Direct Method Research Articles

A Direct Discontinuous Galerkin Method for Time Fractional Diffusion Equations with Fractional Dynamic Boundary Conditions

A Direct Discontinuous Galerkin Method for Time Fractional Diffusion Equations with Fractional Dynamic Boundary Conditions

The thermodynamic properties of diatomic gases. A rigorous analytical approach

The paper reports a closed-form analytical representation of the thermodynamic functions of diatomic molecules in the ground electronic state. For this purpose, analytical formulas for vibrational and rotational partition functions are derived. A rigorous method is proposed for factorizing the vibrational and rotational contributions to the ro-vibrational partition function using an average vibrational quantum number. The behavior of thermodynamic functions as a function of temperature is discussed for carbon monoxide as a test system. The reliability of the analytical method is verified by comparison with experimental data and with the direct summation method, which uses theoretical values of energy levels.

Multichannel deconvolution with a high-frequency structural regularization

The resolution of seismic data determines the ability to characterize stratigraphic features from observed seismic records. Sparse spike inversion, as an important processing method, can effectively improve the band-limited property of the seismic data. However, this approach ignores the spatial information along seismic traces, which causes the unreliability of the reconstructed high-resolution data. In this paper, we develop a high-frequency structure-constrained multichannel deconvolution (HFSC-MD) to alleviate this issue. This method allows the cost function to incorporate high-frequency spatial information in the form of a prediction-error filter (PEF) to regularize the components of the result beyond the original frequency. The PEF, also called a high-frequency reflection structural characterization operator, is estimated from the mapping relationship of low- and high-frequency components. We adopt the alternating direction method of multipliers to solve the cost function in HFSC-MD. Synthetic and field data demonstrate that our method recovers more reliable high-resolution data and enriches reflective structures.

A novel high-precision fiber tractography for nuclear localization in transcranial magnetic resonance-guided focused ultrasound surgery: a pilot study.

In transcranial MR-guided focused ultrasound (TcMRgFUS), fiber tractography using diffusion tensor imaging (DTI) has been proposed as a direct method to identify the ventral intermediate nucleus (Vim), the ventral caudal nucleus (Vc), and the pyramidal tract (PT). However, the limitations of the DTI algorithm affect the accuracy of visualizing anatomical structures due to its low-quality fiber tractography, whereas the application of the generalized q-sampling imaging (GQI) algorithm enables the visualization of high-quality fiber tracts, offering detailed insights into the spatial distribution of motor cortex fibers. This retrospective study aimed to investigate the usefulness of high-precision fiber tractography using the GQI algorithm as a planning image in TcMRgFUS to achieve favorable clinical outcomes. This study included 20 patients who underwent TcMRgFUS. The Clinical Rating Scale for Tremor (CRST) scores and MR images were evaluated pretreatment and at 24 hours and 3-6 months after treatment. Cases were classified based on the presence and adversity of adverse events (AEs): no AEs, mild AEs without additional treatment, and severe AEs requiring prolonged hospitalization. Fiber tractography of the Vim, Vc, and PT was visualized using the DTI and GQI algorithm. The overlapping volume between Vim fibers and the lesion was measured, and correlation analysis was performed. The relationship between AEs and the overlapping volume of the Vc and PT fibers within the lesions was examined. The cutoff value to achieve a favorable clinical outcome and avoid AEs was determined using receiver operating characteristic curve analysis. All patients showed improvement in tremors 24 hours after treatment, with 3 patients experiencing mild AEs and 1 patient experiencing severe AEs. At the 3- to 6-month follow-up, 5 patients experienced recurrence, and 2 patients had persistent mild AEs. Although fiber visualization in the motor cortex using the DTI algorithm was insufficient, the GQI algorithm enabled the visualization of significantly higher-quality fibers. A strong correlation was observed between the overlapping volume that intersects the lesion and Vim fibers and the degree of tremor improvement (r = 0.72). Higher overlapping volumes of Vc and PT within the lesion were associated with an increased likelihood of AEs (p < 0.05); the cutoff volume of Vim fibers within the lesion for a favorable clinical outcome was 401 mm3, while the volume of Vc and PT within the lesion to avoid AEs was 99 mm3. This pilot study suggests that incorporating the high-precision GQI algorithm for fiber tractography as a planning imaging technique for TcMRgFUS has the potential to enhance targeting precision and achieve favorable clinical outcomes.

Multiple Complementary Priors for Multispectral Image Compressive Sensing Reconstruction.

Compressive sensing (CS) techniques using a few compressed measurements have drawn considerable interest in reconstructing multispectral imagery (MSI). Nonlocal-based tensor methods have been widely used for MSI-CS reconstruction, which employ the nonlocal self-similarity (NSS) property of MSI to obtain satisfactory results. However, such methods only consider the internal priors of MSI while ignoring important external image information, for example deep-driven priors learned from a corpus of natural image datasets. Meanwhile, they usually suffer from annoying ringing artifacts due to the aggregation of overlapping patches. In this article, we propose a novel approach for highly effective MSI-CS reconstruction using multiple complementary priors (MCPs). The proposed MCP jointly exploits nonlocal low-rank and deep image priors under a hybrid plug-and-play framework, which contains multiple pairs of complementary priors, namely, internal and external, shallow and deep, and NSS and local spatial priors. To make the optimization tractable, a well-known alternating direction method of multiplier (ADMM) algorithm based on the alternating minimization framework is developed to solve the proposed MCP-based MSI-CS reconstruction problem. Extensive experimental results demonstrate that the proposed MCP algorithm outperforms many state-of-the-art CS techniques in MSI reconstruction. The source code of the proposed MCP-based MSI-CS reconstruction algorithm is available at: https://github.com/zhazhiyuan/MCP_MSI_CS_Demo.git.

HDG Method for Nonlinear Parabolic Integro-Differential Equations

Abstract The hybridizable discontinuous Galerkin (HDG) method has been applied to a nonlinear parabolic integro-differential equation. The nonlinear functions are considered to be Lipschitz continuous to analyze uniform in time a priori bounds. An extended type Ritz–Volterra projection is introduced and used along with the HDG projection as an intermediate projection to achieve optimal order convergence of O ⁢ ( h k + 1 ) O(h^{k+1}) when polynomials of degree k ≥ 0 k\geq 0 are used to approximate both the solution and the flux variables. By relaxing the assumptions in the nonlinear variable, super-convergence is achieved by element-by-element post-processing. Using the backward Euler method in temporal direction and quadrature rule to discretize the integral term, a fully discrete scheme is derived along with its error estimates. Finally, with the help of numerical examples in two-dimensional domains, computational results are obtained, which verify our results.

A State-of-the-Art Review on Direct Welding of Polymer to Metal for Structural Applications: Part 1 — Promising Processes

Structural lightweighting through the effective use of multiple materials has received increasing attention for fulfilling today’s demands for environmental sustainability in transportation systems. Direct dissimilar material joining methods (versus, e.g., traditional adhesive bonding or mechanical fastening) have become increasingly desirable since they offer process simplicity, production efficiency, and hermetic sealing, among others. In this two-part article, we provide a critical assessment of the state-of-the-art research and promising direct dissimilar material joining techniques reported over the last decades, with a particular emphasis on their potential for structural applications in Part I. As such, recent advances in advanced joint design and modeling methods for enabling optimum joint design for jointability and joint performance are presented along with some detailed examples for demonstrating their potential impacts on industrial applications in Part II. Finally, recommendations on future research and development directions are outlined for supporting the industry’s drive towards multi-material lightweighting.

Association of two square difference identity to regular polygons and circles

GeoGebra is a dynamic software that is frequently used and of increasing importance in mathematics teaching processes in our digital age. Accordingly, in this study a new perspective has been brought to the proofs of the “two square difference identity” expressed for the square, which is a flat polygon, made with different approaches. With side lengths a, b, and a&amp;gt;b, it has been shown that the identity given by the equation (difference of area) a&lt;sup&gt;2&lt;/sup&gt;-b&lt;sup&gt;2&lt;/sup&gt;=(a-b)(a+b) is true for other regular polygons as well. In the study, direct proof method was used within the framework of the principle of conservation of measure, which is one of the basic principles of geometry teaching. GeoGebra program, which is a dynamic geometry software, was preferred for drawing geometric shapes used in proofs. In order to generalize the number n, a different fragmentation technique was preferred to the proofs made using different drawings for equilateral triangle and square, which are the simplest regular polygons. It has also been shown that this identity is true for circles viewed as polygons with an infinite number of sides.

Fabrication of micro-ultrasonic phased array electrode via electric field-driven printing method

The electrode is an important part of a micro-ultrasonic phased array, commonly fabricated by magnetron sputtering technology. However, with the expansion of the application field, the structures of the phased arrays are becoming more complex, making the traditional magnetron sputtering method no longer applicable, and the realization of micro-scale phased array electrodes by direct printing technology still faces great challenges. Herein, an electric field-driven direct printing method was proposed to fabricate micro-ultrasonic phased array electrodes. The influence of sintering parameters on the resistivity and the effect of printing parameters on the geometric size and surface morphology of the electrode were comprehensively explored, and the mathematical model between the electrode line width and the printing parameters was established. The final printed electrodes exhibit superior properties with a resistivity of 2.4 × 10−7 Ω⋅m , a surface roughness range of 460–550 nm, and a line width range of 125–480 µm. Additionally, a micro-scale electrode was printed on the piezoelectric ceramic with PZT-5 as the material, and after polarization treatment, the piezoelectric coefficient can reach 405 pC N−1, which proves that this method can be applied in the field of fabricating phased array electrodes.

Direct Control Method for Linear-Rotary Switched Reluctance Motor With Two Radial Windings

For Linear-Rotary Switched Reluctance Motors (LRSRMs) with two radial windings, the coupling between torque and axial force increases the complexity of control. Due to the calculation of current in existing control method for LRSRMs, the complicated derivation of current expression was necessary, and some constraints were also introduced which meant the difficulties on designing the current control algorithm were greatly increased. In order to solve these problems, a direct torque and direct axial force (DTC&DAFC) control method is proposed in this paper. By the way of selecting appropriate space voltage vectors, the proposed method can directly regulate torque and axial force at the same time to achieve rotation and linear motion and significantly reduce the ripple of the torque. Moreover, the mechanism of two-DOF motion and the principle of the proposed method are demonstrated in detail based on a 6/4 LRSRM with two radial windings. Based on MATLAB/Simulink and the experimental prototype, the feasibility of the proposed control method has been verified according to the simulation and experimental results.

Explicit Model Predictive Cascade-Free Direct Force and Torque Control of Quadrotors

High-performance control of quadrotors' force and torques is the basis for achieving excellent trajectory tracking. However, the conventional method indirectly controls the force and torques by controlling four permanent magnet synchronous motors' speeds, resulting in poor dynamic performance and static deviations when pneumatic parameters mismatch. Dedicated to achieving high-dynamic and high-accuracy control of force and torques, an explicit model predictive (EMP) cascade-free direct force and torque control method is proposed. Firstly, defining force current and torque currents, whereby a new structure is proposed. Then, dynamics of the quadrotor and motor are fused to design the linear robust predictive models of force and torques. Accordingly, the EMP controller is constructed to replace the traditional cascaded speed solution module and four speed controllers uniformly. The EMP controller generates the optimal current commands directly from the force and torque commands, realizing direct control without the speed loop. The new structure makes it possible to solve the pneumatic parameter mismatch problem with an auxiliary extended state observer, obtaining offset-free control. The proposed method optimizes the dynamic performance and steady-state accuracy of force and torques and improves the system's robustness. Finally, the proposed method's feasibility is verified by experiments.

Efficient Dynamic Distributed Resource Slicing in 6G Multi-Access Edge Computing Networks With Online ADMM and Message Passing Graph Neural Networks

We consider the problem of resource slicing in the 6 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> generation multi-access edge computing (6G-MEC) network. The network includes many non-stationary space-air-ground-sea nodes with dynamic, unstable connections and resources, where any node can be in one of two hidden states: i) reliable – when the node generates/propagates no data errors; ii) unreliable – when the node can generate/propagate random errors. We show that solving this problem is challenging, since it represents a non-deterministic polynomial-time (NP) hard dynamic combinatorial optimization problem depending on the unknown distribution of hidden nodes' states and time-varying parameters (connections and resources of nodes) which can only be observed locally. To tackle these challenges, we develop a new deep learning (DL) model based on the message passing graph neural network (MPNN) to estimate hidden nodes' states in dynamic network environments. We then propose a novel algorithm based on the integration of MPNN-based DL and online alternating direction method of multipliers (ADMM) – extension of the well-known classical “static” ADMM to dynamic settings, where the slicing problem is solved distributedly, in real time, based on local information. We prove that our algorithm converges to a global optimum of our problem with a superior performance even in the highly-dynamic, unreliable scenarios.

Grouped Multivariate Variational Mode Decomposition With Application to EEG Analysis.

In this paper, a novel extended form of multivariate variational mode decomposition (MVMD) method to multigroup data named as grouped MVMD (GMVMD) is proposed. GMVMD is distinct from MVMD as it extracts common frequencies with strong correlations among regional channels. Firstly, GMVMD utilizes a new clustering algorithm named as frequencies grouping algorithm to classify the nearest common frequencies among all channels to specified groups. Secondly, a generic variational optimization model which is extended from MVMD is formulated. Thirdly, alternating direction method of multipliers (ADMM) is utilized to obtain optimal solution of GMVMD model. The proposed method introduces an extra parameter to decide the number of clusterings which need to be specified by the user. The effectiveness and superiority of the algorithm are demonstrated on a series of experiments. The utility of GMVMD is verified by grouping real-world electroencephalogram (EEG) data having similar center frequencies successfully. GMVMD outperforms MVMD in mode-alignment, signal reduction error and et al. Significance: GMVMD can obtain more accurate center frequencies and less signal reduction error than MVMD.

Toward Quantized Model Parallelism for Graph-Augmented MLPs Based on Gradient-Free ADMM Framework.

While graph neural networks (GNNs) are popular in the deep learning community, they suffer from several challenges including over-smoothing, over-squashing, and gradient vanishing. Recently, a series of models have attempted to relieve these issues by first augmenting the node features and then imposing node-wise functions based on multilayer perceptron (MLP), which are widely referred to as graph-augmented MLP (GA-MLP) models. However, while GA-MLP models enjoy deeper architectures for better accuracy, their efficiency largely deteriorates. Moreover, popular acceleration techniques such as stochastic-version or data-parallelism cannot be effectively applied due to the dependency among samples (i.e., nodes) in graphs. To address these issues, in this article, instead of data parallelism, we propose a parallel graph deep learning Alternating Direction Method of Multipliers (pdADMM-G) framework to achieve model parallelism: parameters in each layer of GA-MLP models can be updated in parallel. The extended pdADMM-G-Q algorithm reduces communication costs by introducing the quantization technique. Theoretical convergence to a (quantized) stationary point of the pdADMM-G algorithm and the pdADMM-G-Q algorithm is provided with a sublinear convergence rate o(1/k) , where k is the number of iterations. Extensive experiments demonstrate the convergence of two proposed algorithms. Moreover, they lead to a more massive speedup and better performance than all state-of-the-art comparison methods on nine benchmark datasets. Last but not least, the proposed pdADMM-G-Q algorithm reduces communication overheads by up to 45% without loss of performance. Our code is available at https://github.com/xianggebenben/pdADMM-G.

Development and benchmarking of space-time kinetics solver in REDAC

A space–time kinetics solver based on finite volume approach has been developed for REactor Dynamics Analysis Code (REDAC). It solves 3D multi-group neutron diffusion equation, coupled with the heat transport equations. Code offers flexibility in terms of reactor geometry, boundary conditions and discretization. Efficient numerical methods have been employed for solving the governing equations. The performance of the code has been demonstrated by solving diverse benchmark problems. It is shown that REDAC is able to predict important static and dynamic core parameters with good accuracy. The framework of REDAC easily lends itself for sensitivity studies with respect to spatial discretization and temporal integration schemes. From these studies, it is observed that spatial resolution could have significant influence on accuracy and the semi-implicit method performs better than the implicit method. Improved quasi-static method requires less computational time compared to the direct method, however, its accuracy depends on the time step size.

The Dark Energy Survey Supernova Program: Cosmological Biases from Host Galaxy Mismatch of Type Ia Supernovae

Redshift measurements, primarily obtained from host galaxies, are essential for inferring cosmological parameters from type Ia supernovae (SNe Ia). Matching SNe to host galaxies using images is nontrivial, resulting in a subset of SNe with mismatched hosts and thus incorrect redshifts. We evaluate the host galaxy mismatch rate and resulting biases on cosmological parameters from simulations modeled after the Dark Energy Survey 5 Yr (DES-SN5YR) photometric sample. For both DES-SN5YR data and simulations, we employ the directional light radius method for host galaxy matching. In our SN Ia simulations, we find that 1.7% of SNe are matched to the wrong host galaxy, with redshift differences between the true and matched hosts of up to 0.6. Using our analysis pipeline, we determine the shift in the dark energy equation of state parameter (Δw) due to including SNe with incorrect host galaxy matches. For SN Ia–only simulations, we find Δw = 0.0013 ± 0.0026 with constraints from the cosmic microwave background. Including core-collapse SNe and peculiar SNe Ia in the simulation, we find that Δw ranges from 0.0009 to 0.0032, depending on the photometric classifier used. This bias is an order of magnitude smaller than the expected total uncertainty on w from the DES-SN5YR sample of ∼0.03. We conclude that the bias on w from host galaxy mismatch is much smaller than the uncertainties expected from the DES-SN5YR sample, but we encourage further studies to reduce this bias through better host-matching algorithms or selection cuts.

An ADMM-enabled robust optimization framework for self-healing scheduling of smart grids integrated with smart prosumers

Enhancing the reliability of energy networks and minimizing downtime is crucial, making self-healing smart grids indispensable for ensuring a continuous power supply and fortifying resilience. As smart grids increasingly incorporate decentralized prosumers, innovative coordination strategies are essential to fully exploit their potential and improve system self-healing capabilities. To address this need, this paper presents a novel bi-level strategy for managing the self-healing process within a smart grid influenced by Hydrogen Refueling Stations (HRSs), Electric Vehicle Charging Stations (EVCSs), and energy hubs. This approach taps into the combined potential of these prosumers to boost system self-healing speed and reliability. In the initial stage, the Smart Grid Operator (SGO) conducts self-healing planning during emergencies, communicating required nodal capacities to prevent forced load shedding and outlining incentives for smart prosumers. Subsequently, prosumers schedule their activities and contribute flexible capacities to the SGO. Bridging the first and second stages, an adaptive Alternating Direction Method of Multipliers (ADMM) algorithm ensures convergence between the SGO and prosumer schedules within a decentralized framework. This strategy underwent implementation on a 118-node distribution system using GAMS. Results demonstrate that the proposed concept reduces Forced Load Shedding (FLS) by 32.04% and self-healing costs by 17.48% through effective utilization of smart prosumers' flexible capacities. Furthermore, outcomes indicate that the SGO reduces FLS by 6.69% by deploying Mobile Electrical Energy Storages (MEESs) and Mobile Fuel Cell Trucks (MFCTs) to critical nodes.

Pseudo-Label Calibration Semi-supervised Multi-Modal Entity Alignment

Multi-modal entity alignment (MMEA) aims to identify equivalent entities between two multi-modal knowledge graphs for integration. Unfortunately, prior arts have attempted to improve the interaction and fusion of multi-modal information, which have overlooked the influence of modal-specific noise and the usage of labeled and unlabeled data in semi-supervised settings. In this work, we introduce a Pseudo-label Calibration Multi-modal Entity Alignment (PCMEA) in a semi-supervised way. Specifically, in order to generate holistic entity representations, we first devise various embedding modules and attention mechanisms to extract visual, structural, relational, and attribute features. Different from the prior direct fusion methods, we next propose to exploit mutual information maximization to filter the modal-specific noise and to augment modal-invariant commonality. Then, we combine pseudo-label calibration with momentum-based contrastive learning to make full use of the labeled and unlabeled data, which improves the quality of pseudo-label and pulls aligned entities closer. Finally, extensive experiments on two MMEA datasets demonstrate the effectiveness of our PCMEA, which yields state-of-the-art performance.

Error estimates of a space–time Legendre spectral method for solving the Korteweg–de Vries equation

In this paper, a space–time spectral method for solving the Korteweg–de Vries equation is considered. The discrete schemes of the method are based on the Legendre–Petrov–Galerkin method in spatial direction and the Legendre-tau method in temporal direction with nonperiodic boundary conditions. Stability analysis results and error estimates are obtained in L2-norm by introducing a cut-off function without Lipschitz condition. The method is also applicable to solve some (2m+1)th-order differential equations. Comparison of our numerical results with those of other spectral methods exhibits the accuracy of our methods for the Korteweg–de Vries equation.

A model-based MR parameter mapping network robust to substantial variations in acquisition settings

Deep learning methods show great potential for the efficient and precise estimation of quantitative parameter maps from multiple magnetic resonance (MR) images. Current deep learning-based MR parameter mapping (MPM) methods are mostly trained and tested using data with specific acquisition settings. However, scan protocols usually vary with centers, scanners, and studies in practice. Thus, deep learning methods applicable to MPM with varying acquisition settings are highly required but still rarely investigated. In this work, we develop a model-based deep network termed MMPM-Net for robust MPM with varying acquisition settings. A deep learning-based denoiser is introduced to construct the regularization term in the nonlinear inversion problem of MPM. The alternating direction method of multipliers is used to solve the optimization problem and then unrolled to construct MMPM-Net. The variation in acquisition parameters can be addressed by the data fidelity component in MMPM-Net. Extensive experiments are performed on R2 mapping and R1 mapping datasets with substantial variations in acquisition settings, and the results demonstrate that the proposed MMPM-Net method outperforms other state-of-the-art MR parameter mapping methods both qualitatively and quantitatively.