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We developed a technique allowing the direct observation of photoinduced charge-transfer states (CTSs)—the weakly coupled electron–hole pairs preceding the completely separated charges in organic photovoltaic (OPV) blends. Quadrature detection of the electron spin echo (ESE) signal enables the observation of an out-of-phase ESE signal of CTS. The out-of-phase Electron–Electron Double Resonance (ELDOR) allows measuring electron–hole distance distributions within CTS and its temporal evolution in the microsecond range. The technique was applied to OPV bulk heterojunction blends of different donor polymers, including the benchmark polymer P3HT and the high-performance polymer PCDTBT, with the fullerene PC61BM acceptor. The corresponding electron–hole distance distributions were obtained using the Tikhonov regularization. It was found that not only the dipolar interaction but also the exchange interaction contributes to the formation of the out-of-phase ELDOR signal. By varying the delay time after photoexcitation, we observed CTSs at different stages of charge separation. The initial distribution of the electron–hole distances for different blends correlates with their photoelectric conversion efficiency, with shorter average thermalization distances found for the blends of PC61BM with the less efficient regiorandom polymer P3HT. Spin-selective recombination of the CTS was unambiguously demonstrated for the blend of regioregular P3HT with PC61BM. It produces characteristic features in the out-of-phase ELDOR trace for small "dipolar" evolution times. These data allow us to estimate the CTS recombination rate for a certain distance between the electron and the hole within the CTS. The proposed method can be used to probe CTS in a variety of OPV active layer materials. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the inverse problem of determining the order ν of the major derivative in the multi-term fractional in time semilinear diffusion equation with memory terms. This framework is relevant to the oxygen transport through capillaries. We obtain an explicit formula reconstructing the order ν for small time state measurements. Appealing to the Tikhonov regularization scheme and the quasi-optimality criterion, we propose the computational algorithm to recovery ν from noisy discrete measurements. We provide several numerical tests illustrating the effectiveness of this algorithm in practice. [ABSTRACT FROM AUTHOR]

Manifold-valued signal- and image processing has received attention due to modern image acquisition techniques. Recently, a convex relaxation of the otherwise non-convex Tikhonov-regularization for denoising circle-valued data has been proposed by Condat (2022). The circle constraints were encoded in a series of low-dimensional, positive semi-definite matrices. Using Schur complement arguments, we showed that the resulting variational model can be simplified while leading to the same solution. The simplified model can be generalized to higher dimensional spheres and to the special orthogonal group SO(3), where we relied on the quaternion representation of the latter. Standard algorithms from convex analysis can be applied to solve the resulting convex minimization problem. As proof-of-the-concept, we used the alternating direction method of multipliers to demonstrate the denoising behavior of the proposed method. In a series of experiments, we demonstrated the numerical convergence of the signal- or image values to the underlying manifold. [ABSTRACT FROM AUTHOR]

This paper is devoted to the determination of the initial data in a one-dimensional degenerate and singular wave equation with degeneracy and singularity occurring at the boundary of the spatial domain. In particular, we address the question of well posedness of the problem. Then, the identification of the initial data is formulated as a minimization problem combining the output least squares method and the Tikhonov regularization. Finally, numerical results are provided to show the performance of the proposed approach. [ABSTRACT FROM AUTHOR]

AbstractIn this article, we investigate both forward and backward problems for coupled systems of time-fractional diffusion equations, encompassing scenarios of strong coupling. For the forward problem, we establish the well-posedness of the system, leveraging the eigensystem of the corresponding elliptic system as the foundation. When considering the backward problem, specifically the determination of initial values through final time observations, we demonstrate a Lipschitz stability estimate, which is consistent with the stability observed in the case of a single equation. To numerically address this backward problem, we refer to the explicit formulation of Tikhonov regularization to devise a multi-channel neural network architecture. This innovative architecture offers a versatile approach, exhibiting its efficacy in multidimensional settings through numerical examples and its robustness in handling initial values that have not been trained. [ABSTRACT FROM AUTHOR]

In recent years error bounds for Tikhonov regularization of linear and non-linear ill-posed problems with oversmoothing penalty received attention. Here, oversmoothing means, that the exact solution does not admit a finite value for the used penalty. This problem was initiated by F. Natterer in "Error bounds for Tikhonov regularization in Hilbert scales". Applicable Anal., 18(1–2):29–37, 1984. In this note we discuss the application of distance functions to derive order optimal error bounds. We specify these bounds in case that the smoothness is given in terms of source conditions. [ABSTRACT FROM AUTHOR]

The improvement in hydrogen fuel cell vehicles markedly aids carbon reduction in transportation. Nevertheless, hydrogen is a highly combustible and explosive gas. Addressing the potential risk that hydrogen leakage from these vehicles in underground garages may pose to the garage structure and occupants, the reliability of the hydrogen leakage model is validated firstly through experimental data. Subsequently, the simulation results of the hydrogen leakage model are integrated with Tikhonov regularization and Bayesian methods to develop an identification model. The model can identify the release rate profiles with constant and decaying release. Finally, the impact of sensor distribution positioning on both the release rate and location identification is determined. It is shown that the relative error in identifying the constant release rate linked to the optimal regularization parameter ranges from 0.32% to 5.80%. Accurately locating a leakage depends more on the average relative error than on sensor arrangement. When the average relative error in predicted concentration is less than 0.03 %, sensors positioned at various locations can correctly identify the leak's location with a median probability of approximately 75%. This research can provide a reliable source information about the occurrence of unexpected hydrogen leaks. • Hydrogen leakages with different release characteristics were investigated. • Optimization regularization parameter influenced identification relative error. • Sensors positioned downstream exhibited better identification performance. • Leakage location identification depends on predicted concentration error. [ABSTRACT FROM AUTHOR]

Recovering the relaxation spectrum, a fundamental rheological characteristic of polymers, from experiment data requires special identification methods since it is a difficult ill-posed inverse problem. Recently, a new approach relating the identification index directly with a completely unknown real relaxation spectrum has been proposed. The integral square error of the relaxation spectrum model was applied. This paper concerns regularization aspects of the linear-quadratic optimization task that arise from applying Tikhonov regularization to relaxation spectra direct identification problem. An influence of the regularization parameter on the norms of the optimal relaxation spectra models and on the fit of the related relaxation modulus model to the experimental data was investigated. The trade-off between the integral square norms of the spectra models and the mean square error of the relaxation modulus model, parameterized by varying regularization parameter, motivated the definition of two new multiplicative indices for choosing the appropriate regularization parameter. Two new problems of the regularization parameter optimal selection were formulated and solved. The first and second order optimality conditions were derived and expressed in the matrix-vector form and, alternatively, in finite series terms. A complete identification algorithm is presented. The usefulness of the new regularization parameter selection rules is demonstrated by three examples concerning the Kohlrausch–Williams–Watts spectrum with short relaxation times and uni- and double-mode Gauss-like spectra with middle and short relaxation times. [ABSTRACT FROM AUTHOR]

Assessing image fusion quality purposefully is a challenging task due to the diversities of fused features. In this work, a specific multi-focus image fusion quality assessment method is proposed based on joint image layering and convolutional sparse representation. Specifically, the proposed method includes two stages: Tikhonov regularization optimization-based joint image layering and convolutional sparse representation-based focus similarity comparison. The first stage decomposes the source images and their fusion result jointly into a common base layer and respective detail layers, and then, the second stage compares the focus similarity between these detail layers with their convolutional sparse features. The main novelty of our work is to assess fusion quality with learning features, rather than with those handcrafted low-level patterns. Consequently, our method has higher reliability and feature-level analytical ability. A large number of objective and subjective experiments demonstrate the effectiveness and specificity of the proposed method. Moreover, the applicability of the general blind natural image quality metrics for image fusion was also examined and discussed. Besides experiments, the feature-level characteristics of multi-focus image fusion were also investigated and analyzed with the proposed method. Our analysis reveals some potential laws that could provide new perspectives for fusion algorithm design and improvement. [ABSTRACT FROM AUTHOR]

Crossover adjustment is a crucial method for processing satellite altimetry data. The primary focus in the research of crossover adjustment has been on correcting time-related systematic errors, while studies on correcting geography-related systematic errors are relatively few. Reducing the impact of geography-related systematic errors can improve the detection of finer ocean signals in sea surface height (SSH). To address this problem, a novel approach introduced in the study divides the process for satellite altimetry crossover adjustment into two steps. Compared to other crossover adjustment, the two-step method incorporates the correcting of geography-related systematic errors using a harmonic expansion model. In the first step, a hybrid polynomial model with time as the variable is used to correct for time-related systematic errors. Unlike the traditional crossover adjustment method that constructs observation equations directly through crossover discrepancies, the first step can be processed for each track individually, and an appropriate hybrid polynomial empirical model is selected based on the number of crossover points on the track, thereby mitigating the impact of rank deficiency. In the second step, a harmonic expansion model with latitude and longitude as variables is used to correct for geography-related systematic errors. The parameter of the harmonic expansion are estimated using the Tikhonov regularization principle and the L-curve method to determine the regularization parameter, thereby resolving the issue of rank deficiency. The completion of the two-step process corrects time-related and geography-related systematic errors in satellite altimeter data. Experiments were conducted in the Arabian Sea region (46°∼80°E, 0°∼30°N) using Archiving, Validation and Interpretation of Satellite Oceanographic data (AVISO) Level 2 plus data of Sentinel-3A, Sentinel-3B. The results indicate that the two-step method can reduce the crossover difference from Sentinel-3A and Sentinel-3B by about 4.1 cm, and reduce the discrepancy between Sentinel-3A/Sentinel-3B and Sentinel-6A/Jason-3 by approximately 1.6 cm to 1.8 cm. The tide gauge data is used to validate the two-step method, showing that the STD of the differences between the SSHs processed using this method and the tide gauge data was reduced by 5.3 % to 17.3 %. [ABSTRACT FROM AUTHOR]

The Modified Ambiguity Function Approach (MAFA) implicitly conducts the search procedure of carrier phase GNSS integer ambiguity resolution (IAR) in the coordinate domain using the integer least squares (ILS) principle, i.e. MAFA-ILS. One of the still open scientific problems is an accurate definition of the search region, especially in the context of instantaneous IAR. In doing so, the float solution results, which encompass float position (FP) and its variance-covariance (VC) matrix, must be improved as these are necessary for defining the search region. For this reason, the ambiguity parameters are separately regularized, and then the baseline parameters are conditioned on regularized float ambiguities. The conditional-regularized estimation is thus designed, obtaining the regularized FP (RFP) and its VC-matrix. This solution is promising because its accuracy is enhanced in the sense of mean squared error (MSE) thanks to the improved precision at the cost of regularized bias. The optimal regularization parameter (RP) values obtained for ambiguity parameters balance the contributions of improved precision and bias in the regularized float baseline solution's MSE. Therefore, the regularized search region is defined accurately in the coordinate domain to contain such approximate coordinates that more frequently give the correct ILS solution. It also contains fewer MAFA-ILS candidates, improving the search procedure's numerical efficiency. The regularized ILS estimator performs well with the presence of bias, increasing the probability of correct IAR in the coordinate domain. [ABSTRACT FROM AUTHOR]

This paper deals with a second order dynamical system with a Tikhonov regularization term in connection to the minimization problem of a convex Fréchet differentiable function. The fact that beside the asymptotically vanishing damping we also consider an implicit Hessian driven damping in the dynamical system under study allows us, via straightforward explicit discretization, to obtain inertial algorithms of gradient type. We show that the value of the objective function in a generated trajectory converges rapidly to the global minimum of the objective function and depending the Tikhonov regularization parameter the generated trajectory converges weakly to a minimizer of the objective function or the generated trajectory converges strongly to the element of minimal norm from the argmin set of the objective function. We also obtain the fast convergence of the velocities towards zero and some integral estimates. Our analysis reveals that the Tikhonov regularization parameter and the damping parameters are strongly correlated, there is a setting of the parameters that separates the cases when weak convergence of the trajectories to a minimizer and strong convergence of the trajectories to the minimal norm minimizer can be obtained. [ABSTRACT FROM AUTHOR]

In this paper, the deconvolution of Electrochemical Impedance Spectroscopy (EIS) data into the Distribution of Relaxation Times (DRTs) is employed to provide a detailed examination of degradation mechanisms in lithium-ion batteries. Using an nth RC model with Gaussian functions, this study achieves enhanced separation of overlapping electrochemical processes where Gaussian functions yield smoother transitions and clearer peak identification than conventional piecewise linear functions. The advantages of employing Tikhonov Regularization (TR) with Gaussian functions over Maximum Entropy (ME) and FFT methods are highlighted as this approach provides superior noise resilience, unbiased analysis, and enhanced resolution of critical features. This approach is applied to LIB cell data to identify characteristic peaks of the DRT plot and evaluate their correlation with battery degradation. By observing how these peaks evolve through cycles of battery aging, insights into specific aging mechanisms and performance decline are obtained. This study combines experimental measurements with DRT peak analysis to characterize the impedance distribution within LIBs which enables accelerated detection of degradation pathways and enhances the predictive accuracy for battery life and reliability. This analysis contributes to a refined understanding of LIB degradation behavior, supporting the development of advanced battery management systems designed to improve safety, optimize battery performance, and extend the operational lifespan of LIBs for various applications. [ABSTRACT FROM AUTHOR]

We consider the monotone inclusion problems in real Hilbert spaces. Proximal splitting algorithms are very popular technique to solve it and generally achieve weak convergence under mild assumptions. Researchers assume the strong conditions like strong convexity or strong monotonicity on the considered operators to prove strong convergence of the algorithms. Mann iteration method and normal S-iteration method are popular methods to solve fixed point problems. We propose a new common fixed point algorithm based on normal S-iteration method using Tikhonov regularization to find common fixed point of non-expansive operators and prove strong convergence of the generated sequence to the set of common fixed points without assuming strong convexity and strong monotonicity. Based on proposed fixed point algorithm, we propose a forward–backward-type algorithm and a Douglas–Rachford algorithm in connection with Tikhonov regularization to find the solution of monotone inclusion problems. Further, we consider the complexly structured monotone inclusion problems which are very popular these days. We also propose a strongly convergent forward–backward-type primal–dual algorithm and a Douglas–Rachford-type primal–dual algorithm to solve the monotone inclusion problems. Finally, we conduct a numerical experiment to solve image deblurring problems. [ABSTRACT FROM AUTHOR]

As a non-invasive imaging method, electrical impedance tomography (EIT) technology has become a research focus for grounding grid corrosion diagnosis. However, the existing algorithms have not produced ideal image reconstruction results. This article proposes an electrical impedance imaging method based on sparse representation, which can improve the accuracy of reconstructed images obviously. First, the basic principles of EIT are outlined, and the limitations of existing reconstruction methods are analyzed. Then, an EIT reconstruction algorithm based on sparse representation is proposed to address these limitations. It constructs constraints using the sparsity of conductivity distribution under a certain sparse basis and utilizes the accelerated Fast Iterative Shrinkage Threshold Algorithm (FISTA) for iterative solutions, aiming to improve the imaging quality and reconstruction accuracy. Finally, the grounding grid model is established by COMSOL simulation software to obtain voltage data, and the reconstruction effects of the Tikhonov regularization algorithm, the total variation regularization algorithm (TV), the one-step Newton algorithm (NOSER), and the sparse reconstruction algorithm proposed in this article are compared in MATLAB. The voltage relative error is introduced to evaluate the reconstructed image. The results show that the reconstruction algorithm based on sparse representation is superior to other methods in terms of reconstruction error and image quality. The relative error of the grounding grid reconstructed image is reduced by an average of 12.54%. [ABSTRACT FROM AUTHOR]

In this paper, we study the influence of analytical regularization used in the generalized function (distribution) space to the Tikhonov regularization procedure utilized in the different versions of Moore–Penrose's inversion. By introducing a new analytical term to the Tikhonov regularization of Moore–Penrose's inversion procedure, we derive new optimization conditions that extend the Tikhonov regularization framework and influence the fitting parameter. This enhancement yields a more robust and accurate reconstruction of physical quantities, demonstrating its potential impact on various studies. We illustrate the significance of new term through schematic examples of physical applications, highlighting its relevance to diverse fields. Our findings provide a valuable tool for improving inversion methods and their applications in physics and beyond. [ABSTRACT FROM AUTHOR]

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]

This paper studies simultaneous inversion of initial value and source term of a time‐fractional Black‐Scholes equation. This problem is ill‐posed, and we use Tikhonov regularization method to solve it. Under the selection rules of a priori and a posteriori regularization parameters, a priori and a posteriori Hölder type error estimates are derived. Finally, numerical experiments demonstrate the stability and effectiveness of the proposed regularization method. [ABSTRACT FROM AUTHOR]

The main focus of our study is the one-dimensional parabolic Cauchy-Stefan inverse problem, which entails the identification of initial condition data. In this context, we explore the application of the method of fundamental solutions. This method is employed iteratively until the optimal initial condition data is determined. It generates an ill-conditioned matrix, which can be addressed through different regularization methods. Our numerical experiments and theoretical analyses of these approaches illustrate that accurate results can be achieved. [ABSTRACT FROM AUTHOR]

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

This paper investigates the inverse problem of determining the merely time-dependent lowest term and the particle concentration in a linear Bi-flux diffusion equation from knowledge of the particle concentration at the left boundary. The unique solvability of this inverse problem is established through the application of the contraction principle for sufficiently small time intervals. To solve this problem computationally, it is reformulated as a nonlinear least-squares minimization problem with simple bounds on the unknown coefficient, and to ensure stability the Tikhonov regularization is employed. The Crank--Nicolson finite-difference scheme is developed to solve the direct problem. On the other hand, the nonlinear least-squares minimization problem is iteratively solved using the built-in subroutine lsqnonlin from the MATLAB optimization toolbox. Numerical findings for two benchmark examples, involving the recovery of smooth and non-smooth time-dependent lowest terms, are presented and analyzed. [ABSTRACT FROM AUTHOR]

Ill-posed problems manifest in a wide range of scientific and engineering disciplines. The solutions to these problems exhibit a high degree of sensitivity to data perturbations. Regularization methods strive to alleviate the sensitivity exhibited by these solutions. This paper presents a fast iterative scheme for addressing linear ill-posed problems, similar to nonstationary iterated Tikhonov regularization. Both the a-priori and a-posteriori choice rules for regularization parameters are provided, and both rules yield error estimates that are order optimal. In comparison to the nonstationary iterated Tikhonov method, the newly introduced method significantly reduces the required number of iterations to achieve convergence based on an appropriate stopping criterion. The numerical computations provide compelling evidence regarding the efficacy of our new iterative regularization method. Furthermore, the versatility of this method extends to image restorations. [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]

Background: 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. Purpose: 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. Materials and methods: 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. Results: 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. Conclusion: 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]

Objectives: The gravity recovery and climate experiment time-variable gravity field model suffers from significant north-south striping errors due to insufficient east-west sampling, background force model errors, and inadequate solving strategies. These errors severely limit its application. While the official decorrelation and denoising kernel （DDK) regularization filtering algorithm is widely used, it has notable shortcomings: (1) The regularization parameters are empirically determined and do not account for monthly variability, using the same parameters for each month. (2) The Tikhonov regularization method over-regularizes low-frequency components and under-regularizes high-frequency components. To address these issues, we propose an adaptive regularization filtering method. Methods: This method estimates low-frequency components using least squares (without regularization), applies Tikhonov regularization to mid-frequency components, and truncates high-frequency components. Each frequency band and its regularization parameters are optimized based on the minimum mean square error criterion and processed separately each month. Results: The proposed method is applied to the ITSG-Grace2018 time-variable gravity field data with a maximum degree of 96, spanning from April 2002 to June 2017. Experimental results show that our proposed method achieves higher spatial resolution of mass anomalies and aligns more closely with the three official mascon products compared to the classic DDK filtering. Conclusions: Simulation experiments further validate that the mass anomalies obtained by this method are closer to the simulated true values. [ABSTRACT FROM AUTHOR]

This paper proposed a new matrix decomposition method ARCIMA to address issues in the CLSMD approach, where hard thresholding could lead to loss of speech signal components or isolated noise problems. Initially, the energy threshold method estimated the rank of the low-rank matrix. Then, considering the structural characteristics of the speech signal subspace matrix, the MBRP method solved the low-rank matrix representing the clean speech signal, reducing the computational load compared to the SVD method. Tikhonov regularization optimized the solution's stability during iterative solving. Experimental results show that this method achieves better PESQ scores in various noisy environments compared to classical methods, and the enhanced speech waveform is closer to the original speech waveform. The method demonstrates superior denoising performance under low signal-to-noise ratio conditions. [ABSTRACT FROM AUTHOR]

The article discusses the use of Jupyter Notebooks for parameter estimation, uncertainty analysis, and optimization with PEST++ in groundwater modeling. It highlights the development and evolution of PEST/PEST++ software, as well as the creation of interactive tutorials by the Groundwater Modeling Decision Support Initiative (GMDSI) and the U.S. Geological Survey. The tutorials cover various aspects of PEST++ concepts, mechanics, and decision support modeling, providing a comprehensive introduction to users with Python scripting knowledge. The article also addresses the potential benefits and areas for improvement in the tutorial notebooks to enhance their educational value in groundwater modeling. [Extracted from the article]

Accurate identification of impact loads is vital for structural assessment and design. Traditional methods rely on complex equipment, such as accelerometers or strain gauge, which can be expensive and prone to failure. This study introduces a non-contact intelligent identification approach incorporating visual sensing technology, providing a convenient means to identify impact loads. Numerical simulations explore the differences in identifying impact forces through acceleration and displacement responses, particularly by considering such variables as measurement noise and number of measurement points. A meticulously designed experiment verified the feasibility of the proposed method for measuring the displacement and velocity of rapidly moving targets, and evaluated its performance in terms of accuracy. A series of impact loading experiments were conducted on a simply supported girder bridge model to validate the effectiveness of the proposed impact force identification method. Results indicate strong agreement between displacement response measurements and percentile meters. The proposed non-contact method accurately identifies single or continuous impact loads, with a minimum peak relative error of 0.2%. This study represents a pioneering application of intelligent visual sensing technology in the field of impact load identification. Moreover, the current research introduces a novel approach to address the challenges faced by conventional methods in identifying impact loads. Future research can leverage the groundwork laid by this study to further optimize and expand the proposed method, enhancing its capabilities and fully harnessing its potential to offer advanced solutions in structural health monitoring. [ABSTRACT FROM AUTHOR]

Dynamic light scattering (DLS) is a highly efficient approach for extracting particle size distributions (PSDs) from autocorrelation functions (ACFs) to measure nanoparticle particles. However, it is a technical challenge to get an exact inversion of the PSD in DLS. Generally, Tikhonov regularization is widely used to address this issue; it uses the L2 norm for both the data fitting term (DFT) and the regularization constraint term. However, the L2 norm's DFT has poor robustness, and its regularization term lacks sparsity, making the solution susceptible to noise and a reduction in accuracy. To solve this problem, the Lp,q norm restrictive model is formulated to examine the impact of various norms in the DFT and regularization term on the inversion results. On this basis, combined with the robustness of DFT and the sparsity of regularization terms, an L1,∞-constrained Tikhonov regularization model was constructed. This model improves the inversion accuracy of PSD and offers a better noise-resistance performance. Simulation tests reveal that the L1,∞ model has strong noise resistance, exceptional inversion precision, and excellent bimodal resolution. The inversion outcomes for the 33 nm unimodal particles, the 55 nm unimodal, and the 33 nm/203 nm bimodal experimental particles show that L1,∞ reduces peak errors by at most 6.06%, 5.46%, and 12.12%/3.94% compared to L2,2, L1,2, and L2,∞ models, respectively. These simulations are validated by experimental data. [ABSTRACT FROM AUTHOR]

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. [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]

Accurate determination of the ionospheric parameters is one of the important objectives of the Ionospheric Connection Explorer (ICON) mission. Recent analyses of the current ICON Level 2.5 (L2.5) data product have shown that the ionospheric parameters (e.g., the peak electron density, n m F 2 , and the peak height, h m F 2 ) that are retrieved from the nighttime OI 135.6 nm emission observed by ICON's Far Ultraviolet (FUV) imager exhibit a systematic bias when compared to external radio measurements. In this study, we demonstrate that the bias was introduced by Tikhonov regularization that was used for the FUV Level 1 data inversion to generate the L2.5 data product. To address the bias, we develop a Bayesian framework for accurate determination of the nighttime ionospheric parameters through the Maximum A Posteriori (MAP) estimation. We show through analysis of synthetic observations that the key to an accurate MAP estimation is to construct a series of prior distributions associated with different h m F 2 using climatological empirical models. Implementation of the MAP estimation with this series of prior distributions to the ICON FUV observations and comparison of the ionospheric retrievals with external radio measurements verify that the Bayesian method can reduce the systematic bias to a negligible level of ∼1% in the retrieved n m F 2 and ∼1 km in the retrieved h m F 2 . Our study provides a novel method for FUV remote sensing data analysis and an improved data set for ionospheric research. [ABSTRACT FROM AUTHOR]

The calculation of the vertical derivatives of potential field methods can be carried out in a stable manner by Tikhonov regularization, but this procedure requires the appropriate selection of a regularization parameter. For this purpose, we introduce a criterion based on Morozov's discrepancy principle that uses a preliminary approximation given by the vertical derivative of the smoothed data. The smoothing may be performed by a physical or a mathematically based low‐pass filter. The filtered data are computed only for estimating the regularization parameter; once it is found, we evaluate the regularized vertical derivative from the original data (not from the smoothed one) in the frequency domain. We verified from experiments with noise‐corrupted synthetic data, as well as gravity and magnetic field data, that the regularized vertical derivative has about the same smoothness as the one obtained from filtered data, but true anomalies are more easily distinguished from noise and the shapes of the anomalies are better preserved. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of second-order ordinary differential equations with Hessian-driven damping and Tikhonov regularization, which arises from the minimization of a smooth convex function in Hilbert spaces. Inspired by Attouch et al. (J Differ Equ 261:5734–5783, 2016), we establish that the function value along the solution trajectory converges to the optimal value, and prove that the convergence rate can be as fast as o (1 / t 2) . By constructing proper energy function, we prove that the trajectory strongly converges to a minimizer of the objective function of minimum norm. Moreover, we propose a gradient-based optimization algorithm based on numerical discretization, and demonstrate its effectiveness in numerical experiments. [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]

Relaxation time and frequency spectra are not directly available by measurement. To determine them, an ill-posed inverse problem must be solved based on relaxation stress or oscillatory shear relaxation data. Therefore, the quality of spectra models has only been assessed indirectly by examining the fit of the experiment data to the relaxation modulus or dynamic moduli models. As the measures of data fitting, the mean sum of the moduli square errors were usually used, the minimization of which was an essential step of the identification algorithms. The aim of this paper was to determine a relaxation spectrum model that best approximates the real unknown spectrum in a direct manner. It was assumed that discrete-time noise-corrupted measurements of a relaxation modulus obtained in the stress relaxation experiment are available for identification. A modified relaxation frequency spectrum was defined as a quotient of the real relaxation spectrum and relaxation frequency and expanded into a series of linearly independent exponential functions that are known to constitute a basis of the space of square-integrable functions. The spectrum model, given by a finite series of these basis functions, was assumed. An integral-square error between the real unknown modified spectrum and the spectrum model was taken as a measure of the model quality. This index was proved to be expressed in terms of the measurable relaxation modulus at uniquely defined sampling instants. Next, an empirical identification index was introduced in which the values of the real relaxation modulus are replaced by their noisy measurements. The identification consists of determining the spectrum model that minimizes this empirical index. Tikhonov regularization was applied to guarantee model smoothness and noise robustness. A simple analytical formula was derived to calculate the optimal model parameters and expressed in terms of the singular value decomposition. A complete identification algorithm was developed. The analysis of the model smoothness and model accuracy for noisy measurements was carried out. The equivalence of the direct identification of the relaxation frequency and time spectra has been demonstrated when the time spectrum is modeled by a series of functions given by the product of the relaxation frequency and its exponential function. The direct identification concept can be applied to both viscoelastic fluids and solids; however, some limitations to its applicability have been pointed out. Numerical studies have shown that the proposed identification algorithm can be successfully used to identify Gaussian-like and Kohlrausch–Williams–Watt relaxation spectra. The applicability of this approach to determining other commonly used classes of relaxation spectra was also examined. [ABSTRACT FROM AUTHOR]

The measurement of particle volume fraction in flow fields is of great significance in scientific research and engineering applications. As one of the particle detection techniques, the light extinction method is widely used in measuring nano-particles volume fraction in flow fields due to its simplicity and non-contact nature. In particular, in complex reactive flow fields like combustion reactions, the volume fraction of soot particulate and other particles can be accurately measured and reconstructed via the light extinction method that based on the Beer–Lambert law. This is crucial for exploring combustion phenomena, understanding their internal mechanisms, and reducing pollutant emissions. However, due to the enormous computational burden, current algebra reconstruction techniques struggle to achieve high-precision three-dimensional (3D) reconstruction of particles. Therefore, this paper originally proposes a 3D reconstruction algorithm based on the Beer–Lambert law physical information neural networks (LB-PINNs). By incorporating physical information as constraints into the particle reconstruction process, it is possible to achieve high-precision 3D reconstruction of particles in complex flow field environments with low computational cost. Meanwhile, to address the trade-off issues of reconstruction accuracy and smooth noise resistance in previous reconstruction algorithms, i.e., Tikhonov regularization, this paper employs dynamically adjusted regularization parameters in the LB-PINN algorithm. This approach ensures smooth noise-resistant processing while maintaining reconstruction accuracy, significantly reducing computation time and resource consumption. According to the experimental results, LB-PINNs demonstrate superior performance compared to previous reconstruction algorithms when reconstructing the soot volume fraction in complex reacting flow fields, i.e., combustion flame scenarios. [ABSTRACT FROM AUTHOR]

In addressing the multifaceted problem of multiple-input multiple-output (MIMO) detection in wireless communication systems, this work highlights the pressing need for enhanced detection reliability under variable channel conditions and MIMO antenna configurations. We propose a novel method that sets a new standard for deep unfolding in MIMO detection by integrating the iterative conjugate gradient method with Tikhonov regularization, combining the adaptability of modern deep learning techniques with the robustness of classical regularization. Unlike conventional techniques, our strategy treats the Tikhonov regularization parameter, as well as the step size and search direction coefficients of the conjugate gradient (CG) method, as trainable parameters within the deep learning framework. This enables dynamic adjustments based on varying channel conditions and MIMO antenna configurations. Detection performance is significantly improved by the proposed approach across a range of MIMO configurations and channel conditions, consistently achieving lower bit error rate (BER) and normalized minimum mean square error (NMSE) compared to well-known techniques like DetNet and CG. The proposed method has superior performance over CG and other model-based methods, especially with a small number of iterations. Consequently, the simulation results demonstrate the flexibility of the proposed approach, making it a viable choice for MIMO systems with a range of antenna configurations, modulation orders, and different channel conditions. [ABSTRACT FROM AUTHOR]

This paper is devoted to an inverse problem of simultaneously determining the spatially dependent source term and the Robin boundary coefficient in a time fractional diffusion equation, with the aid of extra measurement data at a subdomain near the accessible boundary. Firstly, the spatially varying source is uniquely determined in view of the unique continuation principle and Duhamel principle for the fractional diffusion equation. The Hopf lemma for a homogeneous time-fractional diffusion equation is proved and then used to prove the uniqueness of recovering the Robin boundary coefficient. Numerically, based on the theoretical uniqueness result, we apply the classical Tikhonov regularization method to transform the inverse problem into a minimization problem, which is solved by an iterative thresholding algorithm. Finally, several numerical examples are presented to show the accuracy and effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The mesoscale interaction between sea surface temperature (SST) and wind is a crucial factor influencing oceanic and atmospheric conditions. To investigate the mesoscale coupling characteristics of the Yellow Sea and East China Sea, we applied a locally weighted regression filtering method to extract mesoscale signals from Quik-SCAT wind field data and AMSR-E SST data and found that the mesoscale coupling intensity is stronger in the Yellow Sea during the spring and winter seasons. We calculated the mesoscale coupling coefficient to be approximately 0.009 N·m−2/°C. Subsequently, the Tikhonov regularization method was used to establish a mesoscale empirical coupling model, and the feedback effect of mesoscale coupling on the ocean was studied. The results show that the mesoscale SST–wind field coupling can lead to the enhancement of upwelling in the offshore area of the East China Sea, a decrease in the upper ocean temperature, and an increase in the eddy kinetic energy in the Yellow Sea. Diagnostic analyses suggested that mesoscale coupling-induced variations in horizontal advection and surface heat flux contribute most to the variation in SST. Moreover, the increase in the wind energy input to the eddy is the main factor explaining the increase in the eddy kinetic energy. [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]