material parameters by comparison with the direct inverse NN method. When only the direct inverse problem neural network is used for training, all samples are from FE simulations and the simulation time is 50.7 h, and the prediction time is tens of seconds. Besides, our two-way neural network calls the trained forward NN to collect training samples, and all the samples can be obtained in seconds, with which the simulation time is only 78 s. The predicted results are in good agreement with the experimental data, and the comparison shows that our two-way NN is an efficient and proficient method to predict the parameters for other biological soft tissues.

Calibrated Frequency-Division Distorted Born Iterative Tomography for Real-Life Head Imaging.

The clinical use of microwave tomography (MT) requires addressing the significant mismatch between simulated environment, which is used in the forward solver, and real-life system. To alleviate this mismatch, a calibrated tomography, which uses two homogeneous calibration phantoms and a modified distorted Born iterative method (DBIM), is presented. The two phantoms are used to derive a linear model that matches the forward solver to real-life measurements. Moreover, experimental observations indicate that signal quality at different frequencies varies between different antennas due to inevitably inconsistent manufacturing tolerance and variances in radio-frequency chains. An optimum frequency, at which the simulated and measured signals of the antenna present maximum similarity when irradiating the calibrated phantoms, is thus calculated for each antenna. A frequency-division DBIM (FD-DBIM), in which different antennas in the array transmit their corresponding optimum frequencies, is subsequently developed. A clinical brain scanner is then used to assess performance of the algorithm in lab and healthy volunteers' tests. The linear calibration model is first used to calibrate the measured data. After that FD-DBIM is used to solve the problem and map the dielectric properties of the imaged domain. The simulated and experimental results confirm validity of the presented approach and its superiority to other tomographic method.

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 and resource-demanding. 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. Compared to traditional algorithms, the proposed framework can obtain faster calculation speed and higher accuracy. It is believed that the proposed framework offers a new pattern for real-time heat flux inversion.

Multi-variance replica exchange SGMCMC for inverse and forward problems via Bayesian PINN

Physics-informed neural network (PINN) has been successfully applied in solving a variety of nonlinear non-convex forward and inverse problems. However, the training is challenging because of the non-convex loss functions and the multiple optima in the Bayesian inverse problem. In this work, we propose a multi-variance replica exchange stochastic gradient Langevin dynamics method to tackle the challenge of the multiple local optima in the optimization and the challenge of the multiple modal posterior distribution in the inverse problem. Replica exchange methods are capable of escaping from the local traps and accelerating the convergence; two chains with different temperatures are designed where the low temperature chain aims for the local convergence, and the target of the high temperature chain is to travel globally and explore the whole loss function entropy landscape. However, it may not be efficient to solve mathematical inversion problems by using the vanilla replica method directly since the method doubles the computational cost in evaluating the forward solvers (likelihood functions) in the two chains. To address this issue, we propose to make different assumptions on the energy function estimation and this facilities one to use solvers of different fidelities in the likelihood function evaluation. More precisely, one can use a solver with low fidelity in the high temperature chain while using a solver with high fidelity in the low temperature chain. Our proposed method significantly lowers the computational cost in the high temperature chain, meanwhile preserving the accuracy and converging very fast. We give an unbiased estimate of the swapping rate and give an estimation of the discretization error of the scheme. To verify our idea, we design and solve four inverse problems which have multiple modes. The proposed method is also employed to train the Bayesian PINN to solve the forward and inverse problems; faster and more accurate convergence has been observed when compared to the stochastic gradient Langevin dynamics (SGLD) method and vanilla replica exchange methods.

A high precision finite-element forward solver for surface nuclear magnetic resonance incorporating conductivity changes and surface-topography variations

Surface nuclear magnetic resonance (SNMR) is a geophysical method that can be used directly for detecting groundwater resources, and it has attracted the attention of many scholars. In this paper, we propose a new effective algorithm for numerical modeling of 3D SNMR data for arbitrary topography in a conductive medium. We adopt a total-field algorithm for solving the quasi-static variant of Maxwell’s equation and handle a complex-shaped loop source by discretizing the transmitter into electric dipoles, which can be further easily discretized into electric dipoles along the three directions of the Cartesian coordinate system. To solve the 3D SNMR forward-modeling problem quickly and accurately, a new element-integration system based on a new symmetric orthogonal rule is used for calculating the sensitivity (i.e., kernel) functions of all elements. The new rule is based on a special arrangement involving a cubic close-packed lattice structure and is characterized by fast convergence, positive weight, and symmetry. We apply the developed numerical algorithm to SNMR tomography of several typical hydrogeological models. The synthetic results show that higher precision can be achieved with few grids and nodes without increasing the computation time by using the new integration algorithm. In addition, we find that the topography and conductivity can affect the SNMR response, which needs to be considered while interpreting SNMR data.

Nested kernel degeneration-based boundary element method solver for rapid computation of eddy current signals

In this article, the nested kernel degeneration (NKD) based integral equation fast solver is proposed to compress the dense system matrix generated by boundary element method (BEM) for calculating eddy current signals in 3D nondestructive evaluation (NDE) problems. The near and diagonal block interactions are computed and stored by full matrices. However, for the far block interactions, not like in the single level kernel degeneration-based method that all the interaction matrices are compressed by the interpolations, the nested approximation framework makes the expression of the higher-level matrices by the ones at leaf levels and the transfer matrices between neighboring levels. Different integral kernels are degenerated and nested to ensure the performance of NKD based method. The robustness and efficiency of the proposed solver are validated and verified by making numerical predictions and comparing them with the ones achieved by analytical method, numerical methods and experiments. Our numerical experiments show that the NKD solver is more efficient than the butterfly solver (multilevel adaptive cross approximation algorithm (MLACA)) for electrically small eddy current NDE problem. With the NKD based method, O(N) computational complexity is achieved for multiscale low frequency eddy current simulations without sacrificing accuracy, where N is the number of unknowns. So far, to our best knowledge, the proposed method with linear complexity is the most efficient integral equation-based BEM forward solver to predict the eddy current signals from cracks and flaws.

Conservative Finite Element Modeling of EEG and MEG on Unstructured Grids.

For interpretation of electroencephalography (EEG) and magnetoencephalography (MEG) data, multiple solutions of the respective forward problems are needed. In this paper, we assess performance of the mixed-hybrid finite element method (MHFEM) applied to EEG and MEG modeling. The method provides an approximate potential and induced currents and results in a system with a positive semi-definite matrix. The system thus can be solved with a variety of standard methods (e.g. the preconditioned conjugate gradient method). The induced currents satisfy discrete charge conservation law making the method conservative. We studied its performance on unstructured tetrahedral grids for a layered spherical head model as well as a realistic head model. We also compared its accuracy versus the conventional nodal finite element method ( P1 FEM). To avoid modeling singular sources, we completed our computations with a subtraction approach; the derived expression for the MEG response different from earlier published and involves integration of finite quantities only. We conclude that although the MHFEM is more computationally demanding than the P1 FEM, its use is justified for EEG and MEG modeling on low-resolution head models where P1 FEM loses accuracy.

Inverse problem solver for multiple light scattering using modified Born series

The inverse scattering problem, whose goal is to reconstruct an unknown scattering object from its scattered wave, is essential in fundamental wave physics and its wide applications in imaging sciences. However, it remains challenging to invert multiple scattering accurately and efficiently. Here, we exploit the modified Born series to demonstrate an inverse problem solver that efficiently and directly computes inverse multiple scattering without making any assumptions. The inversion process is based on a physically intuitive approach and can be easily extended to other exact forward solvers. We utilize the proposed method in optical diffraction tomography and numerically and experimentally demonstrate 3D reconstruction of optically thick specimens with higher fidelity than those obtained using conventional methods based on the weak scattering approximation.

Contracting Electromagnetic Full-Wave Inversion of 2-D Inhomogeneous Objects With Irregular Shapes Based on the Hybrid SESI Forward Solver

An efficient two-dimensional (2-D) electromagnetic (EM) full-wave inversion (FWI) method based on the hybrid spectral-element spectral-integral (SESI) forward solver is proposed. In the forward scattering computation, the scalar Helmholtz equation is discretized and solved by the spectral element method (SEM). Its computational domain is truncated by a circle on which a 2-D surface integral equation is performed and the spectral-integral method (SIM) is used to accelerate its solution. Transmitters and receivers are placed outside the circular boundary and they are connected to the equivalent current on the circle by Green’s functions. In each iteration of FWI, the sensitivity matrix is updated by the electric field values solved by SESI only in the internal nodes inside the computation domain. Then the conjugate gradient (CG) method is used to solve for the relative permittivity and conductivity values in the corresponding interior elements. Meanwhile, the structural consistency constraint (SCC) algorithm is adopted to gradually compress the inversion domain and guarantee the inversion accuracy. Several numerical experiments are carried out not only to show the computation efficiency of the SESI forward solver but also to verify the correctness of the iterative inversion procedure as well as the effectiveness of SCC.

Deep Learning-Assisted Real-Time Forward Modeling of Electromagnetic Logging in Complex Formations

Higher dimensional (i.e., 2-D and 3-D) modeling is indispensable to correctly evaluate the responses of electromagnetic (EM) logging tools in complex formation environments. However, limited by the high computational cost of rigorous modeling, such as the finite-difference method and the finite-element method, the real-time applications in the well logging industry primarily rely on the 1-D forward solver, which would result in erroneous formation evaluation for complex scenarios. As a result, aiming at realizing fast modeling for EM logging tools in complex formations, this letter proposes a general framework assisted by deep neural networks (DNNs). The framework consists of three modules: earth model classification, parameter extraction, and surrogate construction. Separate DNNs are trained and tested for different modules. The accuracy and efficiency of the DNN-assisted fast modeling are validated by several experiments. This study finds that the fast modeling assisted by DNNs is able to calculate the tool responses and reconstruct the subsurface formations in real time.

A Deep Learning-Based GPR Forward Solver for Predicting B-Scans of Subsurface Objects

The forward full-wave modeling of ground-penetrating radar (GPR) facilitates the understanding and interpretation of GPR data. Traditional forward solvers require excessive computational resources, especially when their repetitive executions are needed in signal processing and/or machine learning algorithms for GPR data inversion. To alleviate the computational burden, a deep learning-based 2D GPR forward solver is proposed to predict the GPR B-scans of subsurface objects buried in the heterogeneous soil. The proposed solver is constructed as a bimodal encoder-decoder neural network. Two encoders followed by an adaptive feature fusion module are designed to extract informative features from the subsurface permittivity and conductivity maps. The decoder subsequently constructs the B-scans from the fused feature representations. To enhance the network's generalization capability, transfer learning is employed to fine-tune the network for new scenarios vastly different from those in training set. Numerical results show that the proposed solver achieves a mean relative error of 1.28%. For predicting the B-scan of one subsurface object, the proposed solver requires 12 milliseconds, which is 22,500x less than the time required by a classical physics-based solver.

Simultaneous Measurement of Radius and Coating Thickness of Finite Length Metal Rods by Eddy Current Surface Probes

The paper proposes an efficient inverse method for simultaneous measurement of radius and coating thickness of a finite length metal rod, using the output signal of a surface eddy current probe. The proposed method adopts a physical model for iterative solution of the inverse problem. It utilizes a hybrid conjugate gradient optimization algorithm in the inversion process, and adopts an efficient forward problem solver to obtain the predicted probe output signal in each iteration. The solution of the forward problem is based on the truncated region eigenfunction expansion method and uses the even/odd parity solution decomposition to treat the governing boundary value problem. The main feature of the proposed inversion method is its robustness in the presence of noise regardless of the choice of the initial values. Experimental and simulation results are provided to demonstrate the validity of the proposed methods for solving the forward and inverse problems. It has been shown that the rod radius and coating thickness can be measured with an error less than 5% of the true value when the probe is located at the half distance from the edge of the rod.

Factorization method for inverse time-harmonic elastic scattering with a single plane wave

<p style='text-indent:20px;'>This paper is concerned with the factorization method with a single far-field pattern to recover an arbitrary convex polygonal scatterer/source in linear elasticity. The approach also applies to the compressional (resp. shear) part of the far-field pattern excited by a single compressional (resp. shear) plane wave. The one-wave factorization is based on the scattering data for a priori given testing scatterers. It can be regarded as a domain-defined sampling method and does not require forward solvers. We derive the spectral system of the far-field operator for rigid disks and show that, using testing disks, the one-wave factorization method can be justified independently of the classical factorization method.</p>

Bayesian Seismic Azimuth-Difference Inversion of Horizontal Transversely Isotropic Media for Low-Frequency Component of Fracture Weaknesses in Laplace-Fourier Domain

Fracture weakness is one of the most important anisotropic parameters used to characterize the fractures and identify the fluids. The model of horizontal transversely isotropic (HTI) medium is usually utilized in seismic azimuthal inversion for the fracture weaknesses. Low-frequency component of fracture weaknesses plays a significant role in seismic fracture characterization and fluid identification due to the deficiency of low-frequency component in acquired seismic azimuthal data. The commonly used approaches to estimate low-frequency component include the smoothing model constraints and the damped wave field in complex-frequency domain. Following the Bayesian framework, we propose a novel approach used for compensating the low-frequency component of fracture weaknesses in Laplace-Fourier domain. Firstly, we reconstruct the seismic forward solver in Laplace-Fourier domain to obtain the low-frequency component of fracture weaknesses. Then, we propose a method of seismic azimuth-difference inversion for fracture weaknesses in Laplace-Fourier domain in a Bayesian framework. Finally, both synthetic and field data examples are used to demonstrate the superiority and stability of the proposed inversion approach. Compared with the conventional inversion approach, the proposed approach can reduce the dependence on the initial model of model parameters and weaken the effect of missing low-frequency components of fracture weaknesses in azimuthal seismic data, and it may help to improve the inversion accuracy of fracture weaknesses and reduce the uncertainty of inversion results.

Rapid Surrogate Modeling of Electromagnetic Data in Frequency Domain Using Neural Operator

The efficiency of solving geophysical inverse problem largely relies on the efficiency of solving the corresponding forward problem. As for electromagnetic (EM) data forward modeling in frequency domain, the conventional numerical methods, e.g. finite difference method (FDM), discretize the governing equations resulting a large linear system which is usually expensive to solve. Meanwhile, for inversion iteration we normally do not need to solve the forward problem in high precision. Thus a rapid surrogate modeling approach which uses the neural network is promising for replacing the forward modeling module in the inversion scheme. Here we proposed an algorithm which uses the neural operator to solve the EM data modeling problem in the frequency domain. To develop a surrogate model for EM data forward problem, we introduce an extended Fourier neural operator (EFNO) that enables the calculation at least 100 times faster than the conventional FDM solver while maintaining good precision. Moreover, by adding a sub-network the proposed neural operator has good generalization which has the capacity of predicting solution at any site locations and frequencies. Due to the discretization-invariance of Fourier neural operator, the neural operator trained on coarse grids can easily transfer to fine grids with only retraining part of parameters, resulting in a super-resolution prediction capability. We test our proposed method with 2-D and 3-D magnetotelluric (MT) data modeling problems, demonstrating that the EFNO has great potentials for severing as a general rapid surrogate forward solver in EM data inversion scheme.

A parallel domain decomposition algorithm for fluid-structure interaction simulations of the left ventricle with patient-specific shape

<abstract> <p>In this paper, we propose a scalable parallel algorithm for simulating the cardiac fluid-structure interactions (FSI) of a patient-specific human left ventricle. It provides an efficient forward solver to deal with the induced sub-problems in solving an inverse problem that can be used to quantify the interested parameters. The FSI between the blood flow and the myocardium is described in an arbitrary Lagrangian-Eulerian (ALU) framework, in which the velocity and stress are assumed being continuous across the fluid-structure interface. The governing equations are discretized by using a finite element method and a fully implicit backward Eulerian formula, and the resulting algebraic system is solved by using a parallel Newton-Krylov-Schwarz algorithm. We numerically show that the algorithm is robust with respect to multiple model parameters and scales well up to 2300 processor cores. The ability of the proposed method to produce qualitatively true prediction is also demonstrated via comparing the simulation results with the clinic data.</p> </abstract>

Deep-Learning-Based Calibration in Contrast Source Inversion Based Microwave Subsurface Imaging

A deep-learning (DL)-based data calibration technique applied to quantitative microwave inverse-scattering analysis is presented. This technique aims at subsurface inspection for the buried object under a concrete road or soil. The inverse-scattering analysis provides a complex permittivity profile, which is useful for object identification such as air gap or water. Contrast source inversion (CSI) is one of the most promising inverse-scattering methods. This method is capable of avoiding the iterative use of highly computational forward solvers. However, when applied to the measured data, an appropriate calibration capable of converting measured data to simulation data is required. In this work, a DL-based calibration suitable for nonlinear inverse problems is proposed. Its efficiency is experimentally demonstrated using a concrete cylinder containing water with different salinities.

On the Analytical Inversion of Solver Matrices Used in Numerical Approximations for the Diffusion Equation

The goal of this paper is to introduce an analytical approach for the inversion of nxn solver matrices, which are typically used in Finite Difference Method approximations. In the present case, they are used to solve the Diffusion Equation numerically, since in many physics and engineering fields, partial differential equations cannot be solved analytically. The method presented in this work is primarily formulated for cylindrical coordinates, which are often used in Gas Release Experiments as those described in [8]. However, it is possible to introduce a generalized method, which also allows solutions for Cartesian solvers. The advantage of having the explicit inverse is considerable, since the computational effort is reduced. In this paper we also carry out an investigation on the eigenvalues of the backward and forward solver matrix in order to determine an optimal range for the discretization parameters.

SymPhas—General Purpose Software for Phase‐Field, Phase‐Field Crystal, and Reaction‐Diffusion Simulations

AbstractThis work develops a new open source application programming interface (API) and software package called SymPhas for simulations of phase‐field, phase‐field crystal, and reaction‐diffusion models, supporting up to three dimensions and an arbitrary number of fields. SymPhas delivers two novel program capabilities: 1) User specification of models from the associated dynamical equations in an unconstrained form and 2) extensive support for integrating user‐developed discrete‐grid‐based numerical solvers into the API. The capability to specify general phase‐field models is primarily achieved by developing a novel symbolic algebra functionality that can formulate mathematical expressions at compile time; is able to apply rules of symbolic algebra such as distribution, factoring, and automatic simplification; and support user‐driven expression tree manipulation. A modular design based on the C++ template meta‐programming paradigm is applied to the symbolic algebra library and general API implementation to minimize application runtime and increase the accessibility of the API for third party development. SymPhas is written in C/C++ and emphasizes high‐performance capabilities via parallelization with OpenMP and the C++ standard library. SymPhas is equipped with a forward Euler solver and a semi‐implicit Fourier spectral solver. Sample implementations and simulations of several phase‐field models are presented, generated using the semi‐implicit Fourier spectral solver.