Solution Of Inverse Problem Research Articles

Spectral properties of radiation for the Helmholtz equation with a random coefficient

Abstract We consider the Helmholtz equation [ Δ + k 0 2 ( 1 + q ( x , ω ) ) ] u = − f in R R 2 and R R 3 , where the coefficient q ( x , ω ) is a Gaussian random field. For any open ball B that includes the support of x ↦ q ( x , ω ) , we approximate and characterize spectrally the source-to-measurement map f ↦ u | ∂ B in the weak scattering/low-frequency setting. To this end, we first analyze the case with a deterministic coefficient q(x), and here we discover and quantify a ‘spectral leakage’ effect caused by the presence of the medium. Furthermore, we compare numerically our first-order asymptotic expansion of the source-to-measurement map with the map achieved by a Finite Element Method implementation of an example radiation problem. Our results are applicable in the analysis of the robustness of the solution of inverse source problems in the presence of deterministic and random media.

Paired autoencoders for likelihood-free estimation in inverse problems

Abstract We consider the solution of nonlinear inverse problems where the forward problem is a discretization of a partial differential equation. Such problems are notoriously difficult to solve in practice and require minimizing a combination of a data-fit term and a regularization term. The main computational bottleneck of typical algorithms is the direct estimation of the data misfit. Therefore, likelihood-free approaches have become appealing alternatives. Nonetheless, difficulties in generalization and limitations in accuracy have hindered their broader utility and applicability. In this work, we use a paired autoencoder framework as a likelihood-free estimator (LFE) for inverse problems. We show that the use of such an architecture allows us to construct a solution efficiently and to overcome some known open problems when using LFEs. In particular, our framework can assess the quality of the solution and improve on it if needed. We demonstrate the viability of our approach using examples from full waveform inversion and inverse electromagnetic imaging.

A parabolic multiscale inverse problem approached via homogenization: A numerical method

A method based on homogenization is studied for the solution of a multiscale inverse problem. We consider a class of parabolic problems with highly oscillatory tensors that vary on a microscopic scale. We assume that the microscopic structure is known and seek to recover a macroscopic scalar parameterization of the multiscale tensor. Classical approaches, such as finite elements methods, would require mesh resolution for the direct problem down to the finest scale, that could lead to computational difficulties when implemented. So, starting from the full fine scale model, we solve the inverse problem for a coarse model obtained by homogenization, both theoretically and numerically. The input data, which consist on measurements from the fluxes and the solutions of the direct problem in a given time, are solely based on the original fine scale model. Uniqueness and stability of the inverse problem obtained via homogenization are established under some natural conditions for the fine scale model, and a link with this latter model is established by means of G-convergence. Error estimates are proven for the method.

Data-Driven Morozov Regularization of Inverse Problems

The solution of inverse problems is crucial in various fields such as medicine, biology, and engineering, where one seeks to find a solution from noisy observations. These problems often exhibit non-uniqueness and ill-posedness, resulting in instability under noise with standard methods. To address this, regularization techniques have been developed to balance data fitting and prior information. Recently, data-driven variational regularization methods have emerged, mainly analyzed within the framework of Tikhonov regularization, termed Network Tikhonov (NETT). This paper introduces Morozov regularization combined with a learned regularizer, termed DD-Morozov regularization. Our approach employs neural networks to define non-convex regularizers tailored to training data, enabling a convergence analysis in the non-convex context with noise-dependent regularizers. We also propose a refined training strategy that improves adaptation to ill-posed problems compared to NETT’s original strategy, which primarily focuses on addressing non-uniqueness. We present numerical results for attenuation correction in photoacoustic tomography, comparing DD-Morozov regularization with NETT using the same trained regularizer, both with and without an additional total variation regularizer.

An inverse problem for the system modeling nonhomogeneous asymmetric fluids

In this paper, we prove the local uniqueness of an inverse problem arising in the nonstationary flow of a nonhomogeneous incompressible asymmetric fluid in a bounded domain with a smooth boundary. The direct problem is an initial boundary value problem for a system where the unknowns are the velocity field of the fluid particles, the angular velocity of rotation of the fluid particles, the mass density of the fluid, and the pressure distribution. The inverse problem consists of recovering external forces to the linear and angular momentum equations by assuming a set of measurements as an integral overspecified condition. We introduce and prove several a priori estimates. We characterize the inverse problem solutions using an operator equation of a second kind, deduced from applying the Helmholtz decomposition. We prove several properties of the associated operator, which implies that the Tikhonov fixed point theorem hypothesis is valid. Then, we deduce the local unique solvability of the inverse problem.

A Physics Informed Neural Network for Solving the Inverse Heat Transfer Problem in Gas Turbine Rotating Cavities

Abstract Recently, Physics-Informed Neural Networks have displayed great potential in delivering swift and accurate solutions for inverse problems. Here, a Physics-Informed Neural Network (PINN) was used to solve the inverse heat conduction problem in the rotating cavities of a aeroengine high-pressure compressor internal air system. The neural network was designed to receive experimentally captured radially distributed temperature profiles as inputs and predict the associated surface heat fluxes. The correctness of these predicted heat fluxes are assessed by numerically solving the direct heat conduction equation using a 2D Finite-Element model, thereby recovering the original temperature profiles. A comparative analysis is conducted between the predicted temperature profiles and the initial inputs. The physics informed neural network is trained using noise free synthetic data, created from a range of radial temperature curve fit coefficients, and subsequently tested on noisy experimental data at engine representative conditions. The predicted temperature values exhibit some good agreement with their respective actual counterparts. Furthermore, the sensitivity of model hyperparameters are explored to showcase the capability of the proposed approach. The results show that Physics- Informed Neural Networks exhibit reduced susceptibility to experimental uncertainties when addressing inverse problems, in contrast to inverse solution methods, and offer a possible new approach for analysis of experimental data if trained on a sufficiently large dataset.

Some Remarks About Forward and Inverse Modelling in Hydrology, Within a General Conceptual Framework

The solution to inverse problems is crucial for model calibration and to provide a good basis for model results to be reliable. This paper is based on a recently proposed conceptual framework for the development and application of mathematical models that require the solution of forward and inverse problems. The focus of this paper is on the discussion of some terminology related to the results of forward problems and their reanalysis, on the use of the proposed framework to revise and generalise some methods of solutions of the inverse problem, and to provide a non-standard insight in some aspects about the Bayesian approach to model calibration.

On the resolution of the non-smooth inverse Cauchy problem by the primal-dual method

On the resolution of the non-smooth inverse Cauchy problem by the primal-dual method

An actuator space optimal kinematic path tracking framework for tendon-driven continuum robots: Theory, algorithm and validation

Path tracking of continuum robots is a fundamental and crucial problem across various applications. In this article, we address this problem by focussing on three aspects. Firstly, we propose an efficient multi-solution inverse kinematics solver for three-section constant curvature robots by bridging the theoretical reduction and the numerical correction. Secondly, we derive a linear tendon-driven actuation model, establishing the connection between the robot configuration space and the actuator space. With this model, we achieve optimal distance planning and optimal time allocation considering the constraints of actuator velocity and acceleration, generating a continuous trajectory directly in the actuator space. Finally, we present our kinematic path tracking framework, which includes offline optimal trajectory planning and online feedforward and feedback control. Experiments are conducted both in simulations and in the real world on our three-section tendon-driven continuum robot. The experiments validate the increased efficiency, higher success rates, and accessibility of multiple solutions offered by our inverse kinematics solver, as well as the optimality in distance planning and time allocation in the actuator space. Performance improvements in tracking accuracy are demonstrated through comparative experiments and the application of our framework in path tracking tasks with obstacles is presented through a case study.

Physics-informed neural networks for weakly compressible flows using Galerkin–Boltzmann formulation

In this work, we study the Galerkin–Boltzmann formulation within a physics-informed neural network (PINN) framework to solve flow problems in weakly compressible regimes. The Galerkin–Boltzmann equations are discretized with second-order Hermite polynomials in microscopic velocity space, which leads to a first-order conservation law with six equations. Reducing the output dimension makes this equation system particularly well suited for PINNs compared with the widely used D2Q9 lattice Boltzmann velocity space discretizations. We created two distinct neural networks to overcome the scale disparity between the equilibrium and non-equilibrium states in collision terms of the equations. We test the accuracy and performance of the formulation with benchmark problems and solutions for forward and inverse problems with limited data. We compared our approach with the incompressible Navier–Stokes equation and the D2Q9 formulation. We show that the Galerkin–Boltzmann formulation results in similar L2 errors in velocity predictions in a comparable training time with the Navier–Stokes equation and lower training time than the D2Q9 formulation. We also solve forward and inverse problems for a flow over a square, try to capture an accurate boundary layer, and infer the relaxation time parameter using available data from a high-fidelity solver. Our findings show the potential of utilizing the Galerkin–Boltzmann formulation in PINN for weakly compressible flow problems.

Thrust estimate method of an on-orbit Hall thruster using Hall drift current

In order to realize the thrust estimation of the Hall thruster during its flight mission, this study establishes an estimation method based on measurement of the Hall drift current. In this method, the Hall drift current is calculated from an inverse magnetostatic problem, which is formulated according to its induced magnetic flux density detected by sensors, and then the thrust is estimated by multiplying the Hall drift current with the characteristic magnetic flux density of the thruster itself. In addition, a three-wire torsion pendulum micro-thrust measurement system is utilized to verify the estimate values obtained from the proposed method. The errors were found to be less than 8% when the discharge voltage ranged from 250 V to 350 V and the anode flow rate ranged from 30 sccm to 50 sccm, indicating the possibility that the proposed thrust estimate method could be practically applied. Moreover, the measurement accuracy of the magnetic flux density is suggested to be lower than 0.015 mT and improvement on the inverse problem solution is required in the future.

Quantitative estimation of optical properties in bilayer media within the subdiffusive regime using a tilted fiber-optic probe in diffuse reflectance spectroscopy, part 1: a theoretical framework for designing probe geometry.

As biological tissues are highly heterogeneous, there is a great interest in developing non-invasive optical approaches capable of characterizing them in a very localized manner. Obtaining accurate absolute values of the local optical properties from the measured reflectance requires finding a probe geometry, which allows us to solve this inverse problem robustly and reliably despite neglecting the higher-order moments of the scattering phase function. Our goal is to develop a theoretical framework for designing tilted-fiber diffuse reflectance probes that allow quantitative estimation of the optical properties corresponding to limited tissue volume (typically a few cubic millimeters). Relationships among probe geometry, sampled tissue volume, and robustness of the inverse solver to calculate optical properties from reflectance are studied using Monte Carlo simulations. The analysis of the number of scattering events of the collected photons leads to the establishment of relationships among the probe geometry, the sampled tissue volume, and the validity of a subdiffusive regime for the reflectance. A methodology is proposed for the design of new compact probes with tilted fiber geometry that can quantitatively estimate the values of the optical coefficients in a localized manner within living biological tissues by recording diffuse reflectance spectra.

Efficient computation of magnetic polarizability tensor spectral signatures for object characterisation in metal detection

PurposeMagnetic polarizability tensors (MPTs) provide an economical characterisation of conducting magnetic metallic objects and their spectral signature can aid in the solution of metal detection inverse problems, such as scrap metal sorting, searching for unexploded ordnance in areas of former conflict and security screening at event venues and transport hubs. In this work, the authors aim to discuss methods for efficiently building large dictionaries for classification approaches.Design/methodology/approachPrevious work has established explicit formulae for MPT coefficients, underpinned by a rigorous mathematical theory. To assist with the efficient computation of MPTs at differing parameters and objects of interest, this work applies new observations about the way the MPT coefficients can be computed. Furthermore, the authors discuss discretisation strategies for hp-finite elements on meshes of unstructured tetrahedra combined with prismatic boundary layer elements for resolving thin skin depths and using an adaptive proper orthogonal decomposition (POD) reduced-order modelling methodology to accelerate computations for varying parameters.FindingsThe success of the proposed methodologies is demonstrated using a series of examples. A significant reduction in computational effort is observed across all examples. The authors identify and recommend a simple discretisation strategy and improved accuracy is obtained using adaptive POD.Originality/valueThe authors present novel computations, timings and error certificates of MPT characterisations of realistic objects made of magnetic materials. A novel postprocessing implementation is introduced and an adaptive POD algorithm is demonstrated.

Method for prediction magnetic silencing of uncertain energy-saturated extended technical objects in prolate spheroidal coordinate system

Aim. Development of method for prediction by energy-saturated extended technical objects magnetic silencing based on magnetostatics geometric inverse problems solution and magnetic field spatial spheroidal harmonics calculated in prolate spheroidal coordinate system taking into account of technical objects magnetic characteristics uncertainties. Methodology. Spatial prolate spheroidal harmonics of extended technical objects magnetic field model calculated as magnetostatics geometric inverse problems solution in the form of nonlinear minimax optimization problem based on near field measurements for prediction far extended technical objects magnetic field magnitude. Nonlinear objective function calculated as the weighted sum of squared residuals between the measured and predicted magnetic field COMSOL Multiphysics software package used. Nonlinear minimax optimization problems solutions calculated based on particle swarm nonlinear optimization algorithms. Results. Results of prediction extended technical objects far magnetic field magnitude based on extended technical objects spatial prolate spheroidal harmonics of the magnetic field model in the prolate spheroidal coordinate system using near field measurements with consideration of extended technical objects magnetic characteristics uncertainty. Originality. The method for prediction by extended technical objects magnetic cleanliness based on spatial prolate spheroidal harmonics of the magnetic field model in the prolate spheroidal coordinate system with consideration of magnetic characteristics uncertainty is developed. Practical value. The important practical problem of prediction extended technical objects magnetic silencing based on the spatial prolate spheroidal harmonics of the magnetic field model in the prolate spheroidal coordinate system with consideration of extended technical objects magnetic characteristics uncertainty solved. References 48, figures 2.

Oracle-Net for Nonlinear Compressed Sensing in Electrical Impedance Tomography Reconstruction Problems

Sparse recovery principles play an important role in solving many nonlinear ill-posed inverse problems. We investigate a variational framework with learned support estimation for compressed sensing sparse reconstructions, where the available measurements are nonlinear and possibly corrupted by noise. A graph neural network, named Oracle-Net, is proposed to predict the support from the nonlinear measurements and is integrated into a regularized recovery model to enforce sparsity. The derived nonsmooth optimization problem is then efficiently solved through a constrained proximal gradient method. Error bounds on the approximate solution of the proposed Oracle-based optimization are provided in the context of the ill-posed Electrical Impedance Tomography problem (EIT). Numerical solutions of the EIT nonlinear inverse reconstruction problem confirm the potential of the proposed method which improves the reconstruction quality from undersampled measurements, under sparsity assumptions.

Source localization of mu-rhythm event related desynchronization in EEG during tactile imagery

Tactile imagery remains a relatively understudied phenomenon in the field of mental imagery research. However, alongside motor imagery, this technique holds promise as an effective tool for sensorimotor rehabilitation following stroke and spinal cord injuries. In this study, conducted with 22 healthy volunteers, we investigated the source localization of mu-rhythm event related desynchronization (Event-Related Desynchronization, ERD) using multi-channel electroencephalogram recordings and subsequent inverse problem solution with the sLORETA method. All participants exhibited ERD during tactile imagery task, as well as under vibrotactile stimulation. It was demonstrated that mu-rhythm ERD during vibrotactile stimulation, as well as tactile imagery, was localized in the sensorimotor areas of the contralateral hemisphere. Within the source space, ERD in the postcentral gyrus was significantly stronger compared to the precentral gyrus. These findings indicate that tactile imagery, akin to the perception of real tactile stimuli, leads to prominent activation of sensorimotor cortical areas, consistent with the general understanding of the shared neural substrate during mental imagery and perception.

Robust estimation of structural orientation parameters and 2D/3D local anisotropic Tikhonov regularization

Understanding the orientation of geologic structures is crucial for analyzing the complexity of the earths’ subsurface. For instance, information about geologic structure orientation can be incorporated into local anisotropic regularization methods as a valuable tool to stabilize the solution of inverse problems and produce geologically plausible solutions. We introduce a new variational method that uses the alternating direction method of multipliers within an alternating minimization scheme to jointly estimate orientation and model parameters in 2D and 3D inverse problems. Specifically, our approach adaptively integrates recovered information about structural orientation, enhancing the effectiveness of anisotropic Tikhonov regularization in recovering geophysical parameters. The paper also discusses the automatic tuning of algorithmic parameters to maximize the new method’s performance. Our algorithm is tested across diverse 2D and 3D examples, including structure-oriented denoising and trace interpolation. The results indicate that the algorithm is robust in solving the considered large and challenging problems, alongside efficiently estimating the associated tilt field in 2D cases and the dip, strike, and tilt fields in 3D cases. Synthetic and field examples show that our anisotropic regularization method produces a model with enhanced resolution and provides a more accurate representation of the true structures.

VSHARP: Variable Splitting Half-quadratic ADMM algorithm for reconstruction of inverse-problems

Medical Imaging (MI) tasks, such as accelerated parallel Magnetic Resonance Imaging (MRI), often involve reconstructing an image from noisy or incomplete measurements. This amounts to solving ill-posed inverse problems, where a satisfactory closed-form analytical solution is not available. Traditional methods such as Compressed Sensing (CS) in MRI reconstruction can be time-consuming or prone to obtaining low-fidelity images. Recently, a plethora of Deep Learning (DL) approaches have demonstrated superior performance in inverse-problem solving, surpassing conventional methods. In this study, we propose vSHARP (variable Splitting Half-quadratic ADMM algorithm for Reconstruction of inverse Problems), a novel DL-based method for solving ill-posed inverse problems arising in MI. vSHARP utilizes the Half-Quadratic Variable Splitting method and employs the Alternating Direction Method of Multipliers (ADMM) to unroll the optimization process. For data consistency, vSHARP unrolls a differentiable gradient descent process in the image domain, while a DL-based denoiser, such as a U-Net architecture, is applied to enhance image quality. vSHARP also employs a dilated-convolution DL-based model to predict the Lagrange multipliers for the ADMM initialization. We evaluate vSHARP on tasks of accelerated parallel MRI Reconstruction using two distinct datasets and on accelerated parallel dynamic MRI Reconstruction using another dataset. Our comparative analysis with state-of-the-art methods demonstrates the superior performance of vSHARP in these applications.

Non-Invasive Assessment of Coronary Microvascular Dysfunction Using Vascular Deformation-Based Flow Estimation.

Non-invasive computation of the index of microcirculatory resistance from coronary computed tomography angiography (CTA), referred to as IMR[Formula: see text], is a promising approach for quantitative assessment of coronary microvascular dysfunction (CMD). However, the computation of IMR[Formula: see text] remains an important unresolved problem due to its high requirement for the accuracy of coronary blood flow. Existing CTA-based methods for estimating coronary blood flow rely on physiological assumption models to indirectly identify, which leads to inadequate personalization of total and vessel-specific flow. To overcome this challenge, we propose a vascular deformation-based flow estimation (VDFE) model to directly estimate coronary blood flow for reliable IMR[Formula: see text] computation. Specifically, we extract the vascular deformation of each vascular segment from multi-phase CTA. The concept of inverse problem solving is applied to implicitly derive coronary blood flow based on the physical constraint relationship between blood flow and vascular deformation. The vascular deformation constraints imposed on each segment within the vascular structure ensure sufficient individualization of coronary blood flow. Experimental studies on 106 vessels collected from 89 subjects demonstrate the validity of our VDFE, achieving an IMR[Formula: see text] accuracy of 82.08 %. The coronary blood flow estimated by VDFE has better reliability than the other four existing methods. Our proposed VDFE is an effective approach to non-invasively compute IMR[Formula: see text] with excellent diagnostic performance. The VDFE has the potential to serve as a safe, effective, and cost-effective clinical tool for guiding CMD clinical treatment and assessing prognosis.

An example of the application of neural networks of a simple architecture to unfocused well electrometry probes

An effective method of finding stable solutions of inverse problems of electric and induction logging along the well is proposed, which allows avoiding the influence of the resistance values of the neighboring formations on the determination of the geoelectrical parameters of the object under study. A highly efficient method was proposed for solving such an unstable inverse problem. This method is based on the application of a neural network with inverse error propagation of a simple architecture. Namely three-layer. The mathematical statement of the problem is given, both the topology of the neural network and all its parameters are described in detail. In the course of the numerical experiment, they were selected as optimal. The process of building a base for training a neural network is described in detail. Namely, how each of the examples of the learning base is built by solving a direct problem. With this cut parameter, the training for each example is chosen arbitrarily, which guarantees a comprehensive range for training the neural network. The number of examples in the training base is one hundred thousand examples. As the activation function, the sigmoid is chosen due to the fact that it is differentiable everywhere. The results of testing the written program are given. The learning rate was estimated to obtain the required small error. It is shown that this approach is stably convergent. For testing, the parameters of the layers of the cut, which are inherent to the geophysical parameters of the cuts of the Dnipro-Donetsk depression, were chosen. A complex of lateral logging sounding was chosen as the electrical logging equipment. Four-probe low-frequency induction logging equipment was chosen as induction logging equipment. Examples for induction and electrical logging are given separately. The obtained results are analyzed in detail. Ways of further improvement of the obtained neural network and its use for other problems of geophysics are given.