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The free vibrations of a rod are governed by a differential equation of the form (a(x)y')' + λa(x)y(x) = 0, where a(x) is the cross sectional area and λ is an eigenvalue parameter. Using the finite element method (FEM) we transform this equation to a generalized matrix eigenvalue problem of the form (K, M)u = 0 and, for given a(x), we correct the eigenvalues Λ of the matrix pair (K - ΛM)u to approximate the eigenvalues of the rod equation. The results show that with step size h the correction technique reduces the error from O(h²i4) to O(h²i²) for the i-th eigenvalue. We then solve the inverse spectral problem by imposing numerical algorithms that approximate the unknown coefficient a(x) from the given spectral data. The cross section is obtained by solving a nonlinear system using Newton's method along with a regularization technique. Finally, we give numerical examples to illustrate the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper we are trying to formalize the definition of gradient for inverse problems and other non-classical optimization problems such as gradient approaches to problems having to do with the calculus of variations. We provide the readers with several examples of converting the problem to the gradient-type problem, describe several properties of the obtained gradient problem, provide analysis on its convexity or smoothness, and proving the convergence of the algorithm. [ABSTRACT FROM AUTHOR]

In this study, a nonlinear degenerate parabolic equation is used to describe a nonlinear-Laplacian equation process that arises in many areas of science and engineering in mechanics, quantum physics, and chemical design. This work has the objective of proving the existence of the local weak solution of a nonlinear p(x)- Laplacian equation by the compactness theorem. The uniformly local characteristics of the solutions for the gradients by estimating the regularization problem and using the Moser iterative techniques. Moreover, some properties of the local solutions depend on uniformly bounded situations and the Lp(x)-norm to the gradient is considered. [ABSTRACT FROM AUTHOR]

Variational regularization, renowned for its sound theoretical foundations and impressive performance, is widely used in image restoration. The traditional regularization models typically use a predefined regularizer to promote smoothness in the solution. However, these models do not explicitly take into account any external information that should be preserved in the restoration. In this paper, we introduce a novel guided regularization model to enhance the efficacy of traditional regularization. Our model incorporates an external guidance regularizer, utilizing a guidance image to bolster the quality of restoration. By integrating this external information into the regularization process, the model is better equipped to preserve specific features or attributes indicated by the guidance image, leading to more accurate and aesthetically pleasing restored images. Furthermore, we demonstrate the convexity of the model and prove the existence and uniqueness of the solution. The alternating direction method of multipliers (ADMM) algorithm is employed to numerically solve the proposed model. In the experimental evaluation, the proposed model is applied to image denoising and deblurring tasks. The experiments successfully validate the proposed model and algorithm. Compared with several state-of-the-art models, the proposed model demonstrates the best performance in terms of peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM). • A novel guided regularization model is proposed, which aims to incorporate additional guidance into regularization process. • We demonstrate the convexity of the model and prove the existence and uniqueness of the solution. • The proposed guided regularization model is applied to image denoising and deblurring tasks, yielding satisfactory results. [ABSTRACT FROM AUTHOR]

Heavy-tailed continuous shrinkage priors, such as the horseshoe prior, are widely used for sparse estimation problems. However, there is limited work extending these priors to explicitly incorporate multivariate shrinkage for regressors with grouping structures. Of particular interest in this article, is regression coefficient estimation where pockets of high collinearity in the regressor space are contained within known regressor groupings. To assuage variance inflation due to multicollinearity we propose the group inverse-gamma gamma (GIGG) prior, a heavy-tailed prior that can trade-off between local and group shrinkage in a data adaptive fashion. A special case of the GIGG prior is the group horseshoe prior, whose shrinkage profile is dependent within-group such that the regression coefficients marginally have exact horseshoe regularization. We establish posterior consistency and posterior concentration results for regression coefficients in linear models and mean parameters in sparse normal means models. The full conditional distributions corresponding to GIGG regression can be derived in closed form, leading to straightforward posterior computation. We show that GIGG regression results in low mean-squared error across a wide range of correlation structures and within-group signal densities via simulation. We apply GIGG regression to data from the National Health and Nutrition Examination Survey for associating environmental exposures with liver functionality. [ABSTRACT FROM AUTHOR]

The soft SVD is a robust matrix decomposition algorithm and a key component of matrix completion methods. However, computing the soft SVD for large sparse matrices is often impractical using conventional numerical methods for the SVD due to large memory requirements. The Rank-Restricted Soft SVD (RRSS) algorithm introduced by Hastie et al. addressed this issue by sequentially computing low-rank SVDs that easily fit in memory. We analyze the convergence of the standard RRSS algorithm and we give examples where the standard algorithm does not converge. We show that convergence requires a modification of the standard algorithm, and is related to non-uniqueness of the SVD. Our modification specifies a consistent choice of sign for the left singular vectors of the low-rank SVDs in the iteration. Under these conditions, we prove linear convergence of the singular vectors using a technique motivated by alternating subspace iteration. We then derive a fixed point iteration for the evolution of the singular values and show linear convergence to the soft thresholded singular values of the original matrix. This last step requires a perturbation result for fixed point iterations which may be of independent interest. [ABSTRACT FROM AUTHOR]

This study aims to explore two classes of ill‐posed problems governed by a nonclassical fractional heat equation with an involution perturbation. To achieve a stable solution, we introduce a modified pseudo‐parabolic regularization method that involves a correction term through a mixed fractional derivation operator, resulting in a sequence of well‐posed problems that depend on a regularization parameter ε$$ \varepsilon $$. We rigorously demonstrate that the resulting approximate problems are well‐posed and present convergence results for this regularization approach. [ABSTRACT FROM AUTHOR]

To obtain high-quality nanocrystalline scanning electron microscopy (SEM) images, this paper proposed a Poisson denoising model that combined the fractional-order total variation (TV) and nuclear norm regularizers. The developed novel model integrated the superiorities of fractional-order TV and nuclear norm constraints, which contributed to significantly improving the accuracy of image restoration while preventing the staircase effect and preserving edge details. By combining the variable separation method and singular value thresholding method, an improved alternating direction method of multipliers was developed for numerical computation. Compared with some existing popular solvers, numerical experiments demonstrated the superiority of the new method in visual effects and quality evaluation. [ABSTRACT FROM AUTHOR]

This paper deals with global well-posedness of the solution to the complex short pulse equation. We first use regularized technology and the approximation argument to prove the local existence and uniqueness of this equation. Then, based on conserved quantities and energy analysis, we show that the solution can be extended globally in time for suitably small initial data. [ABSTRACT FROM AUTHOR]

This article presents a novel approach to image restoration utilizing a unique non-convex l 1 / 2 -TV regularization model. This model integrates the l 1 / 2 -quasi norm as a regularization function, introducing non-convexity to promote sparsity and unevenly penalize elements, thereby enhancing restoration outcomes. To tackle this model, an efficient algorithm named the Alternating Direction Method of Multipliers, based on the Lagrangian multiplier, is introduced. This effectively prevents the penalty parameter from reaching infinity and ensures excellent convergence behavior. The proposed algorithm decomposes the optimization problem into subproblems, for which closed-form solutions are derived, particularly addressing the challenging l 1 / 2 regularization problem. To validate its effectiveness, a comprehensive set of experiments are conducted to compare its performance with existing methods. The experimental results demonstrate that the proposed model performs well in both qualitative and quantitative evaluations. Consequently, the proposed model is not only efficient and stable but also exhibits excellent convergence behavior. • A novel regularization model for image restoration, based on non-convex total variation is proposed in this work. • We propose an effective solution approach to solve the model using the Alternating Direction Method of Multipliers (ADMM). • The proposed solution provides the closed-form thresholding formula for regularization model. • Extensive numerical experiments demonstrate the superiority of the proposed method over compared methods. [ABSTRACT FROM AUTHOR]

Sparse regression and feature extraction are the cornerstones of knowledge discovery from massive data. Their goal is to discover interpretable and predictive models that provide simple relationships among scientific variables. While the statistical tools for model discovery are well established in the context of linear regression, their generalization to nonlinear regression in material modeling is highly problem‐specific and insufficiently understood. Here we explore the potential of neural networks for automatic model discovery and induce sparsity by a hybrid approach that combines two strategies: regularization and physical constraints. We integrate the concept of Lp regularization for subset selection with constitutive neural networks that leverage our domain knowledge in kinematics and thermodynamics. We train our networks with both, synthetic and real data, and perform several thousand discovery runs to infer common guidelines and trends: L2 regularization or ridge regression is unsuitable for model discovery; L1 regularization or lasso promotes sparsity, but induces strong bias that may aggressively change the results; only L0 regularization allows us to transparently fine‐tune the trade‐off between interpretability and predictability, simplicity and accuracy, and bias and variance. With these insights, we demonstrate that Lp regularized constitutive neural networks can simultaneously discover both, interpretable models and physically meaningful parameters. We anticipate that our findings will generalize to alternative discovery techniques such as sparse and symbolic regression, and to other domains such as biology, chemistry, or medicine. Our ability to automatically discover material models from data could have tremendous applications in generative material design and open new opportunities to manipulate matter, alter properties of existing materials, and discover new materials with user‐defined properties. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce a new definition of JA‐module Arens regularity, which extends former standard definition. Then a new module structure on the semigroup algebra l1S as l1ES‐module with nontrivial right and left module actions is constructed. Using this construction, it is also proved that l1S is a Jl1S‐module Arens regular if and only if S/≈ is finite. [ABSTRACT FROM AUTHOR]

The task of link prediction has become a fundamental research problem in the analysis of complex networks. However, most existing non-negative matrix factorization (NMF) methods are decoding models and solely consider a single type of structural information around node pairs in networks, resulting in sub-optimal prediction accuracy. To solve these limitations, we propose a novel model, namely Graph Regularized Autoencoder-inspired Non-negative Matrix Factorization via jointly Clustering information and Biased random walk (GRANMFCB for short), for link prediction. Specifically, GRANMFCB comprises both encoder and decoder components, fully leveraging the advantages of autoencoders. In addition, clustering information and high-order structures are utilized to preserve abundant structural information around node pairs and against sparsity of networks. Finally, effective iterative multi-multiplication updating rules are proposed to optimize the objective function and the convergence is strictly proved. Extensive experimental results on twelve real-world networks show that our proposed model outperforms the state-of-the-art approaches. Codes are available at https://github.com/litongf/GRANMFCB. [ABSTRACT FROM AUTHOR]

The integration of Synthetic Aperture Radar (SAR) imaging technology with deep neural networks has experienced significant advancements in recent years. Yet, the scarcity of high-quality samples and the difficulty of extracting prior information from SAR data have experienced limited progress in this domain. This study introduces an innovative sparse SAR imaging approach using a self-supervised non-local asymmetric pixel-shuffle blind spot network. This strategy enables the network to be trained without labeled samples, thus solving the problem of the scarcity of high-quality samples. Through asymmetric pixel-shuffle downsampling (AP) operation, the spatial correlation between pixels is broken so that the blind spot network can adapt to the actual scene. The network also incorporates a non-local module (NLM) into its blind spot architecture, enhancing its capability to analyze a broader range of information and extract more comprehensive prior knowledge from SAR data. Subsequently, Plug and Play (PnP) technology is used to integrate the trained network into the sparse SAR imaging model to solve the regularization term problem. The optimization of the inverse problem is achieved through the Alternating Direction Method of Multipliers (ADMM) algorithm. The experimental results of the unlabeled samples demonstrate that our method significantly outperforms traditional techniques in reconstructing images across various regions. [ABSTRACT FROM AUTHOR]

We introduce a regularization method with two differential operators for solving a linear ill-posed operator equation system. The existence and uniqueness of regularized solutions to the problem are derived. With an a priori as well as an a posteriori parameter choice strategy, convergence and convergence rates of the regularized solution are also obtained. As an application, we apply the regularization to a simultaneous inversion of the source term and the initial value problem for a heat conduction equation, numerical experiments are presented to demonstrate the effectiveness of the proposed method. [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]

In this paper, a novel non-convex fractional-order image denoising model is proposed to suppress the staircase effect produced by the TV model while maintaining a neat contour. The model combines ℓ q (0 < q < 1) quasi-norm and fractional-order regularization, and employs a diffusion coefficient with a faster convergence rate to preserve more image edges and details. Additionally, an adaptive regularization parameter is designed to adjust the denoising performance of the algorithm. To obtain the optimal approximate solution of the model, an enhanced primal-dual algorithm is adopted and the complexity and convergence of the algorithm are theoretically analyzed. Finally, the effectiveness of the proposed method is demonstrated through numerical experiments. • A new non-convex FOTV model is proposed. • A new diffusion coefficient is introduced to retain more edge details of the image. • The existing primal-dual algorithm is improved, and the convergence of this algorithm is analyzed. • An adaptive regularization parameter is designed. [ABSTRACT FROM AUTHOR]

Despite growing interest in estimating individualized treatment rules, little attention has been given the binary outcome setting. Estimation is challenging with nonlinear link functions, especially when variable selection is needed. We use a new computational approach to solve a recently proposed doubly robust regularized estimating equation to accomplish this difficult task in a case study of depression treatment. We demonstrate an application of this new approach in combination with a weighted and penalized estimating equation to this challenging binary outcome setting. We demonstrate the double robustness of the method and its effectiveness for variable selection. The work is motivated by and applied to an analysis of treatment for unipolar depression using a population of patients treated at Kaiser Permanente Washington. [ABSTRACT FROM AUTHOR]

Nonlocal regularization is frequently used to resolve the mesh dependency issue that is caused by strain softening in finite-element (FE) simulations. Some or all variables that affect strain softening are assumed to depend on the local, neighboring, or both in this method. The weight function is the main component of a regularization method. There are three widely used weight functions, which include the Gaussian distribution (GD), Galavi and Schweiger (GS), and over-nonlocal (ON) functions. All of them could alleviate or eliminate the mesh dependency in simple boundary value problems (BVPs), such as plane strain compression; the evaluation of their performance in real-world BVPs is rare. A detailed comparison of these functions has been carried out based on an anisotropic sand model that accounts for the evolution of anisotropy. The increment of void ratio is assumed nonlocal. All functions give mesh-independent force–displacement relationships in drained and undrained plane strain compression tests. The shear band thickness shows a small variation when the mesh size is smaller than the internal length. None could eliminate the mesh dependency of shear band orientation. The GS method is the most efficient in eliminating the mesh dependency in the strip footing problem. The ON method could give excessive overpredictions of the volume expansion around strip footings, which leads to unrealistic low reaction forces on strip footings at large deformations. All three weight functions give mesh-independent results for the earth pressure that acts on a retaining wall. [ABSTRACT FROM AUTHOR]

In general, it is a worth challenging problem to build a robust classifier for data sets with noises or outliers. Establishing a robust classifier is a more difficult problem for datasets with asymmetric noise distribution. The Fisher-regularized extreme learning machine (Fisher-ELM) considers the statistical knowledge of the data, however, it ignores the impact of noises or outliers. In this paper, to reduce the negative influence of noises or outliers, we first put forward a novel asymmetric Welsch loss function named AW-loss based on asymmetric L 2 -loss function and Welsch loss function. Based on the AW-loss function, we then present a new robust Fisher-ELM called AWFisher-ELM. The proposed AWFisher-ELM not only takes into account the statistical information of the data, but also considers the impact of asymmetric distribution noises. We utilize concave-convex procedure (CCCP) and dual method to solve the non-convexity of the proposed AWFisher-ELM. Simultaneously, an algorithm for AWFisher-ELM is given and a theorem about the convergence of the algorithm is proved. To validate the effectiveness of our algorithm, we compare our AWFisher-ELM with the other state-of-the-art methods on artificial data sets, UCI data sets, NDC large data sets and image data sets by setting different ratios of noises. The experimental results are as follows, the accuracy of AWFisher-ELM is the highest in the artificial data sets, reaching 98.9%. For the large-scale NDC data sets and the image data sets, the accuracy of AWFisher-ELM is also the highest. For the ten UCI data sets, the accuracy and F 1 value of AWFisher-ELM are the highest in most data sets expect for Diabetes. In terms of training time, our AWFisher-ELM has almost the same training time with RHELM and CHELM, but it takes longer time than OPT-ELM, WCS-SVM, Fisher-SVM, Pinball-FisherSVM, and Fisher-ELM. This is because AWFisher-ELM, RHELM, and CHELM need to solve a convex quadratic subprogramming problem in each iteration. In conclusion, our method exhibits excellent generalization performance expect for the longer training time. [ABSTRACT FROM AUTHOR]

For an entire transcendental function f and a sequence (λn) of positive numbers increasing to +∞ a series A(z) = ∑∞n=1 anf(λnz) in the system {f(λnz)} is said to be regularly convergent in C if M(r,A) = ∑∞n=1 |an|Mf(rλn) < +∞ for all r (0,+∞), where Mf (r) = max{|f(z)|: |z| = r}. Conditions are found on (λn) and f, under which ln M(r,A) ~ ln μ(r,A) as r → +∞, where μ(r,A) = max{|an|Mf (rλn): n ≥ 1} is the maximal term of the series. A formula for finding the lower generalized order... is obtained, where the functions α and β are positive, continuous and increasing to +∞. The open problems are formulated. [ABSTRACT FROM AUTHOR]

Decentralized distributed learning has recently attracted significant attention in many applications in machine learning and signal processing. To solve a decentralized optimization with regularization, we propose a Multi-consensus Decentralized Primal-Dual Fixed Point (MD-PDFP) algorithm. We apply multiple consensus steps with the gradient tracking technique to extend the primal-dual fixed point method over a network. The communication complexities of our procedure are given under certain conditions. Moreover, we show that our algorithm is consistent under general conditions and enjoys global linear convergence under strong convexity. With some particular choices of regularizations, our algorithm can be applied to decentralized machine learning applications. Finally, several numerical experiments and real data analyses are conducted to demonstrate the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper proposes a multivariate form of generalized information criterion (MGIC) for the multivariate response bridge regression (multi-bridge) model. Also, we prove the identifiability of the multi-bridge as a prerequisite for model selection. We introduce the general form of MGIC for regularization parameter selection in the multi-bridge model. We assess the performance of MGIC variants from three viewpoints: consistency of the obtained models, analysis of high-dimensional data, and comparison to other criteria. Based on the numerical study, we reach better performance for MGIC in comparison to other common criteria (cross-validation and GCV) using simulated and real datasets. [ABSTRACT FROM AUTHOR]

Hyperspectral images are easily contaminated by multi-modal noise in the process of data acquisition. Generally, hyperspectral signatures and noise appear unexpectedly scattered in the spatial-spectral domain. Conversely, in the transform domain, the principle components of an image are gathered in the low-frequency bands, while the details and noise often occur in high-frequency parts. The existing denoising methods usually divide the whole frequency domain into two parts, that is, high frequency and low frequency, which can not make the most of the knowledge contained at different frequencies or scales, especially in the multiscale transform domain. Furthermore, aiming to restore a clean image from the noise-corrupted data, hyperspectral image denoising often amounts to an ill-posed problem, in which reshaping cubic data into matrix form results in the loss of structure information. To address the aforementioned issues, this paper incorporates the reweighted smoothing regularization in the transform domain into the noise-perturbed degradation model to reformulate a novel optimization problem. First, the multiscale curvelet is leveraged to transform the spatial matrix into frequency domain, where the coefficients are carefully designed to weight the components at different scales and orientations. Second, ${l_1}$ l 1 -norm is imposed on the abundance term to promote sparsity in the transform domain for performance enhancement. Finally, using proximal alternating optimization, an efficient algorithm is well developed to obtain the closed-form solutions to the proposed denoising problem. The experimental results on several synthetic and real datasets, illustrate that the proposed method improves PSNR by at least 1.5 dB and reduces ERGAS error by more than 35% than the state-of-the-art approaches under the multi-modal noise mixed environments, which verifies the validity of regularization terms in hyperspectral images denoising. [ABSTRACT FROM AUTHOR]

We consider the problem of recovering elements of a low-dimensional model from under-determined linear measurements. To perform recovery, we consider the minimization of a convex regularizer subject to a data fit constraint. Given a model, we ask ourselves what is the 'best' convex regularizer to perform its recovery. To answer this question, we define an optimal regularizer as a function that maximizes a compliance measure with respect to the model. We introduce and study several notions of compliance. We give analytical expressions for compliance measures based on the best-known recovery guarantees with the restricted isometry property. These expressions permit to show the optimality of the |$\ell ^{1}$| -norm for sparse recovery and of the nuclear norm for low-rank matrix recovery for these compliance measures. We also investigate the construction of an optimal convex regularizer using the examples of sparsity in levels and of sparse plus low-rank models. [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 a hyperspectral image, there is a close correlation between spectra and a certain degree of correlation in the pixel space. However, most existing low-rank representation (LRR) methods struggle to utilize these two characteristics simultaneously to detect anomalies. To address this challenge, a novel low-rank representation with dual graph regularization and an adaptive dictionary (DGRAD-LRR) is proposed for hyperspectral anomaly detection. To be specific, dual graph regularization, which combines spectral and spatial regularization, provides a new paradigm for LRR, and it can effectively preserve the local geometrical structure in the spectral and spatial information. To obtain a robust background dictionary, a novel adaptive dictionary strategy is utilized for the LRR model. In addition, extensive comparative experiments and an ablation study were conducted to demonstrate the superiority and practicality of the proposed DGRAD-LRR method. [ABSTRACT FROM AUTHOR]

This paper proposes a fractional structure and texture aware Retinex (FSTAR) model for image decomposition with application to low-light enhancement. First, a novel structure aware measure called Maximum Fractional Difference (MFD) is introduced, which is the maximum of fractional differences of the input image in eight symmetric directions. Then fractional structure and texture aware maps are built based on MFD and Huber function from robust statistics, which are used as weighted matrices in the regularization terms of reflectance and illumination components. To ensure the intrinsic physical constraints of Retinex on two components, two extra penalty terms are incorporated into the objective function of FSTAR to penalize deviations of the estimated components from the corresponding constraints. The FSTAR is solved via an alternating iterative algorithm; i.e., the objective function is minimized with respect to one variable with another variable fixed, and this process is done alternately until termination condition is met. Qualitative and quantitative evaluations of FSTAR on three low-light image datasets, compared to ten state-of-the-art methods, show its superior performance in Retinex decomposition and low-light image enhancement. • Present texture aware measure based on fractional directional difference for texture extraction. • Propose fractional structure-texture aware Retinex model with constraints on two components. • Model is faithful to intrinsic physical meaning of Retinex in theory and easy to solve numerically. • Model has outstanding performance in terms of image decomposition and low-light enhancement. [ABSTRACT FROM AUTHOR]

We present SQUIC, a fast and scalable package for sparse precision matrix estimation. The algorithm employs a second-order method to solve the \(\ell_{1}\) -regularized maximum likelihood problem, utilizing highly optimized linear algebra subroutines. In comparative tests using synthetic datasets, we demonstrate that SQUIC not only scales to datasets of up to a million random variables but also consistently delivers runtimes that are significantly faster than other well-established sparse precision matrix estimation packages. Furthermore, we showcase the application of the introduced package in classifying microarray gene expressions. We demonstrate that by utilizing a matrix form of the tuning parameter (also known as the regularization parameter), SQUIC can effectively incorporate prior information into the estimation procedure, resulting in improved application results with minimal computational overhead. [ABSTRACT FROM AUTHOR]

In this paper we construct a new type of noise of fractional nature that has a strong regularizing effect on differential equations. We consider an equation driven by a highly irregular vector field and study the effect of this noise on such dynamical systems. We employ a new method to prove existence and uniqueness of global strong solutions, where classical methods fail because of the "roughness" and non-Markovianity of the driving process. In addition, we prove the rather remarkable property that such solutions are infinitely many times classically differentiable with respect to the initial condition in spite of the vector field being discontinuous. The technique used in this article corresponds, in a certain sense, to the Nash–Moser iterative scheme in combination with a new concept of "higher order averaging operators along highly fractal stochastic curves". This approach may provide a general principle for the study of regularization by noise effects in connection with important classes of partial differential equations. [ABSTRACT FROM AUTHOR]

This study operates the Gradient Method to control the leading coefficient function of a heat equation and presents a Maplet Application which facilitates the computation of control function. The control is the heat conductivity function and this function is controlled by aiming the desired value approximation of final heat. After mentioning the existence and uniqueness of the control, the application is submitted by MAPLE mathematical software program and the results are tested on a problem. [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]

We consider the regularized Landau-Pekar equations with positive speed of sound and prove the existence of subsonic traveling waves. We provide a definition of the effective mass for the regularized Landau-Pekar equations based on the energy-velocity expansion of subsonic traveling waves. Moreover we show that this definition of the effective mass agrees with the definition based on an energy-momentum expansion of low energy states. [ABSTRACT FROM AUTHOR]

Full-waveform inversion (FWI) is an advanced geophysical inversion technique. FWI provides images of subsurface structures with higher resolution in fields such as oil exploration and geology. The conventional algorithm minimizes the misfit error by calculating the least squares of the wavefield solutions between observed data and simulated data, followed by gradient direction and model update increment. Since the gradient is calculated by forward and backward wavefields, the high-accuracy model update relies on accurate forward and backward wavefield modelling. However, the quality of wavefield solutions obtained in practical situations could be poor and does not meet the requirements of high-resolution FWI. Specifically, the low-frequency wavefield is easily affected by noise and downsampling, which influences data quality, whereas the high-frequency wavefield is susceptible to spatial aliasing effects that produce imaging artefacts. Therefore, we propose using an algorithm called sparse relaxation regularized regression to optimize the wavefield solution in frequency-domain FWI, which is the forward and backward wavefield obtained from the Helmholtz equation, thus improving FWI's accuracy. The sparse relaxation regularized regression algorithm combines sparsity and regularization, allowing the broadband FWI to reduce the effects of noise and outliers, which can provide data supplementation in the low-frequency band and anti-aliasing in the high-frequency band. Our numerical examples demonstrate the wavefield optimization effect of the sparse relaxation regularized regression-based algorithm in various cases. The improved algorithm's accuracy and stability are verified compared to the Tikhonov regularization algorithm. [ABSTRACT FROM AUTHOR]

This paper focuses on the minimization of a sum of a twice continuously differentiable function f and a nonsmooth convex function. An inexact regularized proximal Newton method is proposed by an approximation to the Hessian of f involving the ϱ th power of the KKT residual. For ϱ = 0 , we justify the global convergence of the iterate sequence for the KL objective function and its R-linear convergence rate for the KL objective function of exponent 1/2. For ϱ ∈ (0 , 1) , by assuming that cluster points satisfy a locally Hölderian error bound of order q on a second-order stationary point set and a local error bound of order q > 1 + ϱ on the common stationary point set, respectively, we establish the global convergence of the iterate sequence and its superlinear convergence rate with order depending on q and ϱ . A dual semismooth Newton augmented Lagrangian method is also developed for seeking an inexact minimizer of subproblems. Numerical comparisons with two state-of-the-art methods on ℓ 1 -regularized Student's t-regressions, group penalized Student's t-regressions, and nonconvex image restoration confirm the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In a Hilbertian framework, for the minimization of a general convex differentiable function f, we introduce new inertial dynamics and algorithms that generate trajectories and iterates that converge fastly towards the minimizer of f with minimum norm. Our study is based on the non-autonomous version of the Polyak heavy ball method, which, at time t, is associated with the strongly convex function obtained by adding to f a Tikhonov regularization term with vanishing coefficient ε (t) . In this dynamic, the damping coefficient is proportional to the square root of the Tikhonov regularization parameter ε (t) . By adjusting the speed of convergence of ε (t) towards zero, we will obtain both rapid convergence towards the infimal value of f, and the strong convergence of the trajectories towards the element of minimum norm of the set of minimizers of f. In particular, we obtain an improved version of the dynamic of Su-Boyd-Candès for the accelerated gradient method of Nesterov. This study naturally leads to corresponding first-order algorithms obtained by temporal discretization. In the case of a proper lower semicontinuous and convex function f, we study the proximal algorithms in detail, and show that they benefit from similar properties. [ABSTRACT FROM AUTHOR]

In this paper, we have considered the problem of reconstructing the time dependent potential term for the third order time fractional pseudoparabolic equation from an additional data at the left boundary of the space interval. This is very challenging and interesting inverse problem with many important applications in various fields of engineering, mechanics and physics. The existence of unique solution to the problem has been discussed by means of the contraction principle on a small time interval and the unique solvability theorem is proved. The stability results for the inverse problem have also been presented. However, since the governing equation is yet ill-posed (very slight errors in the additional input may cause relatively significant errors in the output potential), the regularization of the solution is needed. Therefore, to get a stable solution, a regularized objective function is to be minimized for retrieval of the unknown coefficient of the potential term. The proposed problem is discretized using the cubic B-spline (CB-spline) collocation technique and has been reshaped as a non-linear least-squares optimization of the Tikhonov regularization function. The stability analysis of the direct numerical scheme has also been presented. The MATLAB subroutine $lsqnonlin$ tool has been used to expedite the numerical computations. Both perturbed data and analytical are inverted and the numerical outcomes for two benchmark test examples are reported and discussed. [ABSTRACT FROM AUTHOR]

The randomized sparse Kaczmarz (RSK) method is an algorithm used to calculate sparse solutions for the basis pursuit problem. In this paper, we propose an algorithm framework for computing sparse solutions of linear systems, which includes the sparse Kaczmarz and sparse block Kaczmarz algorithms. In order to overcome the limitations of the l1 penalty, we design an effective and new randomized sparse Kaczmarz algorithm (RSK-MCP) based on the non-convex minimax-concave penalty (MCP) in sparse signal reconstruction. Additionally, we prove that the RSK-MCP algorithm is equivalent to the randomized coordinate descent method for the corresponding dual problem. Based on this result, we demonstrate that the RSK-MCP algorithm exhibits linear convergence, meaning it converges to a sparse solution of the MCP model when the regularization of MCP is a strongly convex function. Numerical experiments indicate that the RSK-MCP algorithm outperforms RSK-L1 in terms of both efficiency and accuracy [ABSTRACT FROM AUTHOR]

Video stabilization plays a pivotal role in enhancing video quality by eliminating unwanted jitter in shaky videos. This paper introduces a novel video stabilization algorithm that leverages low‐rank constraint and trajectory optimization to effectively eliminate undesirable motion and generate stabilized videos. In the proposed algorithm, a low‐rank constraint regularization term is incorporated to enhance the smoothness of motion trajectories. Additionally, a predictive path smoothness term is integrated to ensure the consistency of motion across neighbouring frames. To address the problem of excessive cropping resulting from aggressive smoothing, a flexible local window strategy that emphasizes local motion relationships within the trajectories is introduced. The experimental results show that, compared to other excellent video stabilization algorithms, the proposed algorithm improves the stability metric by approximately 2.3%. Furthermore, in the stabilized videos generated by the algorithm, an approximate improvement of 2.18 dB in average image temporal fidelity and a 5.7% increase in average structural similarity between adjacent frames are achieved. The code that implements the proposed method is publicly accessible at https://github.com/CZS0319/VS_Low‐Rank_Constraint_Trajectory_Optimization. [ABSTRACT FROM AUTHOR]

Distributed algorithms are receiving renewed attention across multiple disciplines due to the dramatically increasing demand of big data processing. We consider a class of consensus optimization problems over a static network system of multiple agents, where each of local cost functions is a sum of a smooth function and a non-Lipschitz regularization term. This kind of problems is widely found in scientific and engineering areas such as machine learning and data analysis. Inspired by Bian and Chen (SIAM J. Opt. 23(3), 1718–1741 2017), we propose a decentralized smoothing quadratic regularization algorithm (abbreviated as D-SQRA) for solving the composite consensus problem with non-Lipschitz singularities. To some extent, D-SQRA can be seen as an extension in the decentralized setup of the smoothing quadratic regularization algorithm (SQRA) proposed in (SIAM J. Opt. 23(3), 1718–1741 2017). Our main contribution is to show that D-SQRA can inherit the theoretical properties of its centralized counterpart, i.e., SQRA, in both sides of convergence and worst-case iteration complexity to achieve an ϵ scaled stationary point. We also present some numerical examples on sparse sensing problems based on synthetic and real datasets to corroborate the effectiveness of the proposed decentralized algorithm. [ABSTRACT FROM AUTHOR]

Standard beams are mainly used for the calibration of strain sensors using their load reconstruction models. However, as an ill-posed inverse problem, the solution to these models often fails to converge, especially when dealing with dynamic loads of different frequencies. To overcome this problem, a piecewise Tikhonov regularization method (PTR) is proposed to reconstruct dynamic loads. The transfer function matrix is built both using the denoised excitations and the corresponding responses. After singular value decomposition (SVD), the singular values are divided into submatrices of different sizes by utilizing a piecewise function. The regularization parameters are solved by optimizing the piecewise submatrices. The experimental result shows that the MREs of the PTR method are 6.20% at 70 Hz and 5.86% at 80 Hz. The traditional Tikhonov regularization method based on GCV exhibits MREs of 28.44% and 29.61% at frequencies of 70 Hz and 80 Hz, respectively, whereas the L-curve-based approach demonstrates MREs of 29.98% and 18.42% at the same frequencies. Furthermore, the PREs of the PTR method are 3.54% at 70 Hz and 3.73% at 80 Hz. The traditional Tikhonov regularization method based on GCV exhibits PREs of 27.01% and 26.88% at frequencies of 70 Hz and 80 Hz, respectively, whereas the L-curve-based approach demonstrates PREs of 29.50% and 15.56% at the same frequencies. All in all, the method proposed in this paper can be extensively applied to load reconstruction across different frequencies. [ABSTRACT FROM AUTHOR]

Cold forming of polycarbonate films results in the formation of shear bands in the necking zone. The numerical results obtained from standard viscoplastic material models exhibit mesh size dependency, requiring mathematical regularization. For this purpose, we present in this work a large deformation gradient theory for a viscoplastic isotropic material model published before. We extend our model to a micromorphic model by introducing a new micromorphic variable as an additional degree of freedom along with its first gradient. This variable represents a microequivalent plastic strain. The relation between the macroequivalent plastic strain and the micromorphic variable is accomplished by a micromorphic coupling modulus. This coupling forces proximity between the macro- and microvariables, leading to the targeted regularization effect. The micromorphic model is implemented as a three-dimensional initial boundary value problem in an in-house finite element tool. The analysis is performed for both uniaxial and biaxial specimens. The provided numerical examples show the ability of our model to regularize shear bands within the specimens and address the issue of localization. [ABSTRACT FROM AUTHOR]

This study explores the effects of using space-fractional derivatives and adding multiplicative noise, modeled by a Wiener process, on the solutions of the space-fractional stochastic regularized long wave equation. New fractional stochastic solutions are constructed, and the consistency of the obtained solutions is examined using the transition between phase plane orbits. Their bifurcation and dependence on initial conditions are investigated. Some of these solutions are shown graphically, illustrating both the individual and combined influences of fractional order and noise on selected solutions. These effects appear as alterations in the amplitude and width of the solutions, and as variations in their smoothness. [ABSTRACT FROM AUTHOR]

The problem of signal desynchronization in passive imaging based on noise correlation for defect detection in elastic plates is investigated. Although a post-processing resynchronization process relying on the symmetry of noise correlation functions can be applied prior to the imaging algorithm, perfect synchronization might not be achieved experimentally. Effect of such residual synchronization errors on the defect detection performance is quantified as a function of their probability density function. A mathematical regularization process is then proposed to reduce the standard deviation of the resynchronization errors by a factor of N − 1 / N (N is the number of sensors), which results in a significant improvement in the detection performance. Finally, these theoretical results are validated through a simple flexural-wave propagation simulation. [ABSTRACT FROM AUTHOR]

Distributed learning has attracted considerable attention in recent years due to its power to deal with big data in various science and engineering problems. Based on a divide-and-conquer strategy, this paper studies the distributed robust regression algorithm associated with correntropy losses and coefficient regularization in the scheme of kernel networks, where the kernel functions are not required to be symmetric or positive semi-definite. We establish explicit convergence results of such distributed algorithm depending on the number of data partitions, robustness and regularization parameters. We show that with suitable parameter choices the distributed robust algorithm can obtain the optimal convergence rate in the minimax sense, and simultaneously reduce the computational complexity and memory requirement in the standard (non-distributed) algorithms. [ABSTRACT FROM AUTHOR]

This paper presents novel way to achieve fast and accurate Artificial Neural Network (ANN) modeling of Radio Frequency (RF) front end Low Noise Amplifier (LNA). Multilayer perceptron neural network is implemented to model high gain, ultra-low noise pseudomorhic High Electron Mobility Transistor (pHEMT) based LNA. Datasets are developed for input and output that is obtained from the Electro-Magnetic simulator. Different neural networks such as PatternNet, FitNet, and CascadeForwardNet with different algorithms are trained, tested, and compared to learn the behavior of the amplifier, and the best model is analyzed. Neural networks are modeled for LNA S-parameters and NF. It is observed that greater than 99% accuracy is achieved for PatternNet Bayesian regularization algorithm with less number of hidden layers and also it is found that the simulation results are almost similar to the developed ANN. [ABSTRACT FROM AUTHOR]

This research examines the impact of three widely utilized regularization approaches - data augmentation, weight decay, and dropout -on mitigating overfitting, as well as various amalgamations of these methods. Employing a Convolutional Neural Network (CNN), the study assesses the performance of these strategies using two distinct datasets: a flower dataset and the CIFAR-10 dataset. The findings reveal that dropout outperforms weight decay and augmentation on both datasets. Additionally, a hybrid of dropout and augmentation surpasses other method combinations in effectiveness. Significantly, integrating weight decay with dropout and augmentation yields the best performance among all tested method blends. Analyses were conducted in relation to dataset size and convergence time (measured in epochs). Dropout consistently showed superior performance across all dataset sizes, while the combination of dropout and augmentation was the most effective across all sizes, and the triad of weight decay, dropout, and augmentation excelled over other combinations. The epoch-based analysis indicated that the effectiveness of certain techniques scaled with dataset size, with varying results. [ABSTRACT FROM AUTHOR]

Contrastive learning has actively been used for unsupervised graph representation learning owing to its success in computer vision. Most graph contrastive learning methods use instance discrimination. It treats each instance as a distinct class against a query instance as the pretext task. However, such methods inevitably cause a class collision problem because some instances may belong to the same class as the query. Thus, the similarity shared through instances from the same class cannot be reflected in the pre-training stage. To address this problem, we propose graph contrastive learning with consistency regularization (GCCR), which introduces a consistency regularization term to graph contrastive learning. Unlike existing methods, GCCR can obtain a graph representation that reflects intra-class similarity by introducing a consistency regularization term. To verify the effectiveness of the proposed method, we performed extensive experiments and demonstrated that GCCR improved the quality of graph representations for most datasets. Notably, experimental results in various settings show that the proposed method can learn effective graph representations with better robustness against transformations than other state-of-the-art methods. • We propose a graph contrastive learning method to address the class collision problem. • We suggest a novel loss function that adds a consistency regularization term. • Our method obtains effective graph representation reflecting intra-class similarity. • Experimental results show that our method outperforms previous state-of-the-art methods. [ABSTRACT FROM AUTHOR]