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Inverse problems are commonly used in many fields as they enable the estimation of parameters that cannot be experimentally measured. However, the complex nature of inverse problems requires a strong background in data and signal processing. Moreover, ill-posed problems yield parameters that have a strong linear dependence on the problem. The ill-posed nature of these problems lead to many errors in numerical computations that can make parameter identification nearly impossible. In this paper, a new data processing tool is proposed to maximize the sensitivity of the model to the parameters of interest, while reducing the correlation between them. The effectiveness of the toll is demonstrated through a given inverse problem example using Periodically Pulsed Photothermal Radiometry (PPTR). [ABSTRACT FROM AUTHOR]

We investigate a time-dependent inverse source problem for a parabolic partial differential equation with an integral constraint and subject to Neumann boundary conditions in a domain of R d , d ≥ 1 . We prove the well-posedness as well as higher regularity of solutions in Hölder spaces. We then develop and implement an algorithm that we use to approximate solutions of the inverse problem by means of a finite element discretization in space. Due to instability in inverse problems, we apply Tikhonov regularization combined with the discrepancy principle for selecting the regularization parameter in order to obtain a stable reconstruction. Our numerical results show that the proposed scheme is an accurate technique for approximating solutions of this inverse problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse problem for identifying the source term and the initial value of time-fractional diffusion equation with Caputo-like counterpart hyper-Bessel operator. Firstly, we prove that the problem is ill-posed and give the conditional stability result. Then, we choose the Tikhonov regularization method to solve this ill-posed problem, and give the error estimates under a priori and a posteriori regularization parameter selection rules. Finally, we give numerical examples to illustrate the effectiveness of this method. [ABSTRACT FROM AUTHOR]

Image restoration is a large-scale discrete ill-posed problem, which can be transformed into a Tikhonov regularization problem that can approximate the original image. Kronecker product approximation is introduced into the Tikhonov regularization problem to produce an alternative problem of solving the generalized Sylvester matrix equation, reducing the scale of the image restoration problem. This paper considers solving this alternative problem by applying the conjugate gradient least squares (CGLS) method which has been demonstrated to be efficient and concise. The convergence of the CGLS method is analyzed, and it is demonstrated that the CGLS method converges to the least squares solution within the finite number of iteration steps. The effectiveness and superiority of the CGLS method are verified by numerical tests. [ABSTRACT FROM AUTHOR]

In this paper, we study an inverse problem for a linear third-order pseudoparabolic equation. The investigated inverse problem consists of determining a space-dependent coefficient of the right-hand side of a pseudoparabolic equation. As an additional information a final overdetermination condition is considered. Under the suitable conditions on the data of the problem, the unique solvability of the considered inverse problem is established. The stability of solutions is also proved. The established results are also true for inverse problems for parabolic equations, which could be obtained as a regularization of the studied pseudoparabolic equation. In addition, the pseudoparabolic problem is discretized using the cubic B-spline functions and recast as a nonlinear least-squares minimization of the Tikhonov regularization function. Numerically, this is effectively solved using the MATLAB subroutine lsqnonlin. Both exact and noisy data are inverted. Numerical results for three benchmark test examples are presented and discussed. Moreover, the von Neumann stability analysis is also discussed. [ABSTRACT FROM AUTHOR]

This paper proposes a novel structural damage identification approach coupling the Mayfly algorithm (MA) with static displacement-based response surface (RS). Firstly, a hybrid optimal objective function is established that simultaneously considers the sensitivity-based residual errors of static damage identification equation and the static displacement residual. In the objective function, the static damage identification equation is addressed by the Tikhonov regularization technique. The MA is subsequently employed to conduct an optimal search and pinpoint the location and intensity of damages at the structural element level. To handle the inconformity of the static loading points and the measurement points of displacements, the model reduction and displacement extension techniques are implemented to reconstruct the static damage identification equation. Meanwhile, the static displacement-based RS is constructed to calculate the displacement residual in the hybrid objective function, thereby circumventing the time-consuming finite element calculations and improving computational efficiency. The identification results of the numerical box girder bridge demonstrate that the proposed method outperforms the particle swarm optimization, differential evolution, Jaya and whale optimization algorithms about both convergence rate in optimal searching and identification accuracy. The proposed method enables more accurate damage identification compared to methods solely based on the indicator of the residual of static damage identification equations or displacement residual. The results of identifying damage in the 21 element-truss structure and the static experiments on identifying damage in an aluminum alloy cantilever beam confirm the high efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

We evaluate a decadal ozone profile record derived from the Suomi National Polar‐orbiting Partnership (SNPP) Ozone Mapping and Profiler Suite (OMPS) Limb Profiler (LP) satellite instrument. In 2023, the OMPS LP data were re‐processed with the new version 2.6 retrieval algorithm that combines measurements from the ultraviolet (UV) and visible (VIS) parts of the spectra. It employs the second order Tikhonov regularization to retrieve a single vertical ozone profile between 12.5 km (or cloud tops) and 57.5 km with a vertical resolution of about 1.9–2.5 km between 20 and 55 km. The algorithm uses radiances measured at six UV ozone‐sensitive wavelengths (295, 302, 306, 312, 317, and 322 nm) paired with 353 nm, and one VIS wavelength at 606 nm combined with 510 and 675 nm to form a triplet. Each wavelength pair or triplet is used over a limited range of tangent altitudes where the sensitivity to ozone change is strongest. A new aerosol correction scheme was implemented based on a gamma‐function particle size distribution. In addition, numerous calibration changes that affected ozone retrievals were applied to measured LP radiances, including updates in altitude registration, radiometric calibration, stray light, and spectral registration. The key version 2.6 enhancement is the improved stability of the LP ozone record confirmed by the reduction in relative drifts between LP ozone and correlative measurements, linked previously to a drift in the version 2.5 LP altitude registration. Plain Language Summary: The Montreal Protocol protects the Earth's ozone layer by regulating the production and usage of ozone‐depleting substances. As a result of this international treaty, we expect stratospheric ozone to increase over time. A series of OMPS Limb Profilers (LP) was designed to ensure continuous spaceborne capabilities for detecting changes in the stratospheric ozone distribution over several decades. The ozone record from the first OMPS LP on board the Suomi NPP mission spans more than 12 years, from April 2012 to the present. A statistically significant positive drift in the previous version 2.5 of LP ozone data, linked to a drift in the LP altitude registration, has compromised its fitness to accurately detect ozone trends. In this paper, we introduce the new version 2.6 OMPS LP ozone data set with improved stability. We found a substantial reduction in relative drifts between LP ozone and correlative measurements. Therefore, the version 2.6 LP ozone can be used with higher confidence for monitoring and quantifying stratospheric ozone recovery. Key Points: The ozone record from the first OMPS Limb Profiler that spans more than 12 years has been reprocessed with the new version 2.6 algorithmThe key v2.6 improvement is the reduction in relative drift that was linked previously to a drift in the v2.5 LP altitude registrationThe study demonstrates the importance of continuous improvements in calibration and retrieval to ensure accuracy and stability of LP ozone [ABSTRACT FROM AUTHOR]

Total least squares (TLS), also named as errors in variables in statistical analysis, is an effective method for solving linear equations with the situations, when noise is not just in observation data but also in mapping operations. Besides, the Tikhonov regularization is widely considered in plenty of ill-posed problems. Moreover, the structure of mapping operator plays a crucial role in solving the TLS problem. Tensor operators have some advantages over the characterization of models, which requires us to build the corresponding theory on the tensor TLS. This paper proposes tensor regularized TLS and structured tensor TLS methods for solving ill-conditioned and structured tensor equations, respectively, adopting a tensor-tensor-product. Properties and algorithms for the solution of these approaches are also presented and proved. Based on this method, some applications in image and video deblurring are explored. Numerical examples illustrate the effectiveness of our methods, compared with some existing methods. [ABSTRACT FROM AUTHOR]

In a Hilbert setting we aim to study a second order in time differential equation, combining viscous and Hessian-driven damping, containing a time scaling parameter function and a Tikhonov regularization term. The dynamical system is related to the problem of minimization of a nonsmooth convex function. In the formulation of the problem as well as in our analysis we use the Moreau envelope of the objective function and its gradient and heavily rely on their properties. We show that there is a setting where the newly introduced system preserves and even improves the well-known fast convergence properties of the function and Moreau envelope along the trajectories and also of the gradient of Moreau envelope due to the presence of time scaling. Moreover, in a different setting we prove strong convergence of the trajectories to the element of minimal norm from the set of all minimizers of the objective. The manuscript concludes with various numerical results. [ABSTRACT FROM AUTHOR]

This report documents evaluation of simultaneous estimation of multiple interfacial heat transfer coefficients (HTCs) using transient measurements from an experiment designed for steady-state operation. The design of a mirror system for directing X-rays under cryogenic conditions requires knowledge of the interfacialHTC (contact conductance) between silicon and indium. An experimental apparatus was constructed to measure temperatures in a stack of five 7.62-mm thick pucks of silicon separated by 0.1-mm thick sheets of indium, which is operated under cryogenic temperatures in vacuum. Multiple pucks and interfaces are incorporated into the apparatus to allow evaluation of HTCs for surfaces of different roughness from a single experiment. Analysis of the sensitivity of each of the measured temperatures to each of the unknown HTCs reveals lack of linear independence of these sensitivities and suggests the recovery of the HTCs will be challenging. Artificially noised "data" were created from two different computational models by solving for temperatures and adding random Gaussian noise with a specified standard deviation. These data are subsequently analyzed using two different iterative parameter estimation methods: a Levenberg scheme and a Tikhonov iterative scheme. The required sensitivity matrix is computed using forward finite difference approximations. The results for the heat transfer coefficients for this model problem suggest that coefficients cannot be estimated independently, but the ratios relative to one of the unknowns can be recovered. [ABSTRACT FROM AUTHOR]

The bridge influence line can effectively reflect its overall structural stiffness, and it has been used in the studies of safety assessment, model updating, and the dynamic weighing of bridges. To accurately obtain the influence line of a bridge, an Empirical and Variational Mixed Modal Decomposition (E-VMD) method is used to remove the dynamic component from the vehicle-induced deflection response of a bridge, which requires the preset fundamental frequency of the structure to be used as the cutoff frequency for the intrinsic modal decomposition operation. However, the true fundamental frequency is often obtained from the picker, and the testing process requires the interruption of traffic to carry out the mode decomposition. To realize the rapid testing of the influence lines of bridges, a new method of indirectly identifying the operational modal frequency and deflection influence lines of bridge structures from the axle dynamic response is proposed as an example of cable-stayed bridge structures. Based on the energy method, an analytical solution of the first-order frequency of vertical bending is obtained for a short-tower cable-stayed bridge, which can be used as the initial base frequency to roughly measure the deflection influence line of the cable-stayed bridge. The residual difference between the deflection response and the roughly measured influence line under the excitation of the vehicle is operated by Fast Fourier Transform, from which the operational fundamental frequency identification of the bridge is realized. Using the operational fundamental frequency as the cutoff frequency and comparing the influence-line identification equations, the empirical variational mixed modal decomposition, and the Tikhonov regularization to establish a more accurate identification of the deflection influence line, the deflection influence line is finally identified. The accuracy and practicality of the proposed method are verified by real cable-stayed bridge engineering cases. The results show that the relative error between the recognized bridge fundamental frequency and the measured fundamental frequency is 0.32%, and the relative error of the recognized deflection influence line is 0.83%. The identification value of the deflection influence line has a certain precision. [ABSTRACT FROM AUTHOR]

This paper is concerned with developing an iterative tensor Krylov subspace method to solve linear discrete ill-posed systems of equations with a particular tensor product structure. We use the well-known Frobenius inner product for two tensors and the n-mode matrix-product of a tensor with a matrix to define tensor QR decomposition and alternative Arnoldi algorithms. Moreover, we illustrate how the tensor alternative Arnoldi process can be exploited to solve ill-posed problems by recovering blurry color images and videos in conjunction with the Tikhonov regularization technique, to derive approximate regularized solutions. We also review a generalized cross-validation technique for selecting the regularization parameter in the Tikhonov regularization. Theoretical properties of this method are demonstrated and applications including image deblurring and video processing are investigated. Numerical examples compare the proposed method with several other methods and illustrate the potential superiority of mentioned methods. [ABSTRACT FROM AUTHOR]

Background Purpose Materials and methods Results Conclusion Intraoperative 2D quantitative angiography (QA) for intracranial aneurysms (IAs) has accuracy challenges due to the variability of hand injections. Despite the success of singular value decomposition (SVD) algorithms in reducing biases in computed tomography perfusion (CTP), their application in 2D QA has not been extensively explored. This study seeks to bridge this gap by investigating the potential of SVD‐based deconvolution methods in 2D QA, particularly in addressing the variability of injection durations.Building on the identified limitations in QA, the study aims to adapt SVD‐based deconvolution techniques from CTP to QA for IAs. This adaptation seeks to capitalize on the high temporal resolution of QA, despite its two‐dimensional nature, to enhance the consistency and accuracy of hemodynamic parameter assessment. The goal is to develop a method that can reliably assess hemodynamic conditions in IAs, independent of injection variables, for improved neurovascular diagnostics.The study included three internal carotid aneurysm (ICA) cases. Virtual angiograms were generated using computational fluid dynamics (CFD) for three physiologically relevant inlet velocities to simulate contrast media injection durations. Time‐density curves (TDCs) were produced for both the inlet and aneurysm dome. Various SVD variants, including standard SVD (sSVD) with and without classical Tikhonov regularization, block‐circulant SVD (bSVD), and oscillation index SVD (oSVD), were applied to virtual angiograms. The method was applied on virtual angiograms to recover the aneurysmal dome impulse response function (IRF) and extract flow related parameters such as Peak Height PHIRF, Area Under the Curve AUCIRF, and Mean transit time MTT. Next, correlations between QA parameters, injection duration, and inlet velocity were assessed for unconvolved and deconvolved data for all SVD methods. Additionally, we performed an in vitro study, to complement our in silico investigation. We generated a 2D DSA using a flow circuit design for a patient‐specific internal carotid artery phantom. The DSA showcases factors like x‐ray artifacts, noise, and patient motion. We evaluated QA parameters for the in vitro phantoms using different SVD variants and established correlations between QA parameters, injection duration, and velocity for unconvolved and deconvolved data.The different SVD algorithm variants showed strong correlations between flow and deconvolution‐adjusted QA parameters. Furthermore, we found that SVD can effectively reduce QA parameter variability across various injection durations, enhancing the potential of QA analysis parameters in neurovascular disease diagnosis and treatment.Implementing SVD‐based deconvolution techniques in QA analysis can enhance the precision and reliability of neurovascular diagnostics by effectively reducing the impact of injection duration on hemodynamic parameters. [ABSTRACT FROM AUTHOR]

The distortions of measured beta spectra are addressed by means of unfolding algorithms. Two different approaches, the Maximum-Likelihood ExpectationMaximization and the Tikhonov regularization, are tested on various simulated spectra, for which the initial spectrum to retrieve is known, and on a 99Tc spectrum measured with our dedicated setup. Statistical uncertainties of distorted measured spectra are propagated by determining the covariance matrices. Both algorithms provide satisfactory results but Tikhonov performs overall better for most of the studied radionuclides. Highlight is made on the necessity to employ at least two independent methods to ensure the accuracy of the unfolded spectra and to estimate the internal bias of each algorithm. [ABSTRACT FROM AUTHOR]

This study examines a class of time-dependent constitutive equations used to describe viscoelastic materials under creep in solid mechanics. In nonlinear elasticity, the strain response to the applied stress is expressed via an implicit graph allowing multi-valued functions. For coercive and maximal monotone graphs, the existence of a solution to the quasi-static viscoelastic problem is proven by applying the Browder–Minty fixed point theorem. Moreover, for quasi-linear viscoelastic problems, the solution is constructed as a semi-analytic formula. The inverse viscoelastic problem is represented by identification of a design variable from non-smooth measurements. A non-empty set of optimal variables is obtained based on the compactness argument by applying Tikhonov regularization in the space of bounded measures and deformations. Furthermore, an illustrative example is given for the inverse problem of isotropic kernel identification. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'. [ABSTRACT FROM AUTHOR]

Objective. In quantitative dynamic positron emission tomography (PET), time series of images, reflecting the tissue response to the arterial tracer supply, are reconstructed. This response is described by kinetic parameters, which are commonly determined on basis of the tracer concentration in tissue and the arterial input function. In clinical routine the latter is estimated by arterial blood sampling and analysis, which is a challenging process and thus, attempted to be derived directly from reconstructed PET images. However, a mathematical analysis about the necessity of measurements of the common arterial whole blood activity concentration, and the concentration of free non-metabolized tracer in the arterial plasma, for a successful kinetic parameter identification does not exist. Here we aim to address this problem mathematically. Approach. We consider the identification problem in simultaneous pharmacokinetic modeling of multiple regions of interests of dynamic PET data using the irreversible two-tissue compartment model analytically. In addition to this consideration, the situation of noisy measurements is addressed using Tikhonov regularization. Furthermore, numerical simulations with a regularization approach are carried out to illustrate the analytical results in a synthetic application example. Main results. We provide mathematical proofs showing that, under reasonable assumptions, all metabolic tissue parameters can be uniquely identified without requiring additional blood samples to measure the arterial input function. A connection to noisy measurement data is made via a consistency result, showing that exact reconstruction of the ground-truth tissue parameters is stably maintained in the vanishing noise limit. Furthermore, our numerical experiments suggest that an approximate reconstruction of kinetic parameters according to our analytic results is also possible in practice for moderate noise levels. Significance. The analytical result, which holds in the idealized, noiseless scenario, suggests that for irreversible tracers, fully quantitative dynamic PET imaging is in principle possible without costly arterial blood sampling and metabolite analysis. [ABSTRACT FROM AUTHOR]

Doppler positioning, as an early form of positioning, has regained significant research interest in the context of Low Earth Orbit (LEO) satellites.Given the LEO mega-constellation scenario, the objective function of Doppler positioning manifests significant nonlinearity, leading to ill-conditioning challenges for prevalent algorithms like iterative least squares (LS) estimation, especially in cases where inappropriate initial values are selected. In this study, we investigate the causes of ill-posed problems from two perspectives. Firstly, we analyze the linearization errors of the Doppler observation equations in relation to satellite orbital altitude and initial value errors, revealing instances where traditional algorithms may fail to converge. Secondly, from an optimization theory perspective, we demonstrate the occurrence of convergence to locally non-unique solutions for Doppler positioning. Subsequently, to address these ill-conditioning issues, we introduce Tikhonov regularization terms in the objective function to constrain algorithm divergence, with a fitted model for the regularization coefficient. Finally, we conduct comprehensive simulation experiments in both dynamic and static scenarios to validate the performance of the proposed algorithm. On the one hand, when the initial values are set to 0, our algorithm achieves high-precision positioning, whereas the iterative LS fails to converge. On the other hand, in certain simulation scenarios, the iterative LS converges to locally non-unique solutions, resulting in positioning errors exceeding 50 km in the north and east directions, several hundred kilometers in the vertical direction, and velocity errors surpassing 120 m/s. In contrast, our algorithm demonstrates typical errors of a position error of 6.8462 m, velocity error of 0.0137 m/s, and clock drift error of 8.3746 × 10 − 6 s/s. This work provides an effective solution to the sensitivity issue of initial points in Doppler positioning and can serve as a reference for the algorithm design of Doppler positioning receivers with LEO mega-constellations. [ABSTRACT FROM AUTHOR]

Objective. Magnetic particle imaging (MPI) is an emerging medical tomographic imaging modality that enables real-time imaging with high sensitivity and high spatial and temporal resolution. For the system matrix reconstruction method, the MPI reconstruction problem is an ill-posed inverse problem that is commonly solved using the Kaczmarz algorithm. However, the high computation time of the Kaczmarz algorithm, which restricts MPI reconstruction speed, has limited the development of potential clinical applications for real-time MPI. In order to achieve fast reconstruction in real-time MPI, we propose a greedy regularized block Kaczmarz method (GRBK) which accelerates MPI reconstruction. Approach. GRBK is composed of a greedy partition strategy for the system matrix, which enables preprocessing of the system matrix into well-conditioned blocks to facilitate the convergence of the block Kaczmarz algorithm, and a regularized block Kaczmarz algorithm, which enables fast and accurate MPI image reconstruction at the same time. Main results. We quantitatively evaluated our GRBK using simulation data from three phantoms at 20 dB, 30 dB, and 40 dB noise levels. The results showed that GRBK can improve reconstruction speed by single orders of magnitude compared to the prevalent regularized Kaczmarz algorithm including Tikhonov regularization, the non-negative Fused Lasso, and wavelet-based sparse model. We also evaluated our method on OpenMPIData, which is real MPI data. The results showed that our GRBK is better suited for real-time MPI reconstruction than current state-of-the-art reconstruction algorithms in terms of reconstruction speed as well as image quality. Significance. Our proposed method is expected to be the preferred choice for potential applications of real-time MPI. [ABSTRACT FROM AUTHOR]

In this paper, we propose two parameter choice rules for the discretizing Tikhonov regularization via multiscale Galerkin projection for solving linear ill-posed integral equations. In contrast to previous theoretical analyses, we introduce a new concept called the projection noise level to obtain error estimates for the approximate solutions. This concept allows us to assess how noise levels change during projection. The balance principle and Hanke–Raus rule are modified by incorporating the error estimates of the projection noise level. We demonstrate the convergence rate of these two modified parameter choice rules through rigorous proof. In addition, we find that the error between the approximate solution and the exact solution improves as the noise frequency increases. Finally, numerical experiments are provided to illustrate the theoretical findings presented in this paper. [ABSTRACT FROM AUTHOR]

In 2021, Z. Fu, Y. Chen and B. Han introduced an inexact Newton regularization (REGINN-IT) using an idea involving the non-stationary iterated Tikhonov regularization scheme for solving nonlinear ill-posed operator equations. In this paper, we suggest a simplified version of the REGINN-IT scheme by using the Bregman distance, duality mapping and a suitable parameter choice strategy to produce an approximate solution. The method is comprised of inner and outer iteration steps. The outer iterates are stopped by a Morozov-type stopping rule, while the inner iterate is executed by making use of the non-stationary iterated Tikhonov scheme. We have studied convergence of the proposed method under some standard assumptions and utilizing tools from convex analysis. The novelty of the method is that it requires computation of the Fréchet derivative only at an initial guess of an exact solution and hence can be identified as more efficient compared to the method given by Z. Fu, Y. Chen and B. Han. Further, in the last section of the paper, we discuss test examples to inspect the proficiency of the method. [ABSTRACT FROM AUTHOR]

Identification of physical properties of materials is very important because they are in general unknown. Furthermore, their direct experimental measurement could be costly and inaccurate. In such a situation, a cheap and efficient alternative is to mathematically formulate an inverse, but difficult, problem that can be solved, in general, numerically; the challenge being that the problem is, in general, nonlinear and ill-posed. In this paper, the reconstruction of a lower-order unknown time-dependent coefficient in a Cahn–Hilliard-type fourth-order equation from an additional integral observation, which has application to characterizing the nonlinear saturation of the collisional trapped-ion mode in a tokamak, is investigated. The local existence and uniqueness of the solution to such inverse problem is established by utilizing the Rothe method. Moreover, the continuous dependence of the unknown coefficient upon the measured data is derived. Next, the Tikhonov regularization method is applied to recover the unknown coefficient from noisy measurements. The stability estimate of the minimizer is derived by investigating an auxiliary linear fourth-order inverse source problem. Henceforth, the variational source condition can be verified. Then the convergence rate is obtained under such source condition. [ABSTRACT FROM AUTHOR]

Full waveform inversion (FWI) is a high-resolution technique to estimate the parameters of dielectric permittivity (ϵ) and electrical conductivity (σ) and identify the structure of the subsurface for Ground Penetrating Radar (GPR) application. However, permittivity and conductivity parameters can be coupled in bi-parameter GPR inversion. This coupling effect leads to the crosstalk in the FWI result. To solve this problem, we propose a novel approach to use the multi-scale FWI with the hybrid regularization method, which combines Tikhonov and total-variation (TV) regularizers that simultaneously invert the ϵ and the σ parameters, which improve the inversion accuracy and reduce the crosstalk effect. The multi-scale strategy uses the Wiener filtering to process the GPR data in different frequency ranges. Then, the low frequencies signal updates the bottom part and subsequently increases the frequencies to invert for the shallow areas. The Tikhonov regularization stabilizes the reconstruction of the smoothly varying background part. In contrast, Total Variation (TV) regularization can recover the large contrasts associated with the LNAPL model. The new Tikhonov-TV (TT) regularization can mitigate the crosstalk caused by the parameter coupling effect. Numerical tests with typical GPR models demonstrate that the proposed multi-scale TT-FWI strategy can effectively eliminate the crosstalk and improve the reconstruction accuracy when the model parameters have a different structure. [ABSTRACT FROM AUTHOR]

This study presents a Bayesian maximum a posteriori (MAP) framework for dynamical system identification from time-series data. This is shown to be equivalent to a generalized Tikhonov regularization, providing a rational justification for the choice of the residual and regularization terms, respectively, from the negative logarithms of the likelihood and prior distributions. In addition to the estimation of model coefficients, the Bayesian interpretation gives access to the full apparatus for Bayesian inference, including the ranking of models, the quantification of model uncertainties, and the estimation of unknown (nuisance) hyperparameters. Two Bayesian algorithms, joint MAP and variational Bayesian approximation, are compared to the least absolute shrinkage and selection operator (LASSO), ridge regression, and the sparse identification of nonlinear dynamics (SINDy) algorithms for sparse regression by application to several dynamical systems with added Gaussian or Laplace noise. For multivariate Gaussian likelihood and prior distributions, the Bayesian formulation gives Gaussian posterior and evidence distributions, in which the numerator terms can be expressed in terms of the Mahalanobis distance or "Gaussian norm" | | y − y ^ | | M − 1 2 = (y − y ^) ⊤ M − 1 (y − y ^) , where y is a vector variable, y ^ is its estimator, and M is the covariance matrix. The posterior Gaussian norm is shown to provide a robust metric for quantitative model selection for the different systems and noise models examined. [ABSTRACT FROM AUTHOR]

We investigate the strong convergence properties of a Nesterov type algorithm with two Tikhonov regularization terms in connection to the minimization problem of a smooth convex function f. We show that the generated sequences converge strongly to the minimal norm element from argmin f . We also show fast convergence for the potential energies f (x n) - min f and f (y n) - min f , where (x n) , (y n) are the sequences generated by our algorithm. Further we obtain fast convergence to zero of the discrete velocity and some estimates concerning the value of the gradient of the objective function in the generated sequences. Via some numerical experiments we show that we need both Tikhonov regularization terms in our algorithm in order to obtain the strong convergence of the generated sequences to the minimum norm minimizer of our objective function. [ABSTRACT FROM AUTHOR]

The hypothesis of homogeneous treatment effects is central to the instrumental variables literature. This assumption signifies that treatment effects are constant across all subjects. It allows to interpret instrumental variable estimates as average treatment effects over the whole population of the study. When this assumption does not hold, the bias of instrumental variable estimators can be greater than that of naive estimators ignoring endogeneity. This article develops two tests for the assumption of homogeneous treatment effects when the treatment is endogenous and an instrumental variable is available. The tests leverage a covariable that is (jointly with the error terms) independent of a coordinate of the instrument. This covariate does not need to be exogenous. The first test assumes that the potential outcomes are linear in the regressors and is computationally simple. The second test is nonparametric and relies on Tikhonov regularization. The treatment can be either discrete or continuous. We show that the tests have asymptotically correct level and asymptotic power equal to one against a range of alternatives. Simulations demonstrate that the proposed tests attain excellent finite sample performances. The methodology is also applied to the evaluation of returns to schooling and demand estimation in a fish market. [ABSTRACT FROM AUTHOR]

This article explores the generalized Gauss-Weierstrass transform associated with the space-fractional Bessel diffusion equation. Explicit inversion formulae for this transform are developed using best approximation methods and reproducing kernel theory. To address the inherent ill-posedness of this transform, Tikhonov regularization is implemented. Furthermore, the convergence rate of the regularized solutions is rigorously established. [ABSTRACT FROM AUTHOR]

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The hole-drilling method is one of the most widespread techniques to measure residual stresses. Since the introduction of the Integral Method to evaluate non-uniform stress distributions, there has been a considerable improvement in the instrumentation technology, as step increments of about 10 microns are now achievable. However, that spatial resolution makes the ill-posedness of the problem stand out among other sources of uncertainty. As the solution becomes totally dominated by noise, an additional regularization of the problem is needed to obtain meaningful results. Tikhonov regularization is the most common option, as it is also prescribed by the hole-drilling ASTM E837 standard, but it has only been studied in the reference case of a hole with no eccentricity with respect to the strain rosette. A recent work by Schajer addresses the eccentricity problem by defining a correction strategy that transforms strain measurements, allowing one to obtain the solution with the usual decoupled equations. In this work, Tikhonov regularization is applied to the eccentric hole case through the influence functions approach, in order to avoid the introduction of new error-compensating functions and bias-prone interpolations. Some useful general considerations for a practical implementation of the procedure and an experimental test case on an aluminum specimen are presented. [ABSTRACT FROM AUTHOR]

Tau Toolbox is a mathematical library for the solution of integro-differential problems, based on the spectral Lanczos' Tau method. Over the past few years, a class within the library, called polynomial, has been developed for approximating functions by classical orthogonal polynomials and it is intended to be an easy-to-use yet efficient object-oriented framework. In this work we discuss how this class has been designed to solve linear ill-posed problems and we provide a description of the available methods, Tikhonov regularization and truncated singular value expansion. For the solution of the Fredholm integral equation of the first kind, which is built from a low-rank approximation of the kernel followed by a numerical truncated singular value expansion, an error estimate is given. Numerical experiments illustrate that this approach is capable of efficiently compute good approximations of linear discrete ill-posed problems, even facing perturbed available data function, with no programming effort. Several test problems are used to evaluate the performance and reliability of the solvers. The final product of this paper is the numerical solution of a first-kind integral equation, which is constructed using only two inputs from the user: the kernel and the right-hand side. [ABSTRACT FROM AUTHOR]

The recently developed data-driven eigenmatrix method shows very promising reconstruction accuracy in sparse recovery for a wide range of kernel functions and random sample locations. However, its current implementation can lead to numerical instability if the threshold tolerance is not appropriately chosen. To incorporate regularization techniques, we have proposed to regularize the eigenmatrix method by replacing the computation of an ill-conditioned pseudo-inverse by the solution of an ill-conditioned least squares system, which can be efficiently treated by Tikhonov regularization. Extensive numerical examples confirmed the improved effectiveness of our proposed method, especially when the noise levels were relatively high.

Gravity inversion methods play a fundamental role in subsurface exploration, facilitating the characterization of geological structures and economic deposits. In this study, we conduct a comparative analysis of two widely used regularization methods, Tikhonov (L2) and Sparse (L1) regularization, within the framework of gravity inversion. To assess their performance, we constructed two distinct synthetic models by implementing tensor meshes, considering station spacing to discretize the subsurface environment precisely. Both methods have proven ability to recover density distributions while minimizing the inherent non-uniqueness and ill-posed nature of gravity inversion problems. Tikhonov regularization yields stable results, presenting smooth model parameters even with limited prior information and noisy data. Conversely, sparse regularization, utilizing sparsity-promoting penalties, excels in capturing sharp geological features and identifying anomalous regions, such as mineralized zones. Applying these methodologies to real gravity data from the Safu manganese deposit in northwest Iran, we assess their efficacy in recovering the geometry of dense ore deposits. Sparse regularization demonstrates superior performance, yielding lower misfit values and sharper boundaries during individual inversions. This underscores its capacity to provide a more accurate representation of the depth and edges of anomalous targets in this specific case. However, both methods represent the same top depth of the target in the real case study, but the lower depth and density distribution were not the same in the XZ cross-sections. Inversion results imply the presence of a near-surface deposit characterized by a high-density contrast and linear distribution, attributed to the high grade of manganese mineralization.

This article explores the generalized Gauss-Weierstrass transform associated with the space-fractional Bessel diffusion equation. Explicit inversion formulae for this transform are developed using best approximation methods and reproducing kernel theory. To address the inherent ill-posedness of this transform, Tikhonov regularization is implemented. Furthermore, the convergence rate of the regularized solutions is rigorously established.

Accurate identification of moving vehicle loads on bridges is one of the challenging tasks in bridge structural health monitoring, but lacks of intensive investigations to merge the heterogeneous data of vision-based vehicle spatiotemporal information (VVSI) and vehicle-induced bridge responses for moving force identification (MFI) in the existing time domain methods (TDM). In this study, a novel MFI method is proposed by integrating instantaneous VVSI and an improved TDM (iTDM). At first, a novel VVSI method combining background subtraction with template matching is presented to accurately track moving vehicles on bridges. With the calibration technique and camera perspective transformation model, the distribution of vehicles (DOV) on bridges is obtained and used as a priori information in the subsequent MFI. Then, the iTDM is developed based on the MFI equation re-formed in the form of instantaneous VVSI instead of the constant speed vehicle crossing bridges assumed in the traditional TDM. Finally, based on the redundant dictionary matrix composed of Haar functions for a moving load, the MFI problem is converted to explore a solution to the atom vectors and then solved by the Tikhonov regularization method. Experimental verifications in laboratory and a comparative study with the existing three methods are conducted to assess the feasibility of the proposed method. The results show that the proposed MFI method outperforms the existing methods and can effectively identify the moving vehicle loads with a higher and acceptable accuracy. It is successful for the proposed method to replace the assumption of constant speed vehicle crossing bridge in the traditional TDM with the instantaneous VVSI in the MFI problem. [ABSTRACT FROM AUTHOR]

The directional loudspeaker array has various applications due to its capability to direct sound generation towards the target listener and reduce noise pollution. Differential beamforming has recently been applied to the loudspeaker line array to produce a broadside frequency-invariant radiation pattern. However, the existing methods cannot achieve a compromise between robustness and broadband frequency-invariant beampattern preservation. This paper proposed a robust broadband differential beamforming design to allow the loudspeaker line array to radiate broadside frequency-invariant radiation patterns with robustness. Specifically, we propose a method to determine the ideal broadside differential beampattern by combining multiple criteria, namely null positions, maximizing the directivity factor, and achieving a desired beampattern with equal sidelobes. We derive the above ideal broadside differential beampattern into the target beampattern in the modal domain. We propose a robust modal matching method with Tikhonov regularization to optimize the loudspeaker weights in the modal domain. Simulations and experiments show improved frequency-invariant broadside beamforming over the 250–4k Hz frequency range compared with the existing modal matching and null-constrained methods. [ABSTRACT FROM AUTHOR]

Tikhonov regularization is a widely used technique in solving inverse problems that can enforce prior properties on the desired solution. In this paper, we propose a Krylov subspace based iterative method for solving linear inverse problems with general-form Tikhonov regularization term xT Mx, where M is a positive semidefinite matrix. An iterative process called the preconditioned Golub--Kahan bidiagonalization (pGKB) is designed, which implicitly utilizes a proper preconditioner to generate a series of solution subspaces with desirable properties encoded by the regularizer xT Mx. Based on the pGKB process, we propose an iterative regularization algorithm via projecting the original problem onto small dimensional solution subspaces. We analyze the regularization properties of this algorithm, including the incorporation of prior properties of the desired solution into the solution subspace and the semiconvergence behavior of the regularized solution. To overcome instabilities caused by semiconvergence, we further propose two pGKB based hybrid regularization algorithms. All the proposed algorithms are tested on both small-scale and large-scale linear inverse problems. Numerical results demonstrate that these iterative algorithms exhibit excellent performance, outperforming other state-of-the-art algorithms in some cases. [ABSTRACT FROM AUTHOR]

Tikhonov regularization technique is widely recognized as one of the most prevalent and well-established approaches for solving linear discrete ill-posed problems. The present study introduces two novel randomized iterative algorithms for the computation of numerical solutions to large-scale Tikhonov regularization problems. The first one applies the randomized Kaczmarz algorithm to an augmented regularized normal system of equations, the second one is an accelerated version of the first one by means of greedy probability criterion. In theory, we establish some convergence results for these two algorithms. Numerical experiments demonstrate the convergence properties and illustrate the performances of these two algorithms. [ABSTRACT FROM AUTHOR]

The recently developed data-driven eigenmatrix method shows very promising reconstruction accuracy in sparse recovery for a wide range of kernel functions and random sample locations. However, its current implementation can lead to numerical instability if the threshold tolerance is not appropriately chosen. To incorporate regularization techniques, we have proposed to regularize the eigenmatrix method by replacing the computation of an ill-conditioned pseudo-inverse by the solution of an ill-conditioned least squares system, which can be efficiently treated by Tikhonov regularization. Extensive numerical examples confirmed the improved effectiveness of our proposed method, especially when the noise levels were relatively high. [ABSTRACT FROM AUTHOR]

In order to achieve an effective approximation solution for solving discrete ill-conditioned problems, Golub, Hansen, and O'Leary used Tikhonov regularization and the total least squares (TRTLS) method, where the bidiagonal technique is considered to deal with computational aspects. In this paper, the generalized singular value decomposition (GSVD) technique is used for computational aspects, and then Tikhonov regularized total least squares based on the generalized singular value decomposition (GTRTLS) algorithm is proposed, whose time complexity is better than TRTLS. For medium- and large-scale problems, the randomized GSVD method is adopted to establish the randomized GTRTLS (RGTRTLS) algorithm, which reduced the storage requirement, and accelerated the convergence speed of the GTRTLS algorithm. [ABSTRACT FROM AUTHOR]

The entropy error function has been widely used in neural networks. Nevertheless, the network training based on this error function generally leads to a slow convergence rate, and can easily be trapped in a local minimum or even with the incorrect saturation problem in practice. In fact, there are many results based on entropy error function in neural network and its applications. However, the theory of such an algorithm and its convergence have not been fully studied so far. To tackle the issue, this works proposes a novel entropy function with smoothing l 0 regularization for feed-forward neural networks. An empirical evaluation has been conducted on real-world datasets to demonstrate that the newly conceived algorithm allows us to substantially improve the prediction performance of the considered neural networks. More importantly, the experimental results also show that the proposed function brings in more precise classifications, compared to well-founded baselines. The work is novel as it enables neural networks to learn effectively, producing more accurate predictions compared to state-of-the-art algorithms. In this respect, it is expected that the algorithm will contribute to existing studies in the field, advancing research in Machine Learning and Deep Learning. [ABSTRACT FROM AUTHOR]

The uneven distribution of global navigation satellite system (GNSS) continuous stations in the Yellow River Basin, combined with the sparse distribution of GNSS continuous stations in some regions and the weak far-field load signals, poses challenges in using GNSS vertical displacement data to invert terrestrial water storage changes (TWSCs). To achieve the inversion of water reserves in the Yellow River Basin using unevenly distributed GNSS continuous station data, in this study, we employed the Tikhonov regularization method to invert the terrestrial water storage (TWS) in the Yellow River Basin using vertical displacement data from network engineering and the Crustal Movement Observation Network of China (CMONOC) GNSS continuous stations from 2011 to 2022. In addition, we applied an inverse distance weighting smoothing factor, which was designed to account for the GNSS station distribution density, to smooth the inversion results. Consequently, a gridded product of the TWS in the Yellow River Basin with a spatial resolution of 0.5 degrees on a daily scale was obtained. To validate the effectiveness of the proposed method, a correlation analysis was conducted between the inversion results and the daily TWS from the Global Land Data Assimilation System (GLDAS), yielding a correlation coefficient of 0.68, indicating a strong correlation, which verifies the effectiveness of the method proposed in this paper. Based on the inversion results, we analyzed the spatial–temporal distribution trends and patterns in the Yellow River Basin and found that the average TWS decreased at a rate of 0.027 mm/d from 2011 to 2017, and then increased at a rate of 0.010 mm/d from 2017 to 2022. The TWS decreased from the lower-middle to lower reaches, while it increased from the upper-middle to upper reaches. Furthermore, an attribution analysis of the terrestrial water storage changes in the Yellow River Basin was conducted, and the correlation coefficients between the monthly average water storage changes inverted from the results and the monthly average precipitation, evapotranspiration, and surface temperature (AvgSurfT) from the GLDAS were 0.63, −0.65, and −0.69, respectively. This indicates that precipitation, evapotranspiration, and surface temperature were significant factors affecting the TWSCs in the Yellow River Basin. [ABSTRACT FROM AUTHOR]

This research studies the inverse boundary value problem for fractional elliptic equation of Tricomi–Gellerstedt–Keldysh type and obtains a condition stability result. To recover the continuous dependence of the solution on the measurement data, a generalized Tikhonov regularization method based on ill-posedness analysis is constructed. Under the a priori and a posterior selection rules for the regularization parameter, corresponding Hölder type convergence results are obtained. On this basis, this thesis verifies the simulation effect of the generalized Tikhonov method through numerical examples. The examples show that the method performs well in dealing with the problem under consideration. [ABSTRACT FROM AUTHOR]

The electrical capacitance tomography (ECT) is a visual nondestructive testing technology. The relative positional distribution between the electrodes and the phantom object affects the accuracy of the reconstructed image. To solve this problem, an image reconstruction method and image fusion algorithm of ECT system based on rotating electrodes are proposed. First, 4 image reconstruction algorithms are employed to reconstruct the experimental model, the Landweber iterative algorithm based on Tikhonov regularization presents the best performance. Then, by rotation the 12 electrodes 4 times, we can obtain 5 sets of capacitance data, and obtain 5 images. Finally, the fusion results can be obtained by performing the adaptive weighted fusion on these 5 images. Results show that the adaptive weighted image fusion method based on rotation electrodes improves the quality of reconstructed images and effectively reduces the artefacts. [ABSTRACT FROM AUTHOR]

This paper surveys an important class of methods that combine iterative projection methods and variational regularization methods for large-scale inverse problems. Iterative methods such as Krylov subspace methods are invaluable in the numerical linear algebra community and have proved important in solving inverse problems due to their inherent regularizing properties and their ability to handle large-scale problems. Variational regularization describes a broad and important class of methods that are used to obtain reliable solutions to inverse problems, whereby one solves a modified problem that incorporates prior knowledge. Hybrid projection methods combine iterative projection methods with variational regularization techniques in a synergistic way, providing researchers with a powerful computational framework for solving very large inverse problems. Although the idea of a hybrid Krylov method for linear inverse problems goes back to the 1980s, several recent advances on new regularization frameworks and methodologies have made this field ripe for extensions, further analyses, and new applications. In this paper, we provide a practical and accessible introduction to hybrid projection methods in the context of solving large (linear) inverse problems. [ABSTRACT FROM AUTHOR]

In this paper, we deal with a class of second-order evolutionary history-dependent variational–hemivariational inequalities with constraint. The unique solvability of the considered second-order evolutionary inequality problem is established via a surjectivity result combined with a fixed point theorem. Moreover, we construct a regularized problem for such second-order evolutionary history-dependent variational–hemivariational inequality and show the convergence of the regularized solutions toward the solution of the original problem under some mild assumptions. [ABSTRACT FROM AUTHOR]

This article deals with an inverse problem of determining two space-time-dependent sources in an integerfractional mobile-immobile two-region solute transport system by additional Dirichlet-Neumann data. The unique existence of a solution to the forward problem is obtained by the method of Laplace transform, and a dynamical system connecting the known data with the unknown sources is established by variational method and boundary homogenization. The dynamical system is discretized to a linear system at a given time in a homogenous polynomial space, and the sources are reconstructed by alternative iterations and Tikhonov regularization. Numerical examples are presented to illustrate the validity of the inversion algorithm. [ABSTRACT FROM AUTHOR]

Interstory drift response is one of most important quantities to quickly assess performance and damages in buildings. Nevertheless, direct measurement of interstory drift is difficult and expensive, because a stationary reference is required to attach measurement devices. With the goal of accurate and fast reference-free estimation, this paper proposes a new strategy to determine dynamic interstory drifts using accelerations. In particular, a Tikhonov regularization is adopted in a generalized minimization problem to achieve an efficient and stable finite impulse response (FIR) filter. Furthermore, due to independent clocks in wireless sensors, accurate time synchronization of the records is critical; consequently, a strategy for accurate synchronization is also presented. Finally, the proposed strategy has been deployed on edge devices for onboard real-time interstory drift estimation. The proposed method for dynamic interstory drift estimation is validated, first, by numerical simulation using earthquake records as base excitation of linear and nonlinear buildings, as well as through laboratory shake table experiments. Both numerical and lab test results show good agreement of dynamic interstory drifts between the measured value and estimated results, demonstrating the efficacy of the proposed method to estimate the dynamic displacements of seismically excited structures. [ABSTRACT FROM AUTHOR]

To systematically and thoroughly analyze the sensitivity and reliability of the MSBAS regularization for the estimation of surface deformation over a mine in combination with an application example, this study processed 101 Sentinel-1A/B SAR images, constructed and solved the 2D deformation models using SVD and Tikhonov regularization methods with different orders and parameters, and estimated the vertical and east-west surface deformation time series in a mine of China. Then, this study collected the leveling-monitoring vertical surface deformation data on three leveling points, and compared and analyzed the sensitivity and reliability of the MSBAS regularization methods for estimating vertical surface deformation. The results indicate that different regularization orders and parameters can lead to thousands of times differences in condition numbers and significant differences in illposed degree of the deformation models. The zero-order Tikhonov regularized deformation model with regularization parameter of 0.1 has the minimum condition number and the equation is not ill-posed. The first-order Tikhonov regularized deformation model with regularization parameter of 0.001 has the maximum condition number and the equation is seriously illposed. As a result, the estimates of 2D surface deformation models with different parameters and orders are also different in terms of numerical values and accuracy. Compared with the leveling-monitoring data, the first- and second-order MSBAS regularization methods with parameter 0.1 have the minimum fluctuation and the maximum correlation coefficients between the estimated values and the leveling-monitoring values, and are also closest to the leveling-monitoring results with the highest accuracy. [ABSTRACT FROM AUTHOR]

Tikhonov Regularization is the most widely used method for geophysical inversion problems. The result of previous and current research has shown that how to estimate the regularization parameter has a dramatic effect on inversion results. In the present research, conventional methods, including L-curve, Discrepancy principle, GCV, and ACB are compared with an innovative technique called Unbiased Predictive Risk Estimator (UPRE) in the inversion of 2D magnetotelluric data. For this purpose, MT2DInvMatlab is applied as the main program. It uses the Levenberg–Marquardt method as the inversion core and the ACB method to estimate the regularization parameter. Then, this program was developed in a way that it could estimate the regularization parameter using all of the above-mentioned methods. Next, a relatively complex model consisting of two layers and three blocks was used as a synthetic model. Comparing the results of all methods in TM, TE, and joint modes showed that the UPRE method, which previously provided desirable results in the inversion of potential field data in terms of convergence rate, time, and accuracy of results, here along with the ACB method, presented more acceptable results in the same indicators. Therefore, these two methods were used in a geothermal case in the North-West of Iran as a real test. In this case, the UPRE presented results at the same level as the ACB method and better than it in terms of some indicators. So, the UPRE method, especially in large-scale problems, could be a suitable alternative to the ACB method. [ABSTRACT FROM AUTHOR]

The gravity inversion methods play a fundamental role in subsurface exploration, facilitating the characterization of geological structures and economic deposits. In this study, we conduct a comparative analysis of two widely used regularization methods, Tikhonov (L2) and Sparse (L1) regularization, within the framework of the gravity inversion. To assess their performance, we constructed two distinct synthetic models by implementing tensor meshes, considering station spacing to discretize the subsurface environment precisely. Both methods have proven ability to recover density distributions, while minimizing the inherent non-uniqueness and ill-posed nature of the gravity inversion problems. The Tikhonov regularization yields stable results, presenting smooth model parameters even with limited prior information and noisy data. Conversely, the Sparse regularization, utilizing sparsity-promoting penalties, excels in capturing sharp geological features and identifying anomalous regions, such as mineralized zones. Applying these methodologies to real gravity data from the Safu manganese deposit in northwest Iran, we assess their efficacy in recovering the geometry of dense ore deposits. The Sparse regularization demonstrates superior performance, yielding lower misfit values and sharper boundaries during individual inversions. This underscores its capacity to provide a more accurate representation of the depth and edges of anomalous targets in this specific case. However, both methods represent the same top depth of the target in the real case study, but the lower depth and density distribution were not the same in the XZ cross-sections. The inversion results imply the presence of a near-surface deposit characterized by a high-density contrast and linear distribution, attributed to the high grade of manganese mineralization. [ABSTRACT FROM AUTHOR]