Structure-preserving Algorithm Research Articles

Structure-Preserving Algorithm and Its Error Estimate for the Relativistic Charged-Particle Dynamics Under the Strong Magnetic Field

Structure-Preserving Algorithm and Its Error Estimate for the Relativistic Charged-Particle Dynamics Under the Strong Magnetic Field

A structure-preserving algorithm for Vlasov–Darwin system conserving charge, canonical momentum, and Hamiltonian

In this paper, a conservative numerical method for the Vlasov–Darwin system is proposed. The Darwin model was assumed to be valid only in the Coulomb gauge, but recently this model has also been extended to the Lorenz gauge [1]. The Darwin model, based on the Lorenz gauge, exhibits a good symmetry between scalar and vector potentials, making the proofs of physical constraints relatively easy. In addition, the total energy was believed to be one of the conservative quantities of the Vlasov–Darwin system. However, the improved theory suggests that the Hamiltonian is the conservative quantity rather than the total energy. The structure-preserving scheme proposed in this paper exactly maintains the Lorenz gauge and the conservation laws of charge, canonical momentum, and Hamiltonian in discrete form.

A new structure-preserving algorithm based on the semi-tensor product of matrices for split quaternion matrix LDU decomposition and its applications

In this paper, we put forward a new structure-preserving algorithm based on the semi-tensor product of matrices for the split quaternion matrix LDU decomposition, which makes full use of high-level operations, and relation of operations between split quaternion matrices and their L-representation matrices. The algorithm is more stable and efficient than the complex structure-preserving algorithm proposed in Wang et al. (2021). Numerical experiments are provided to verify the efficiency of the new structure-preserving algorithm.

Thermodynamically consistent hybrid computational models for fluid-particle interactions

We introduce a novel computational framework designed to explore the dynamic interactions between fluid and solid particles or structures immersed in a viscous fluid medium adhering to the generalized Onsager principle. This innovative framework harnesses the power of the phase-field-embedding method, in which each solid component, whether rigid or elastic, is characterized by a volume-preserving phase field. The unified velocity within the fluid-solid ensemble governs the movement of both solid particles and the surrounding fluid, specifically for passive particles. Active particles, however, are not only influenced by this unified velocity but are also driven by their self-propelling velocities. To capture exclusive volume interactions among particles and between particles and boundaries, we employ repulsive potential forces at a coarser scale. These forces effectively model repulsion and collision effects. Rigid particles maintain structural integrity by enforcing a zero velocity gradient tensor within their spatial domains, necessitating the introduction of a constraining stress tensor. In contrast, elastic particles are governed by a quasi-linear constitutive equation describing the elastic stress within their domains, allowing for accurate modeling of their deformations. The motion of solid particles is tracked by monitoring the dynamics of their centers of mass. This approach facilitates the development of a hybrid, thermodynamically consistent hydrodynamic model applicable to both rigid and elastic particles. To numerically solve this thermodynamically consistent model for elastic particles, we present a structure-preserving numerical algorithm. Notably, in the limit of an infinite elastic modulus, this algorithm converges to the one employed for modeling rigid particles. Finally, we substantiate the effectiveness, accuracy, and stability of our proposed scheme through a series of numerical experiments. These experiments not only validate the computational framework but also showcase its capabilities, reinforcing the reliability of our approach.

Construction and Analysis of Structure-Preserving Numerical Algorithm for Two-Dimensional Damped Nonlinear Space Fractional Schrödinger equation

Construction and Analysis of Structure-Preserving Numerical Algorithm for Two-Dimensional Damped Nonlinear Space Fractional Schrödinger equation

A novel structure-preserving algorithm for the singular value decomposition of biquaternion matrices

In this paper, we study the singular value decomposition of biquaternion matrices. We prove that the singular value decomposition of biquaternion matrices can be equivalently converted to the singular value decomposition of quaternion adjoint matrices. By means of the Cheng product, we propose a novel representation called Q -representation and design a Q structure-preserving algorithm to deal with the singular value decomposition of quaternion matrices, which makes use of high-level operations. The algorithm is more efficient than that in quaternion toolbox for matlab using quaternion arithmetics and real structure-preserving algorithm. The Q structure-preserving algorithm for singular value decomposition of biquaternion matrices is presented through the application of the Q structure-preserving algorithm for singular value decomposition of quaternion matrices. Numerical experiments are provided to demonstrate the efficiency of the Q structure-preserving algorithm. We apply the Q structure-preserving algorithm of quaternion singular value decomposition to improve the image denoising process.

Algebraic method for LU decomposition of dual quaternion matrix and its corresponding structure-preserving algorithm

Algebraic method for LU decomposition of dual quaternion matrix and its corresponding structure-preserving algorithm

A coarse-to-fine unsupervised domain adaptation method based on metric learning

Domain adaptation solves the challenge of inadequate labeled samples in the target domain by leveraging the knowledge learned from the labeled source domain. Most existing approaches aim to reduce the domain shift by performing some coarse alignments such as domain-wise alignment and class-wise alignment. To circumvent the limitation, we propose a coarse-to-fine unsupervised domain adaptation method based on metric learning, which can fully utilize more geometric structure and sample-wise information to obtain a finer alignment. The main advantages of our approach lie in four aspects: (1) it employs a structure-preserving algorithm to automatically select the optimal subspace dimension on the Grassmannian manifold; (2) based on coarse distribution alignment using maximum mean discrepancy, it utilizes the smooth triplet loss to leverage the supervision information of samples to improve the discrimination of data; (3) it introduces structure regularization to preserve the geometry of samples; (4) it designs a graph-based sample reweighting method to adjust the weight of each source domain sample in the cross-domain task. Extensive experiments on several public datasets demonstrate that our method achieves remarkable superiority over several competitive methods (more than 1.5% improvement of the average classification accuracy over the best baseline).

A structure-preserving doubling algorithm for the square root of regular M-matrix

<p>The matrix square root is widely encountered in many fields of mathematics. In this paper, based on the properties of M-matrix and quadratic matrix equations, we study the square root of M-matrix, and prove that for a regular M-matrix there always exists a regular M-matrix as its square root. In addition, a structure-preserving doubling algorithm is proposed to compute the square root. Theoretical analysis and numerical experiments are given to show that our method is feasible and is effective under certain conditions.</p>

A structure-preserving doubling algorithm for the quadratic matrix equation with $M$-matrix

A structure-preserving doubling algorithm for the quadratic matrix equation with $M$-matrix

TWO ALGEBRAIC ALGORITHMS FOR THE LU DECOMPOSITION OF COMMUTATIVE QUATERNION MATRICES AND THEIR APPLICATIONS

The paper proposes two algebraic algorithms for the LU decomposition of com- mutative quaternion matrices. The first of them is an algebraic algorithm based on a complex representation matrix, and the second is a complex structure-preserving algorithm based on Gaussian elimination of commutative quaternion matrices. With their help, it was established that the LU decomposition of commutative quaternion matrices is not unique. The paper also presents the results of numerical implementations of a mathematical model for color image restoration using the proposed algorithms, which demonstrated the higher efficiency of the first algorithm.

A REAL STRUCTURE-PRESERVING ALGORITHM FOR THE LOW- RANK DECOMPOSITION OF PURE IMAGINARY QUATERNION MATRICES AND ITS APPLICATIONS IN SIGNAL PROCESSING

With the increasing use of quaternions in fields such as quantum mechanics, rigid body rotation, signal and color image processing, and aerospace engineering, three- dimensional signal models represented by pure imaginary quaternion matrices have been cre- ated. This model treats the 3D information as a whole, and the processing preserves the intrinsic connections between the different channels. However, the computational process of quaternion algebra often generates real parts, which are inevitable. How to ensure the pure imaginary properties of 3D signal models is also a research priority. In this paper, we inves- tigate the low-rank decomposition of pure imaginary quaternion matrices using least squares iteration and give a real structure-preserving algorithm for the low-rank decomposition based on the isomorphic real representation. In addition, this matrix decomposition algorithm is applied to color image compression and 3D wave signal denoising problems. Numerical ex- periments show the effectiveness of the algorithm in this paper.

An efficient real structure-preserving algorithm for the quaternion weighted least squares problem with equality constraint

An efficient real structure-preserving algorithm for the quaternion weighted least squares problem with equality constraint

A complex structure-preserving algorithm for computing the singular value decomposition of a quaternion matrix and its applications

A complex structure-preserving algorithm for computing the singular value decomposition of a quaternion matrix and its applications

Factorized Doubling Algorithm for Large-Scale High-Ranked Riccati Equations in Fractional System

In real-life control problems, such as power systems, there are large-scale high-ranked discrete-time algebraic Riccati equations (DAREs) from fractional systems that require stabilizing solutions. However, these solutions are no longer numerically low-rank, which creates difficulties in computation and storage. Fortunately, the potential structures of the state matrix in these systems (e.g., being banded-plus-low-rank) could be beneficial for large-scale computation. In this paper, a factorized structure-preserving doubling algorithm (FSDA) is developed under the assumptions that the non-linear and constant terms are positive semidefinite and banded-plus-low-rank. The detailed iteration scheme and a deflation process for FSDA are analyzed. Additionally, a technique of partial truncation and compression is introduced to reduce the dimensions of the low-rank factors. The computation of residual and the termination condition of the structured version are also redesigned. Illustrative numerical examples show that the proposed FSDA outperforms SDA with hierarchical matrices toolbox (SDA_HODLR) on CPU time for large-scale problems.

Direct estimation of regional lung volume change from paired and single CT images using residual regression neural network.

Chest computed tomography (CT) enables characterization of pulmonary diseases by producing high-resolution and high-contrast images of the intricate lung structures. Deformable image registration is used to align chest CT scans at different lung volumes, yielding estimates of local tissue expansion and contraction. We investigated the utility of deep generative models for directly predicting local tissue volume change from lung CT images, bypassing computationally expensive iterative image registration and providing a method that can be utilized in scenarios where either one or two CT scans areavailable. A residual regression convolutional neural network, called Reg3DNet+, is proposed for directly regressing high-resolution images of local tissue volume change (i.e., Jacobian) from CT images. Image registration was performed between lung volumes at total lung capacity (TLC) and functional residual capacity (FRC) using a tissue mass- and structure-preserving registration algorithm. The Jacobian image was calculated from the registration-derived displacement field and used as the ground truth for local tissue volume change. Four separate Reg3DNet+ models were trained to predict Jacobian images using a multifactorial study design to compare the effects of network input (i.e., single image vs. paired images) and output space (i.e., FRC vs. TLC). The models were trained and evaluated on image datasets from the COPDGene study. Models were evaluated against the registration-derived Jacobian images using local, regional, and global evaluationmetrics. Statistical analysis revealed that both factors - network input and output space - were significant determinants for change in evaluation metrics. Paired-input models performed better than single-input models, and model performance was better in the output space of FRC rather than TLC. Mean structural similarity index for paired-input models was 0.959 and 0.956 for FRC and TLC output spaces, respectively, and for single-input models was 0.951 and 0.937. Global evaluation metrics demonstrated correlation between registration-derived Jacobian mean and predicted Jacobian mean: coefficient of determination (r2 ) for paired-input models was 0.974 and 0.938 for FRC and TLC output spaces, respectively, and for single-input models was 0.598 and 0.346. After correcting for effort, registration-derived lobar volume change was strongly correlated with the predicted lobar volume change: for paired-input models r2 was 0.899 for both FRC and TLC output spaces, and for single-input models r2 was 0.803 and 0.862, respectively. Convolutional neural networks can be used to directly predict local tissue mechanics, eliminating the need for computationally expensive image registration. Networks that use paired CT images acquired at TLC and FRC allow for more accurate prediction of local tissue expansion compared to networks that use a single image. Networks that only require a single input image still show promising results, particularly after correcting for effort, and allow for local tissue expansion estimation in cases where multiple CT scans are not available. For single-input networks, the FRC image is more predictive of local tissue volume change compared to the TLCimage.

A high-order structure-preserving difference scheme for generalized fractional Schrödinger equation with wave operator

This paper focuses on the construction and analysis of the structure-preserving algorithm for generalized fractional Schrödinger equation with wave operator. A fourth-order energy-conserving difference scheme is developed for the resulting equivalent system based on scalar auxiliary variable approach. The discrete energy conservation law, boundedness and convergence of difference solutions are proved in detail. Numerical experiments are performed to verify our theoretical analysis results.

Color image watermarking based on a fast structure-preserving algorithm of quaternion singular value decomposition

This paper presents a new color image watermarking algorithm based on a fast structure-preserving algorithm of quaternion singular value decomposition (QSVD). We first propose F structure-preserving algorithm of QSVD by using quaternion Householder transformation, then apply it to color image watermarking, finally compare it with other algorithms and get better invisibility, security and almost robustness.

Thermodynamically Consistent Models for Coupled Bulk and Surface Dynamics.

We present a constructive paradigm to derive thermodynamically consistent models coupling the bulk and surface dynamics hierarchically following the generalized Onsager principle. In the model, the bulk and surface thermodynamical variables are allowed to be different and the free energy of the model comprises the bulk, surface, and coupling energy, which can be weakly or strongly non-local. We illustrate the paradigm using a phase field model for binary materials and show that the model includes the existing thermodynamically consistent ones for the binary material system in the literature as special cases. In addition, we present a set of such phase field models for a few selected mobility operators and free energies to show how boundary dynamics impart changes to bulk dynamics and vice verse. As an example, we show numerically how reactive transport on the boundary impacts the dynamics in the bulk using a reactive transport model for binary reactive fluids by adopting a structure-preserving algorithm to solve the model equations in a rectangular domain.

High-performance computing of structure-preserving algorithm for the coupled BBM system formulated by weighted compact difference operators

A compact difference operator is a valuable technique which has been used to develop numerical schemes capable of solving various types of differential equations. Advantages of compact procedures include obtaining high accuracy with a relatively small number of grid points. In this paper, we introduce weighted parameters in standard compact difference operators. Therefore, applications of both high-order compact finite difference operators for spatial derivatives and the Crank–Nicolson/Adams–Bashforth method for temporal derivatives have been used to solve coupled BBM equations. As the consequence system of equations is uncoupled during computing, it requires less CPU processing time. In addition, when producing numerical solutions, the system equations obtained at each time step are constant-coefficient matrices, gaining a remarkable number of computational advantages. In addition, deriving an a priori estimate of numerical solutions obtains convergence and stability analyses. Simultaneously, the algorithm is globally structure-preserving as to the negligence of nonphysical behavior. Numerical results demonstrate that the proposed weighted compact finite difference scheme can improve the computational efficiency of standard second-order methods. Additionally, appropriately weighted parameters can refine numerical accuracy. Lastly, an investigation of the relevant properties of numerically computed solutions is conducted, including a finite time blow-up of a head-on collision and a two-way wave propagation.