Splitting Algorithm Research Articles

Single-particle reconstruction in cryo-EM based on three-dimensional weighted nuclear norm minimization

Single-particle reconstruction (SPR) in cryogenic electron microscopy (cryo-EM) aims at aligning and averaging two-dimensional micrographs to reconstruct a three-dimensional particle.How to reconstruct micrographs from heavy noise is a crucial point for achieving better micrograph quality, and thus many methods focus on noise removal. However, new problems such as over-smoothing often occur in their results due to failure in handling heavy noise well. This paper proposes a three-dimensional weighted nuclear norm minimization (3DWNNM) model for SPR in the cryo-EM task to address these issues. Specifically, we design a minimization solver based on the forward-backward splitting algorithm to tackle our model efficiently. Under certain conditions, this solution has an energy-decaying feature and performs exceptionally well in reconstruction. Numerical experiments fully demonstrate the effectiveness and the robustness of the proposed method.

SALSA-Net: Explainable Deep Unrolling Networks for Compressed Sensing.

Deep unrolling networks (DUNs) have emerged as a promising approach for solving compressed sensing (CS) problems due to their superior explainability, speed, and performance compared to classical deep network models. However, the CS performance in terms of efficiency and accuracy remains a principal challenge for approaching further improvements. In this paper, we propose a novel deep unrolling model, SALSA-Net, to solve the image CS problem. The network architecture of SALSA-Net is inspired by unrolling and truncating the split augmented Lagrangian shrinkage algorithm (SALSA) which is used to solve sparsity-induced CS reconstruction problems. SALSA-Net inherits the interpretability of the SALSA algorithm while incorporating the learning ability and fast reconstruction speed of deep neural networks. By converting the SALSA algorithm into a deep network structure, SALSA-Net consists of a gradient update module, a threshold denoising module, and an auxiliary update module. All parameters, including the shrinkage thresholds and gradient steps, are optimized through end-to-end learning and are subject to forward constraints to ensure faster convergence. Furthermore, we introduce learned sampling to replace traditional sampling methods so that the sampling matrix can better preserve the feature information of the original signal and improve sampling efficiency. Experimental results demonstrate that SALSA-Net achieves significant reconstruction performance compared to state-of-the-art methods while inheriting the advantages of explainable recovery and high speed from the DUNs paradigm.

Dumpy: A Compact and Adaptive Index for Large Data Series Collections

Data series indexes are necessary for managing and analyzing the increasing amounts of data series collections that are nowadays available. These indexes support both exact and approximate similarity search, with approximate search providing high-quality results within milliseconds, which makes it very attractive for certain modern applications. Reducing the pre-processing (i.e., index building) time and improving the accuracy of search results are two major challenges. DSTree and the iSAX index family are state-of-the-art solutions for this problem. However, DSTree suffers from long index building times, while iSAX suffers from low search accuracy. In this paper, we identify two problems of the iSAX index family that adversely affect the overall performance. First, we observe the presence of a proximity-compactness trade-off related to the index structure design (i.e., the node fanout degree), significantly limiting the efficiency and accuracy of the resulting index. Second, a skewed data distribution will negatively affect the performance of iSAX. To overcome these problems, we propose Dumpy, an index that employs a novel multi-ary data structure with an adaptive node splitting algorithm and an efficient building workflow. Furthermore, we devise Dumpy-Fuzzy as a variant of Dumpy which further improves search accuracy by proper duplication of series. Experiments with a variety of large, real datasets demonstrate that the Dumpy solutions achieve considerably better efficiency, scalability and search accuracy than its competitors.

FEM treatments for MHD highly mixed convection flow within partially heated double-lid driven odd-shaped enclosures using ternary composition nanofluids

A highly mixed convection due to the movements of two perpendicular walls of odd-shaped enclosures is examined. The upper and right boundaries are partially heated, and several heating situations based on the positions of the active parts are considered. Also, several directions of the lid-walls movements are examined. The inner medium is a heat generating and the considered Lorentz force is inclined with angle Φ. The mathematical formulations are based on the Brinkman-extended non-Darcy model. The containers are filled with ternary composition nanofluids. The components of the ternary nanofluids are spherical Al2O3 nanoparticles, MWCNT cylindrical nanoparticles and platelet graphene nanoparticles and suitable correlations for the thermophysical properties are introduced. The numerical simulations are carried out using the finite element method (FEM) based on the characteristics–based split (CBS) algorithm. The major outcomes revealed that the ternary composition nanofluids cause an enhancement in the heat transfer rate by 17.65% comparing to the mono-nanofluids. Also, to get the best mixed convection situation, it is recommended to locate the heat sources in the middle of the outer perpendicular walls, set Re≥100 and make the lower perpendicular walls move in anticlockwise direction to get the best mixed convection situation.

An Efficient Splitting Algorithm for Solving the CDT Subproblem

The CDT subproblem, which minimizes a quadratic function over the intersection of two ellipsoids, is a classical quadratic programming problem. In this paper, we study a method of solving CDT with a positive Lagrangian duality gap. An efficient splitting algorithm is proposed for finding the global optimal solutions. A cutting plane is firstly added to divide the feasible set of CDT into two subsets, and then two new quadratic programming problems with ellipsoidal and linear constraints are generated accordingly. Using the newly developed technique — second-order cone constraints to enhance the efficiencies of the SDP relaxation-based algorithms on the two subproblems, an optimal solution of CDT can be acquired by comparing the objective values of the two subproblems. Numerical experiments show that the new algorithm outperforms the two recent SDP relaxation-based algorithms, the two-parameter eigenvalue-based algorithm and the solver Gurobi 9.5 for certain types of CDT.

An Improved Mixture Model of Gaussian Processes and Its Classification Expectation–Maximization Algorithm

The mixture of experts (ME) model is effective for multimodal data in statistics and machine learning. To treat non-stationary probabilistic regression, the mixture of Gaussian processes (MGP) model has been proposed, but it may not perform well in some cases due to the limited ability of each Gaussian process (GP) expert. Although the mixture of Gaussian processes (MGP) and warped Gaussian process (WGP) models are dominant and effective for non-stationary probabilistic regression, they may not be able to handle general non-stationary probabilistic regression in practice. In this paper, we first propose the mixture of warped Gaussian processes (MWGP) model as well as its classification expectation–maximization (CEM) algorithm to address this problem. To overcome the local optimum of the CEM algorithm, we then propose the split and merge CEM (SMC EM) algorithm for MWGP. Experiments were done on synthetic and real-world datasets, which show that our proposed MWGP is more effective than the models used for comparison, and the SMCEM algorithm can solve the local optimum for MWGP.

Split monotone variational inclusion with errors for image-feature extraction with multiple-image blends problem

In this paper, we introduce a new iterative forward–backward splitting algorithm with errors for solving the split monotone variational inclusion problem of the sum of two monotone operators in real Hilbert spaces. We suggest and analyze this method under some mild appropriate conditions imposed on the parameters such that another strong convergence theorem for this problem is obtained. We also apply our main result to image-feature extraction with the multiple-image blends problem, the split minimization problem, and the convex minimization problem, and provide numerical experiments to illustrate the convergence behavior and show the effectiveness of the sequence constructed by the inertial technique.

Proximal Splitting Algorithms for Convex Optimization: A Tour of Recent Advances, with New Twists

Convex nonsmooth optimization problems, whose solutions live in very high dimensional spaces, have become ubiquitous. To solve them, the class of first-order algorithms known as proximal splitting algorithms is particularly adequate: they consist of simple operations, handling the terms in the objective function separately. In this overview, we demystify a selection of recent proximal splitting algorithms: we present them within a unified framework, which consists in applying splitting methods for monotone inclusions in primal-dual product spaces, with well-chosen metrics. Along the way, we easily derive new variants of the algorithms and revisit existing convergence results, extending the parameter ranges in several cases. In particular, we emphasize that when the smooth term in the objective function is quadratic, e.g., for least-squares problems, convergence is guaranteed with larger values of the relaxation parameter than previously known. Such larger values are usually beneficial for the convergence speed in practice.

A three-operator splitting algorithm with deviations for generalized DC programming

In this paper, we introduce a three-operator splitting algorithm with deviations for solving the minimization problem composed of the sum of two convex functions minus a convex and smooth function in a real Hilbert space. The main feature of the proposed method is that two per-iteration deviation vectors provide additional degrees of freedom. We propose one-step and two step inertial three-operator splitting algorithms by selecting the deviations along a momentum direction. A numerical experiment for DC regularized sparse recovery problems shows that the proposed algorithms have better performance than the original three-operator splitting algorithm.

A second order fractional step hybrid numerical algorithm for time delayed singularly perturbed 2D convection-diffusion problems

The objective of this work is to provide an efficient and higher-order numerical scheme for the solution of a singularly perturbed 2D time-delayed parabolic convection-diffusion problem. Each time step is split into two partial time steps by using the Peaceman-Rachford splitting algorithm. The splitting is done over the idea of the Crank-Nicholson scheme. The hybrid scheme is a combination of the central difference scheme in the layer region and the mid-point upwind scheme in the outer region defined on a Shishkin mesh for the discretization in the spatial direction. This makes the order of convergence to two (up to a logarithmic factor) in space. Again, the presence of the logarithmic effect is rectified by the use of the Bakhvalov-Shishkin mesh. The proposed method is proved to be parameter uniform and also it attains second-order spatial accuracy in the discrete supremum norm. Numerical tests validate the efficacy of the proposed scheme.

Modeling of particle-laden flows with n-sided polygonal smoothed finite element method and discrete phase model

In this work, an n-sided polygonal smoothed finite element method (nSFEM) using mean value shape function for concave polygonal element coupled with the discrete phase model (DPM) is developed to solve particle-laden flows. The fluid phase is solved using nSFEM stabilized by the well-developed characteristic-based split (CBS) algorithm. The concave polygonal element is successfully constructed via the intrinsic characteristic of mean value shape function and a new smoothing domain (SD) construction with more physical meaning; thus, addressing the severely distorted local mesh, as well as retaining good geometric adaptability. In the meantime, the modeling of the coupling between the fluid and the particle is achieved by reusing the mean value interpolation to obtain the fluid drag force acting on the particle. The accurate capture of fluid velocities at particle positions within arbitrary polygonal elements via mean value interpolation ensures fluid drag with high-fidelity. Several classical numerical examples, including particle-laden flow around a circular cylinder, are presented to demonstrate the high accuracy and good robustness, and the capability of the proposed method for predicting particle motion in the analysis of complex particle-laden flows.

Two New Modified Regularized Methods for Solving the Variational Inclusion and Null Point Problems

In this article, based on the regularization techniques, we construct two new algorithms combining the forward-backward splitting algorithm and the proximal contraction algorithm, respectively. Iterative sequences of the new algorithms can converge strongly to a common solution of the variational inclusion and null point problems in real Hilbert spaces. Multi-inertial extrapolation steps are applied to expedite their convergence rate. We also give some numerical experiments to certify that our algorithms are viable and efficient.

Physics-informed sinogram completion for metal artifact reduction in CT imaging

Objective. Metal artifacts in the computed tomography (CT) imaging are unavoidably adverse to the clinical diagnosis and treatment outcomes. Most metal artifact reduction (MAR) methods easily result in the over-smoothing problem and loss of structure details near the metal implants, especially for these metal implants with irregular elongated shapes. To address this problem, we present the physics-informed sinogram completion (PISC) method for MAR in CT imaging, to reduce metal artifacts and recover more structural textures. Approach. Specifically, the original uncorrected sinogram is firstly completed by the normalized linear interpolation algorithm to reduce metal artifacts. Simultaneously, the uncorrected sinogram is also corrected based on the beam-hardening correction physical model, to recover the latent structure information in metal trajectory region by leveraging the attenuation characteristics of different materials. Both corrected sinograms are fused with the pixel-wise adaptive weights, which are manually designed according to the shape and material information of metal implants. To furtherly reduce artifacts and improve the CT image quality, a post-processing frequency split algorithm is adopted to yield the final corrected CT image after reconstructing the fused sinogram. Main results. We qualitatively and quantitatively evaluated the presented PISC method on two simulated datasets and three real datasets. All results demonstrate that the presented PISC method can effectively correct the metal implants with various shapes and materials, in terms of artifact suppression and structure preservation. Significance. We proposed a sinogram-domain MAR method to compensate for the over-smoothing problem existing in most MAR methods by taking advantage of the physical prior knowledge, which has the potential to improve the performance of the deep learning based MAR approaches.

On Douglas-Rachford splitting that generally fails to be a proximal mapping: a degenerate proximal point analysis

Based on a degenerate proximal point analysis, we show that the Douglas-Rachford splitting can be reduced to a well-defined resolvent, but generally fails to be a proximal mapping. This extends the recent result of [Bauschke, Schaad and Wang. Math. Program. 2018;168:55–61] to more general setting. The related concepts and consequences are also discussed. In particular, the results regarding the maximal and cyclic monotonicity are instrumental for analysing many operator splitting algorithms.

Gearbox fault diagnosis based on generalized multivariate logarithmic regularization

Feature extraction based on sparse representation is widely applied in the field of mechanical fault diagnosis. L1 norm regularization is a classical sparse regularization method, but this method has sparse underestimation for large-value features. A signal sparse representation method based on generalized multivariate logarithmic regularization is proposed in this paper. First, the sparse penalty term in the proposed method is designed according to the minimum convolution and logarithmic function, namely the generalized logarithmic non-convex penalty function. Then, the convexity condition of the objective function is studied to verify the feasibility of the method. The applicability of the method is also improved by analyzing the parameter constraint relation in the objective function. Finally, the sparse optimal solution is obtained by the forward-back splitting algorithm. Experiments show that compared with other non-convex sparse models, the proposed method can solve the problem of sparse underestimation more effectively and improve the reliability of gearbox fault diagnosis.

Importance of shear dilation to two-phase flow in naturally fractured geological media: A numerical study using zero-thickness interface elements

In this paper, a hydro-mechanical coupling model is proposed to investigate the geomechanical effects on two-phase flow in fractured rocks. The discrete fracture model (DFM) and modified Barton-Bandis's model are used to model two-phase fluid flow and mechanical behavior of fractures, respectively. The coupled problem is solved by a mixed finite volume/finite element method, together with the fixed-stress split algorithm. Our model is able to capture the complex coupled behavior induced by the interaction between pore pressure and in-situ stress loadings. Then the proposed model is applied to investigate the two-phase flow behaviors in the fracture networks with different fracture configurations under various in-situ stress conditions. The results illustrate significant influences of shear dilation on the hydraulic properties of fractured rock. With the increase of differential stress, fracture aperture gradually increases due to shear dilation, which accordingly enhances the equivalent permeability of the fractured medium. The degree of shear dilation is highly dependent on the fracture orientation and length distribution. Consequently, channelized flow is formed through the dilated fractures, which results in early water breakthrough and reduces the water sweep efficiency. This study highlights the necessity of considering hydro-mechanical coupling effect for accurate prediction of saturation distributions in fractured rocks.

A comparative study of split advection algorithms on the moment-of-fluid (MOF) method for incompressible flow

The moment-of-fluid method (MOF) is known as an extension of the volume-of-fluid method with piecewise linear interface construction (VOF-PLIC). In this study, several directional splitting advection algorithms are extended from the VOF-PLIC method to the MOF method. These methods, along with some existing directional splitting algorithms, are presented in detail, especially on the geometrical and non-geometrical nature of the advection algorithm. In addition, a simple and efficient analytic form to calculate the volume and centroid of the intersection polyhedron in 3D is proposed. Several numerical tests, including both 2D and 3D tests, are carried out to investigate mass conservation and geometrical error. Numerical results suggest that the mixed EI and LE scheme (EILE2D) (Aulisa et al., 2003) has the overall best performance in 2D, and Weymouth and Yue’s scheme (WY) has the overall best performance in 3D.

Modified Picard-like Method for Solving Absolute Value Equations

We present a modified Picard-like method to solve absolute value equations by equivalently expressing the implicit fixed-point equation form of the absolute value equations as a two-by-two block nonlinear equation. This unifies some existing matrix splitting algorithms and improves the efficiency of the algorithm by introducing the parameter ω. For the choice of ω in the new method, we give a way to determine the quasi-optimal values. Numerical examples are given to show the feasibility of the proposed method. It is also shown that the new method is better than those proposed by Ke and Ma in 2017 and Dehghan and Shirilord in 2020.

Exact Model Order Reduction for the Full-System Finite Element Solution of Thermal Elastohydrodynamic Lubrication Problems

The derivation of fast, reliable, and accurate modeling procedures for the solution of thermal elastohydrodynamic lubrication problems is a topic of significant interest in the Tribology community. In this paper, a novel model order reduction technique is introduced for the analysis of thermal elastohydrodynamic lubrication problems. The method uses static condensation to reduce the size of the linear elasticity part within the overall matrix system, followed by a splitting algorithm to avoid the burden of solving a semi-dense matrix system. The results reveal the exactness of the proposed methodology, which does not introduce any additional model-reduction approximations to the overall solution. They also reveal the reduction in computational times, which is in the order of 10–20% for line contacts, while it is in excess of 50% for circular contacts. The robustness of the proposed method is displayed by using it to model some relatively highly loaded contacts whose numerical solution is known to be rather challenging.

A Primal-Dual Forward-Backward Splitting Algorithm for Distributed Convex Optimization

Motivated by modern large-scale information processing problems in engineering, this paper concentrates on studying distributed constrained convex optimization problems over a connected undirected network. The problem involves a sum of a differentiable convex function with Lipschitz continuous gradient and two non-smooth convex functions with a linear operator. To solve such a problem, we propose a novel distributed primal-dual forward-backward splitting algorithm, called D-PDFBS. Each agent locally computes the Lipschitz continuous gradient and two proximal operators, and exchanges information with its neighbors. D-PDFBS adopts non-identical stepsizes, and we reveal the relationship between selection of stepsizes and parameters of objective functions. The simulation results verify the feasibility of D-PDFBS and the correctness of theoretical findings.