L1 Algorithm Research Articles

A fast compact difference scheme on graded meshes for solving the time-fractional nonlinear KdV equation

The fractional KdV equation has significant physical implications, the study of its efficient method has great scientific significance. A fast compact difference (FCD) scheme on graded meshes is used to address the time-fractional nonlinear KdV equation with an initial singularity. The temporal discretization employs the fast L1 algorithm, while a fourth-order compact scheme is utilized for spatial discretization. The Newton's method is used to approximate the nonlinear term. The fast L1 algorithm can greatly shorten the CPU calculation time without losing the precision of numerical solution. We verify the existence and uniqueness of numerical solutions, while also provide unconditional stability and convergence analysis. Theoretical analysis is validated by numerical results, indicating the FCD scheme on graded meshes has high-accuracy and time-saving performance.

Poisson noise removal based on non-convex hybrid regularizers

The presence of TV regularizer always induces an unsatisfactory staircase effect. To overcome the staircase while better sustaining edge information, this work proposes a novel model for Poisson noise removal. The model is based on non-convex mixed regularizers, which involves introducing a non-convex penalty into a composition of the total variation and the higher-order total variation. The iterative reweighted l1 algorithm was used to convert the non-convex model into a convex one. The classic alternating direction method of multipliers was then employed to obtain approximate solutions of the model. When applying this model to degraded images contaminated by Poisson noise of medium to high intensity, its performance in noise suppression was tested. The regularizer was compared with others in the terms of the visual effect of the picture, time cost, and several commonly accepted quantitative indicators for evaluation, such as peak signal-to-noise ratio, feature similarity index and structural similarity index. Numerical experiments showed that the present model not only eliminates block artifacts but also retains sharp edges.

Optimizing heat transfer with nano additives: A mathematical approach

This study is carried out to investigate heat transfer of hybrid nanofluid between two concentric cylinders using the fractional Maxwell model. The fractional model is solved numerically; fractional derivatives are integrated using the L1 algorithm of Caputo derivative and the integer order derivatives are discretized using the Crank–Nicolson numerical method. The findings show that skin friction decreases when the relaxation time parameter increases. The thermal performance of the working fluid increased up to 5% when hybrid nanoparticles were suspended to it.

On choosing initial values of iteratively reweighted ℓ1 algorithms for the piece-wise exponential penalty

Computing the proximal operator of the sparsity-promoting piece-wise exponential (PiE) penalty [Formula: see text] with a given shape parameter [Formula: see text], which is treated as a popular nonconvex surrogate of [Formula: see text]-norm, is fundamental in feature selection via support vector machines, image reconstruction, zero-one programming problems, compressed sensing, neural networks, etc. Due to the nonconvexity of PiE, for a long time, its proximal operator is frequently evaluated via an iteratively reweighted [Formula: see text] algorithm, which substitutes PiE with its first-order approximation, however, the obtained solutions only are the critical point. Based on the exact characterization of the proximal operator of PiE, we explore how the iteratively reweighted [Formula: see text] solution deviates from the true proximal operator in certain regions, which can be explicitly identified in terms of [Formula: see text], the initial value and the regularization parameter in the definition of the proximal operator. Moreover, the initial value can be adaptively and simply chosen to ensure that the iteratively reweighted [Formula: see text] solution belongs to the proximal operator of PiE.

Low-rank reconstruction for mid-infrared spectroscopy using broadband ultra-wide-angle photonic filters and IReg algorithm

In this study, a broadband mid-infrared photonic filter is designed. It has the advantages of polarization insensitivity, ultrawide angle, high manufacturability, and easy integration. For infrared spectrum reconstruction, the use of filters as spectral sampling channels can eliminate the influence of slits or complex optical splitting elements in spectral imaging technology. Further, we propose a low-rank matrix logarithm regularization algorithm based on a three-segment filter function (IReg algorithm) combined with the designed filter for a variety of mid-infrared detector front ends, which can significantly suppress noise interference. Here, we demonstrate that the stability of the spectral reconstruction accuracy under the large disturbance environment obtained by the IReg algorithm is much higher than those of the L2 and L1 algorithms. In the low-dimensional matrix state, the accuracy and stability of the ill-conditioned linear equations are guaranteed, the anti-noise ability is enhanced, the effect of data dimensionality reduction is realized, and the accuracy of mid-infrared spectral reconstruction is improved.

An optimal sparse sensing approach for scanning point selection and response reconstruction in full-field structural vibration testing

Non-contact vibration measurements, such as 3D scanning laser Doppler vibrometry (3D SLDV), are becoming more prevalent in testing next-generation lightweight aerospace structures. This approach reduces the impact of attached sensors and improves measurement reliability. Acquiring precise measurement data for the whole area is feasible, albeit it would demand a considerable amount of time and storage space for testing. The concept of compressed sensing has been recently approved as an effective way to exploit signal sparsity and achieve full response reconstruction with very few measurements. The objective of this work is to enhance the efficiency of non-contact vibration testing by utilizing the state-of-the-art compressive sensing approach. In contrast to conventional sensor placement methods that rely on effective independence, modal kinetic energy, or modal assurance criterion matrix as targets, this paper proposes a novel sensor placement methodology from the perspective of dynamic response reconstruction. The scanning points are chosen with a minimal number to reduce testing time and are well-placed such that full-field vibration responses of the test structure can be reconstructed accurately. This allows for the spatially-detailed vibration responses to be obtained efficiently and accurately with optimal sparse sensing placement and effective response reconstruction through ℓ1 algorithm. Two case studies will be presented in this work to demonstrate and validate the methodology. The first case study is focused on a simplified cantilever beam using the numerical data from the FE analysis to demonstrate the methodology. The second case study is focused on using 3D SLDV experimental testing data from a full-scale industrial fan blade. Based on the results, it is evident that the proposed approach can significantly decrease the scanning points required for a full-field dynamic response reconstruction during full-field vibration testing.

Mixed Gaussian-impulse noise removal using non-convex high-order TV penalty

To restore images with clear edge details and no staircase artifacts from degraded versions, this paper incorporates the ℓ2 plus ℓ0 data fidelity and non-convex high-order total variation regularizer to establish an optimization model for eliminating mixed Gaussian-impulse noise. Among them, the ℓ2 fidelity is adopted to suppress Gaussian noise, while the ℓ0-norm is more suitable for detecting and removing impulse noise. In addition, the non-convex regularization displays excellent performance in overcoming the staircase effect and preserving edge details. Computationally, by using the iteratively reweighted ℓ1 algorithm and variable splitting method, this work designs a modified alternating minimization method to solve the optimization problem we construct. In theory, the convergence proof of our resulting algorithm is also presented. Finally, in the experimental section, we conduct extensive numerical experiments on degraded images and compare with other existing techniques. From the intuitive effects and restoration accuracy, it follows that our newly proposed method is effective and competitive for image deblurring and mixed noise removal.

Variable time steps in the numerical implementation of viscoelastic fractional models for laminated glass

Abstract We elaborate numerical approaches to calculate the rheological response of laminated glass beams, whose viscoelastic interlayer is modelled via fractional calculus. This mathematical description is very effective when the relaxation function of the polymer can be expressed by continuously connected branches of power-laws, as is the case for most materials used to laminate glass. The classical approach uses the Grünwald-Letnikov approximation of fractional derivatives, but it requires constant time steps, which would become very large to reasonably cover the entire observation time, thus losing accuracy. We propose to use the L1 algorithm with increasing time steps, which is well suited to the power law character of the relaxation function. This allows to follow the long-term creep response, providing a better approximation when needed. The method is implemented for beams laminated with viscolastic interlayers whose relaxation is described by four branches of power laws, to cover most practical cases. Numerical experiments shows its advantages over the Grünwald-Letnikov approach for characterizing the long-term structural response.

Numerical investigation on flow, heat and mass transfer performance of fractional Oldroyd-B hybrid nanofluid as a coolant for power battery

This paper introduced for the first time a viscoelastic hybrid nanofluid as the coolant for direct contact cooling power battery. The governing boundary layer equations were established by adopting fractional Oldroyd-B model and fractional Buongiorno’s model. Second-order velocity slip boundary conditions were also considered. Then the solutions were numerically acquired by finite difference coupled with L1 algorithm. Impact of main physical parameters on the flow, heat and mass transfer of the viscoelastic hybrid nanofluid on the cylindrical battery was graphically presented and detailly discussed. Outcomes show that the heat transfer is improved by both Brownian motion(Nb) and thermophoresis(Nt) to different degrees. When Nb grows from 0.05 to 0.1, the average Nusselt number increases by 2.2%, higher than 0.027% of Nt. The slip behavior only affects the velocity distribution near the individual cell and slightly enhances heat and mass transfer. The velocity relaxation fractional derivative contributes to convection, heat and mass transfer on the cell wall, while velocity retardation fractional derivative behaves just the opposite. The proposed viscoelastic hybrid nanofluid with appropriate volume fractions of nanoparticles enhances heat transfer on the cell wall and is strongly recommended as a candidate for power battery coolant.

Efficient numerical scheme for studying dual-phase chemical reactions in unsteady Sisko fluid flow with relaxation times

This research article investigated the effects of dual-phase chemical reactions on the unsteady flow of Sisko fluid. Utilizing Caputo's fractional derivatives, the model captured the fractional behavior of the fluid and integrated relaxation times to reflect the viscoelastic nature of the Sisko fluid. The model also considered unequal diffusivities of the chemical species. The resulting equations were discretized using the finite difference method, complemented by the L1 algorithm. Our simulations underscored how memory parameters, β and γ, significantly shaped the concentration trajectories by emphasizing the impact of past flow behaviors. Moreover, the study identified the heterogeneous reaction parameter and the diffusion coefficients ratio as critical determinants in the reaction dynamics, underscoring their role in the system's behavior. This study illuminates the intricate matrix of relationships in the realm of chemical reactions and viscoelastic fluid flow, offering valuable insights for fields ranging from chemical processing to environmental remediation.

Non-convex TGV regularized [formula omitted]-norm fidelity model for impulse noise removal

Aiming at preserving the structural features of images from degradations contaminated by impulse noise, this article proposes a novel strategy that inherits the characteristics of ℓ0-norm data fidelity and non-convex total generalized variation regularizer. The ℓ0-norm fidelity is more conducive to removing impulse noise, while the non-convex total generalized variation penalty term is both good at eliminating the staircase artifacts and maintaining neat contours. In addition, a modified alternating minimization algorithm is constructed to optimize the solution of the developed model, which merges the popular iteratively reweighted ℓ1 algorithm and primal-dual approach. Meanwhile, the convergence property of the designed algorithm is briefly presented. Finally, in the experimental part, exhaustive simulation experiments are carried out to demonstrate the effectiveness of the introduced scheme for removing impulse noise. As observed in the numerical results, our scheme possesses obvious superiorities in both visual effects and quantitative comparisons, in contrast to some existing popular image restoration methods.

Impulse noise removal by using a nonconvex TGV regularizer and nonconvex fidelity

With the aim of acquiring high-quality restored images, this paper derives a novel nonconvex variational model for the removal of impulse noise. The investigated solver integrates the superiorities of nonconvex second-order total generalized variation (TGV) regularization and nonconvex data fidelity. More precisely, the usage of a nonconvex TGV regularizer helps to eliminate the staircase artifacts and simultaneously preserve edge details. Nonconvex fidelity, which enhances sparsity, is adopted to effectively detect impulse noise. Computationally, incorporating the popular iteratively reweighted ℓ1 algorithm and variable splitting method, we propose to adopt an efficient alternating direction method of multipliers for the purpose of rapidly resolving the designed optimization problem. Additionally, simulation examples have been executed to compare our method with several state-of-the-art techniques. Experimental results and quantitative comparisons confirm that the developed strategy outperforms other competitors for suppressing impulse noise (even high density) in both visual outcomes and recovery accuracy.

NON-CONVEX HYBRID TOTAL VARIATION FOR RESTORING MEDICAL IMAGE CORRUPTED BY POISSON NOISE

Abstract. In this work, we proposed the hybrid non-convex regularizers for Poisson noise removal on medical images. The model is built by a combination of non-convex total variation and non-convex fractional total variation. The proposed model allows for avoiding the annoying staircase artifacts and obtaining the reconstruction results with sharp and neat edges during the noise removal process. For handling the minimization problem, we employ the alternating minimization method associated with the iteratively reweighted l1 algorithm. Numerical experiments illustrate the efficiency of the proposed model and corresponding algorithm.

A hyperautomative human behaviour recognition algorithm based on improved residual network

ABSTRACT When dealing with the mutual storage relationship of behavioral features in long time sequence video, the convolutional neural network is easy to miss important feature information. To solve the above problems, this paper proposes a super automatic algorithm combining nonlocal convolution and three-dimensional convolution neural network. The algorithm uses sparse sampling to segment the long time sequence video to reduce the amount of redundant information, and integrates non-local convolution into the residual neural network, thus forming a super automatic full variational - L1 algorithm. Experimental results show that the proposed method can significantly improve the efficiency of behavior recognition.

Effects of stretching velocity on double fractional Jeffreys fluids with rheological synergistic heat conductivity

Abstract Double fractional Jeffreys fluids are widely used in production and life. In this paper, the effects of stretching velocity on the flow and heat transfer of double fractional Jeffreys fluid are studied. Three types of stretching velocity are considered, i.e., (i) uniform velocity; (ii) acceleration; and (iii) deceleration. The rheological synergistic thermal conductivity model introduced to the energy equation is formulated based on experiments. The governing equations are solved by using a combination of the finite difference technique and the L1 algorithm. Results show that there is an inflection point on each velocity profile which divides the velocity field into two sections, convex (the elasticity plays a primary effect) and concave (the viscosity plays a primary effect). As the stretching velocity parameter increases, the thickness of the region where the elasticity plays a major role does not change in case (i), however, it reduces in case (ii) and grows in case (iii). We also found that, compared with uniform stretching, accelerated stretching can lead to higher heat transfer, while decelerated stretching causes less heat transfer. And for uniform velocity stretching, the stretching velocity parameter has little effect on the temperature field. In the case of accelerated stretching, increasing the stretching velocity parameter enhances heat transfer, however, for decelerated stretching, it weakens heat transfer. These results are instructive for industrial design.

Two dimensional MHD nanofluid flow analysis of fractional dual-phase-lag heat conduction between inclined cylinders with variable thickness

PurposeThis study aims to investigate the two-dimensional magnetohydrodynamic flow and heat transfer of a fractional Maxwell nanofluid between inclined cylinders with variable thickness. Considering the cylindrical coordinate system, the constitutive relation of the fractional viscoelastic fluid and the fractional dual-phase-lag (DPL) heat conduction model, the boundary layer governing equations are first formulated and derived.Design/methodology/approachThe newly developed finite difference scheme combined with the L1 algorithm is used to numerically solve nonlinear fractional differential equations. Furthermore, the effectiveness of the algorithm is verified by a numerical example.FindingsBased on numerical analysis, the effects of parameters on velocity and temperature are revealed. Specifically, the velocity decreases with the increase of the fractional derivative parameter α owing to memory characteristics. The temperature increase with the increase of fractional derivative parameter ß due to a decrease in thermal resistance. From a physical perspective, the phase lag of the heat flux vector and temperature gradients τq and τT exhibit opposite trends to the temperature. The ratio τT/τq plays an important role in controlling different heat conduction behaviors. Increasing the inclination angle θ, the types and volume fractions of nanoparticles Φ can increase velocity and temperature, respectively.Originality/valueFractional Maxwell nanofluid flows from a fixed-thickness pipe to an inclined variable-thickness pipe, and the fractional DPL heat conduction model based on materials is considered, which provides a basis for the safe and efficient transportation of high-viscosity and condensable fluids in industrial production.

A survey of numerical algorithms that can solve the Lasso problems

AbstractIn statistics, the least absolute shrinkage and selection operator (Lasso) is a regression method that performs both variable selection and regularization. There is a lot of literature available, discussing the statistical properties of the regression coefficients estimated by the Lasso method. However, there lacks a comprehensive review discussing the algorithms to solve the optimization problem in Lasso. In this review, we summarize five representative algorithms to optimize the objective function in Lasso, including iterative shrinkage threshold algorithm (ISTA), fast iterative shrinkage‐thresholding algorithms (FISTA), coordinate gradient descent algorithm (CGDA), smooth L1 algorithm (SLA), and path following algorithm (PFA). Additionally, we also compare their convergence rate, as well as their potential strengths and weakness.This article is categorized under: Statistical Models > Linear Models Algorithms and Computational Methods > Numerical Methods Algorithms and Computational Methods > Computational Complexity

A Symmetric Fractional-order Reduction Method for Direct Nonuniform Approximations of Semilinear Diffusion-wave Equations

We introduce a symmetric fractional-order reduction (SFOR) method to construct numerical algorithms on general nonuniform temporal meshes for semilinear fractional diffusion-wave equations. By using the novel order reduction method, the governing problem is transformed to an equivalent coupled system, where the explicit orders of time-fractional derivatives involved are all \(\alpha /2\) \((1<\alpha <2)\). The linearized L1 scheme and Alikhanov scheme are then proposed on general time meshes. Under some reasonable regularity assumptions and weak restrictions on meshes, the optimal convergence is derived for the two kinds of difference schemes by \(H^2\) energy method. An adaptive time stepping strategy which based on the (fast linearized) L1 and Alikhanov algorithms is designed for the semilinear diffusion-wave equations. Numerical examples are provided to confirm the accuracy and efficiency of proposed algorithms.

Numerical simulation of the fractional Maxwell fluid flow in locally narrow artery

Research on hemorheology and blood flow behavior in non-uniform vessels is of extreme significance for diagnosis and treatment of many cardiovascular diseases. The aim of this study is to reveal the hemodynamics in stenotic vessels, and provide a reference for formulating a clinical operation plan. A set of rheological data of human blood at 37° is utilized in the paper to construct the fractional Maxwell constitutive equation of blood. Consequently, the continuity and momentum equations of a fractional Maxwell fluid passing through a stenosis artery in a two-dimensional cylindrical coordinate system are established. With the help of the vorticity and stream function, the finite difference method combined with the fractional order derivative L1 algorithm is applied to acquire the numerical solutions of the velocity, wall shear stress and intravascular pressure gradient, and the validity of the algorithm is verified. Furthermore, the effects of the stenosis degree, stenosis shoulder length, various Reynolds numbers and fractional parameter α on the blood flow characteristics in stenosis are analyzed.

Non-convex fractional-order TV model for impulse noise removal

For the pursuit of high-quality restoration effects, we introduce a novel non-convex extended scheme to reconstruct blurred images under impulse noise in this paper. Because of the remarkable advantages of the combination of non-convex ℓp-norm and fractional-order total variation regularization, our developed strategy possesses a brilliant ability to overcome the staircase-like aspects and preserve neat contours. To obtain the minimizer of the optimization problem, we adopt the alternating direction method of multipliers closely integrating with the iteratively reweighted ℓ1 algorithm. Theoretically, a convergence proof is provided for the developed algorithm. By considering salt-and-pepper noise and random-valued impulse noise, comprehensive simulation comparisons with some popular methods are carried out for image restoration. And the experimental results illustrate that our proposed model behaves better with respect to visual comparison and quantitative measurement.