Algorithms For Inverse Problems Research Articles

A U-Net architecture as a surrogate model combined with a geostatistical spectral algorithm for transient groundwater flow inverse problems

Characterizing groundwater flow parameters is crucial for understanding complex aquifer systems, and inverse techniques play a fundamental role in modeling hydrogeological parameters and assessing their uncertainties. Nonetheless, the use of a forward model in these methods can be highly time-consuming, especially with an increasing number of model parameters. To address this issue, we propose a surrogate model based on a U-Net architecture that replaces the transient groundwater flow model, reducing runtime and enabling a fast quantification of uncertainties related to key parameters, including heterogeneous hydraulic conductivity, boundary conditions, specific storage, and pumping rate. The surrogate is trained using limited evaluations of the forward model to learn the physical relationship between hydraulic conductivity fields and transient hydraulic heads measured on-site. The physical principles of the studied problem, including boundary conditions, specific storage, and source terms, are also mapped and introduced as inputs to the model to enhance its understanding of the governing equation of transient groundwater flow. To speed up learning using image-image regression, the previously predicted transient hydraulic heads also serve as an input to predict the transient heads at the current time step. Once the model is trained, we use a spectral geostatistical method to solve the inverse problem, a pumping test of 12 h, using the surrogate model in place of the forward model. Our study demonstrates that the trained U-Net accurately reproduces the state variables corresponding to a specific parameter field, and in terms of computational demand, using U-Net as a surrogate model reduces the required computational time by approximately an order of magnitude for the defined problem. The proposed approach offers an efficient and accurate method for groundwater flow parameter characterization and uncertainty quantification in complex aquifer systems.

NF-ULA: Normalizing Flow-Based Unadjusted Langevin Algorithm for Imaging Inverse Problems

NF-ULA: Normalizing Flow-Based Unadjusted Langevin Algorithm for Imaging Inverse Problems

An improved memetic algorithm for multi-objective resource-constrained flexible job shop inverse scheduling problem: An application for machining workshop

Resource-constrained flexible job shop scheduling problems are commonly encountered in some manufacturing industries, and have been widely studied in recent years. However, traditional resource constrained flexible job shop scheduling problem rarely consider the uncertainties in actual manufacturing systems, which may make the original schedule become suboptimal or even unfeasible. Therefore, a resource constrained flexible job shop inverse scheduling problem (RCFJISP) is proposed in this paper, which aims to cope with uncertain events by simultaneously adjusting the machine, worker and process parameters of the original schedule. A multi-objective optimization model is constructed to minimize the makespan, worker cost, machine energy consumption and deviation index. Furthermore, an improved memetic algorithm (IMA) is developed for solving the proposed problem. In IMA, a novel double-layer encoding mechanism is designed to enhance the capacity in exploring new solution’s domains. Three initialization strategies utilizing original scheduling information are designed to improve the quality of initial solutions. An adaptive mutation strategy and a local search mechanism are designed to enhance exploration and exploitation ability of the algorithm. And a crowding operator is proposed to reflect the diversity of the population effectively. In computational experiments, 28 extended benchmarks are constructed, and the effectiveness of the proposed strategy and algorithm is verified by comparing IMA with its 4 variants and other 4 widely used algorithms. Finally, two inverse scheduling problems of a real-world hydraulic cylinder machining workshop under two uncertain situations are studied. The results demonstrate that IMA can effectively solve the actual inverse scheduling problem. With a slight adjustment to the original scheduling, it can reduce the makespan by 11.5%, the worker cost by 8.1% and the machine energy consumption by 27.9% on average.

Non-asymptotic analysis of ensemble Kalman updates: effective dimension and localization

Abstract Many modern algorithms for inverse problems and data assimilation rely on ensemble Kalman updates to blend prior predictions with observed data. Ensemble Kalman methods often perform well with a small ensemble size, which is essential in applications where generating each particle is costly. This paper develops a non-asymptotic analysis of ensemble Kalman updates, which rigorously explains why a small ensemble size suffices if the prior covariance has moderate effective dimension due to fast spectrum decay or approximate sparsity. We present our theory in a unified framework, comparing several implementations of ensemble Kalman updates that use perturbed observations, square root filtering and localization. As part of our analysis, we develop new dimension-free covariance estimation bounds for approximately sparse matrices that may be of independent interest.

A novel ant lion optimizer-salp swarm algorithm for inverse heat conduction problem in pipeline fluid temperature recognition

A novel ant lion optimizer-salp swarm algorithm for inverse heat conduction problem in pipeline fluid temperature recognition

A greedy sensor selection algorithm for hyperparameterized linear Bayesian inverse problems with correlated noise models

We consider optimal sensor placement for a family of linear Bayesian inverse problems characterized by a deterministic hyper-parameter. The hyper-parameter describes distinct configurations in which measurements can be taken of the observed physical system. To optimally reduce the uncertainty in the system's model with a single set of sensors, the initial sensor placement needs to account for the non-linear state changes of all admissible configurations. We address this requirement through an observability coefficient which links the posteriors' uncertainties directly to the choice of sensors. We propose a greedy sensor selection algorithm to iteratively improve the observability coefficient for all configurations through orthogonal matching pursuit. The algorithm allows explicitly correlated noise models even for large sets of candidate sensors, and remains computationally efficient for high-dimensional forward models through model order reduction. We demonstrate our approach on a large-scale geophysical model of the Perth Basin, and provide numerical studies regarding optimality and scalability with regard to classic optimal experimental design utility functions.

A review of regularization strategies and solution techniques for ill-posed inverse problems, with application to inverse heat transfer problems

With the presence of a large number of inversion algorithms for inverse heat transfer problems (IHTPs) and non-IHTPs, a need for review to have a holistic view is seen. An exhaustive literature review, with the motivation of selecting the inversion technique best fit for a given problem, was made for a general inverse problem. For ill-posedness, a classification of available regularization algorithms namely Tikhonov’s regularization, Bayesian regularization, mollification method, Beck’s sequential approach and Alifanov’s iterative approach, has been provided. Inversion methods like singular value decomposition, truncated singular value decomposition, Tikhonov regularization and total variation regularization are explained. Optimization methods namely steepest descent method, conjugate gradient method, Newton method, Levenberg–Marquardt method, Lagrange method, adjoint method, function specification method, genetic algorithm, differential evolution and particle swarm optimization (PSO) are reviewed. Further, a technique based on neural networks is studied, and wavelet methods like shrinking and wavelet vaguelette decomposition are reviewed. Associated literature has also been listed, highlighting the gaps. The usability of various algorithms in IHTP, starting from the golden section search method, for retrieval of a single parameter, to the regularized versions of the inversion technique, for retrieval of multiple parameters with uncertainty, demonstrates real-life applications to fins in IHTP. An inversion algorithm capable to handle every kind of nonlinearity is sought in literature, whose absence raises the research question, “Is there a technique that works globally for every inverse problem?”, is asked prior to, “What if the available techniques were not utilized to an extent that they should?” is posed. In lieu of this gap, a general comparative framework is developed, such that an efficient technique is selected, based on the total minimum error, which can be used in any field of interest.

Determination of rigid inclusions immersed in an isotropic elastic body from boundary measurement

We study the determination of some rigid inclusions immersed in an isotropic elastic medium from overdetermined boundary data. We propose an accurate approach based on the topological sensitivity technique and the reciprocity gap concept. We derive a higher-order asymptotic formula, connecting the known boundary data and the unknown inclusion parameters. The obtained formula is interesting and useful tool for developing accurate and robust numerical algorithms in geometric inverse problems.

Optimal algorithms for integer inverse obnoxious p-median location problems on tree networks

Optimal algorithms for integer inverse obnoxious p-median location problems on tree networks

Retraction Note to: A fast adaptive algorithm for nonlinear inverse problems with convex penalty

Retraction Note to: A fast adaptive algorithm for nonlinear inverse problems with convex penalty

Analytical algorithms for direct and inverse problems pertaining to the electromagnetic excitation of a layered medium by N dipoles

The direct problem of exciting a multilayered spherical medium by a distribution of arbitrarily located (internal and external) electric dipoles is considered. A formulation based on excitation operators is adopted. The solution of the direct problem is obtained by means of an overall superposition method that combines elements of the T‐matrix approach, the Sommerfeld's method, and Green's functions' techniques. Approximations are extracted for the obtained far‐field patterns in the long‐wavelength (low‐frequency) regime. A set of source‐localization inverse problems is considered. Analytical algorithms are devised in which nonlinear systems are obtained from specifically selected low‐frequency far‐field measurements. These systems are subsequently solved explicitly including all special cases. The developed algorithms are noninvasive and can determine accurately the characteristics of internal sources.

Efficient Algorithms for Bayesian Inverse Problems with Whittle–Matérn Priors

Efficient Algorithms for Bayesian Inverse Problems with Whittle–Matérn Priors

Estimation and Uncertainty Quantification for Piecewise Smooth Signal Recovery

This paper presents a sparse Bayesian learning (SBL) algorithm for linear inverse problems with a high order total variation (HOTV) sparsity prior. For the problem of sparse signal recovery, SBL often produces more accurate estimates than maximum a posteriori estimates, including those that rely on l1 regularization. Moreover, rather than a single signal estimate, SBL yields a full posterior density estimate which can be used for uncertainty quantification. However, SBL is only immediately applicable to problems having a direct sparsity prior, or to those that can be formed via synthesis. This paper demonstrates how a problem with an HOTV sparsity prior can be formulated via synthesis, and then develops a corresponding Bayesian learning method. This expands the class of problems available to Bayesian learning to include, e.g., inverse problems dealing with the recovery of piecewise smooth functions or signals from data. Numerical examples are provided to demonstrate how this new technique is effectively employed.

RETRACTED ARTICLE: A fast adaptive algorithm for nonlinear inverse problems with convex penalty

RETRACTED ARTICLE: A fast adaptive algorithm for nonlinear inverse problems with convex penalty

Machine-learning design of graphene nanoribbon waveguide side-coupled absorber

Machine learning is emerging as a new approach that provides more options for solving complex problems involving electromagnetic phenomena. This paper evaluates the application of machine learning to the design of graphene-based absorbers, which is a research challenge. Five machine learning algorithms — [Formula: see text]-nearest neighbor regression (kNN), artificial neural network (ANN), decision tree (DT), extremely randomized trees (ETs) and random forest (RF) — are applied to realize the transmission spectrum prediction and reverse design of a graphene nanoribbon waveguide side-coupled absorber. The results show that all five algorithms are effective, with RF being the most accurate in the inverse design. Compared with previous work, the application of machine learning in the intelligent design of graphene absorbers is evaluated more comprehensively, providing a reference for the selection of machine learning algorithms for future inverse design problems.

Potential risk quantification from multiple biological factors via the inverse problem algorithm as an artificial intelligence tool in clinical diagnosis.

The inverse problem algorithm (IPA) uses mathematical calculations to estimate the expectation value of a specific index according to patient risk factor groups. The contributions of particular risk factors or their cross-interactions can be evaluated and ranked by their importance. This paper quantified the potential risks from multiple biological factors by integrated case studies in clinical diagnosis via the IPA technique. Acting as artificial intelligence field component, this technique constructs a quantified expectation value from multiple patients' biological index series, e.g., the optimal trigger timing for CTA, a particular drug in blood concentration data, the risk for patients with clinical syndromes. Common biological indices such as age, body surface area, mean artery pressure, and others are treated as risk factors upon their normalization to the range from -1.0 to +1.0, with a non-dimensional zero point 0.0 corresponding to the average risk factor index. The patients' quantified indices are re-arranged into a large data matrix. Next, the inverse and column matrices of the compromised numerical solution are constructed. This paper discusses quasi-Newton and Rosenbrock analyses performed via the STATISTICA program to solve the above inverse problem, yielding the specific expectation value in the form of a multiple-term nonlinear semi-empirical equation. The extensive background, including six previous publications of these authors' team on IPA, was comprehensively re-addressed and scrutinized, focusing on limitations, stumbling blocks, and validity range of the IPA approach as applied to various tasks of preventive medicine. Other key contributions of this study are detailed estimations of the effect of risk factors' coupling/cross-interactions on the IPA computations and the convergence rate of the derived semi-empirical equation viz. the final constant term. The main findings and practical recommendations are considered useful for preventive medicine tasks concerning potential risks of patients with various clinical syndromes.

A CCBM-based generalized GKB iterative regularization algorithm for inverse Cauchy problems

This paper examines inverse Cauchy problems that are governed by a kind of elliptic partial differential equation. The inverse problems involve recovering the missing data on an inaccessible boundary from the measured data on an accessible boundary, which is severely ill-posed. By using the coupled complex boundary method (CCBM), which integrates both Dirichlet and Neumann data into a single Robin boundary condition, we reformulate the underlying problem into an operator equation. Based on this new formulation, we study the solution existence issue of the reduced problem with noisy data. A Golub–Kahan bidiagonalization (GKB) process together with Givens rotation is employed for iteratively solving the proposed operator equation. The regularizing property of the developed method, called CCBM-GKB, and its convergence rate results are proved under a posteriori stopping rule. Finally, a linear finite element method is used for the numerical realization of CCBM-GKB. Various numerical experiments demonstrate that CCBM-GKB is a kind of accelerated iterative regularization method, as it is much faster than the classic Landweber method.

Reconstructing Loads in Nanoplates from Dynamic Data

It was recently proved that the knowledge of the transverse displacement of a nanoplate in an open subset of its mid-plane, measured for any interval of time, allows for the unique determination of the spatial components {fm(x,y)}m=1M of the transverse load ∑m=1Mgm(t)fm(x,y), where M≥1 and {gm(t)}m=1M is a known set of linearly independent functions of the time variable. The nanoplate mechanical model is built within the strain gradient linear elasticity theory, according to the Kirchhoff–Love kinematic assumptions. In this paper, we derive a reconstruction algorithm for the above inverse source problem, and we implement a numerical procedure based on a finite element spatial discretization to approximate the loads {fm(x,y)}m=1M. The computations are developed for a uniform rectangular nanoplate clamped at the boundary. The sensitivity of the results with respect to the main parameters that influence the identification is analyzed in detail. The adoption of a regularization scheme based on the singular value decomposition turns out to be decisive for the accuracy and stability of the reconstruction.

The Search for a Generalized Analytical Solution for the Inverse Problem; Some Surprisingly Simple Approximate Methods for Problems of Practical Importance

Advances in numerical algorithms for Inverse problems has led to significant leaps in our ability to model complex situations of importance. Nonetheless, there is still a strong need for analytical approaches that can be used to calibrate/validate numerical simulations since the old adage of Trust, but Verify is sound advice. Moreover, access to Inverse codes is often limited despite a strong need. Fortunately, generalized Inverse formulations capable of good approximations can be obtained for linear problems by using a known Direct solution. Duhamel’s Integral (convolution) and a systems Unit Response are first employed to derive a Direct solution, with generality maintained via an arbitrary function with coefficients such as polynomials to described the time-dependent boundary condition. For the ensuing Inverse problem, the Direct solution in the form of a more complex polynomial with its coefficients now considered unknown are then used to enforce remotely measured data using the Levenberg–Marquardt algorithm. Once the coefficients are determined, the original function (polynomial or alternative) now describes the unknown boundary condition. Intriguingly, an approximate Inverse Laplace Transform method applied to the Convolution Integral allows for relatively simple formulations for both the Direct and Inverse problems; for this approximation, only the remotely measured data along with the Unit Response as the solitary system information are required. However, the predicted boundary temperature history tends to be shifted forward in time indicating the influence of the thermal thickness and will also reflect any errors in the data. Accuracy can be enhanced by correcting (back-shifting) for the penetration time to the measurement depth and/or using least-squares smoothing. When the various generalized solutions were used for a plate and a thick-cylinder with a time-dependent thermal boundary-condition on one surface and convection on the other, good-to-excellent Inverse predictions were observed; the methods can be adapted so that remotely measured and near instantaneous surface-strains can also be used to determine an unknown surface temperature history. Finally, the convolution-based approach can easily be adapted for 1-degree of freedom vibration problems.

A Compound Gaussian Least Squares Algorithm and Unrolled Network for Linear Inverse Problems

A Compound Gaussian Least Squares Algorithm and Unrolled Network for Linear Inverse Problems