Numerical Inversion Methods Research Articles

Physics-informed neural networks for acoustic boundary admittance estimation

Acoustic simulations often face significant uncertainties due to limited knowledge of acoustic boundary conditions. While measuring the boundary admittance in situ is challenging in practical applications, numerical inverse methods can be used to characterize the boundary conditions based on sound pressure data. However, conventional inverse methods require a validated forward model and can become impractical for computationally expensive simulation models. Over the past years, machine learning approaches have emerged as promising methods for scientific computing and data-driven modeling. Physics-informed neural networks incorporate physical prior knowledge into a neural network by adding the residual of the underlying partial differential equation to the loss function. Training the neural network minimizes the loss function, allowing the network to learn a solution that not only fits the training data but also satisfies the corresponding boundary value problem. In this study, physics-informed neural networks are trained to learn the sound pressure solution within two numerical examples governed by the Helmholtz equation without explicitly specifying the boundary conditions at selected boundaries. After training, the neural network’s prediction of the boundary admittance is evaluated and compared to the ground truth, initially assigned in the finite element reference solution. Additionally, the proposed method is validated using experimental data obtained from an acoustic impedance tube measurement. The results show that physics-informed neural networks can accurately learn the sound pressure field and implicitly solve the inverse problem by providing an accurate estimate of the underlying boundary admittance, even in the case of spatially varying boundary conditions.

Modeling of Chemical Tracers for Two-Phase Flow in Advective-Dominated Porous Media at Core Scale

Summary Chemical tracer modeling in porous media plays a key role in subsurface applications including oil recovery, aquifer remediation, and geothermal energy production. In oil reservoirs, chemical tracers are critical to quantifying the remaining oil saturation in porous media after displacing processes, enabling the correct evaluation of the sweep efficiency of recovery methods at the field scale. Even though the transport of solutes under single-phase flow has been modeled extensively with numerous solutions, there are no existing mathematical approaches to examine the displacement of solutes in two-phase flow conditions. Therefore, we present in this research work the first analytical solutions derived to model the transport of ideal and partitioning tracers in porous media with mobile water and oil phases. The models presented are derived from the classic study of fluid displacement by viscous forces and the analysis of dynamic phase distribution in porous media, where key transformation variables are introduced to simplify the nonlinear advection-dispersion equation (ADE) into a conventional partial differential expression. In our derivation process, it is recognized that the dispersion effect can be superimposed onto an ideal concentration front via a singular perturbation expansion, resulting in practical solutions that do not require complex numerical calculations or inversion methods. The solutions derived are verified with numerical simulations and validated with experimental data under different flow conditions for the transport of ideal and partitioning tracers, demonstrating that the complex mechanisms of hydrodynamic dispersion, partitioning, and adsorption are accurately modeled under two-phase flow. Thus, our solutions can be used to rapidly evaluate tracer transport under the existing flow conditions in porous media, significantly reducing the number of experiments and simulations to characterize and select the correct tracer to be used in field applications.

Modular design for optimized acoustic properties of multilayer system

It has been demonstrated that multilayer systems have the potential for excellent sound absorption, effectively attenuating sound across a wide range of frequencies and offering a versatile solution for addressing acoustic challenges in various environments. A key consideration is the design of a multi-layer structure and the parameters of each layer to achieve optimal sound absorption performance. Modular design is a crucial aspect of this work, involving the creation of smaller, units that can be customized and combined to achieve specific sound absorption properties. This study applies numerical optimization, inversion, and matrix methods to multilayer sound absorber systems, aiming to provide customized parameters for the modular design of fibrous materials with exceptional sound absorption properties. The effectiveness of the modular design will be validated and improved through impedance tube measurements of the manufactured multilayer systems, ensuring that the resulting sound absorbers meet the desired acoustic performance across diverse real-world scenarios.

Interdiffusion and atomic mobility in FCC Ag–Cu–Ni alloys

In the present work, the diffusion behavior of the Ag–Cu–Ni alloys was comprehensively studied for the first time. Twelve solid-solid diffusion couples were prepared and annealed at 1023, 1123, and 1223K, respectively. Based on the measured composition-distance profiles, the interdiffusion coefficients in FCC Ag–Cu–Ni alloys at different temperatures were then calculated by both the Matano-Kirkaldy (MK) and numerical inversion (NI) methods. The atomic mobilities of the Ag–Cu–Ni alloys were also optimized. The results indicate that the model-predicted diffusion behavior shows reasonable agreement with the experimental data, validating the reliability of the two versions of atomic mobility obtained in this work and demonstrating the accuracy and efficiency of the NI method. Ag diffuses faster than Ni in the Cu matrix alloy. The pre-exponential factor D0Ag and the activation energy QAgAgCu show a slight change with the Ag, but significantly increase with the Ni. The variation of the pre-exponential factor D0Ni and the activation energy QNiNiCu with the concentration of Ag and Ni were presented with a 3-D planes and discussed in detail. The accurate diffusivities and atomic mobilities of FCC Ag–Cu–Ni alloys obtained in this work will provide crucial input for the development and design of multi-component low-Ag brazing filler metals.

Quantitative assessment of the potential of optimal estimation for aerosol retrieval from geostationary weather satellites in the frame of the iAERUS‐GEO algorithm

AbstractSatellite remote sensing enables the study of atmospheric aerosols at large spatial scales, with geostationary platforms making this possible at sub‐daily frequencies. High‐temporal‐resolution aerosol observations can be made from geostationary data by using robust numerical inversion methods such as the widely‐used optimal estimation (OE) theory. This is the case of the instantaneous Aerosol and surfacE Retrieval Using Satellites in GEOstationary orbit (iAERUS‐GEO) algorithm, which successfully retrieves aerosol optical depth (AOD) maps from the Meteosat Second Generation weather satellite based on a simple implementation of the OE approach combined with the Levenberg–Marquardt method. However, the exact gain in inversion performances that can be obtained from the multiple and more advanced possibilities offered by OE is not well documented in the current literature. Against this background, this article presents the quantitative assessment of OE for the future improvement of the iAERUS‐GEO algorithm. To this end, we use a series of comprehensive experiments based on AOD maps retrieved by iAERUS‐GEO using different OE implementations, and ground‐based observations used as reference data. First, we assess the varying importance in the inversion process of satellite observations and a priori information according to the content of satellite aerosol information. Second, we quantify the gain of AOD estimation in log space versus linear space in terms of accuracy, AOD distribution and number of successful retrievals. Finally, we evaluate the accuracy improvement of simultaneous AOD and surface reflectance retrieval as a function of the regions covered by the Meteosat Earth's disk.

Numerical modelling of advection diffusion equation using Chebyshev spectral collocation method and Laplace transform

In this article a numerical method for numerical modelling of advection diffusion equation is developed. The proposed method is based on Laplace transform (LT) and Chebyshev spectral collocation method (CSCM). The LT is used for time-discretization and the CSCM is used for discretization of spatial derivatives. The LT is used to transform the time variable and avoid the finite difference time stepping method. In time stepping technique the accuracy is achieved for very small time step which results in a very high computational time. The spatial operators are discretized using CSCM to achieve high accuracy as compared to other methods. The method is composed of three primary stages: firstly the given problem is transformed into a corresponding inhomogeneous elliptic problem by using the LT; secondly the CSCM used to solve the transformed problem in LT domain; finally the solution obtained in LT domain is converted to time domain via numerical inverse LT. The inversion of LT is generally an ill-posed problem and due to this reason various numerical inversion methods have been developed. In this article we have utilized the contour integration method which is one of the most efficient methods. The most important feature of this approach is that it handles the time derivative with the Laplace transform rather than the finite difference time stepping approach, avoiding the untoward impact of time steps on stability and accuracy of the method. Five test problems are used to validate the efficiency and accuracy of the proposed numerical scheme.

High-throughput exploration of composition-dependent elasto-plastic and diffusion properties of refractory multi-element Ti-Nb-Zr-W alloys

A combinatorial material synthesis and analysis method can offer a great promise of mapping the elasto-plastic and diffusion properties in refractory multi-element alloys of Ti-Nb-Zr-W system with various compositions. Six groups of Ti-Nb-Zr-W quaternary diffusion couples annealed at 1273 K for 25 h were prepared and measured by electron probe microanalysis and nanoindentation tests. Subsequently, the composition-dependent elasto-plastic properties in a wide composition range of Ti-Nb-Zr-W system were obtained by using two analytical methods including Oliver-Pharr analysis and reverse analysis algorithms. Moreover, elasto-plastic property and interdiffusion coefficient databanks of BCC Ti-Nb-Zr-W system were established by using the machine learning and pragmatic numerical inverse methods, respectively. A hot workability parameter defined by the ratio of Young's modulus to main interdiffusion coefficient was proposed to evaluate the behavior during the hot working. Finally, wear resistance of refractory multi-element alloys was also discussed and verified by the experimental results. The results reveal that Ti-rich refractory multi-element alloys with the low W content have high hardness, good wear resistance, high stress, and excellent processability during the hot working, while the relatively high W content cannot improve the elasto-plastic properties of Ti-rich refractory alloys.

Performance Analysis of Zhao and Durbin Numerical Inversion Methods of Laplace Transform

The Laplace transform is essential to satisfy the independence of time for the analysis of the transient response of the composite or functionally-graded materials. The time independent boundary value problem may be solved then either by numerically or analytically. The solutions should be inverted to the physical plane using inverse Laplace transform. Therefore, the selected numerical inversion method may be crucial to obtain the high accuracy throughout the whole analysis steps. In the present study, Zhao’s Method I, Zhao’s Method II, Durbin’s Method and Modified Durbin’s Method are applied to dynamic loading conditions. The analysis results show that the accurate and stable solutions even for long time inversion have been obtained by Modified Durbin’s Method and Zhao’s Methods. However, compared with the methods of Zhao’s, the computational and programing load of Durbin’s Methods are minimum.

Computational Approach for Differential Equations with Local and Nonlocal Fractional-Order Differential Operators

Laplace transform has been used for solving differential equations of fractional order either PDEs or ODEs. However, using the Laplace transform sometimes leads to solutions in Laplace space that are not readily invertible to the real domain by analytical techniques. Therefore, numerical inversion techniques are then used to convert the obtained solution from Laplace domain into time domain. Various famous methods for numerical inversion of Laplace transform are based on quadrature approximation of Bromwich integral. The key features are the contour deformation and the choice of the quadrature rule. In this work, the Gauss–Hermite quadrature method and the contour integration method based on the trapezoidal and midpoint rule are tested and evaluated according to the criteria of applicability to actual inversion problems, applicability to different types of fractional differential equations, numerical accuracy, computational efficiency, and ease of programming and implementation. The performance and efficiency of the methods are demonstrated with the help of figures and tables. It is observed that the proposed methods converge rapidly with optimal accuracy without any time instability.

Simple and efficient algorithms based on Volterra equations to compute memory kernels and projected cross-correlation functions from molecular dynamics.

Starting from the orthogonal dynamics of any given set of variables with respect to the projection variable used to derive the Mori-Zwanzig equation, a set of coupled Volterra equations is obtained that relate the projected time correlation functions between all the variables of interest. This set of equations can be solved using standard numerical inversion methods for Volterra equations, leading to a very convenient yet efficient strategy to obtain any projected time correlation function or contribution to the memory kernel entering a generalized Langevin equation. Using this strategy, the memory kernel related to the diffusion of tagged particles in a bulk Lennard-Jones fluid is investigated up to the long-term regime to show that the repulsive-attractive cross-contribution to memory effects represents a small but non-zero contribution to the self-diffusion coefficient.

A Novel Performance Evaluation Method for Gas Reservoir-Type Underground Natural Gas Storage

The regulation of the seasonal energy supply for natural gas and the storage of fossil energy are important to society. To achieve it, storing a large amount of natural gas in porous underground media is one of the government’s choices. Due to the successful lesson learned from the oil and gas industry, natural gas storage in underground porous media has been regarded as the most potential long−term energy storage method. In this paper, we developed a new workflow to evaluate the performance of gas reservoir−type underground natural gas storage (UGS). The theoretical background of this workflow includes the correction of the average formation pressure (AFP) and gas deviation factor by error theory and the analytical mathematical model of UGS wells. The Laplace transform, line source function, and Stehfest numerical inversion methods were used to obtain pressure solutions for typical vertical and horizontal wells in UGS. The pressure superposition principle and weighting method of the gas injection−withdrawal rate were used to obtain the AFP. Through the correction of the AFP and gas deviation factor in the material balance equation, the parameters for inventory, effective inventory (the movable gas volume at standard condition), working gas volume (the movable gas volume is operated from the upper limit pressure to the lower limit pressure), and effective gas storage volume (the available gas storage volume at reservoir condition) were determined. Numerical data from the numerical simulator was used to verify the proposed model pressure solution. Actual data from China’s largest Hutubi UGS was used to illustrate the reliability of the proposed workflow in UGS performance evaluation. The results show that large−scale gas injection and withdrawal rates lead to composite heterogeneity in gas storage wells. The nine injection and production cycles’ pressure and effective inventory changes from Hutubi UGS can be divided into a period of rapid pressure rise and a period of slow pressure increase. The final AFP is 32.8 MPa. The final inventory of the Hutubi UGS is 100.1 × 108 m3, with a capacity filling rate (the ratio of effective inventory to designed gas storage capacity) of 93.6%. The effective inventory is 95.3 × 108 m3, and the inventory utilization ratio (the ratio of effective inventory to inventory) is 95.2%. The working gas volume is 40.3 × 108 m3. This study provides a new method for inventory evaluation of the gas reservoir−type UGS.

A Novel Method to Enhance the Inversion Speed and Precision of the NMR T2 Spectrum by the TSVD Based Linearized Bregman Iteration

The low-field nuclear magnetic resonance (NMR) technique has been used to probe the pore size distribution and the fluid composition in geophysical prospecting and related fields. However, the speed and accuracy of the existing numerical inversion methods are still challenging due to the ill-posed nature of the first kind Fredholm integral equation and the contamination of the noises. This paper proposes a novel inversion algorithm to accelerate the convergence and enhance the precision using empirical truncated singular value decompositions (TSVD) and the linearized Bregman iteration. The L1 penalty term is applied to construct the objective function, and then the linearized Bregman iteration is utilized to obtain fast convergence. To reduce the complexity of the computation, empirical TSVD is proposed to compress the kernel matrix and determine the appropriate truncated position. This novel inversion method is validated using numerical simulations. The results indicate that the proposed novel method is significantly efficient and can achieve quick and effective data solutions with low signal-to-noise ratios.

On the Approximation of Fractal-Fractional Differential Equations Using Numerical Inverse Laplace Transform Methods

Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects. Using the Laplace transform for solving differential equations, however, sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analytical means. Thus, we need numerical inversion methods to convert the obtained solution from Laplace domain to a real domain. In this paper, we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with order . Our proposed numerical scheme is based on three main steps. First, we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense, and then into Caputo sense. Secondly, we transform the fractional differential equation in Caputo sense to an equivalent equation in Laplace space. Then the solution of the transformed equation is obtained in Laplace domain. Finally, the solution is converted into the real domain using numerical inversion of Laplace transform. Three inversion methods are evaluated in this paper, and their convergence is also discussed. Three test problems are used to validate the inversion methods. We demonstrate our results with the help of tables and figures. The obtained results show that Euler’s and Talbot’s methods performed better than Stehfest’s method.

Fast and Robust Inverse Kinematics of Serial Robots Using Halley’s Method

This paper proposes a novel numerical inverse kinematics algorithm called the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Quick Inverse Kinematics</i> or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">QuIK</i> method. The QuIK method is a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">third-order</i> algorithm that uses both the first- and second-order derivative information to iteratively converge to a solution. Numerical inverse kinematics methods are readily implemented on any serial robot and do not rely on joint alignment. However, they typically are slower and less robust. The second-order derivative term allows the QuIK algorithm to converge more rapidly and more robustly than existing algorithms. A damped extension to the QuIK method is also proposed to increase reliability near singularities. The QuIK methods are tested in terms of evaluation speed, reliability, and singularity robustness against the Newton–Raphson method and several other modern algorithms. The proposed QuIK methods outperform all other tested algorithms in terms of speed and robustness, and have strong performance near singularities. The QuIK algorithms are proposed as faster and more robust “drop-in” replacements to the Newton–Raphson methods in inverse kinematics. C++ and MATLAB codebases are made available.

Material Parameter Identification of Elastoplastic Constitutive Models Using Machine Learning Approaches

Today, the vast majority of design tasks are based on simulation tools. However, the success of the simulation depends on the accurate identification of the constitutive parameters of materials, i.e., its calibration. The classical parameter identification strategy, which relies on homogeneous tests, does not provide accurate and robust results required by the automotive and aerospace industry. Recently, numerical inverse methods, such as the Finite Element Model Updating (FEMU) and the Virtual Fields Method (VFM), have been developed for identifying constitutive parameters based on heterogeneous tests. Although these methods have proven effective for linear and non-linear models, the parameter identification process is complex, making it computationally expensive. In this work, a machine learning algorithm (XGBoost) is used to pursue the goal of parameter identification of non-linear models using heterogeneous tests. A statistical analysis is conducted to identify the correlation between the training dataset size, mechanical tests results and the material parameters. The goal is to understand the importance of the different inputs and to reduce the computational time.

Uncertainty Quantification of a High-Throughput Profilometry-Based Indentation Plasticity Test of Al 7075 T6 Alloy

The quantification of spatially variable mechanical response in structural materials remains a challenge. Additive manufacturing methods result in increased spatial property variations—the effect of which on component performance is of key interest. To assist iterative design of additively manufactured prototypes, lower-cost benchtop test methods with high precision and accuracy will be necessary. Profilometry-based indentation plastometry (PIP) promises to improve upon the instrumented indentation test in terms of the measurement uncertainty. PIP uses an isotropic Voce hardening model and inverse numerical methods to identify plasticity parameters. The determination of the baseline uncertainty of PIP test is fundamental to its use in characterizing spatial material property variability in advanced manufacturing. To quantify the uncertainty of the PIP test, ninety-nine PIP tests are performed on prepared portions of a traditionally manufactured Al 7075 plate sample. The profilometry data and the Voce parameter predictions are examined to distinguish contributions of noise, individual measurement uncertainty, and additional set-wide variations. Individual measurement uncertainty is estimated using paired profilometry measurements that are taken from each indentation. Principal component analysis is used to analyze and model the measurement uncertainty. The fitting procedure used within the testing device software is employed to examine the effect of profile variations on plasticity predictions. The expected value of the error in the plasticity parameters is given as a function of the number of tests taken, to support rigorous use of the PIP method. The modeling of variability in the presence of measurement uncertainty is discussed.

A mathematical framework for multiphase poromechanics in multiple porosity media

Unconventional geomaterials often exhibit multi-modal pore size distribution. We have developed a comprehensive framework for porous media exhibiting multiple porosity scales that are saturated with one or two types of fluids using mixture theory. Both the governing equations and constitutive laws have been clearly derived and identified, respectively. The effective stress σ′ emerged from the energy balance equation is adoptable for both elastic and elastoplastic deformations, in which pore fractions and saturations play a central role. The proposed model is general in a sense that it works for both uncoupled simulation and coupled simulation. The field equations for uncoupled flow simulation are solved using the Laplace transform and numerical Laplace inversion methods. By visualizing the dimensionless results, we can gain a quantitative insight of the different stages in the depletion process of a naturally fractured reservoir. For coupled flow and geomechanics simulation, a strip load problem and a two-phase flow in a deformable 3D reservoir problem illustrate the impacts of plasticity, multiple porosity, inter-porosity exchange, and capillary pressure on the system response.

Enhancing MAX-DOAS atmospheric state retrievals by multispectral polarimetry – studies using synthetic data

Abstract. Ground-based multi-axis differential optical absorption spectroscopy (MAX-DOAS) is a widely used measurement technique for the remote detection of atmospheric aerosol and trace gases. The technique relies on the analysis of ultra-violet and visible radiation spectra of scattered sunlight (skylight) to obtain information on different atmospheric parameters. From an appropriate set of spectra recorded under different viewing directions (typically a group of observations at different elevation angles) the retrieval of aerosol and trace gas vertical distributions is achieved through numerical inversion methods. It is well known that the polarisation state of skylight is particularly sensitive to atmospheric aerosol content as well as aerosol properties and that polarimetric measurements could therefore provide additional information for MAX-DOAS profile retrievals; however, such measurements have not yet been used for this purpose. To address this issue, we have developed the RAPSODI (Retrieval of Atmospheric Parameters from Spectroscopic Observations using DOAS Instruments) algorithm. In contrast to existing MAX-DOAS algorithms, it can process polarimetric information, and it can retrieve simultaneously profiles of aerosols and various trace gases at multiple wavelengths in a single retrieval step; a further advantage is that it contains a Mie scattering model, allowing for the retrieval of aerosol microphysical properties. The forward-model component in RAPSODI is based on a linearised vector radiative transfer model with Jacobian facilities, and we have used this model to create a database of synthetic measurements in order to carry out sensitivity analyses aimed at assessing the potential of polarimetric MAX-DOAS observations. We find that multispectral polarimetry significantly enhances the sensitivity, particularly to aerosol-related quantities. Assuming typical viewing geometries, the degrees of freedom for signal (DOFS) increase by about 50 % and 70 % for aerosol vertical distributions and aerosol properties, respectively, and by approximately 10 % for trace gas vertical profiles. For an idealised scenario with a horizontally homogeneous atmosphere, our findings predict an improvement in the inversion results' accuracy (root-mean-square deviations to the true values) of about 60 % for aerosol vertical column densities (VCDs) as well as for aerosol surface concentrations and by 40 % for aerosol properties. For trace gas VCDs, very little improvement is apparent, although the accuracy of trace gas surface concentrations improves by about 50 %.

Heat Transfer in Biological Spherical Tissues during Hyperthermia of Magnetoma.

Simple SummaryHeat transport in biological tissue is mediated through a variety of phenomenological processes, involving tissue heat exchange, blood-tissue convection, blood perfusion or advection and diffusion across microvascular beds, and metabolic heat production. In recent years, many physicians and engineers have taken an interest in applying computational and mathematical techniques to model biological systems. The objective of the current paper is to provide an analytical solution to the modified Pennes bioheat conduction equation with a single relaxation time. The suggested model is used to examine heat transport in biological tissues as an infinite concentric spherical region during magnetic fluid hyperthermia. This method is used to investigate the influence of heat generation through heat treatment on a skin tumor a spherical layered structure. The present model can explain the effect of different therapeutic approaches such as cryotherapy sessions, laser therapy, and physical occurrences including transfer, metabolism support, blood perfusion, and other similar treatments.Hyperthermia therapy is now being used to treat cancer. However, understanding the pattern of temperature increase in biological tissues during hyperthermia treatment is essential. In recent years, many physicians and engineers have studied the use of computational and mathematical models of heat transfer in biological systems. The rapid progress in computing technology has intrigued many researchers. Many medical procedures also use engineering techniques and mathematical modeling to ensure their safety and assess the risks involved. One such model is the modified Pennes bioheat conduction equation. This paper provides an analytical solution to the modified Pennes bioheat conduction equation with a single relaxation time by incorporating in it the (MGT) equation. The suggested model examines heat transport in biological tissues as forming an infinite concentric spherical region during magnetic fluid hyperthermia. To investigate thermal reactions caused by temperature shock, specifically the influence of heat generation through heat treatment on a skin tumor [AEGP9], the Laplace transformation, and numerical inverse transformation methods are used. This model was able to explain the effects of different therapeutic approaches such as cryotherapy sessions, laser therapy, and physical occurrences, transfer, metabolism support, and blood perfusion. Comparison of the numerical results of the suggested model with those in the literature confirmed the validity of the model’s numerical results.

3‐D Magnetic Unstructured Inversion

AbstractMagnetic inversion methods based on structured grids have been used extensively in geomagnetic research. Nevertheless, structured grids limit the modeling capability and accuracy of models with arbitrary geometries. To address these challenges, 3‐D magnetic numerical forward modeling and inversion methods using an unstructured tetrahedral grid based on a partial differential equation (PDE) framework are proposed. The methods are derived from Maxwell's partial equations and constructed using the finite element method. Arbitrary undulating topographies, complex geometries, and demagnetization effects can be represented exactly, leading to high‐accuracy forward modeling and inversion solutions. The proposed methods are suitable for both local‐ and global‐scale magnetic data interpretation, for example, in mineral exploration and tectonic research. A synthetic example and real airborne magnetic survey data collected from Mount Iliamna, Alaska, USA, are tested. The results demonstrate that the novel methods significantly improve the capability and accuracy of magnetic data interpretation.