General Inversion Algorithm Research Articles

Linear Time Electromigration Analysis Based on Physics-Informed Sparse Regression

In this work, we propose a novel physics-informed sparse regression (PISR) framework to solve stress evolution (described by Korhonen’s equations) in general multi-segment wires using an unsupervised learning scheme. Unlike the existing physics-informed neural network framework (PINN), the PISR method trains the trainable weights through the Moore-Penrose generalized inverse algorithm used in extreme learning machine (ELM), which is extremely faster than the back-propagation algorithm. To improve the accuracy of PISR for complex multi-segment interconnects, we employ domain decomposition schemes in both space and time. For each subdomain, we use different trainable weights but the same shared neural network to represent each subsolution, which leads to more efficient memory usage. Furthermore, we propose to use sparse matrix techniques to accelerate the training speed of the PISR method and prove that the resulting PISR has linear time complexity for analyzing tree-structured interconnects. Finally, we divide the time into many time internals and apply an autoregressive model to simulate each time interval in sequence to further improve scalability and reduce memory cost so that the PISR method can perform EM analysis for large-scale multi-segment interconnects. Experimental results on different kinds of interconnect structures show that the proposed PISR method has the same accuracy level as the numerical methods. The results on NT T-junctions interconnect trees show that the proposed PISR method indeed demonstrates true linear time complexity. Furthermore, PISR can deliver 8.9×, 20.6×, and 1284× speedups over the recently proposed semi-analytic method (ASOV), finite difference method accelerated with model order reduction (FDM-MOR), FDM for the interconnect with NT= 5000, respectively. Furthermore, we show that PISR also achieves an 818× speedup in training over the plain PINN method based on the traditional back-propagation algorithm.

Generalized inverse matrix-recurrent neural network data processing algorithm for multiwavelength pyrometer

A generalized inverse matrix-recurrent neural network (GIM-RNN) data processing algorithm for unknown emissivity was proposed to measure high-temperature by multiwavelength pyrometer (MWP). First, emissivity classification was realized quickly according to a solution from generalized inverse algorithm to underdetermined multiwavelength equation group. According to the relationship between the emissivity of 2 adjacent channels of the 6 channel thermometer used in this paper, 243 emissivity models (1 * 3 * 3 * 3 * 3 * 3) were designed for classification. Twelve of them were shown on the schematic diagram in the text. To make the figure clear and easy to observe, the prediction results of six common emissivity models were listed as the experimental results. Then, it was input into the corresponding RNN subnetwork to inverse temperature precisely. Simulation results showed that in the range of 1500 to 3000 K temperature, the relative error of the test set was within 1.0% for the network trained after classification by GIM, whereas the relative error of the test set was within 1.2% for the network trained without GIM classification. After 5.0% random noise was added to the inputting data, the relative error still was controlled within 1.5%, which reflected the good antinoise performance of the algorithm. Multispectral measurement data of rocket engine plumes is processed by the proposed algorithm in this manuscript. The inversion results are consistent with the theoretical results. It is indicated that the proposed algorithm has good adaptability to different materials. It is expected to become a general data processing algorithm for MWP.

NEURAL NETWORK INVERSION OF RESISTIVITY DATA FOR DETERMINATION OF DISTANCE TO A BED BOUNDARY

Accurate real-time estimation of a distance to the nearest bed boundary simplifies the steering of directional wells. For estimation of that distance, we propose an approach of pointwise inversion of resistivity data using neural networks based on two-layer resistivity formation model. The model parameters are determined from the tool responses using a cascade of neural networks. The first network calculates the resistivity of the layer containing the tool measure point. The subsequent networks take as input the tool responses and the model parameters determined with the previous networks. All networks are trained on the same synthetic database. The samples of that database consist of the pairs of model parameters and corresponding noisy tool responses. The results of the proposed approach are close to the results of the general inversion algorithm based on the method of the most-probable parameter combination. At the same time, the performance of the proposed inversion is several orders faster.

GMDS-ZNN Model 3 and its Ten-Instant Discrete Algorithm for Time-Variant Matrix Inversion Compared With Other Multiple-Instant Ones

The online time-variant matrix inversion problem has attracted extensive attention and study, because of its considerable appearance and application in scientific research and industrial production. For various control optimization problems, the demand for the high-precision and rapid-convergence of matrix inversion algorithm is increasing. It remains an ongoing challenge due to the rigorous requirements of precision and convergence of the algorithm. In this paper, on the basis of our previous works, by using Zhang neural network (ZNN) method, a continuous time-variant matrix inversion model, which is also a Getz-Marsden dynamic system (i.e., GMDS-ZNN model 3), is proposed. Besides, a general ten-instant Zhang et al discretization (ZeaD) formula is presented and investigated, with corresponding theoretical results being provided. Next, by applying this general formula to discretize the continuous time-variant matrix inversion model, a general ten-instant discrete time-variant matrix inversion (DTVMI) algorithm with the sixth-order precision is proposed. For comparison, four other DTVMI algorithms, with the second-, third-, fourth-, and fifth-order precisions, are also proposed and presented, respectively, by using other ZeaD formulas to discretize the continuous time-variant matrix inversion model. Besides, for the situation of coefficient matrix derivative being unknown, we provided the formula of estimating it with the fifth-order precision. With the help of the proposed matrix derivative estimation formula, the actual application field of GMDS-ZNN model 3 is expanded evidently. Finally, theoretical analyses and simulation experiment results highlight the effectiveness and accuracy of the proposed GMDS-ZNN model 3 and DTVMI algorithms.

The Inverse of a Multivector: Beyond the Threshold $$\varvec{p+q=5}$$ p + q = 5

The algorithm of finding an inverse multivector (MV) numerically and symbolically is of paramount importance in the applied Clifford geometric algebra (GA) $$ Cl _{p,q}$$ . The first general MV inversion algorithm was based on matrix representation of MV. The complexity of calculations and the size of the answer in a symbolic form grow exponentially with the dimension $$n=p+q$$ . The breakthrough occurred when Lundholm and then Dadbeh found compact inverse formulas up to dimension $$n\le 5$$ . The formulas were constructed in a form of Clifford product of the initial MV and carefully chosen grade-negation counterparts. In this report we show that the grade-negated self-product method can be extended beyond the threshold $$n=5$$ if, in addition, properly constructed linear combinations of such MV products are used. In particular, we present compact explicit MV inverse formulas for algebras of vector space of dimension $$n=6$$ and show that they embrace all lower dimensional cases as well. For readers convenience, we have also given various MV formulas in a form of grade negations when $$n\le 5$$ .

Fast Nonlinear Generalized Inversion of Gravity Data with Application to the Three-Dimensional Crustal Density Structure of Sichuan Basin, Southwest China

Generalized inversion is one of the important steps in the quantitative interpretation of gravity data. With appropriate algorithm and parameters, it gives a view of the subsurface which characterizes different geological bodies. However, generalized inversion of gravity data is time consuming due to the large amount of data points and model cells adopted. Incorporating of various prior information as constraints deteriorates the above situation. In the work discussed in this paper, a method for fast nonlinear generalized inversion of gravity data is proposed. The fast multipole method is employed for forward modelling. The inversion objective function is established with weighted data misfit function along with model objective function. The total objective function is solved by a dataspace algorithm. Moreover, depth weighing factor is used to improve depth resolution of the result, and bound constraint is incorporated by a transfer function to limit the model parameters in a reliable range. The matrix inversion is accomplished by a preconditioned conjugate gradient method. With the above algorithm, equivalent density vectors can be obtained, and interpolation is performed to get the finally density model on the fine mesh in the model domain. Testing on synthetic gravity data demonstrated that the proposed method is faster than conventional generalized inversion algorithm to produce an acceptable solution for gravity inversion problem. The new developed inversion method was also applied for inversion of the gravity data collected over Sichuan basin, southwest China. The established density structure in this study helps understanding the crustal structure of Sichuan basin and provides reference for further oil and gas exploration in this area.

Operational modal analysis of three-dimensional structures by second-order blind identification and least square generalized inverse

Most complex engineering structures are three-dimensional in practice. The process of one-dimensional extending to three-dimensional is a challenge that must be conquered by Operational Modal Analysis (OMA) methods when these methods are applied to complex engineering applications supported by scientific researches. This study put forward a new three-dimensional structure OMA method based on Second-Order Blind Identification (SOBI) and general reversion of least square. Firstly, modal coordinates decomposition of one-dimensional structural vibration response signal with SOBI. Secondly, the reasons that modal parameters identified by SOBI including energy uncertainty, order uncertainty and modal missing are explained in theory. Thirdly, the SOBI algorithm is used to decompose the response signals of displacement of a direction whose vibration response is the largest, then the other two directions are calculated by using the least square generalized inverse algorithm, and the modal parameters of three-dimensional structures are identified by the matrix assembly method. Numerical simulation results in a cylindrical shell demonstrated that this novel method is practical and effective by applied to practice in OMA of three-dimensional structures, and robustness to Gauss measurement noise disturbances.

Efficient computation of the genomic relationship matrix and other matrices used in single‐step evaluation

Genomic evaluations can be calculated using a unified procedure that combines phenotypic, pedigree and genomic information. Implementation of such a procedure requires the inverse of the relationship matrix based on pedigree and genomic relationships. The objective of this study was to investigate efficient computing options to create relationship matrices based on genomic markers and pedigree information as well as their inverses. SNP maker information was simulated for a panel of 40 K SNPs, with the number of genotyped animals up to 30 000. Matrix multiplication in the computation of the genomic relationship was by a simple 'do' loop, by two optimized versions of the loop, and by a specific matrix multiplication subroutine. Inversion was by a generalized inverse algorithm and by a LAPACK subroutine. With the most efficient choices and parallel processing, creation of matrices for 30 000 animals would take a few hours. Matrices required to implement a unified approach can be computed efficiently. Optimizations can be either by modifications of existing code or by the use of efficient automatic optimizations provided by open source or third-party libraries.

Generalized Born inversion of seismic reflection data

Summary. Born inverse methods give accurate and stable results when the source wavelet is impulsive. However, in many practical applications (reflection seismology) an impulsive source cannot be realized and the inversion needs to be generalized to include an arbitrary source function. In this paper, we present a Born solution to the seismic inverse problem which can accommodate an arbitrary source function and give accurate and stable results. It is shown that the form of the generalized inversion algorithm reduces to a Wiener shaping ***filter, which is solved efficiently using a Levinson recursion algorithm. Numerical examples of synthetic and real field data illustrate the validity of our method.

Wavenumber Domain Generalized Inverse Algorithm for Potential Field Downward Continuation

AbstractDownward continuation is an important operation to potential field data processing and inversion, but its instability problem influences its application in many processing and inversion techniques. In this paper, through viewing the downward continuation of potential field as an inverse problem of upward continuation, we obtain a convolution type linear integral equation for downward continuation. Making use of the orthogonal symmetry characteristic of Fourier transform matrix, and combining the principles of singular value decomposition of matrix and generalized inverse, we proposed a stable generalized inverse method for downward continuation of potential field, called wavenumber domain generalized inverse algorithm, which doesn't need the computation for inverse matrix. It resolves the instability of potential field downward continuation of large depth. The applications of the method to the downward continuation of 3D theoretical model data and real magnetic field data give ideal results.

Two-Stage Algorithm for Inverting Structure Parameters of the Horizontal Multilayer Soil

The multilayer soil structure model is the fundament to design grounding systems. A fast algorithm is presented to invert the structure parameters of the horizontal multilayer soil. The procedure is divided into two independent stages. First, Fredholm equation of the first kind with respect to the apparent resistivity is solved by using the technology of decay spectrum, which can reduce the computation time greatly. Second, the structure parameter of soil is determined by the generalized Newton-Kantorovich method, which is more robust and less noise sensitive because of using the generalized inverse algorithm to solve the nonlinear equation system. The numerical results show the validities and main features of the proposed approach. The numerical results by using the proposed method are in a good agreement with ones from geophysical exploration.

Fast algorithm for inverting structure parameters of the horizontal multi-layer soil*

A fast algorithm is presented to invert the structure parameter of the horizontal multi-layer soil. The procedure is divided into two independent stages. First, Fredholm equation of the first kind with respect to the apparent resistivity is solved by the technology of decay spectrum to reduce computation time greatly. Second, the structure parameter of soil is determined by the generalized Newton-Kantorovich method, which is more robust and less noise sensitive because of using the generalized inverse algorithm to solve the nonlinear equation group. The numerical results show the validities and main features of the proposed approach.

Determination of three-layer earth model from Wenner four-probe test data

Based on the solution of the electric field inverse problem, an efficient method for determining three-layer earth models and parameters from Wenner's four-probe test data is presented, i.e., from a set of voltage values measured on the earth surface due to the current injected into the earth. The speed of the iterative convergence is increased by employing the generalized inverse algorithm for solving the relevant nonlinear system of equations, and using a suitable method for determining initial values for the iterative process. The earth structure is the field model for simulating grounding electrodes or grounding grids of substations. The determination of the structure is essential in the simulation and the design of grounding grids.

Pn wave velocities beneath the Tanzania Craton and adjacent rifted mobile belts, east Africa

P wave travel times from regional earthquakes recorded by the Tanzania Broadband Seismic Experiment have been inverted for long wavelength (>100 km) Pn velocity variations beneath Tanzania using a generalized inverse algorithm. Pn velocities, on average, are 8.40 to 8.45 km/s beneath the center of the Tanzania Craton, 8.30–8.35 km/s beneath the terminus of the Eastern Branch of the rift system, and 8.35–8.40 km/s beneath the Western Branch. These velocities indicate that there are no broad (>100 km wide) thermal anomalies in the uppermost mantle beneath areas of rifting in Tanzania, and suggest that thermal anomalies present deeper in the mantle have not yet reached the base of the crust.

Electrical impedance tomography reconstruction algorithm based on general inversion theory and finite element method.

A strict EIT reconstruction algorithm, the general inversion algorithm (GIA) is presented. To improve the noise performance, the algorithm is modified by attenuating the condition number of the forward matrix F and implemented using an improved FEM scheme, to obtain the 2D image of impedance change (dynamic image). This modified general inversion algorithm (MGIA) can be used on a larger dimension FEM model (248 elements) and is more practical than the GIA. When implementing this algorithm in computer simulation and in a physical phantom, it is found that the MGIA has a smaller reconstruction error than the currently used algorithms (equipotential-back-projection algorithm and filtered spectral expansion algorithm). With 0.1% white noise in the data, the algorithm can still reconstruct images of a complicated model. Further improvements are also discussed.

Inversion of absorption spectral data for relaxation matrix determination. II. Application to Q-branch line mixing in HCN, C2H2, and N2O

Experimental absorption spectral data from Q-branch line mixing in HCN, C2H2, and N2O are inverted to extract their respective relaxation W matrices. The formulation makes use of a general iterative inversion algorithm based upon first-order sensitivity analysis and Tikhonov regularization. The algorithm, previously applied to R-branch line mixing in HCN, is reformulated to explicitly require detailed balance for the real, off-diagonal W matrix elements. As with the HCN R-branch case, the W matrices recovered typically were found to describe line mixing much better than those derived from the fitting laws currently in use, and the inversion algorithm usually converged within just three iterations.

Inversion of absorption spectral data for relaxation matrix determination. I. Application to line mixing in the 106←000 overtone transition of HCN

A new method of extracting the relaxation matrix directly from absorption spectral data is formulated and applied to R-branch line mixing in HCN. The formulation makes use of a general iterative inversion algorithm based upon first-order sensitivity analysis and Tikhonov regularization. The recovered relaxation matrices describe line mixing much better than those derived from the fitting laws currently in use, and the inversion algorithm usually converges within just three iterations. This formulation presents the first known method for extracting the imaginary, off-diagonal elements of the relaxation matrix.

Use of generalized inverses in a renal optimization problem

The inverse problem in kidney modeling is a severely under determined problem. Imposing a smoothing condition on the basic variables and realistic bounds on the parameters, leads to a constrained minimization problem. A method to solve this problem by a generalized inverse algorithm is given. Computational results are given showing the significance of the smoothing condition.

Determination of diabatic coupling potentials from the inversion of laboratory inelastic scattering data: Application to C4++He→C2++He2+

Experimental inelastic cross section data are used to successfully recover the diabatic coupling potential for the C4++He system via a general iterative inversion algorithm based on first-order functional sensitivity analysis and Tikhonov regularization. The coupling potential was found to converge to a curve with a distinctly different shape from that of an earlier fitted-parameter model potential. Cross sections calculated from the new potential reproduce the relative heights and shapes of the peaks in the experimental cross sections much better than those obtained from the parametrized potential.

Recent Advances of Diffraction Tomography in Geophysics, Ultrasonics, and Optics

AbstractA tutorial review of diffraction tomography is given along with an overview of recent advances of this technique in borehole geophysics, ultrasonic imaging, and optical microscopy. First, the basic principles of diffraction tomography are presented. Then, we discuss a generalized inversion algorithm, valid for irregularly spaced data and a non‐uniform background, and present reconstructions based on ultrasonic water tank data and underground georadar data. Next, we discuss a hybrid filtered backpropagation algorithm for ultrasonic and optical imaging. Quantitative images, based on experimental data, are presented of objects embedded in water or in biological tissue and probed by ultrasound, and of fibers embedded in an index matching liquid and probed by laser light.