Solution Of Inverse Problem Research Articles

Deep learning solutions for inverse problems in advanced biomedical image analysis on disease detection

Inverse problems in biomedical image analysis represent a significant frontier in disease detection, leveraging computational methodologies and mathematical modelling to unravel complex data embedded within medical images. These problems include deducing the unknown properties of biological structures or tissues from the observed imaging data, presenting a unique challenge in decoding intricate biological phenomena. Regarding disease detection, this technique has played a critical role in optimizing diagnostic efficiency by extracting meaningful insights from different imaging modalities like molecular imaging, MRI, and CT scans. Inverse problems contribute to uncovering subtle abnormalities by employing iterative optimization techniques and sophisticated algorithms, enabling precise and early disease detection. Deep learning (DL) solutions have emerged as robust mechanisms for addressing inverse problems in biomedical image analysis, especially in disease recognition. Inverse problems involve reconstructing unknown structures or parameters from observed data, and the DL model excels in learning complex representations and mappings. This study develops a DL Solution for Inverse Problems in the Advanced Biomedical Image Analysis on Disease Detection (DLSIP-ABIADD) technique. The DLSIP-ABIADD technique exploits the DL approach to solve inverse problems and detect the presence of diseases on biomedical images. To solve the inverse problem, the DLSIP-ABIADD technique uses a direct mapping approach. Bilateral filtering (BF) is used for image preprocessing. Besides, the MobileNetv2 model derives feature vectors from the input images. Moreover, the Henry gas solubility optimization (HGSO) method is applied for optimal hyperparameter selection of the MobileNetv2 model. Furthermore, a bidirectional long short-term memory (BiLSTM) model is deployed to identify diseases in medical images. Extensive simulations have been involved to illustrate the better performance of the DLSIP-ABIADD technique. The experimentation outcomes stated that the DLSIP-ABIADD technique performs better than other models.

On diagnostics of the properties of functional-gradient composites considering rheology

Abstract The solution of inverse problems on determining the variable rheological properties of functional-gradient beams using additional information about the angle of rotation of the end section is presented within the framework of the most used models: Euler–Bernoulli and Timoshenko. Within the framework of the concept of complex modules, nonlinear operator equations are obtained that connect the given and sought functions (instantaneous and long-term module, relaxation time). The solution is based on the linearization method and the formulation of an iterative process, at each step of which it is necessary to solve complex-valued boundary value problems and invert completely continuous operators with complex-valued kernels. Functions reflecting the laws of change in long-term and instantaneous modules have been restored. The results of computational experiments in some special cases are presented.

Validation of a multilayer perceptron for rapid, direct solution of the electrical impedance tomography inverse problem

Validation of a multilayer perceptron for rapid, direct solution of the electrical impedance tomography inverse problem

Forward and Inverse Modeling of Ice Sheet Flow Using Physics‐Informed Neural Networks: Application to Helheim Glacier, Greenland

AbstractPredicting the future contribution of the ice sheets to sea level rise over the next decades presents several challenges due to a poor understanding of critical boundary conditions, such as basal sliding. Traditional numerical models often rely on data assimilation methods to infer spatially variable friction coefficients by solving an inverse problem, given an empirical friction law. However, these approaches are not versatile, as they sometimes demand extensive code development efforts when integrating new physics into the model. Furthermore, this approach makes it difficult to handle sparse data effectively. To tackle these challenges, we use the Physics‐Informed Neural Networks (PINNs) to seamlessly integrate observational data and governing equations of ice flow into a unified loss function, facilitating the solution of both forward and inverse problems within the same framework. We illustrate the versatility of this approach by applying the framework to two‐dimensional problems on the Helheim Glacier in southeast Greenland. By systematically concealing one variable (e.g., ice speed, ice thickness, etc.), we demonstrate the ability of PINNs to accurately reconstruct hidden information. Furthermore, we extend this application to address a challenging mixed inversion problem. We show how PINNs are capable of inferring the basal friction coefficient while simultaneously filling gaps in the sparsely observed ice thickness. This unified framework offers a promising avenue to enhance the predictive capabilities of ice sheet models, reducing uncertainties, and advancing our understanding of poorly constrained physical processes.

Adaptive Bregman–Kaczmarz: an approach to solve linear inverse problems with independent noise exactly

We consider the block Bregman–Kaczmarz method for finite dimensional linear inverse problems. The block Bregman–Kaczmarz method uses blocks of the linear system and performs iterative steps with these blocks only. We assume a noise model that we call independent noise, i.e. each time the method performs a step for some block, one obtains a noisy sample of the respective part of the right-hand side which is contaminated with new noise that is independent of all previous steps of the method. One can view these noise models as making a fresh noisy measurement of the respective block each time it is used. In this framework, we are able to show that a well-chosen adaptive stepsize of the block Bregman–Kaczmarz method is able to converge to the exact solution of the linear inverse problem. The plain form of this adaptive stepsize relies on unknown quantities (like the Bregman distance to the solution), but we show a way how these quantities can be estimated purely from given data. We illustrate the finding in numerical experiments and confirm that these heuristic estimates lead to effective stepsizes.

Topology optimization of blazed gratings under conical incidence

A topology optimization method is presented and applied to a blazed diffraction grating in reflection under conical incidence. This type of grating is meant to disperse the incident light on one particular diffraction order, and this property is fundamental in spectroscopy. Conventionally, a blazed metallic grating is made of a sawtooth profile designed to work with the ±1st diffraction order in reflection. In this paper, we question this intuitive triangular pattern and look for optimal opto-geometric characteristics using topology optimization based on finite element modelling of Maxwell’s equations. In practical contexts, the grating geometry is mono-periodic, but it is enlightened by a 3D plane wave with a wave vector outside of the plane of invariance. Consequently, this study deals with the resolution of direct and inverse problems using the finite element method in this intermediate state between 2D and 3D: the so-called conical incidence. A multi-wavelength objective is used in order to obtain a broadband blazed effect. Finally, several numerical experiments are detailed. Our numerical results show that it is possible to reach a 98% diffraction efficiency on the −1st diffraction order if the optimization is performed on a single wavelength, and that the reflection integrated over the [400,1500] nm wavelength range can be 29% higher in absolute terms, 56% in relative terms, than that of the sawtooth blazed grating when using a multi-wavelength optimization criterion (from 52% to 81%).

A direct and analytical method for inverse problems under uncertainty in energy system design: combining inverse simulation and Polynomial Chaos theory

This article introduces and formalizes a novel stochastic method that combines inverse simulation with the theory of generalized Polynomial Chaos (gPC) to solve and study inverse problems under uncertainty in energy system design applications. The method is particularly relevant to design tasks where only a deterministic forward model of a physical system is available, in which a target design quantity is an input to the model that cannot be obtained directly, but can be quantified reversely via the outputs of the model. In this scenario, the proposed method offers an analytical and direct approach to invert such system models. The method puts emphasis on user-friendliness, as it enables its users to conduct the inverse simulation under uncertainty directly in the gPC domain by redefining basic algebra operations for computations. Moreover, the method incorporates an optimization-based approach to integrate supplementary constraints on stochastic quantities. This feature enables the solution of inverse problems bounding the statistical moments of stochastic system variables. The authors exemplify the application of the proposed method with proof-of-concept tests in energy system design, specifically performing uncertainty quantification and sensitivity analysis for a Multi-Energy System (MES). The findings demonstrate the high accuracy of the method as well as clear advantages over conventional sampling-based methods when dealing with a small number of stochastic variables in a system or model. However, the case studies also highlight the current limitations of the proposed method such as slow execution speed due to the optimization-based approach and the challenges associated with, for example, the curse of dimensionality in gPC.

Using torsional wave elastography to evaluate spring pot parameters in skin tumor mimicking phantoms

Estimating the tissue parameters of skin tumors is crucial for diagnosis and effective therapy in dermatology and related fields. However, identifying the most sensitive biomarkers require an optimal rheological model for simulating skin behavior this remains an ongoing research endeavor. Additionally, the multi-layered structure of the skin introduces further complexity to this task. In order to surmount these challenges, an inverse problem methodology, in conjunction with signal analysis techniques, is being employed. In this study, a fractional rheological model is presented to enhance the precision of skin tissue parameter estimation from the acquired signal from torsional wave elastography technique (TWE) on skin tumor-mimicking phantoms for lab validation and the estimation of the thickness of the cancerous layer. An exhaustive analysis of the spring-pot model (SP) solved by the finite difference time domain (FDTD) is conducted. The results of experiments performed using a TWE probe designed and prototyped in the laboratory were validated against ultrafast imaging carried out by the Verasonics Research System. Twelve tissue-mimicking phantoms, which precisely simulated the characteristics of skin tissue, were prepared for our experimental setting. The experimental data from these bi-layer phantoms were measured using a TWE probe, and the parameters of the skin tissue were estimated using inverse problem-solving. The agreement between the two datasets was evaluated by comparing the experimental data obtained from the TWE technique with simulated data from the SP- FDTD model using Pearson correlation, dynamic time warping (DTW), and time-frequency representation. Our findings show that the SP-FDTD model and TWE are capable of determining the mechanical properties of both layers in a bilayer phantom, using a single signal and an inverse problem approach. The ultrafast imaging and the validation of TWE results further demonstrate the robustness and reliability of our technology for a realistic range of phantoms. This fusion of the SP-FDTD model and TWE, as well as inverse problem-solving methods has the potential to have a considerable impact on diagnoses and treatments in dermatology and related fields.

Velocity And Concentration Measurements Of Supersonic Underexpanded Jets

The release of high-speed gas jets due to a tank leakage can pose a substantial safety hazard, as it can give rise to flammable, explosive, or toxic mixtures. Hence, understanding jet dispersion in real-world conditions is crucial for designing effective safety measures. Existing targeted or traceable studies provide velocity field data, but limitations in source-driven-pressures, nozzle geometries, or cross-flow conditions restrict their representativeness under real conditions. Indeed, an exhaustive experimental dataset for supersonic underexpanded gas jet dispersion under the influence of an Atmospheric Boundary Layer (ABL) is currently lacking. As part of a collaboration with the CEA, the von Karman Institute is continuing research efforts towards the development of high quality nonintrusive, optical measurement techniques for the characterization of the velocity and the concentration of a supersonic underexpanded jet submitted to different cross-flow conditions. The research explicitly targets measurements far from the release point within the self-similar region. Such measurements have already been tried on a supersonic jet without cross-flow and with a uniform velocity low-speed cross-flow [2], but never under the influence of an ABL. By filling this gap, the objective of the present work is to further improve the dispersion characterization by combining different techniques, such as Laser Mie Scattering and Light Extinction Spectroscopy (LES). To evaluate the dispersion within the self-similar region up to x/D ≈ 100, a significant FOV 1 is required. Therefore, Large-Scale Particle Image Velocimetry (LS-PIV) has been adapted and successfully applied to measure the velocity field. Prominent results have been obtained and comparable to standard PIV measurements. The pronounced variation in velocity magnitude observed while transitioning from the potential core to the self-similar region requires collecting multiple datasets to cover the broad spectrum of velocities comprehensively. A Mie scattering theory-based technique has been used to retrieve a relative concentration field. It is a non-intrusive technique that operates on the same experimental images produced by the LS-PIV campaign, assuming that light scattered by particles within a specific volume is proportional to the number of particles within this volume. The algorithm for concentration retrieval was adapted from and consequently improved for the present case. Discrepancies in the scaled concentration evolution among different nozzle diameters highlighted the necessity for refining the technique. This emphasizes accurately determining the reference concentration in a region free from intricate phenomena such as multiple scattering. The absolute and independent concentration measurement is done via Light Extinction Spectroscopy (LES). The LES technique measures the transmittance spectrum T of a collimated light beam passing through the particle-laden flow. This method has successfully been applied to several types of flows with similar seeding particles and concentration quantities (e.g. PSD 2 ). The extrapolation of absolute concentration parameters relies on the regularized solution of an ill-conditioned inverse problem. This solution is derived using the transmittance recorded during the test as the primary input parameter. The system to solve exhibits high sensitivity to small perturbations, making it challenging to employ numerical techniques confidently. Due to the substantial impact of minor perturbations in initial conditions on the system solution, a meticulous adaptation of the technique to the demands of compressible flows is necessary and still ongoing. The setup consists of a Deuterium-Tungsten Halogen UV-NIR light source emitting a divergent beam of light towards a 90◦ off-axis parabolic mirror, which collimates and deviates it into a beam to the jet. After passing through the jet, the resulting attenuated light beam goes into a collector mirror, redirecting it into a composite grating spectrometer UV-NIR with a 200 − 1100 nm wavelength range with 0.5 nm optical resolution. The results will examine the dispersion pattern and feature in terms of velocity and concentration of a supersonic underexpanded jet produced by an upstream tank with total pressure ranging from 5 to 9 bar without cross-flows. The latter will provide a foundation for successive studies on supersonic underexpanded jets subjected to an ABL.

Iteratively Refined Image Reconstruction with Learned Attentive Regularizers

We propose a regularization scheme for image reconstruction that leverages the power of deep learning while hinging on classic sparsity-promoting models. Many deep-learning-based models are hard to interpret and cumbersome to analyze theoretically. In contrast, our scheme is interpretable because it corresponds to the minimization of a series of convex problems. For each problem in the series, a mask is generated based on the previous solution to refine the regularization strength spatially. In this way, the model becomes progressively attentive to the image structure. For the underlying update operator, we prove the existence of a fixed point. As a special case, we investigate a mask generator for which the fixed-point iterations converge to a critical point of an explicit energy functional. In our experiments, we match the performance of state-of-the-art learned variational models for the solution of inverse problems. Additionally, we offer a promising balance between interpretability, theoretical guarantees, reliability, and performance.

Deep-Learning-Based Simulation and Inversion of Transient Electromagnetic Sounding Signals in Permafrost Monitoring Problem

Abstract —This article presents a novel approach to addressing the challenges in permafrost monitoring through the integration of deep-learning techniques with conventional electromagnetic sounding methods. Our methodology comprises a forward finite element method (FEM) solver, augmented with the Sumudu transform, and an artificial neural network (ANN) solver trained on FEM-generated data. Remarkably, the ANN solver demonstrates similar accuracy to the FEM solver but operates at a superior speed that is nearly 10,000 times faster. Furthermore, we introduce an inverse problem solution drawing on the PARS algorithm. In addition, we present an ANN-based inverse solver, where the input and output roles are inverted. The ANN inverse solver is trained on the same data, thereby offering an alternative approach to solving the inverse problem. In a computational experiment, we compare the numerical inversion results using the PARS algorithm with those obtained from the ANN forward solver, ANN inversion, and a linear combination of these solutions. This comprehensive analysis sheds light on the effectiveness of our deep-learning-based approach in permafrost monitoring, providing insights for future applications in geophysics and environmental science.

A random walk based methodology to calculate the sensitivity coefficients in inverse radiant boundary design problems

Radiative heating furnaces are widely used for heat treatment applications in manufacturing industries. Strict requirements on heat treatment characteristics of the workpieces require precise control of furnace heater temperatures. Numerical techniques for the solution of inverse heat transfer problems can be used for the optimization of the radiant heater temperatures. In gradient-based optimization techniques, the optimization is carried out using the sensitivity coefficients between the heater and workpiece temperatures. In this study, we present a methodology for calculating the sensitivity coefficients for the solution of inverse boundary design problems involving transient heating of solid workpieces in a three-dimensional radiant furnace. In our methodology, the sensitivity coefficients on the selected design points within the workpieces are calculated by analytical differentiation of design point temperature formulations derived utilizing the floating random walk method and radiative heat flux formulation with exchange factors. The methodology for calculating sensitivity coefficients has been verified through applications to both a one-dimensional scenario and the three-dimensional radiant furnace model. The developed methodology was integrated into a gradient-based optimization algorithm in two test cases, to determine the furnace heater temperatures required to achieve the desired temperature histories at design points within the workpieces. First test case focuses reconstruction of an assumed heater temperature and examines the impact of design point configuration and input data noise on convergence behavior and calculated estimation. The findings reveal that increasing the number of design points in the workpieces increases solution accuracy, with transverse positioning of the design points in the workpieces resulting in heater temperatures closest to the assumed values. Second test case focused on estimating required heater temperatures for spatially uniform heating pattern on the workpieces. With 72 transversely positioned design points, we achieved uniform heating at these points with a maximum relative temperature deviation of 0.26%.

Electrical conductivity of the suboceanic upper mantle constrained by satellite-derived tidal magnetic fields: three-dimensional inversion, validation and interpretation

SUMMARY We present the first 3-D upper-mantle conductivity models obtained by an inversion of the satellite-derived tidally induced magnetic fields (TIMFs). We primarily use the M$_2$ period, but the potential benefit of the O$_1$ period is also inspected. The inverse-problem solution is found using the recently developed frequency-domain, spherical harmonic finite-element method based on the adjoint approach. We tested two different TIMF data sets derived from the satellite measurements of the Swarm mission and two different regularizations; the solution is either required to be sufficiently smooth or reasonably close to the a priori 3-D conductivity model WINTERC-e Wd-emax. The reconstructed conductivity models are locally compared with the 1-D conductivity profiles from other studies. If we use one of the available TIMF data sets, the smooth reconstructed model gravitates towards Wd-emax and the TIMF-adjusted Wd-emax model is closer to the reference conductivity profiles than the original Wd-emax model. Finally, we use the obtained 3-D conductivity distributions to calculate the corresponding 3-D water distribution in the upper mantle using thermodynamical and compositional models coupled to the electrical-conductivity laboratory measurement of individual mantle constituents.

Complex profile metrology via physical symmetry enhanced small angle x-ray scattering

Small angle x-ray scattering (SAXS) stands out as a promising solution in semiconductor metrology. The critical issue of SAXS metrology is to solve the SAXS inverse problem. With the increasing complexity of semiconductor devices, traditional strategies will face problems such as long iteration time and multiple solutions. To address these challenges, we develop a physical symmetry enhanced method to speed up the solution of the SAXS inverse problem for complex nanostructures. We incorporate the physical symmetry into a deep learning model, and a combined loss function is proposed to determine the correct structure in each step of training, which can continuously correct errors and make the model converge faster. The results show that the proposed method achieves high accuracy in determining the critical structural parameters of the complex profile gratings. Compared to traditional strategies, our method performs better in accuracy and does not require time-consuming iterations during reconstruction. The physical symmetry enhanced method provides a feasible way for achieving real-time reconstruction of complex profile nanostructures and is expected to promote the development of SAXS metrology.

Towards accuracy improvement in solution of inverse kinematic problem in redundant robot: A comparative analysis

AbstractThe redundant manipulators have more DOFs (degree of freedoms) than it requires to perform specified task. The inverse kinematic (IK) of such robots are complex and high nonlinear with multiple solutions and singularities. As such, modern Artificial Intelligence (AI) techniques have been used to address these problems. This study proposed two AI techniques based on Neural Network Genetic Algorithm (NNGA) and Particle Swarm Optimization (PSO) algorithm to solve the inverse kinematics (IK) problem of 3DOF redundant robot arm. Firstly, the forward kinematics for 3 DOF redundant manipulator has been established. Secondly, the proposed schemes based on NNGA and PSO algorithm have been presented and discussed for solving the inverse kinematics of the suggested robot. Thirdly, numerical simulations have been implemented to verify the effectiveness of the proposed methods. Three scenarios based on triangle, circular, and sine-wave trajectories have been used to evaluate the performances of the proposed techniques in terms of accuracy measure. A comparison study in performance has been conducted and the simulated results showed that the PSO algorithm gives 7% improvement compared to NNGA technique for triangle trajectory, while 2% improvement has been achieved by the PSO algorithm for circular and sine-wave trajectories.

Genetic Algorithm as the Solution of Non-Linear Inverse Heat Conduction Problems: A Novel Sequential Approach

Abstract Direct measurement of surface heat flux could be extremely challenging, if not impossible, in numerous applications. In such cases, the use of temperature measurement data from subsurface locations can facilitate the determination of surface heat flux and temperature through the solution of the inverse heat conduction problem (IHCP). Several different techniques have been developed over the years for solving IHCPs, with different levels of complexity and accuracy. The filter coefficient technique has proved to be a promising approach for solving IHCPs. Inspired by the filter coefficient approach, a novel method is presented in this paper for solving one-dimensional IHCPs in a domain with temperature-dependent material properties. A test case is developed in COMSOL Multiphysics where the front side of a slab is subject to known transient heat flux and the temperature distributions within the domain are calculated through numerical simulation. The IHCP solution in the form of filter coefficients is defined and a genetic algorithm (GA) is used for the calculation of filter matrix. The number of significant filter coefficients required to evaluate surface heat flux at each time-step is determined through trial and error and the optimal number is selected for evaluating the solution. The structure of the filter matrix is assessed and discussed. The resulting filter coefficients are then used to evaluate the surface heat flux for several different test cases and the performance of the proposed approach is assessed in detail. The results showed that the presented approach is robust and can result in finding optimal filter coefficients that can accurately estimate various types of surface heat flux profiles using temperature data from a limited number of time steps and in a near real-time fashion.

Nonintrusive thermal contact conductance estimation in double-layered pipelines: A reciprocity functional method perspective

Thermal contact conductance (TCC) is a critical factor in various engineering applications involving heat transfer between solid surfaces. Accurate knowledge of TCC enables a more precise thermal system analysis, leading to improved efficiency, reduced energy consumption, and prevention of thermal damages. Moreover, TCC estimation can help to identify material discontinuities or failures. A recent tool for TCC estimation is the Reciprocity Functional Method (RFM) coupled with the Classical Integral Transform Technique (CITT). This method allows the solution of inverse boundary value problems without the need of iterative methods or intrusive measurements, making it valuable for nondestructive testing of interfacial flaws between solid materials. Furthermore, it enables the computation of TCCs using analytical expressions, resulting in a computationally fast procedure. The technique involves the solution of two auxiliary problems: one for the temperature discontinuity estimation, and the other for the interfacial heat flux estimation, both at the inaccessible interface between the two contacting materials. In this study, we applied the RFM and the CITT to estimate different types of TCCs, using a three-dimensional double-layer hollow cylinder geometry as a representative model, like double-layer pipelines used in the oil industry. Temperature measurements for solving the inverse problem were considered available on the outer surface of the pipe which, in the case of a real experiment, could be obtained via an infrared camera. The results demonstrated a good estimate for various analyzed test-cases, even when the measurements were contaminated with Gaussian noises.

Алгоритм визначення параметрів включення при розв’язуванні обернених задач геоелектророзвідки методом про

The paper aims to develop an algorithm for recognizing the physical and geometric parameters of inclusion, using indirect methods of boundary, near-boundary, and partially-boundary elements based on the data of the potential field. Methodology. The direct and inverse two-dimensional problems of the potential theory concerning geophysics were solved when modeling the earth's crust with a piecewise-homogeneous half-plane composed of a containing medium and inclusion that are an ideal contact. To construct the integral representation of the solution of the direct problem, a special fundamental solution for the half-plane (Green's function) of Laplace's equation, which automatically satisfies the zero-boundary condition of the second kind on the day surface, and a fundamental solution for inclusion were used. To find the intensities of unknown sources introduced in boundary, near-boundary, or partially-boundary elements, the collocation technique was used, i.e. the conditions of ideal contact are satisfied in the middle of each boundary element. After solving the obtained SLAE, the unknown potential in the medium and inclusion and the flow through their boundaries are found, considering that the medium and inclusion are considered as completely independent domains. Results. The computational experiment for the task of electric prospecting with a constant artificial field using the resistance method, in particular, electrical profiling, was carried out, while focusing on the physical and geometric interpretation of the data. Initial approximations for the electrical conductivity of the inclusion, its center of mass, orientation and dimensions are determined by the nature of the change in apparent resistivity. To solve the inverse problem two cascades of iterations are organized: the first one is to specify the location of the local heterogeneity and its approximate dimensions, the second one is to specify its shape and orientation in space. At the same time, the minimization of the functional considered on the section of the boundary, where an excess of boundary conditions is set, is carried out. Originality. The method of boundary integral equations is shown to be effective for constructing numerical solutions of direct and inverse problems of potential theory in a piecewise homogeneous half-plane, using indirect methods of boundary, near-boundary, and partial-boundary elements as variants. Practical significance. The proposed approach for solving the inverse problem of electrical exploration with direct current is effective because it allows fora step-by-step, "cascade" recognition of the shape, size, orientation, and electrical conductivity of the inclusion. We follow the principle of not using all the details of the model and not attempting to recognize parameters with little effect on the result, especially with imprecise initial approximations.

Solution of Inverse Geometric Problems Using a Non-Iterative MFS

Solution of Inverse Geometric Problems Using a Non-Iterative MFS

Deep Learning Electromagnetic Inversion Solver Based on a Two-Step Framework for High-Contrast and Heterogeneous Scatterers

Deep Learning Electromagnetic Inversion Solver Based on a Two-Step Framework for High-Contrast and Heterogeneous Scatterers