Inverse Equations Research Articles

Comparison of Classical and Inverse Calibration Equations in Chemical Analysis.

Chemical analysis adopts a calibration curve to establish the relationship between the measuring technique's response and the target analyte's standard concentration. The calibration equation is established using regression analysis to verify the response of a chemical instrument to the known properties of materials that served as standard values. An adequate calibration equation ensures the performance of these instruments. There are two kinds of calibration equations: classical equations and inverse equations. For the classical equation, the standard values are independent, and the instrument's response is dependent. The inverse equation is the opposite: the instrument's response is the independent value. For the new response value, the calculation of the new measurement by the classical equation must be transformed into a complex form to calculate the measurement values. However, the measurement values of the inverse equation could be computed directly. Different forms of calibration equations besides the linear equation could be used for the inverse calibration equation. This study used measurement data sets from two kinds of humidity sensors and nine data sets from the literature to evaluate the predictive performance of two calibration equations. Four criteria were proposed to evaluate the predictive ability of two calibration equations. The study found that the inverse calibration equation could be an effective tool for complex calibration equations in chemical analysis. The precision of the instrument's response is essential to ensure predictive performance. The inverse calibration equation could be embedded into the measurement device, and then intelligent instruments could be enhanced.

Household Durable Goods Spending in Urban Areas: A New Keynesian Micro Perspective

In the New Keynesian model, durable goods are typically assumed to be more responsive to monetary policy than non-durable goods. While this assumption is generally based on macroeconomic policy studies, this research aims to identify whether monetary authorities need to consider household microeconomic aspects such as dependents and long-term saving behavior when formulating policies to stimulate durable goods spending. The study employs two inverse semilogarithmic equations with micro data on urban households. Research findings confirm the existing assumption of the New Keynesian model. Durable goods spending remains sensitive to macroeconomic policies, particularly interest rate adjustments by monetary authorities. Furthermore, the research provides evidence supporting the life-cycle hypothesis for durable goods consumption among urban households. Therefore, this study underscores the pivotal role of durable goods expenditures in ensuring sustainable economic stability.

Review of physics-informed machine-learning inversion of geophysical data

We review five types of physics-informed machine-learning (PIML) algorithms for inversion and modeling of geophysical data. Such algorithms use the combination of a data-driven machine-learning (ML) method and the equations of physics to model or invert geophysical data (or both). By incorporating the constraints of physics, PIML algorithms can effectively reduce the size of the solution space for ML models, enabling them to be trained on smaller data sets. This is especially advantageous in scenarios in which data availability may be limited or expensive to obtain. In this review, we restrict the physics to be that from the governing wave equation, either as a constraint that must be satisfied or by using numerical solutions of the wave equation for modeling and inversion. This approach ensures that the resulting models adhere to physical principles while leveraging the power of ML to analyze and interpret complex geophysical data. There are several potential benefits of PIML compared to standard numerical modeling or inversion of seismic data computed by, for example, finite-difference solutions to the wave equation. 1) Empirical tests suggest that PIML algorithms constrained by the physics of wave propagation can sometimes resist getting stuck in a local minima compared with standard full-waveform inversion (FWI). 2) After the weights of the neural network are found by training, then the forward and inverse operations by PIML can be more than several orders of magnitude more efficient than FWI. However, the computational cost for general training can be enormous. 3) If the ML inversion operator [Formula: see text] is locally trained on a small portion of the recorded data [Formula: see text], then there is sometimes no need for millions of training examples that aim for global generalization of [Formula: see text]. The benefit is that the locally trained [Formula: see text] can be used to economically invert the remaining test data [Formula: see text] for the true velocity [Formula: see text], where [Formula: see text] can comprise more than 90% of the recorded data.

Application of inverse heat transfer to fatigue fracture

Cyclic actuation tends to cause self-heating in the material as the structure experiences fatigue, where the movement and coalescence of defects lead to crack formation, propagation, and eventual fracture. This study explores the modeling aspects of the self-heating phenomenon using a one-dimensional inverse heat problem to analyze heat generation and dissipation to quantify the plastic work rates needed for predicting fatigue life. The approach is based on analyzing the surface thermography obtained using an infrared camera and numerically solving an inverse Fourier heat conduction equation. Formulation of the inverse problem via constrained optimization and method of solution for analyzing the fatigue behavior of CS 1018 flat dog bone specimens subjected to fully reversed bending fatigue are presented. The proposed model demonstrates superior accuracy in predicting the location of maximum heat generation and identifying potential fracture zones compared to traditional methods.

Particle swarm optimization method for parameter selecting in Tikhonov regularization method for solving inverse problems

In this article the Tikhonov method used as a regularization technique for solving the inverse problem liner operator equation of the first kind with noisy and noise-free data. We also provide the Tikhonov method's essential analysis for tackling inverse problems. Following that, we go over how to choose the Tikhonov regularization parameter by implementing the Particle Swarm Optimization (PSO) technique. The effectiveness of combining these two approaches particle swarm optimization and Tikhonov is demonstrated to be highly practicable.

Forward and inverse river contaminant transport modeling using group preserving scheme

Environmental concerns have necessitated the development of computational models predicting pollutant dispersion in natural water systems. Due to the ill-posed nature of the inverse contaminant transport equation, solving this equation using all stable and convergent inverse methods is impossible. Factors such as river geometry, unsteady and non-uniform flow, and tidal influences add to the complexity of the inverse problem. These factors have prompted the evaluation of Group Preserving Scheme for environmental applications. The inverse solution method derives a general equation for solving ordinary differential equations by addressing a dynamical system at negative time steps, ensuring convergence. Three test cases have been presented to evaluate Backward Group Preserving Scheme (BGPS). These have included validation using observational data from Missouri River, inverse simulation in a tidal river, and sensitivity analysis of parameters such as pollutant patterns, advection, dispersion, and decay coefficients. The dataset includes calibrated data from Missouri River, which demonstrated high accuracy, with Mean Relative Error (MRE) ranging from 2.8% to 5.0% for the inverse model. Under tidal conditions, accuracy decreases over time but remains robust, with a Nash–Sutcliffe efficiency of 0.77–0.96 and an MRE of 0.9%–5.9%. Sensitivity analysis revealed optimal model performance for Péclet numbers greater than 500. The model performed best with gradual, wide-peaked pollutant patterns and moderate decay rates (Damköhler number between 5 and 10). BGPS proves effective for transport simulation and concentration history reconstruction in complex rivers, including those with tidal influences, offering a robust tool for water contamination analysis in various flow conditions.

One-Port Coaxial Line Sample Holder Characterisation Method of Dielectric Spectra.

A technique for solving the one-port closed coaxial transmission line sample holder scattering equation for complex permittivity inversion for lossy materials is presented. A non-linear least-squares procedure is used for the determination of parameters for the specification of the spectral functional form of the complex permittivity. The method allows for accurate retrieval of many low- and high-permittivity dielectric materials in the frequency range of 1 GHz to 3 GHz inserted into the coaxial cell. Using this method, the complex permittivity of a number of liquids and a Maltese soil known as Bajjad soil have been extracted by measurements using a short terminated coaxial transmission line sample holder. The proposed novel inversion method is mainly based on the reflection coefficient of the test material. The measured results of the complex permittivity of liquid dielectrics such as ethanol, methanol, and TX100 are validated and compared with previously published data obtained from measurements made by the National Physical Laboratory (NPL) using a two-port measurement setup made with the same commercial coaxial transmission line sample holder used in the one-port setup. Since the technique allows broadband measurements, it has been used to characterise the soil dielectric spectrum in the frequency range of 1-3 GHz, which is also compared with results from a two-port setup of the same coaxial line. The experimental results are a validation of the proposed approach for different types of materials.

3D airborne electromagnetic data inversion based on the block coordinate descent method

Airborne electromagnetic (AEM) surveys usually cover a large area and create a large amount of data. This has limited the application of 3D AEM inversions. To make 3D AEM data inversion at a large scale possible, the local mesh method has been developed to avoid solving large matrix equations in 3D AEM modeling. However, the local mesh only saves the computational cost and memory during forward modeling and Jacobian calculations. When the survey area is very large, the cost for storing and solving the inversion equations can be very high. This brings significant challenges to practical 3D AEM inversions. To solve this problem, we develop a 3D scheme based on the block coordinate descent (BCD) method for inversions of large-scale AEM data. The BCD method divides the inversion for large models into series of small-local inversions, so that we can avoid solving the large matrix equations. Numerical experiments determine that the BCD method can obtain very similar results to those from the existing inversion methods but saves huge amounts of memory.

On the differential equations of frozen Calogero-Moser-Sutherland particle models

Multivariate Bessel and Jacobi processes describe Calogero-Moser-Sutherland particle models. They depend on a parameter k and are related to time-dependent classical random matrix models like Dysom Brownian motions, where k has the interpretation of an inverse temperature. There are several stochastic limit theorems for k→∞ were the limits depend on the solutions of associated ordinary differential equations (ODEs) where these ODEs admit particular simple solutions which are connected with the zeros of the classical orthogonal polynomials. In this paper we show that these solutions attract all solutions. Moreover we present a connection between the solutions of these multivariate ODEs with associated one-dimensional inverse heat equations. These inverse heat equations are used to compute the expectations of some determinantal formulas for the multivariate Bessel and Jacobi processes.

I/O-efficient iterative matrix inversion with photonic integrated circuits

Photonic integrated circuits have been extensively explored for optical processing with the aim of breaking the speed and energy efficiency bottlenecks of digital electronics. However, the input/output (IO) bottleneck remains one of the key barriers. Here we report a photonic iterative processor (PIP) for matrix-inversion-intensive applications. The direct reuse of inputted data in the optical domain unlocks the potential to break the IO bottleneck. We demonstrate notable IO advantages with a lossless PIP for real-valued matrix inversion and integral-differential equation solving, as well as a coherent PIP with optical loops integrated on-chip, enabling complex-valued computation and a net inversion time of 1.2 ns. Furthermore, we estimate at least an order of magnitude enhancement in IO efficiency of a PIP over photonic single-pass processors and the state-of-the-art electronic processors for reservoir training tasks and multiple-input and multiple-output (MIMO) precoding tasks, indicating the huge potential of PIP technology in practical applications.

Power-enhanced residual network for function approximation and physics-informed inverse problems

In this study, we investigate how the updating of weights during forward operation and the computation of gradients during backpropagation impact the optimization process, training procedure, and overall performance of the neural network, particularly the multi-layer perceptrons (MLPs). This paper introduces a novel neural network structure called the Power-Enhancing residual network, inspired by highway network and residual network, designed to improve the network's capabilities for both smooth and non-smooth functions approximation in 2D and 3D settings. By incorporating power terms into residual elements, the architecture enhances the stability of weight updating, thereby facilitating better convergence and accuracy. The study explores network depth, width, and optimization methods, showing the architecture's adaptability and performance advantages. Consistently, the results emphasize the exceptional accuracy of the proposed Power-Enhancing residual network, particularly for non-smooth functions. Real-world examples also confirm its superiority over plain neural network in terms of accuracy, convergence, and efficiency. Moreover, the proposed architecture is also applied to solving the inverse Burgers' equation, demonstrating superior performance. In conclusion, the Power-Enhancing residual network offers a versatile solution that significantly enhances neural network capabilities by emphasizing the importance of stable weight updates for effective training in deep neural networks. The codes implemented are available at: https://github.com/CMMAi/ResNet_for_PINN.

Double‐difference constrained reflection tomography in two‐dimensional elastic media

AbstractThe inverted velocity model obtained from the reflection tomography based on the angle domain common‐image gathers has a certain fuzziness. The inverted velocity model's stratigraphic interface is always not clear enough in areas with complex stratigraphic structure. In order to improve the accuracy and resolution of the inverted velocity model, a double‐difference constraint condition is added on the basis of minimizing the absolute travel‐time residual at the subsurface imaging points. This constraint makes the inverted velocity model local structure information more refined by minimizing the differential travel‐time residual at adjacent imaging points (i.e. closely spaced points within the same layer) and makes the variation of velocity model information within a certain range more accurate. The method in this paper is based on angle domain common‐image gathers, the tomography inversion equation is established by using the ray tracing method, and the conversion relationship between the traveltime residual and the residual curvature of the angle domain common‐image gathers. Then, by adding differential constraint and double‐differential constraint conditions and using the least squares QR decomposition method to solve the set of equations, the inverted velocity model can be obtained through multiple iterations, which provides a high‐precision velocity field for the migration and improves the accuracy of seismic imaging. Numerical experiments on both one typical model and a field data example demonstrate the effectiveness of the proposed double‐difference constrained elastic reflection tomography in generating high‐precision velocity models.

Design and analysis of a novel 3-DOF planar parallel manipulator with two kinematic chains

In this work, to enlarge the workspace, a novel 3-DOF planar parallel manipulator (PPM) with two identical RRR kinematic chains is proposed, and this manipulator has a significant advantage in orienting the moving platform (MP) across a wide range of angles. Resorting to the closed-loop vector approach, the inverse position equation and Jacobian matrix of the novel manipulator are established. Payload performance, dexterity performance, and singularity of the proposed manipulator are analyzed based on the Jacobian matrix. To investigate the distribution characteristics of the constant orientation workspace, orientation workspace, and reachable workspace of the proposed manipulator, a numerical discretization method is employed. Under the same working mode and structural parameters, constant orientation workspace and reachable workspace comparisons between the proposed manipulator and another two traditional PPMs are conducted. Focused on three kinds of performance indices, link lengths optimization of the proposed manipulator is carried out. A 4-DOF pick-and-place equipment is fabricated, and the rotational and translational capabilities are verified based on two examples.

Prestack sparse envelope seismic inversion method adopting the L0−L2-norm regularization

The high drilling costs limit the possibility of obtaining reliable low-frequency constrained models in marine environments or areas with sparse well-log data, making it difficult to constrain lateral continuity and deep data. To enhance stability and accuracy in prestack seismic inversion in these areas, an elastic parameter estimation approach using sparse envelope inversion with [Formula: see text]-norm regularization is proposed. The method combines signal sparse representation and modulation theories to derive a new formula for sparse envelope extraction at lower frequencies. By applying [Formula: see text]-norm regularization to the sparse envelope, we obtain parameter inversion results with smoothed trends, augmenting low-wavenumber information for improved model constraints. In addition, taking the envelope inversion results obtained by the [Formula: see text] norm as the model constraint, the sparsest inversion results with obvious block-like characteristics are obtained by regularizing the inversion equation with the [Formula: see text] norm. Notably, the new method effectively suppresses wavelet side lobes, resulting in stable and accurate inversions without the need for initial low-frequency models. A synthetic example is used to illustrate the feasibility and stability of proposed approach and further demonstrate its practicality in reservoir parameter estimation through a field case study of gas exploration.

A differential equation for inverse point kinetics

The reactivity that represents the percentage departure of a nuclear reactor from its criticality condition is perhaps the most important individual property since it is directly related to the nuclear reactor’s safe operation. Control systems are used to limit the rate of increase in reactivity. The development of systems for monitoring the reactivity, from the nuclear power data acquired from the plant's onboard electronics, requires a constant improvement of the nuclear kinetic models.The transient situations that emerge in nuclear reactors can be predicted from the changes in the neutron flux and, therefore, it is possible to make a sufficiently accurate prediction about the consequences of the disturbances relating to the magnitude of the neutron flux, which varies over time, by using the integral parameters that represent the nuclear reactor core. The point kinetic equations relate these parameters and, in this way, allow the transient situations that can occur in a nuclear reactor to be studied.The reactivity is usually determined by solving the inverse point kinetic equation, which is an integral equation that depends on the nuclear power variation history. In this paper, we present a method to calculate the reactivity through the solution of an ordinary differential equation that does not depend on the nuclear power variation history, then allows the prompt recovery of the reactimeter in case of loss of the nuclear power data acquisition or malfunction of such equipment (Zarei, 2022; Ansari, 1991).The presented paper is organized as follows: Section 1 contains the basic concepts of point kinetics equations and the integral formalism for the reactivity calculation. The mathematical formalism that leads to the differential equation for the reactivity is presented in Section 2. The prompt recovery of the reactivity using the proposed method is discussed in Section 3. The results are shown in Section 4 and the conclusion is presented in Section 5.

Influence of BRDF Models and Solar Zenith Angles on Forest Above-Ground Biomass Derived from MODIS Multi-Angular Indices

Multi-angular remote sensing observation contains crucial information on forest structure parameters. Here, our goal is to examine the ability of multi-angular indices, which are constructed by the typical-angular reflectances in red and NIR bands from MODIS observations, for the retrieval of forest biomass based on the field-measured above-ground biomass (AGB) data. Specifically, we employed the updated version of the MCD43A1 BRDF parameter product as an input for BRDF models to reconstruct the MODIS typical-angular reflectances. Furthermore, we evaluated the effects of different configurations of BRDF models and solar zenith angles (SZA) on forest AGB estimation using our developed multi-angular indices. The semivariogram analysis strategy combined with Landsat ground-surface reflectance data was employed to determine the MODIS pixel heterogeneity; the survey data from field sites of homogeneous pixels was used in our analysis and validation. The results show that our developed multi-angular indices based on a hot-revised BRDF model, under a SZA of 45°, when combined with forest cover information, can account for up to 72% of the variation forest AGB, with an RMSE = 45 Mg/ha. We also found that different kernels for the BRDF models influenced the weight parameters of the biomass inversion equation but did not significantly affect the estimated AGB. In conclusion, our method can enable the better usage of MODIS multi-angular observations for forest AGB estimation.

Acoustic tomographic inversion of 3D temperature fields with mesoscale anomaly in the South China Sea

Acoustic tomographic inversion is based on travel times measured along the transmission paths between all station pairs to reconstruct three-dimensional temperature structures with mesoscale anomalies. In this study, tomographic simulation experiments were designed based on the Hybrid Coordinate Ocean Model (HYCOM) reanalysis data to reconstruct mesoscale phenomena from travel time data obtained from five, seven, and nine stations in the South China Sea over a domain of 100 × 100 km. The travel times for each station pair were calculated in the vertical section using the Bellhop acoustic ray simulation method. Six Empirical orthogonal function (EOF) modes of sound speed along the sound transmission paths in a vertical slice were used to formulate the inversion equations. The horizontal-slice distributions of temperature in the tomography domain were reconstructed using the grid-segmented method for each depth layer. For station-to-station distances greater than 100 km, the performance of inversion was best for the seven-station case rather than for the nine-station case, with the highest horizontal resolution of the three cases. This case study concluded that the seven-station case rather than the nine-station case provided an optimal station number for reconstructing the three-dimensional temperature fields.

Vegetation Classification and a Biomass Inversion Model for Wildfires in Chongli Based on Remote Sensing Data

Vegetation classification, biomass assessment, and wildfire dynamics are interconnected wildfire-ecosystem components. The Chongli District, located in Zhangjiakou City, was the venue for skiing at the 2022 Winter Olympics. Its high mountains and dense forests create a unique environment. The establishment of alpine ski resorts highlighted the importance of comprehensive forest surveys. Understanding vegetation types and their biomass is critical to assessing the distribution of local forest resources and predicting the likelihood of forest fires. This study used satellite multispectral data from the Sentinel-2B satellite to classify the surface vegetation in the Chongli District through K-means clustering. By combining this classification with a biomass inversion model, the total biomass of the survey area can be calculated. The biomass inversion equation established based on multispectral remote sensing data and terrain information in the Chongli area have a strong correlation (shrub forest R2 = 0.811, broadleaf forest R2 = 0.356, coniferous forest R2 = 0.223). These correlation coefficients are key indicators for our understanding of the relationship between remote sensing data and actual vegetation biomass, reflecting the performance of the biomass inversion model. Taking shrubland as an example, a correlation coefficient as high as 0.811 shows the model’s ability to accurately predict the biomass of this type of vegetation. In addition, through multiple linear regression, the optimal shrub, broadleaf, and coniferous forest biomass models were obtained, with the overall accuracy reaching 93.58%, 89.56%, and 97.53%, respectively, meeting the strict requirements for survey accuracy. This study successfully conducted vegetation classification and biomass inversion in the Chongli District using remote sensing data. The research results have important implications for the prevention and control of forest fires.

Arctic Sea Ice Surface Temperature Inversion Using FY-3D/MWRI Brightness Temperature Data

The Arctic plays a crucial role in the intricate workings of the global climate system. With the rapid development of information technology, satellite remote sensing technology has emerged as the main method for sea ice surface temperature (IST) observation. To obtain Arctic IST, we used the FengYun-3D Microwave Radiation Imager (FY-3D/MWRI) brightness temperature (Tb) data for IST inversion using multiple linear regressions. Measured data on IST parameters in the Arctic are difficult to obtain. We used the Moderate-Resolution Imaging Spectroradiometer (MODIS) MYD29 IST data as the baseline to obtain the coefficients for the MWRI IST inversion function. The relation between MWRI Tb data and MODIS MYD29 IST product was established and the microwave IST inversion equation was obtained for the months of January to December 2019. Based on the R2 results and the IST inversion results, we compared and analyzed the MWRI IST data from the months of January to April, November, and December with the Operation IceBridge KT19 IR Surface Temperature data and the Northern High Latitude Level 3 Sea and Sea Ice Surface Temperature (NHL L3 SST/IST). We found that compared MWRI IST with NHL L3 IST, the correlation coefficients (Corr) > 0.72, mean bias ranged from −1.82°C to −0.67 °C, and the standard deviation (Std) ranged from 3.61 °C to 4.54 °C; comparing MWRI IST with KT19 IST, the Corr was 0.69, the bias was 0.51 °C, and the Std was 4.34 °C. The obtained error conforms to the precision requirement. From these results, we conclude that the FY-3D/MWRI Tb data are suitable for IST retrieval in the Arctic using multiple linear regressions.

Waveform Imaging Based on Linear Forward Representations for Scalar Wave Seismic Data

The current reverse-time migration, which is based on wave equations for imaging wavefields, employs an imaging formula derived from Claerbout’s imaging principle. This imaging formula is only valid for plane waves with small incident angles on the perfectly flat reflecting surface. However, the complexity of seismic wave propagation may lead to situations that do not meet this requirement. Therefore, this paper divides the subsurface into local scattering and reflecting bodies. It proposes linear forward representations for scattering and reflection data based on perturbations in the physical parameters and wave impedance, respectively. To further describe the effect on the reflecting body boundary, the local reflection coefficient is defined and the linear forward representation for the reflection data based on it is obtained. After that, the proposed linear forward representations are used as the forward equations for the linear inverse of the seismic data, and the seismic data waveform imaging method is developed based on linear inversion theory. At the same time, the specific waveform imaging calculation formulas for scalar wave scattering data and scalar wave reflection data are provided and validated via numerical experiments. Compared with the current reverse-time migration, waveform migration not only has the correct phase and higher resolution in theory but also does not increase the computational complexity. To some extent, it improves the deficiencies of the current structural imaging and provides a basis for subsurface lithological imaging.