Inverse Problem Research Articles

Numerical Tests to Evaluate the Effect of Constraining the Spectral Shape of Reference Events on Source Parameter Scaling

ABSTRACT Extracting source parameters from recorded spectra requires correction for attenuation effects. In consideration of the trade-off between source and propagation effects, various strategies have been proposed to constrain the inverse problem with a priori assumptions. The objective of this study is to assess the impact of constraining the source spectra of reference earthquakes in an attempt to remove a common unknown term from the spectral decomposition results. We perform numerical analyses to simulate the outcomes of the decomposition by generating source spectra for different stress drop versus seismic moment scaling, considering a large population of earthquakes with moment magnitudes between 1.8 and 6.5. Following the strategy of constraining the corner frequency of reference events, we evaluate the error of the retrieved source parameters when the applied constraint shows different levels of discrepancy with respect to the assumption used to generate the synthetics. The numerical tests show that an assumption that differs from the correct one can introduce a magnitude-dependent bias that could alter the scaling of the corner frequency with earthquake size. Furthermore, the source spectral shape for large events is also influenced by the constraints applied to the reference earthquakes. As a consequence, inferences about the self-similarity of the rupture process across the scales, or the selection of the most appropriate source model based on the goodness of the spectral fit, may be strongly biased by the constraint imposed on the reference earthquakes.

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.

Flexible Krylov Methods for Group Sparsity Regularization

Abstract This paper introduces new solvers for efficiently computing solutions to large-scale inverse problems with group sparsity regularization, including both non- overlapping and overlapping groups. Group sparsity regularization refers to a type of structured sparsity regularization, where the goal is to impose additional structure in the regularization process by assigning variables to predefined groups that may represent graph or network structures. Special cases of group sparsity regularization include l1 and isotropic total variation regularization. In this work, we develop hybrid projection methods based on flexible Krylov subspaces, where we first recast the group sparsity regularization term as a sequence of 2-norm penalization terms using adaptive regularization matrices in an iterative reweighted norm fashion. Then, we exploit flexible preconditioning techniques to efficiently incorporate the weight updates. The main advantages of these methods are that they are computationally efficient (leveraging the advantages of flexible methods), they are general (and therefore very easily adaptable to new regularization term choices), and they are able to select the regularization parameters automatically and adaptively (exploiting the advantages of hybrid methods). Extensions to multiple regularization terms and solution decomposition frameworks (e.g., for anomaly detection) are described, and a variety of numerical examples demonstrate both the efficiency and accuracy of the proposed approaches compared to existing solvers.

Comparative Analysis of Machine Learning Techniques for Identifying Multiple Force Systems from Accelerometer Measurements.

The knowledge of the forces acting on a structure enables, among many other factors, assessments of whether the component's useful life is compromised by the current machine condition. In many cases, a direct measurement of those forces becomes unfeasible, and an inverse problem must be solved. Among the solutions developed, machine learning techniques have stood out as powerful predictive tools increasingly applied to engineering problem-solving. This study evaluates the ability of different machine learning models to identify parameters of multi-force systems from accelerometer measurements. The models were assessed according to their prediction potential based on correlation coefficient (R2), mean relative error (MRE), and processing time. A computational numerical model using the finite element method was generated and validated by vibration measurements performed using accelerometers in the laboratory. A robust database created by the response surface methodology in conjunction with Design of Experiment (DOE) was used for the evaluation of the ability of machine learning models to predict the position, frequency, magnitude, and number of forces acting on a structure. Among the six machine learning models evaluated, k-NN was able to predict with a 0.013% error, and Random Forests showed a maximum error of 0.2%. The innovation of this study lies in the application of the proposed method for identifying parameters of multi-force systems.

Tractable optimal experimental design using transport maps*

Abstract We present a flexible method for computing Bayesian optimal experimental designs (BOEDs) for inverse problems with intractable posteriors. The approach is applicable to a wide range of BOED problems and can accommodate various optimality criteria, prior distributions and noise models. The key to our approach is the construction of a transport-map-based surrogate to the joint probability law of the design, observational and inference random variables. This order-preserving transport map is constructed using tensor trains and can be used to efficiently sample from (and evaluate approximate densities of) conditional distributions that are required in the evaluation of many commonly-used optimality criteria. The algorithm is also extended to sequential data acquisition problems, where experiments can be performed in sequence to update the state of knowledge about the unknown parameters. The sequential BOED problem is made computationally feasible by preconditioning the approximation of the joint density at the current stage using transport maps constructed at previous stages. The flexibility of our approach in finding optimal designs is illustrated with some numerical examples inspired by disease modeling and the reconstruction of subsurface structures in aquifers.

Efficient, multimodal, and derivative-free bayesian inference with Fisher–Rao gradient flows

Abstract In this paper, we study efficient approximate sampling for probability distributions known up to normalization constants. We specifically focus on a problem class arising in Bayesian inference for large-scale inverse problems in science and engineering applications. The computational challenges we address with the proposed methodology are: (i) the need for repeated evaluations of expensive forward models; (ii) the potential existence of multiple modes; and (iii) the fact that gradient of, or adjoint solver for, the forward model might not be feasible.

While existing Bayesian inference methods meet some of these challenges individually, we propose a framework that tackles all three systematically. Our approach builds upon the Fisher-Rao gradient flow in probability space, yielding a dynamical system for probability densities that converges towards the target distribution at a uniform exponential rate. This rapid convergence is advantageous for the computational burden outlined in (i). We apply Gaussian mixture approximations with operator splitting techniques to simulate the flow numerically; the resulting approximation can capture multiple modes thus addressing (ii). Furthermore, we employ the Kalman methodology to facilitate a derivative-free update of these Gaussian components and their respective weights, addressing the issue in (iii).

The proposed methodology results in an efficient derivative-free posterior approximation method, flexible enough to handle multi-modal distributions: Gaussian Mixture Kalman Inversion (GMKI). The effectiveness of GMKI is demonstrated both theoretically and numerically in several experiments with multimodal target distributions, including proof-of-concept and two-dimensional examples, as well as a large-scale application: recovering the Navier-Stokes initial condition from solution data at positive times.



An Inverse Problem for Hypersonic Flow Past a Curved Cone

An Inverse Problem for Hypersonic Flow Past a Curved Cone

An efficient infinity norm minimization algorithm for under-determined inverse problems

The problem of solving under-determined systems of linear equations with minimum peak magnitude (ℓ∞ norm) has numerous applications in signal processing. These include Peak-to-Average Power Ratio (PAPR) reduction in MIMO-OFDM systems, vector quantization, approximate nearest neighbor search, optimal control in robotics, and power grid optimization. Several methods have been proposed to address this problem, but they often face limitations in computational speed or representation accuracy. Some methods also impose constraints on the frame matrix, such as restrictions on the type of its entries or its aspect ratio. In this paper, we present the Fast Iterative Peak Shrinkage Algorithm (FIPSA), which iterates over feasible solutions to consistently reduce peak magnitude and provably converge to near-optimal solutions. Our experimental results, conducted across various frame matrix types and aspect ratios, demonstrate that FIPSA consistently achieves near-minimal ℓ∞ norm values. In addition, it operates 1.3 to 7.3 times faster than previous methods, while maintaining an average representation error of 10−15. Notably, these advancements are achieved without imposing any constraints on the frame matrix.

ISRSL0: compressed sensing MRI with image smoothness regularized-smoothed \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\:{\\ell}_{0}$$\\end{document} norm

The reconstruction of MR images has always been a challenging inverse problem in medical imaging. Acceleration of MR scanning is of great importance for clinical research and cutting-edge applications. One of the primary efforts to achieve this is using compressed sensing (CS) theory. The CS aims to reconstruct MR images using a small number of sampled data in k-space. The CS-MRI techniques face challenges, including the potential loss of fine structure and increased computational complexity. We introduce a novel framework based on a regularized sparse recovery problem and a sharpening step to improve the CS-MRI approaches regarding fine structure loss under high acceleration factors. This problem is solved via the Half Quadratic Splitting (HQS) approach. The inverse problem for reconstructing MR images is converted into two distinct sub-problems, each of which can be solved separately. One key feature of the proposed approach is the replacement of one sub-problem with a denoiser. This regularization assists the optimization of the Smoothed \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\:{\\ell}_{0}$$\\end{document} (SL0) norm in escaping local minimums and enhances its precision. The proposed method consists of smoothing, feature modification, and Smoothed \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\:{\\ell}_{0}$$\\end{document} cost function optimization. The proposed approach improves the SL0 algorithm for MRI reconstruction without complicating it. The convergence of the proposed approach is illustrated analytically. The experimental results show an acceptable performance of the proposed method compared to the network-based approaches.

Asymptotic behavior of the indicator function in the inverse problem of the wave equation for media with multiple types of cavities

This paper discusses an inverse problem of recovering information about cavities by measuring the solutions of a wave equation. When the medium has multiple types of cavities and the distances from the observation site to the ‘positive cavity’ and to the ‘negative cavity’ coincide, the sign of the indicator function of the enclosure method cannot be determined. Thus, sign cancellation may occur, and information may be lost. In this study, by using an asymptotic solution and examining the top term of the indicator function in detail, it is shown that the shortest distance to the cavity can be obtained, even in the above case.

An Inverse recursive algorithm to retrieve the shape of the inaccessible dielectric objects

A regularized electromagnetic iterative inverse algorithm is formulated and implemented to reconstruct the shape of 2D dielectric objects using the far-field pattern of the scattered field data. To achieve this, an integral operator that maps the unknown boundary of the object onto the far-field pattern of the scattered field is defined and solved for the unknown boundary. The addressed inverse problem has an ill-posed nature and inherits nonlinearity. To overcome these, the proposed solution is linearized via Newton and regularized by Tikhonov in the sense of least squares. Besides, the dominance of the shadow region in the inverse-imaging process is exceeded by considering the superposition of multi-incoming plane waves, leading to less computational cost and a very fast inversion process. Comprehensive numerical analyses are carried out to ascertain the algorithm's feasibility, revealing that it is very efficient and promising.

Global Lipschitz stability for an inverse coefficient problem for a mean field game system

For an inverse coefficient problem of determining a state‐varying factor in the corresponding Hamiltonian for a mean field game system, we prove the global Lipschitz stability by spatial data of one component and interior data in an arbitrarily chosen subdomain over a time interval. The proof is based on Carleman estimates with different norms.

A Fractional Tikhonov Regularization Method for Identifying a Time-Independent Source in the Fractional Rayleigh–Stokes Equation

The aim of this paper is to identify a time-independent source term in the Rayleigh–Stokes equation with a fractional derivative where additional data are considered at a fixed time point. This inverse problem is proved to be ill-posed in the sense of Hadamard. By using a fractional Tikhonov regularization method, we construct a regularized solution. Then, according to a priori and a posteriori regularization parameter selection rules, we prove the convergence estimates of the regularization method. Finally, we provide some numerical examples to prove the effectiveness of the proposed method.

Improved Bald Eagle Search Optimization Algorithm for the Inverse Kinematics of Robotic Manipulators.

The inverse kinematics of robotic manipulators involves determining an appropriate joint configuration to achieve a specified end-effector position. This problem is challenging because the inverse kinematics of manipulators are highly nonlinear and complexly coupled. To address this challenge, the bald eagle search optimization algorithm is introduced. This algorithm combines the advantages of evolutionary and swarm techniques, making it more effective at solving nonlinear problems and improving search efficiency. Due to the tendency of the algorithm to fall into local optima, the Lévy flight strategy is introduced to enhance its performance. This strategy adopts a heavy-tailed distribution to generate long-distance jumps, thereby preventing the algorithm from becoming trapped in local optima and enhancing its global search efficiency. The experiments first evaluated the accuracy and robustness of the proposed algorithm based on the inverse kinematics problem of manipulators, achieving a solution accuracy of up to 10-18 m. Subsequently, the proposed algorithm was compared with other algorithms using the CEC2017 test functions. The results showed that the improved algorithm significantly outperformed the original in accuracy, convergence speed, and stability. Specifically, it achieved over 70% improvement in both standard deviation and mean for several test functions, demonstrating the effectiveness of the Lévy flight strategy in enhancing global search capabilities. Furthermore, the practicality of the proposed algorithm was verified through two real engineering optimization problems.

Geostatistical Inversion of Frequency-Domain Electromagnetic Data for Near Surface Modelling

The detailed characterization of near-surface deposits is important for both environmental and economic reasons. These shallow subsurface systems can be very complex and heterogenous due to natural dynamics and anthropogenic interferences. Modeling techniques based exclusively on direct sampling generate limited informed three-dimensional models of the near-surface. Geophysical methods provide valuable and additional information to model the spatial distribution of the near-surface for locations where direct observations are not available. From this set of methods, frequency-domain electromagnetic induction (FDEM) has been successfully applied to image complex near-surface deposits. Yet, predicting the spatial distribution of relevant subsurface properties from geophysical data, and the integration of direct observations, is not straightforward. It requires solving a challenging geophysical inversion problem. Geostatistical modelling tools have been effectively applied to couple direct observations with geophysical data such as seismic reflection. We propose an iterative geostatistical FDEM inversion method able to integrate data from direct measurements of the near-surface with surface loop-loop FDEM measurements to simultaneously predict high-resolution models of electrical conductivity and magnetic susceptibility, and their associated uncertainty. The iterative geostatistical inversion method is based on stochastic sequential simulation and co-simulation as model perturbation and update techniques. The iterative optimization is based on the local data misfit between observed and simulated FDEM data, weighted by the sensitivity of the acquisition equipment. The proposed method is first demonstrated for a synthetic landfill data set created based on real data collected at a mine tailing disposal site in Portugal, and on a real data set collected at a site with archaeological features in Knowlton, UK. The results show the ability of the proposed method to accurately predict and characterize the spatial distribution of electrical conductivity and magnetic susceptibility down to the depth of interest while reproducing the recorded FDEM data.

Evaluating Models of the Ageing BOLD Response.

Neural activity cannot be directly observed using fMRI; rather it must be inferred from the hemodynamic responses that neural activity causes. Solving this inverse problem is made possible through the use of forward models, which generate predicted hemodynamic responses given hypothesised underlying neural activity. Commonly-used hemodynamic models were developed to explain data from healthy young participants; however, studies of ageing and dementia are increasingly shifting the focus toward elderly populations. We evaluated the validity of a range of hemodynamic models across the healthy adult lifespan: from basis sets for the linear convolution models commonly used to analyse fMRI studies, to more advanced models including nonlinear fitting of a parameterised hemodynamic response function (HRF) and nonlinear fitting of a biophysical generative model (hemodynamic modelling, HDM). Using an exceptionally large sample of participants, and a sensorimotor task optimized for detecting the shape of the BOLD response to brief stimulation, we first characterised the effects of age on descriptive features of the response (e.g., peak amplitude and latency). We then compared these to features from more complex nonlinear models, fit to four regions of interest engaged by the task, namely left auditory cortex, bilateral visual cortex, left (contralateral) motor cortex and right (ipsilateral) motor cortex. Finally, we validated the extent to which parameter estimates from these models have predictive validity, in terms of how well they predict age in cross-validated multiple regression. We conclude that age-related differences in the BOLD response can be captured effectively by models with three free parameters. Furthermore, we show that biophysical models like the HDM have predictive validity comparable to more common models, while additionally providing insights into underlying mechanisms, which go beyond descriptive features like peak amplitude or latency, and include estimation of nonlinear effects. Here, the HDM revealed that most of the effects of age on the BOLD response could be explained by an increased rate of vasoactive signal decay and decreased transit rate of blood, rather than changes in neural activity per se. However, in the absence of other types of neural/hemodynamic data, unique interpretation of HDM parameters is difficult from fMRI data alone, and some brain regions in some tasks (e.g., ipsilateral motor cortex) can show responses that are more difficult to capture using current models.

Robot inverse kinematics solution for center selection battle royale optimization algorithm

In order to solve the problems of high time complexity and insufficient global exploration ability of the battle royale optimization algorithm, this paper proposes an improved battle royale optimization algorithm based on chaos mapping, center selection and elite adaptive strategy. Through benchmark function testing, compared to particle swarm optimization algorithm, whale optimization algorithm, and the battle royale optimization algorithm, the improved algorithm significantly reduces time complexity and notably enhances convergence precision, speed, and stability. The improved algorithm is applied to the inverse kinematics problem of robots. The accuracy and The stability of the improved algorithm is better than those of the traditional battle royale optimization algorithm, it demonstrates superior solving precision and stability, proving its practicality and potential for development in solving robot inverse kinematics problems.

Investigating Enhanced Convection Heat Transfer in 3D Micro-Ribbed Tubes Using Inverse Problem Techniques

The improved heat dissipation observed in 3D micro-ribbed tubes is primarily influenced by the intricate interplay of multiple structural parameters. Nevertheless, research into the coupling mechanisms of these multi-structural parameters remains constrained by the absence of effective methodology in numerical solutions. In the present work, a new 3D micro-rib structure based on discrete adjoint method is established. Firstly, the research examines the interplay of different parameters (such as arrangement, relative roughness height, angle of attack, and circumferential rows) on the thermo-hydraulic performance. It is noted that the heat transfer efficiency is notably impacted by the relative roughness height. And the arrangement methodology dictates the optimal positioning for heat transfer efficiency. An increase in the number of circumferential rows enhances fluid mixing, while the angle of attack plays a crucial role in the formation of longitudinal vortices. Secondly, the coupling optimization technique is employed to obtain the optimal structure featuring non-uniform relative roughness height by the developed numerical solution. Overall, in comparison to the smooth tube, the optimized ribbed tube exhibits a remarkable 64.9% enhancement in performance evaluation criteria. Finally, a notable enhancement of 10.65–22.78% is observed when comparing with the prevailing micro-rib structures.

Epsilon-Greedy Thompson Sampling to Bayesian Optimization

Abstract Bayesian optimization (BO) has become a powerful tool for solving simulation-based engineering optimization problems thanks to its ability to integrate physical and mathematical understandings, consider uncertainty, and address the exploitation–exploration dilemma. Thompson sampling (TS) is a preferred solution for BO to handle the exploitation–exploration trade-off. While it prioritizes exploration by generating and minimizing random sample paths from probabilistic models—a fundamental ingredient of BO—TS weakly manages exploitation by gathering information about the true objective function after it obtains new observations. In this work, we improve the exploitation of TS by incorporating the e-greedy policy, a well-established selection strategy in reinforcement learning. We first delineate two extremes of TS, namely the generic TS and the sample-average TS. The former promotes exploration, while the latter favors exploitation. We then adopt the e-greedy policy to randomly switch between these two extremes. Small and large values of e govern exploitation and exploration, respectively. By minimizing two benchmark functions and solving an inverse problem of a steel cantilever beam, we empirically show that e-greedy TS equipped with an appropriate e is more robust than its two extremes, matching or outperforming the better of the generic TS and the sample-average TS.

On the inverse problem of parameter identification in the steady Oseen model with unilateral and frictional‐type boundary conditions

AbstractRecently, Migórski‐Dudek investigated a steady Oseen flow for a generalized Newtonian incompressible fluid with unilateral and frictional‐type boundary conditions. They established an existence theorem for the steady Oseen model when the divergence‐free convection field and acting volume force are known. However, prior knowledge of and is often impossible in practical engineering applications. This leads to the question of whether the divergence‐free convection field and acting volume force can be determined. The main objective of this paper is to provide a positive response to this challenging and intriguing question. Specifically, we aim to formulate an inverse problem for identifying and , characterized by a nonlinear and nonsmooth regularized optimization problem. Initially, we introduce a variational selection mapping for the steady Oseen model. We then establish the boundedness and generalized continuity of this variational selection. Finally, by leveraging optimization theory, convex analysis, and nonsmooth analysis, we develop an abstract regularization framework for the considered inverse problem and establish the existence of a solution.