Structure-preserving Property Research Articles

Variational Approximation for a Non-Isothermal Coupled Phase-Field System: Structure-Preservation & Nonlinear Stability

Abstract A Cahn–Hilliard–Allen–Cahn phase-field model coupled with a heat transfer equation, particularly with full non-diagonal mobility matrices, is studied. After reformulating the problem with respect to the inverse of temperature, we proposed and analysed a structure-preserving approximation for the semi-discretisation in space and then a fully discrete approximation using conforming finite elements and time-stepping methods. We prove structure-preserving property and discrete stability using relative entropy methods for the semi-discrete and fully discrete case. The theoretical results are illustrated by numerical experiments.

Leveraging tensor kernels to reduce objective function mismatch in deep clustering

Objective Function Mismatch (OFM) occurs when the optimization of one objective has a negative impact on the optimization of another objective. In this work we study OFM in deep clustering, and find that the popular autoencoder-based approach to deep clustering can lead to both reduced clustering performance, and a significant amount of OFM between the reconstruction and clustering objectives. To reduce the mismatch, while maintaining the structure-preserving property of an auxiliary objective, we propose a set of new auxiliary objectives for deep clustering, referred to as the Unsupervised Companion Objectives (UCOs). The UCOs rely on a kernel function to formulate a clustering objective on intermediate representations in the network. Generally, intermediate representations can include other dimensions, for instance spatial or temporal, in addition to the feature dimension. We therefore argue that the naïve approach of vectorizing and applying a vector kernel is suboptimal for such representations, as it ignores the information contained in the other dimensions. To address this drawback, we equip the UCOs with structure-exploiting tensor kernels, designed for tensors of arbitrary rank. The UCOs can thus be adapted to a broad class of network architectures. We also propose a novel, regression-based measure of OFM, allowing us to accurately quantify the amount of OFM observed during training. Our experiments show that the OFM between the UCOs and the main clustering objective is lower, compared to a similar autoencoder-based model. Further, we illustrate that the UCOs improve the clustering performance of the model, in contrast to the autoencoder-based approach. The code for our experiments is available at https://github.com/danieltrosten/tk-uco.

Large time-stepping, delay-free, and invariant-set-preserving integrators for the viscous Cahn–Hilliard–Oono equation

Stabilization techniques are frequently used to construct efficient and stable schemes for stiff and nonlinear systems. However, the introduced errors may lead to severe delay effects when facing large time-steps or stabilization parameters. Considering a fourth-order-in-space viscous Cahn–Hilliard–Oono equation that consists of both short- and long-range interactions, and equipped with either a polynomial or logarithmic potential, we analyze its invariant sets by proposing a novel reformulation. Additionally, we devise a family of temporal integrators up to eighth-order, along with a suitable stabilization parameter, to ensure the preservation of these invariant sets for any time-step. To eliminate the lagging phenomenon, we propose rewriting the method into a parametric two-step Runge–Kutta (TSRK) formulation, quantifying the rescaled time-step, and finally performing a relaxation technique. We demonstrate that the s-stage, pth-order parametric relaxation TSRK integrators retain the order of the underlying TSRK Butcher tableau, and promote the strong-stability-preserving coefficient to at most s when p>2. To further reduce the stabilization parameter, we propose a parametric relaxation integrating factor TSRK framework that treats the stiff linear operator as an integrating factor. The proposed framework has mild restrictions on the underlying TSRK tableau, i.e., the coefficients are non-negative and the parametric abscissas are non-decreasing. Numerical experiments demonstrate the accuracy and structure-preserving property of the proposed schemes, and are used to investigate the effect of parameters on microphase separations in diblock copolymer melts.

A new open-source library based on novel high-resolution structure-preserving convection schemes

This work develops a new finite volume convection scheme and open-source library for convection-dominated problems on unstructured grids. The formulated reconstruction operators can realize high-resolution and structure-preserving properties by adjusting numerical dissipation and anti-dissipation errors in a unified normalized-variable diagram. Then, this work develops a new open-source library based on the devised schemes within one of the most widely used general-purpose open-source software whose solution quality is intrinsically limited by the low accuracy of the conventional total variation diminishing schemes. The performance of the new library is first verified by some fundamental problems including pure advection and inviscid Euler equations. The numerical results of the accuracy tests show that the developed interpolation library can significantly reduce numerical errors with a minor increased computational cost compared with the conventional second-order schemes. The complex profile advection test demonstrates that the new library can offer an improved structure-preserving property that gives essentially oscillation-free solutions and preserves the structures of the passive transported scalar. Furthermore, the robustness of the new library is validated by compressible high-speed flow simulations with irregular geometries. Thus, this work presents a new library of high-resolution structure-preserving schemes for the accurate and efficient discretization of the convection term which plays an essential role in a wide range of computational fluid dynamics. The source code of the new scheme and library are released via DOI:10.5281/zenodo.8399000.

High-order unconditionally maximum-principle-preserving parametric integrating factor Runge-Kutta schemes for the nonlocal Allen-Cahn equation

We utilize the second-order quadrature-based finite difference method and the high-order parametric integrating factor Runge-Kutta (pIFRK) integrators to construct efficient and accurate schemes for solving the nonlocal Allen-Cahn equation. These schemes preserve the maximum principle for any time-step, and exhibit up to fourth-order accuracy in the temporal direction. We establish a rigorous error estimate and an asymptotic compatibility analysis for the pIFRK schemes. Numerical experiments demonstrate the accuracy and structure-preserving property of the proposed schemes, verify their asymptotic compatibility, and investigate the discontinuity of the nonlocal Allen-Cahn equation under certain conditions.

An accurate and practical numerical solver for simulations of shock, vortices and turbulence interaction problems

Numerical simulations of high-speed compressible flows remain challenging in engineering as the appearance of shock waves poses difficulties for high-resolution schemes as well as explicit large eddy simulation methods. Therefore, in this work, we propose a practical numerical solver for simulations of shock-vortex and shock-turbulence problems with the implicit large eddy simulation approach and newly developed low-dissipative schemes. In order to handle the multi-scale raised in the shock-vortex and shock-turbulence problems, it is required that the flow solver can simultaneously solve the large-scale flow structures such as shock waves with numerical oscillation-free, the resolvable flow structures above the grid scale with low-dissipation, and the under-resolve isotropic sub-grid scale with numerical stability. To this end, a low-dissipative, structure-preserving scheme with rigorously adjusted numerical dissipation is employed in the proposed solver. The structure-preserving property of this scheme can ensure that the large-scale structure is solved without numerical oscillations and the low-dissipative property of this scheme can produce high-resolution results for the resolvable flow scales. Moreover, the sub-grid scale is solved and stabilized by the inherent numerical dissipation in the shock-capturing scheme. The proposed numerical solver is then applied to simulate a wide range of shock-vortex and shock-turbulence interaction problems including supersonic planar jets, transonic flows past a deep cavity and impingement of a supersonic jet on a cone mounted on a flat plate. A comparison is also made with the solver using conventional total variation diminishing schemes. The numerical results of the supersonic planar jet have demonstrated that the proposed solver can resolve the small-scale structure such as Kelvin–Helmholtz instability involved in shock and shear layer with high-resolution. Through the results of transonic flow past a deep cavity and comparisons with the experimental data, it is verified that the proposed solver can reproduce the turbulence statistical data. Furthermore, the proposed solver resolves the complex turbulent fluid mechanical phenomena in the impingement of a supersonic jet on a cone, which demonstrates the proposed solver can robustly handle irregular geometry. The proposed solver also enjoys simplicity without using the explicit sub-grid scale model and involving the complexity of very high-order schemes. Thus, this work provides an accurate and practical numerical solver for shock-vortex and shock-turbulence problems in high-speed flows.

Large-eddy simulation of bluff-body stabilized premixed flames with low-dissipative, structure-preserving convection schemes

Large eddy simulation (LES) has the potential to predict turbulent combustion phenomena in modern practical combustors. As errors from sub-grid models may be comparable to the numerical errors in the LES approach, mitigating the impact of the numerical errors is as important as constructing accurate sub-grid models. Therefore, a low-dissipative, structure-preserving ROUND (Reconstruction Operators on Unified Normalized-variable Diagram) scheme is tested for the LES of reacting flows in this study. The high efficiency of this scheme is demonstrated by evaluating its accuracy, central processing unit cost, and structure-preserving property by simulating the convection–diffusion process of a passive scalar. Simulations of two bluff body stabilized flames are studied using this scheme. For low turbulence intensity, the current scheme improves the numerical resolution of the instantaneous and averaged flow fields. The prediction of flow statistics is also improved by the ROUND schemes compared to the conventional schemes. Moreover, the ROUND schemes preserve the axisymmetry of the averaged flow better than the conventional schemes for the cases investigated here. For the high turbulence intensity case, the ROUND scheme avoids nonphysical numerical oscillations. The flow and flame statistics obtained using this scheme compare well with measurements. Therefore, this work demonstrates the advantages of using ROUND schemes for LES of reacting flows.

Locally structure-preserving div-curl operators for high order discontinuous Galerkin schemes

We propose a novel Structure-Preserving Discontinuous Galerkin (SPDG) operator that recovers at the discrete level the algebraic property related to the divergence of the curl of a vector field, which is typically referred to as div-curl problem. A staggered Cartesian grid is adopted in 3D, where the vector field is naturally defined at the corners of the control volume, while its curl is evaluated as a cell-centered quantity. Firstly, the curl operator is rewritten as the divergence of a tensor, hence allowing compatible finite difference schemes to be devised and to be proven to mimic the algebraic div-curl property. Successively, a high order DG divergence operator is built upon integration by parts, so that the structure-preserving finite difference div-curl operator is exactly retrieved for first order discretizations. We further demonstrate that the novel SPDG schemes are capable of obtaining a zero div-curl identity with machine precision from second up to sixth order accuracy. Curl-grad operators follow as a consequence. In a second part, we show the applicability of these SPDG methods by solving the incompressible Navier-Stokes equations written in vortex-stream formulation. This hyperbolic system deals with divergence-free involutions related to the velocity and vorticity field as well as to the stream function, thus it provides an ideal setting for the validation of the novel schemes. A compatible discretization of the numerical viscosity is also proposed in order to maintain the structure-preserving property of the div-curl DG operators even in the presence of artificial or physical dissipative terms. Finally, to overcome the time step restriction dictated by the viscous sub-system, Implicit-Explicit (IMEX) Runge-Kutta time stepping techniques are tailored to handle the SPDG framework. Numerical examples show that the theoretical order of convergence is reached by this new class of methods, and prove the exact preservation of the incompressibility constraint.

VPNets: Volume-preserving neural networks for learning source-free dynamics

We propose volume-preserving networks (VPNets) for learning unknown source-free dynamical systems using trajectory data. We propose three modules and combine them to obtain two network architectures, coined R-VPNet and LA-VPNet. The distinct feature of the proposed models is that they are intrinsic volume-preserving. In addition, the corresponding approximation theorems are proved, which theoretically guarantee the expressivity of the proposed VPNets to learn source-free dynamics. The effectiveness, generalization ability and structure-preserving property of the VP-Nets are demonstrated by numerical experiments.

Generalized Hessian-Schatten Norm Regularization for Image Reconstruction

Regularization plays a crucial role in reliably utilizing imaging systems for scientific and medical investigations. It helps to stabilize the process of computationally undoing any degradation caused by physical limitations of the imaging process. In the past decades, total variation regularization played a dominant role in the literature. Two forms of total variation regularizations, namely the first-order and the second-order total variation (TV-1 and TV-2) have been widely used. TV-1 has a disadvantage: it reconstructs images in the form of piece-wise constants when the noise and/or under-sampling is severe, while TV-2 reconstructs natural-looking images under such scenarios. On the other hand, TV-1 can recover sharp jumps better than TV-2. Two forms of generalizations, namely Hessian-Schatten norm (HSN) regularization, and total generalized variation (TGV) regularization, have been proposed and have become significant developments in the area of regularization for imaging inverse problems owing to their performance. While the strength of TGV is that it can combine the advantages of TV-1 and TV-2, HSN has better structure-preserving property. Here, we develop a novel regularization for image recovery that combines the strengths of TGV and HSN. We achieve this by restricting the maximization space in the dual form of HSN in the same way that TGV is obtained from TV-2. We call the new regularization the generalized Hessian-Schatten norm regularization (GHSN). We develop a novel computational method for image reconstruction using the new form of regularization based on the well-known framework called the alternating direction method of multipliers (ADMM). We demonstrate the strength of the GHSN using some reconstruction examples.

A conservative Allen–Cahn equation with a curvature-dependent Lagrange multiplier

In the present study, we propose a novel conservative Allen–Cahn (CAC) equation with a curvature-dependent Lagrange multiplier. The proposed CAC equation has a superior structure-preserving property. Unlike the conventional CAC equations which have motion by mean curvature with area or volume constraint, the proposed model has minimum dynamics of motion by mean curvature and only has smoothing property of interface transition layer. Therefore, it can be utilized as a building block equation for modeling conservative phase-field applications such as two-phase fluid flows. Several computational tests are conducted to confirm the superior performance of the proposed CAC equation in terms of structure-preserving property.

Positivity-preserving symplectic methods for the stochastic Lotka–Volterra predator-prey model

In this paper, positivity-preserving symplectic numerical approximations are investigated for the 2d-dimensional stochastic Lotka–Volterra predator-prey model driven by multiplicative noises, which plays an important role in ecosystem. The model is shown to possess both a unique positive solution and a stochastic symplectic geometric structure, and hence can be interpreted as a stochastic Hamiltonian system. To inherit the intrinsic biological characteristic of the original system, a class of stochastic Runge–Kutta methods is presented, which is proved to preserve positivity of the numerical solution and possess the discrete stochastic symplectic geometric structure as well. Uniform boundedness of both the exact solution and the numerical one are obtained, which are crucial to derive the conditions for convergence order one in the \(\mathbb {L}^1(\varOmega )\)-norm. Numerical examples illustrate the stability and structure-preserving property of the proposed methods over long time.

A Structure-Preserving Parametric Finite Element Method for Surface Diffusion

We propose a structure-preserving parametric finite element method (SP-PFEM) for discretizing the surface diffusion of a closed curve in two dimensions (2D) or surface in three dimensions (3D). Here the "structure-preserving" refers to preserving the two fundamental geometric structures of the surface diffusion flow: (i) the conservation of the area/volume enclosed by the closed curve/surface, and (ii) the decrease of the perimeter/total surface area of the curve/surface. For simplicity of notations, we begin with the surface diffusion of a closed curve in 2D and present a weak (variational) formulation of the governing equation. Then we discretize the variational formulation by using the backward Euler method in time and piecewise linear parametric finite elements in space, with a proper approximation of the unit normal vector by using the information of the curves at the current and next time step. The constructed numerical method is shown to preserve the two geometric structures and also enjoys the good property of asymptotic equal mesh distribution. The proposed SP-PFEM is "weakly" implicit (or almost semi-implicit) and the nonlinear system at each time step can be solved very efficiently and accurately by the Newton's iterative method. The SP-PFEM is then extended to discretize the surface diffusion of a closed surface in 3D. Extensive numerical results, including convergence tests, structure-preserving property and asymptotic equal mesh distribution, are reported to demonstrate the accuracy and efficiency of the proposed SP-PFEM for simulating surface diffusion in 2D and 3D.

Self-reinforcing Unsupervised Matching

Remarkable gains in deep learning usually benefit from large-scale supervised data. Ensuring the intra-class modality diversity in training set is critical for generalization capability of cutting-edge deep models, but it burdens human with heavy manual labor on data collection and annotation. In addition, some rare or unexpected modalities are new for the current model, causing reduced performance under such emerging modalities. Inspired by the achievements in speech recognition, psychology and behavioristics, we present a practical solution, self-reinforcing unsupervised matching (SUM), to annotate the images with 2D structure-preserving property in an emerging modality by cross-modality matching. Specifically, we propose a dynamic programming algorithm, dynamic position warping (DPW), to reveal the underlying element correspondence relationship between two matrix-form data in an order-preserving fashion, and devise a local feature adapter (LoFA) to allow for cross-modality similarity measurement. On these bases, we develop a two-tier self-reinforcing learning mechanism on both feature level and image level to optimize the LoFA. The proposed SUM framework requires no any supervision in emerging modality and only one template in seen modality, providing a promising route towards incremental learning and continual learning. Extensive experimental evaluation on two proposed challenging one-template visual matching tasks demonstrate its efficiency and superiority.

Fast neural Poincaré maps for toroidal magnetic fields

Poincaré maps for toroidal magnetic fields are routinely employed to study gross confinement properties in devices built to contain hot plasmas. In most practical applications, evaluating a Poincaré map requires numerical integration of a magnetic field line, a process that can be slow and that cannot be easily accelerated using parallel computations. We propose a novel neural network architecture, the HénonNet, and show that it is capable of accurately learning realistic Poincaré maps from observations of a conventional field-line-following algorithm. After training, such learned Poincaré maps evaluate much faster than the field-line integration method. Moreover, the HénonNet architecture exactly reproduces the primary physics constraint imposed on field-line Poincaré maps: flux preservation. This structure-preserving property is the consequence of each layer in a HénonNet being a symplectic map. We demonstrate empirically that a HénonNet can learn to mock the confinement properties of a large magnetic island by using coiled hyperbolic invariant manifolds to produce a sticky chaotic region at the desired island location. This suggests a novel approach to designing magnetic fields with good confinement properties that may be more flexible than ensuring confinement using KAM tori.

Symmetry breaking of infinite-dimensional dynamic system

The relationship between the symmetry breaking and the energy dissipation of dynamic systems is the foundation of the geometric mechanics, the investigation of which will establish a bridge between the structure-preserving theory and the assessment approach of the structure-preserving property for the employed numerical scheme. In this letter, two typical factors inducing the symmetry breaking for the infinite-dimensional dynamic system, including the symmetry breaking of the coefficient matrices and the space–time dependence of the Hamiltonian function, are investigated in detail. Based on the multi-symplectic theory, the local energy variations for dynamic systems with the mentioned symmetry breaking factors are deduced and the specific forms of which for a flexible cantilever with the variable bending rigidity under an external excitation are presented, which shows the local energy dissipation explicitly and provides the possibility of reproduction the local energy dissipation for the infinite-dimensional dynamic system in the numerical simulation.

Structure-Preserving Method for Reconstructing Unknown Hamiltonian Systems From Trajectory Data

We present a numerical approach for approximating unknown Hamiltonian systems using observation data. A distinct feature of the proposed method is that it is structure-preserving, in the sense that it enforces conservation of the reconstructed Hamiltonian. This is achieved by directly approximating the underlying unknown Hamiltonian, rather than the right-hand-side of the governing equations. We present the technical details of the proposed algorithm and its error estimate in a special case, along with a practical de-noising procedure to cope with noisy data. A set of numerical examples are then presented to demonstrate the structure-preserving property and effectiveness of the algorithm.

Sensing Matrix Design for Compressive Spectral Imaging via Binary Principal Component Analysis.

Compressive spectral imaging (CSI) is a framework that captures coded-and-multiplexed low-dimensional projections of spectral data-cubes. In general, the sensing process in many CSI architectures is described using binary matrices, so-called sensing/projection matrices, whose elements can be either random or designed. However, some characteristics of the spectral data, such as the ℓ2-norm or the second moment statistics, can be lost when this dimensionality reduction is performed. Similarly, principal component analysis (PCA) is a data dimensionality reduction technique that minimizes the least-squared error between the spectral data and its low-dimensional projection, but preserving its structure or variance. Thus, PCA can be used to guide the CSI acquisition process by designing the binary sensing matrix. Nonetheless, PCA requires to know the spectral image a-priori, and also, its associated projection matrix is not binary, as required by CSI optical architectures. Therefore, in this paper, an algorithm to design CSI sensing matrices by exploiting the structure-preserving property of the PCA projection is proposed. First, a set of compressive measurements obtained with random sensing matrices is used to rapidly estimate the covariance matrix associated with the spectral data. Then, a new sensing matrix is designed by solving a non-convex optimization problem that finds a set of binary vectors that approximate the principal components of the covariance matrix, thus maximizing the explanation of the data variance. Experimental results show an improvement of up to 3 dB in image reconstruction quality, in terms of the peak signal to noise ratio (PSNR), when the binary PCA-based sensing matrices are used and compared with conventional random sensing matrices and state-of-art designed matrices based on PCA.

Remarks on energy methods for structure-preserving finite difference schemes – Small data global existence and unconditional error estimate

In the previous article (Yoshikawa, 2017), the author proposes the energy method for structure-preserving finite difference schemes, which enable us to show global existence and uniqueness of solution for the schemes and error estimates. In this article, we give two extended remarks of the methods. One is related to the small data global existence results for schemes of which energy is not necessarily bounded from below. The other is an unconditional error estimate which holds globally in time and without smallness condition for split sizes. These results can be shown due to the structure-preserving property.

Multiscale eigenelement method for periodical composites: A review

Based on the eigenvector expansion idea, the Multiscale Eigenelement Method (MEM) was proposed by the author and co-workers. MEM satisfies two equivalent conditions, one condition is the equivalence of strain energy, and the other is the deformation similarity. These two equivalent conditions character the structure-preserving property of a multiscale analysis method. The equivalence of strain energy is necessary for achieving accurate macro behaviors such as lower order frequencies, while the deformation similarity is essential for predicting accurate micro behaviors such as stresses. The MEM has become a powerful multiscale method for the analysis of composite structures because of its high accuracy and efficiency. In this paper, the research advances of MEM are reviewed and all types of eigenelement methods are compared, focusing on superiorities and deficiencies from practical viewpoint. It is concluded that the eigenelement methods with smooth shape functions are more suitable for the analysis of macro behaviors such as lower order frequencies, and the eigenelement methods with piecewise shape functions are suitable for the analysis of both macro and micro behaviors.