In this paper, we tackle air quality forecasting by using machine learning approaches to predict the hourly concentration of air pollutants (e.g., Ozone, PM_2.5 and Sulfur Dioxide). Machine learning, as one of the most popular techniques, is able to efficiently train a model on big data by using large-scale optimization algorithms. Although there exists some works applying machine learning to air quality prediction, most of the prior studies are restricted to small scale data and simply train standard regression models (linear or non-linear) to predict the hourly air pollution concentration. In this work, we propose refined models to predict the hourly air pollution concentration based on meteorological data of previous days by formulating the prediction of 24 hours as a multi-task learning problem. It enables us to select a good model with different regularization techniques. We propose a useful regularization by enforcing the prediction models of consecutive hours to be close to each other, and compare with several typical regularizations for multi-task learning including standard Frobenius norm regularization, nuclear norm regularization, ℓ_2,1 norm regularization. Our experiments show the proposed formulations and regularization achieve better performance than existing standard regression models and existing regularizations.

Background: Modern MEG devices include 102 sensor triplets containing one magnetometer and two planar gradiometers. The first processing step is often a signal space separation (SSS), which provides a powerful noise reduction. A question commonly raised by researchers and reviewers is which data should be employed in source reconstruction: (1) magnetometers only, (2) gradiometers only, (3) magnetometers and gradiometers together. The MEG community is currently divided about the proper answer and strong arguments in favor and against these three approaches often expressed. Methods: First, we provide theoretical evidence that both gradiometers and magnetometers contain the same information after SSS, and argue that they both result from the backprojection of the same SSS components. Then, we compare beamforming source reconstructions from magnetometers and gradiometers in real MEG recordings before and after SSS. Results: Without SSS, the correlation between source time series extracted from magnetometers and gradiometers was high, with Pearson correlation coefficient r=0.5-0.8. After SSS, these correlation values increased dramatically, reaching over 0.90 across all cortical areas. Conclusions: After SSS, almost identical source reconstructions (r>0.9) can be obtained with magnetometers and gradiometers, as long as regularization is selected appropriately to account for the different properties in magnetometers and gradiometers covariance matrices.

The concept of ‘big data’ has been widely discussed, and its value has been illuminated throughout a variety of domains. To quickly mine potential values and alleviate the ever-increasing volume of information, machine learning is playing an increasingly important role and faces more challenges than ever. Because few studies exist regarding how to modify machine learning techniques to accommodate big data environments, we provide a comprehensive overview of the history of the evolution of big data, the foundations of machine learning, and the bottlenecks and trends of machine learning in the big data era. More specifically, based on learning principals, we discuss regularization to enhance generalization. The challenges of quality in big data are reduced to the curse of dimensionality, class imbalances, concept drift and label noise, and the underlying reasons and mainstream methodologies to address these challenges are introduced. Learning model development has been driven by domain specifics, dataset complexities, and the presence or absence of human involvement. In this paper, we propose a robust learning paradigm by aggregating the aforementioned factors. Over the next few decades, we believe that these perspectives will lead to novel ideas and encourage more studies aimed at incorporating knowledge and establishing data-driven learning systems that involve both data quality considerations and human interactions.

In this paper, we relate Poisson’s summation formula to Heisenberg’s uncertainty principle. They both express Fourier dualities within the space of tempered distributions and these dualities are furthermore the inverses of one another. While Poisson’s summation formula expresses a duality between discretization and periodization, Heisenberg’s uncertainty principle expresses a duality between regularization and localization. We define regularization and localization on generalized functions and show that the Fourier transform of regular functions are local functions and, vice versa, the Fourier transform of local functions are regular functions.

Many studies in the current literature annotate patterns in stock prices and use computer vision models to learn and recognize these patterns from stock price-action chart images. Additionally, current literature also use Long Short-Term Memory Networks to predict prices from continuous dollar amount data. In this study, we combine the two techniques. We annotate the consolidation breakouts for a given stock price data, and we use continuous stock price data to predict consolidation breakouts. Unlike computer vision models that look at the image of a stock price action, we explore using the convolution operation on raw dollar values to predict consolidation breakouts under a supervised learning problem setting. Unlike LSTMs that predict stock prices given continuous stock data, we use the continuous stock data to classify a given price window as breakout or not. Finally, we do a regularization study to see the effect of L₁, L₂, and Elastic Net regularization. We hope that combining regression and classification shed more light on stock market prediction studies.

In this paper, we introduce the evolve-then-filter (EF) regularization method for reduced order modeling of convection-dominated stochastic systems. The standard Galerkin projection reduced order model (G-ROM) yield numerical oscillations in a convection-dominated regime. The evolve-then-filter reduced order model (EF-ROM) aims at the numerical stabilization of the standard G-ROM, which uses explicit ROM spatial filter to regularize various terms in the reduced order model (ROM). Our numerical results based on a stochastic Burgers equation with linear multiplicative noise. It shows that the EF-ROM is significantly better results than G-ROM.

In remote sensing applications, one of the key points is the acquisition, real-time pre-processing and storage of information. Due to the large amount of information present in the form of images or videos, compression of this data is necessary. Compressed sensing (CS) is an efficient technique to meet this challenge. It consists in acquiring a signal, assuming that it can have a sparse representation, using a minimal number of non-adaptive linear measurements. After this CS process, a reconstruction of the original signal must be performed at the receiver. Reconstruction techniques are often unable to preserve the texture of the image and tend to smooth out its details. To overcome this problem, we propose in this work, a CS reconstruction method that combines the total variation regularization and the non-local self-similarity constraint. The optimization of this method is performed by the augmented Lagrangian which avoids the difficult problem of non-linearity and non-differentiability of the regularization terms. The proposed algorithm, called denoising compressed sensing by regularizations terms (DCSR), will not only perform image reconstruction but also denoising. To evaluate the performance of the proposed algorithm, we compare its performance with state-of-the-art methods, such as Nesterov's algorithm, group-based sparse representation and wavelet-based methods, in terms of denoising, and preservation of edges, texture and image details, as well as from the point of view of computational complexity. Our approach allows to gain up to 25% in terms of denoising efficiency, and visual quality using two metrics: PSNR and SSIM.

The necessary and sufficient conditions of existence of the nonlinear operator equations' branches of solutions in the neighbourhood of branching points are derived. The approach is based on reduction of the nonlinear operator equations to finite-dimensional problems. Methods of nonlinear functional analysis, integral equations, spectral theory based on index of Kronecker-Poincare, Morse-Conley index, power geometry and other methods are employed. Proposed methodology enables justification of the theorems on existence of bifurcation points and bifurcation sets in the nonstandard models. Formulated theorems are constructive. For a certain smoothness of the nonlinear operator, the asymptotic behaviour of the solutions is analysed in the neighbourhood of the branch points and uniformly converging iterative schemes with a choice of the uniformization parameter enables the comprehensive analysis of the problems details. General theorems are illustrated on the nonlinear integral equations.

Numerous approaches address over-fitting in neural networks: by imposing a penalty on the parameters of the network (L1, L2, etc); by changing the network stochastically (drop-out, Gaussian noise, etc.); or by transforming the input data (batch normalization, etc.). In contrast, we aim to ensure that a minimum amount of supporting evidence is present when fitting the model parameters to the training data. This, at the single neuron level, is equivalent to ensuring that both sides of the separating hyperplane (for a standard artificial neuron) have a minimum number of data points — noting that these points need not belong to the same class for the inner layers. We firstly benchmark the results of this approach on the standard Fashion-MINST dataset, comparing it to various regularization techniques. Interestingly, we note that by nudging each neuron to divide, at least in part, its input data, the resulting networks make use of each neuron, avoiding a hyperplane completely on one side of its input data (which is equivalent to a constant into the next layers). To illustrate this point, we study the prevalence of saturated nodes throughout training, showing that neurons are activated more frequently and earlier in training when using this regularization approach. A direct consequence of the improved neuron activation is that deep networks are now easier to train. This is crucially important when the network topology is not known a priori and fitting often remains stuck in a suboptimal local minima. We demonstrate this property by training a network of increasing depth (and constant width): most regularization approaches will result in increasingly frequent training failures (over different random seeds) whilst the proposed evidence-based regularization significantly outperforms in its ability to train deep networks.

Alcoholism is attributed to regular or excessive drinking of alcohol and leads to the disturbance of the neuronal system in the human brain. This results in certain malfunctioning of neurons that can be detected by an electroencephalogram (EEG) using several electrodes on a human skull at appropriate positions. It is of great interest to be able to classify an EEG activity as that of a normal person or an alcoholic person using data from the minimum possible electrodes (or channels). Due to the complex nature of EEG signals, accurate classification of alcoholism using only a small data is a challenging task. Artificial neural networks, specifically convolutional neural networks (CNN), provide efficient and accurate results in various pattern-based classification problems. In this work, we apply CNN on raw EEG data, and demonstrate how we achieved 98% average accuracy by optimizing a baseline CNN model and outperforming its results in a range of performance evaluation metrics on the UCI-KDD EGG dataset. This article explains the step-wise improvement of the baseline model using the dropout, batch normalization, and kernel regularization techniques, and provides a comparison of the two models that can be beneficial for aspiring practitioners who aim to develop similar classification models in CNN. A performance comparison is also provided with other approaches using the same dataset.

The purpose of feature selection is to find important features from the original high-dimensional space. As atypical feature selection algorithm, Locally linear embedding(LLE)-based feature selection algorithm, which applies the idea of LLE to the graph-preserving feature selection framework, has been received wide attention. However, LLE-based feature selection framework is sensitive to noise and K-nearest neighbors. To address these problems, an improved LLE-based feature selection algorithm, robust LLE (RLLE) vote, is proposed. In this algorithm, $l_1$ and $l_2$ regularization are introduced into the high-dimensional reconstruction model of LLE. Furthermore, RLLE vote also proposes a criterion to measure the difference between the reconstruction features and the original features, and then the importance features can be selected by this criteria. Extensive experiments are carried out on a benchmark fault data set and the bearing data set collected from our own laboratory, and the experimental results demonstrate that RLLE vote achieves the most significant performance compared existing state-of-art methods.

Building outlines extracted from remote sensing images and raster maps often have irregular boundaries, redundant points, inaccurate positions, and unclear turns due to factors such as image quality, complexity of the surrounding environment, and extraction methods. This study proposes a regularization algorithm for right-angled polygon building outlines with jagged edges. First, the minimum bounding rectangle of the building outline is established and populated with a square grid based on the smallest visible length principle. An overlay analysis is then applied to the grid and original building to extract the turning points of the outline. Finally, the building orientation is used as a reference axis to sort the turning points and reconstruct the simplified building outline. Analysis of the experimental results shows that the proposed simplification method enhances the morphological characteristics of building outlines, such as parallelism and orthogonality, while considering simplification principles, such as the preservation of direction, position, area, and shape of the building. The proposed algorithm provides a new regularization method for building outlines with jagged edges.

Classical methods for inverse problems are mainly based on regularization theory. In particular those which are based on optimization of a criterion with two parts: a data-model matching and a regularization term. Different choices for these two terms and great number of optimization algorithms have been proposed. When these two terms are distance or divergence measures, they can have a Bayesian Maximum A Posteriori (MAP) interpretation where these two terms correspond, respectively, to the likelihood and prior probability models.

The last series of Raven's standard progressive matrices (SPM-LS) test were studied with respect to its psychometric properties in a series of recent papers. In this paper, the SPM-LS dataset is analyzed with regularized latent class models (RLCM). For dichotomous item response data, an alternative estimation approach for RLCMs is proposed. For polytomous item responses, different alternatives for performing regularized latent class analysis are proposed. The usefulness of the proposed methods is demonstrated in a simulated data illustration and for the SPM-LS dataset. For the SPM-LS dataset, it turned out the regularized latent class model resulted in five partially ordered latent classes.

The fatigue life of concrete is affected by many interwoven factors whose effect is nonlinear. Be-cause of its unique self-learning ability and strong generalization capability, the Bayesian regu-larized backpropagation neural network (BR-BPNN) is proposed to predict concrete behavior in tensile fatigue. The optimal model was determined through various combinations of network parameters. The average relative impact value (ARIV) was constructed to evaluate the correla-tion between fatigue life and its influencing parameters (maximum stress level Smax, stress ratio R, static strength f, failure probability P). ARIV results were also compared with other factor as-sessment methods (weight equation and multiple linear regression analyses). Using BR-BPNN, S-N curves were then obtained for the combinations of R=0.1, 0.2, 0.5; f=5, 6, 7MPa; P=5%, 50%, 95%. The tensile fatigue results under different testing conditions were finally compared for compatibility. It was concluded that Smax has the most significant negative effect on fatigue life; the degree of influence of R, P, and f, which positively correlate with fatigue life, decreases suc-cessively. ARIV is confirmed as a feasible way to analyze the importance of parameters and could be recommended for future applications. The tensile fatigue performance of plain concrete under different stress states (flexural tension, axial tension, splitting tension) does not differ sig-nificantly. Besides utilizing the valuable fatigue test data scattered in the literature, insights gained from this work could provide a reference for subsequent fatigue test program design and fatigue evaluation.

Traditional soil characterization methods are time consuming, laborious and invasive and do not allow long-term repeatability of measurements. The overall aim of this paper was to assess and model spatial variability of the soil in an olive grove in south Italy by using data from two sensors of different type: a multi-spectral on-board drone radiometer and a hyperspectral visible-near infrared-shortwave infrared (VIS-NIR-SWIR) reflectance radiometer as well as sample data, to arrive at a delineation of homogeneous areas. The hyperspectral data were processed using continuum removal methodology to obtain information about the content and composition of clay. Differently, the multispectral data were firstly upscaled to the support of soil data using geostatistics and taking into account change of support. Secondly, the two-sensor data were integrated with soil granulometric properties by using the multivariate geostatistical techniques of multi-collocated cokriging and factor cokriging, in order to achieve a more exhaustive and finer-scale soil characterisation. The paper shows the impact of change of support on the uncertainty of soil prediction that can have a significant effect on decision making in Precision Agriculture. Moreover, four regionalised factors at two different scales (two per each scale) were retained and mapped. Each factor provided a different delineation of the field with areas characterised by different granulometry and clay composition. The applied method is sufficiently flexible and could be applied to any number and type of sensors.

A regularization approach to model selection, within a generalized HJM framework, is introduced which learns the closest arbitrage-free model to a prespecified factor model. This optimization problem is represented as the limit of a one-parameter family of computationally tractable penalized model selection tasks. General theoretical results are derived and then specialized to affine term-structure models where new types of arbitrage-free machine learning models for the forward-rate curve are estimated numerically and compared to classical short-rate and the dynamic Nelson-Siegel factor models.

Variational Auto-Encoders (VAEs) are deep generative models used for unsupervised learning, however their standard version is not topology-aware in practice since the data topology may not be taken into consideration. In this paper, we propose two different approaches with the aim to preserve the topological structure between the input space and the latent representation of a VAE. Firstly, we introduce InvMap-VAE as a way to turn any dimensionality reduction technique, given an embedding it produces, into a generative model within a VAE framework providing an inverse mapping into original space. Secondly, we propose the Witness Simplicial VAE as an extension of the Simplicial Auto-Encoder to the variational setup using a witness complex for computing the simplicial regularization, and we motivate this method theoretically using tools from algebraic topology. The Witness Simplicial VAE is independent of any dimensionality reduction technique and together with its extension, Isolandmarks Witness Simplicial VAE, preserves the persistent Betti numbers of a data set better than a standard VAE.

The case of a virtual bosonic scalar field phi(x) gravitationally coupled in a conformally flat spacetime is investigated. The action S is known to be Weyl invariant only for specific expressions of the potential V(phi). In the conformally flat FLRW case the harmonic angular frequency omega_k(eta) becomes uniquely temporally independent. Therefore any inertial observers embedded in a conformally flat FLRW spacetime all agree on the choice of virtual vacuum states. We postulate that the wavenumber k must be quantized in order to be able to regulate the vev by zeta regularization. Some fundamental implications of this result is derived, and likely conclusions drawn.

In the era of big data, feature engineering has proved its efficiency and importance in dimensionality reduction and useful information extraction from original features. Feature engineering can be expressed as dimensionality reduction and is divided into two types of methods such as feature selection and feature extraction. Each method has its pros and cons. There are a lot of studies to combine these methods. Sparse autoencoder (SAE) is a representative deep feature learning method that combines feature selection with feature extraction. However, existing SAEs do not consider the feature importance during training. It causes extracting irrelevant information. In this paper, we propose a parallel guiding sparse autoencoder (PGSAE) to guide the information by two parallel guiding layers and sparsity constraints. The parallel guiding layers keep the main distribution using Wasserstein distance which is a metric of distribution difference, and it suppresses the leverage of guiding features to prevent overfitting. We perform our experiments using four datasets that have different dimensionality and number of samples. The proposed PGSAE method produces a better classification performance compared to other dimensionality reduction methods.

Background: Current prostate cancer evaluation can be inaccurate and burdensome. Quantitative evaluation of MRI sequences non-invasively helps prostate tumor assessment. However, including Dynamic Contrast Enhancement in the examined MRI sequence set can add complications, inducing possible side effects from the IV placement or injected contrast material, and prolongs scanning time. Purpose: Predict the risk of developing Clinically Significant (Insignificant) prostate cancer CsPCa (CiPCa) and International Society of Urologic Pathology (ISUP) grade using processed Signal to Clutter Ratio (SCR) derived from spatially registered bi-parametric MRI (SRBP-MRI) and thereby enhance non-invasively management of prostate cancer. Methods: This pilot study retrospectively analyzed 42 consecutive prostate cancer patients from the PI-CAI data collection. BP-MRI (Apparent Diffusion Coefficient, High B-value, T2) were resized, translated, cropped, and stitched to form spatially registered SRBP-MRI. Efficacy of noise reduction was tested by regularizing, eliminating principal components (PC), and minimizing elliptical volume from the covariance matrix to optimize the SCR. MRI guided biopsy (MRBx), Systematic Biopsy (SysBx), combination (MRBx+SysBx), or radical prostatectomy determined the ISUP grade for each patient. ISUP grade >=2 (<2) was judged as CsPCa (CiPCa). Linear and Logistic regression were fitted to ISUP grade and CsPCa/CiPCa SCR. Correlation Coefficients (R) and Area Under the Curves (AUC) for Receiver Operator Curves (ROC) evaluated the performance. Results: High correlation coefficients (R) (>0.55) and high AUC (=1.0) for linear and/or logistic fit from processed SCR and z-score for SRBP-MRI greatly exceed fits using prostate serum antigen, prostate volume, and patient age (R~0.17). Patients assessed with combined MRBx+SysBx and from individual MRI scanners achieved higher R (R=0.207+/-0.118) than using all patients in the fits. Conclusions: In the first study to date to apply and test hyperspectral approaches for assessing tumor aggressiveness on SRBP-MRI, high values of R and exceptional AUC to fit the ISUP grade and CsPCA/CiPCA were comparable or better than those from artificial intelligence in other studies.

Temporal downscaling of gridded geophysical data is essential for improving climate models, weather forecasting, and environmental assessments. However, existing methods often could not accurately capture multi-scale temporal features, affecting their accuracy and reliability. To address this issue, we introduce an Enhanced Residual U-Net architecture for temporal downscaling. The architecture, which incorporates residual blocks, allows for deeper network structures without the risk of overfitting or vanishing gradients, thus capturing more complex temporal dependencies. The U-Net design inherently could capture multi-scale features, making it ideal for simulating various temporal dynamics. Moreover, we implement a flow regularization technique with advection loss to ensure that the model adheres to physical laws governing geophysical fields. Our experimental results across various variables within the ERA5 dataset demonstrate an improvement in downscaling accuracy, outperforming other methods.

Nonnegative matrix factorization (NMF) has been shown to be a strong data representation technique, with applications in text mining, pattern recognition, image processing, clustering and other fields. In this paper, we propose a hypergraph regularized Lp smooth nonnegative matrix factorization (HGSNMF), by incorporating hypergraph regularization and Lp smoothing constraint terms into the standard NMF. The hypergraph regularization term can capture the intrinsic geometry structure of the high dimension space data more comprehensively than simple graph, the Lp smoothing constraint may yield a smooth and more accurate solution to the optimization problem. The updating rules are given using multiplicative update techniques, and the convergence of HGSNMF is theoretically investigated. The experimental results on four different data sets show that the proposed method has better clustering effect than the related state-of-the-art methods in the vast majority of cases.

Cervical cancer is one of the ailments that endangers the health of women all over the world and causes infertility. Fortunately, this disease can be avoided. Both the results and participation rates of the current preventative strategy remain poor. Hence, preventative or early detection methods remain challenging and open. In gynecology and computer science fields, few studies on identifying cervical cancer are based on behavior risk factor and machine learning. Moreover, various social and behavioral aspects might often make it more difficult to detect cervical cancer especially in low‑ and middle‑income countries (LMICs). Accordingly, focus on predicting the presence of cervical cancer through behavioral risk factors is the basic structure of this research. We proposed an approach to classify cervical cancer from a social-behavioral perspective using logistic regression to predict cervical cancer. We got significantly improved outcomes than the existing methods. Moreover, we used feature engineering to formulate less dimensionality as a pre-processing step and optimized parameters to be prepared for classification phase. The results proved that using the logistic regression with regularization type L1 had a promising performance. The accuracy of Logistic Regression (LR) L1 was 97.2%. Such results support the proposed work as a reliable classification predictive tool.

The elastic wave equation with the seismic tensorial force is solved in a homogeneous and isotropic medium (the Earth). Spherical-shell waves are obtained, which are associated to the primary P and S seismic waves. It is shown that these waves produce secondary waves with sources on the plane surface of a half-space, which have the form of abrupt walls with a long tail, propagating in the interior and on the surface of the half-space. These secondary waves are associated to the seismic mainshock. The results, previously reported, are re-derived by using Fourier transformations and specific regularization procedures. The relevance of this seismic motion for the ground motion, the seismographs' recordings and the effect of the inhomogeneities in the medium are discussed.

This paper presents three types of tensor Conjugate-Gradient methods for solving large-scale linear discrete ill-posed problems based on the t-product between third-order tensors. An automatical determination strategy of a suitable regularization parameter is proposed for the tensor conjugate gradient (tCG) method. A truncated version and a preprocessed verion of the tCG method are further presented. The discrepancy principle is employed to determine a suitable regularization parameter. Several numerical examples are given to show the effectiveness of the proposed tCG methods in image and video restoration.

This study employs the phase-field regularized cohesion model (PF-CZM) to simulate crack propagation and damage behavior in porous granite. We investigate how pore radius (r), initial crack-pore distance (D), and pore-crack angle (θ) impact crack propagation. The simulation findings reveal that, with a fixed deflection angle and initial crack-pore distance, larger pores are more likely to induce crack extension under identical loading conditions. Moreover, with r and θ keeping constant, the crack extension can be divided into two stages: from its initiation to the lower edge of the pore and then from the lower edge to the upper boundary of the model. By varying the values of D and r, we derive multiple combinations of different D/r ratios and pore radii. These results demonstrate that with constant r, cracks tend to deflect toward the pore closer to the initial crack. Conversely, when D is maintained constant, cracks will preferentially deflect toward pores with a larger r. In summary, the numerical simulation of rock pores and initial cracks, based on the PF-CZM, exhibits remarkable predictive capabilities and holds significant potential in advancing rock fracture analysis.

This paper studies the dividend optimization problem in the entropy regularization framework by following the same continuous-time reinforcement learning setting as in Wang et al. (2020). The exploratory HJB is established and the optimal exploratory dividend policy is a truncated exponential distribution. We show that, for suitable choices of the maximal dividend paying rate and the temperature parameter, the value function of the exploratory dividend optimization problem could be significantly different from the value function in the classical dividend optimization problem. In particular, the value function of the exploratory dividend optimization problem could be classified into three cases based on its monotonicity. Numerical examples are also presented to show the impact of temperature parameter on the solution.

This work implements a stable algorithm in Field-Programmable Gate Arrays (FPGAs) for which two architectures (unrolling the loop) were developed and analyzed. The algorithm recovers sources located at the boundary separating two homogeneous media that make up a two-dimensional non-homogeneous region from measurements on the boundary of such region. The problem of recovering these sources is an ill-posed inverse problem as small errors in the measurement can produce important changes in the source location. Inverse source problems have many applications in different areas, such as engineering and medicine, making the proposed implementation important. The first architecture (mode one) allows considering different operating speeds, which is an advantage depending on whether we work with fast or slow signals. The second one (mode two) reduces resource consumption by exploiting the characteristics of the source identification algorithm, which is an advantage for multichannel problems such as inverse electrocardiography or electroencephalography. The architectures were tested on four FPGAs of the 7 Series of Xilinx: Spartan-7 xc7s100fgga484, Virtex-7 xc7v585tffg1157, Kintex-7 xc7k70tfbg484, and Artix-7 xc7a35tcpg236. The two FPGA implementations of the algorithm were validated using synthetic examples implemented in MATLAB. The results presented here can be extended to concentric spheres and complex geometries.

This work tackles the problem of image restoration, a crucial task in many fields of applied sciences, focusing on removing degradation caused by blur and noise during the acquisition process. Drawing inspiration from the multi-penalty approach based on the Uniform Penalty principle introduced in [Bortolotti et al. arXiv.math.NA/2309.14163], we develop here a new image restoration model and an iterative algorithm for its effective solution. The model incorporates pixel-wise regularization terms and establishes a rule for parameters selection, aiming to restore images through the solution of a sequence of constrained optimization problems. To achieve this, we present a modified version of the Newton Projection method, adapted to multi-penalty scenarios, and prove its convergence. Numerical experiments demonstrate the efficacy of the method in eliminating noise and blur while preserving the image edges.

In the present paper, we efficiently solve the two-body problem for extreme cases such as those with high eccentricities. The use of numerical methods, with the usual variables, cannot maintain the perihelion passage accurately. In previous articles, we have verified that this problem is treated more adequately through temporal reparametrizations related to the mean anomaly through the partition function. The biparametric family of anomalies, with an appropriate partition function, allows a systematic study of these transformations. In the present work, we consider the elliptical orbit as a meridian section of the ellipsoid of revolution, and the partition function depends on two variables raised to specific parameters. One of the variables is the mean radius of the ellipsoid at the secondary, and the other is the distance to the primary. One parameter regulates the concentration of points in the apoapsis region, and the other produces a symmetrical displacement between the polar and equatorial regions. The three most used geodesy latitude variables are also studied, resulting in one not belonging to the biparametric family. However, it is in the one introduced now, which implies an extension of the biparametric method. The results obtained by the method presented here now allow a causal interpretation of the operation of numerous reparametrizations used in the study of orbital motion.

Fast Field-Cycling (FFC) Nuclear Magnetic Resonance (NMR) relaxometry is a powerful non-destructive magnetic resonance technique that enables, among other things, the investigation of slow molecular dynamics at low magnetic field intensities. FFC-NMR relaxometry measurements provide insight into molecular motion across various timescales within a single experiment. This study focuses on a \textit{model-free} approach, representing the NMRD profile $R_1$ as a linear combination of Lorentzian functions, thereby addressing the challenges of fitting data within an ill-conditioned linear least-squares framework. Tackling this problem, we present a comprehensive review and experimental validation of three regularization approaches to implement the model-free approach to analysing NMRD profiles. These include: (1) \nameUpe, utilizing locally adapted $L_2$ regularization; (2) \nameL, based on $L_1$ penalties; and (3) a hybrid approach combining locally adapted $L_2$ and global $L_1$ penalties. Each method's regularization parameters are determined automatically, according to the Balancing and Uniform Penalty principles. % Our contributions include the implementation and experimental validation of the \nameUpe{} and \nameUp{} algorithms, and the development of a "dispersion analysis" technique to assess the existence range of the estimated parameters. The objective of this work is to delineate the variance in fit quality and correlation time distribution yielded by each algorithm, thus broadening the set of software tools for the analysis of sample structures in FFC-NMR studies. The findings underline the efficacy and applicability of these algorithms in the analysis of NMRD profiles from samples representing different potential scenarios. 1