Laplacian Term Research Articles

Multiscale Spectral Manifold Wavelet Regularizer for Unsupervised Deep Functional Maps

AbstractIn deep functional maps, the regularizer computing the functional map is especially crucial for ensuring the global consistency of the computed pointwise map. As the regularizers integrated into deep learning should be differentiable, it is not trivial to incorporate informative axiomatic structural constraints into the deep functional map, such as the orientation‐preserving term. Although commonly used regularizers include the Laplacian‐commutativity term and the resolvent Laplacian commutativity term, these are limited to single‐scale analysis for capturing geometric information. To this end, we propose a novel and theoretically well‐justified regularizer commuting the functional map with the multiscale spectral manifold wavelet operator. This regularizer enhances the isometric constraints of the functional map and is conducive to providing it with better structural properties with multiscale analysis. Furthermore, we design an unsupervised deep functional map with the regularizer in a fully differentiable way. The quantitative and qualitative comparisons with several existing techniques on the (near‐)isometric and non‐isometric datasets show our method's superior accuracy and generalization capabilities. Additionally, we illustrate that our regularizer can be easily inserted into other functional map methods and improve their accuracy.

Structural graph learning method for hyperspectral band selection

ABSTRACT Recently, graph learning-based hyperspectral band selection algorithms illustrate impressive performance for hyperspectral image (HSI) processing, whose goal is to select an optimal band combination containing less redundancy through the learned graph matrix. However, most of the previous methods work in single spectral domain, which neglects the rich image spatial information. Moreover, they typically model the graph matrix from a global perspective while ignoring the differences in spatial distribution that exist in diverse pixel and band regions. Based on the above considerations, to take full account of structure information in spatial and spectral domains, a structural graph learning method for hyperspectral band selection (SGLM) is designed. Specifically, SGLM constructs two matrices using the correlation between pixels and bands under the local perspective, which can capture the image structure information reasonably. Since the proposed method is modelled in both spatial and spectral dimensions, it is conducive to subsequent band subset generation task. Meanwhile, in order to guarantee local spatial consistency among bands, a Laplacian regularization term associated with the error matrix is introduced to the self-representation model. Additionally, considering that the importance of a band is consistent with its capability to reconstruct the entire band set, an adjacent bands reconstruction strategy is adopted to obtain the ultimate band combination, which can assess the significance of each band effectively. The comprehensive experimental analysis on four datasets demonstrates that the proposed SGLM model has outstanding superiority in comparison with several band selection algorithms.

An adaptive non-uniform L2 discretization for the one-dimensional space-fractional Gray–Scott system

This paper introduces a new numerical method for solving space-fractional partial differential equations (PDEs) on non-uniform adaptive finite difference meshes, considering a fractional order α∈(1,2) in one dimension. The fractional Laplacian in PDE is computed by using Riemann–Liouville (R–L) derivatives, incorporating a boundary condition of the form u=0 in R∖Ω. The proposed approach extends the L2 method to non-uniform meshes for calculating the R–L derivatives. The spatial mesh generation employs adaptive moving finite differences, offering adaptability at each time step through grid reallocation based on previously calculated solutions. The chosen mesh movement technique, moving mesh PDE-5 (MMPDE-5), demonstrates rapid and efficient mesh movement. The numerical solutions are obtained by applying the non-uniform L2 numerical scheme and the MMPDE-5 method for moving meshes automatically. Two numerical experiments focused on the space-fractional heat equation validate the convergence of the proposed scheme. The study concludes by exploring patterns in equations involving the fractional Laplacian term within the Gray–Scott system. It reveals self-replication, travelling wave, and chaotic patterns, along with two distinct evolution processes depending on the order α: from self-replication to standing waves and from travelling waves to self-replication.

An Improved Detail-Enhancement CycleGAN Using AdaLIN for Facial Style Transfer

The rise of comics and games has led to increased artistic processing of portrait photos. With growing commercial demand and advancements in deep learning, neural networks for rapid facial style transfer have become a key research area in computer vision. This involves converting face photos into different styles while preserving content. Face images are more complex than regular images, requiring extensive modification. However, current methods often face issues such as unnatural color transitions, loss of detail in highlighted areas, and noticeable artifacts along edges, resulting in low-quality stylized images. In this study, an enhanced generative adversarial network (GAN) is proposed, which is based on Adaptive Layer Instance Normalization (AdaLIN) + Laplacian. This approach incorporates the AdaLIN normalization method, allowing for the dynamic adjustment of Instance Normalization (IN) and Layer Normalization (LN) parameters’ weights during training. By combining the strengths of both normalization techniques, the model selectively preserves and alters content information to some extent, aiming to strike a balance between style and content. This helps address problems such as unnatural color transitions and loss of details in highlights that lead to color inconsistencies. Furthermore, the introduction of a Laplacian regularization term aids in denoising the image, preventing noise features from interfering with the color transfer process. This regularization also helps reduce color artifacts along the face’s edges caused by noise while maintaining the image’s contour information. These enhancements significantly enhance the quality of the generated face images. To compare our method with traditional CycleGAN and recent algorithms such as XGAN and CariGAN, both subjective and objective evaluations were conducted. Subjectively, our method demonstrates more natural color transitions and superior artifact elimination, achieving higher scores in Mean Opinion Score (MOS) evaluations. Objectively, experiments using our method yielded better scores across three metrics: FID, SSIM, and MS-SSIM. The effectiveness of the proposed methods is validated through both objective and subjective evaluations.

Laplacian regularization of linear regression model for semi-supervised industrial soft sensor development

With more and more data collected from modern industrial processes, data-based soft sensor modeling methods have become popular for online prediction of those difficult-to-measure key variables. As one of the most widely used methods, the linear regression model has been studied for many years. In this paper, a new semi-supervised form of data-based soft sensor is developed upon this simple but commonly used linear regression model, in order to make it more engineering applicable. Particularly, a semi-supervised model structure is formulated by introducing a Laplacian regularization term, through which a lot more unlabeled data samples can be well incorporated for modeling. With an improved estimation for the input data distribution, the real data features can be better captured and thus the model parameters are well restricted, in order to provide a better performance. In addition, the basic Laplacian regularized linear regression model is further extended to nonlinear counterparts by the introductions of basis function and kernel trick in the input data space. For performance evaluation, two industrial case studies are provided, based on which both feasibility and effectiveness of the developed method are confirmed.

A high-order upwind compact difference scheme for solving the streamfunction-velocity formulation of the unsteady incompressible Navier–Stokes equations

In this paper, we propose an upwind compact difference method with fourth-order spatial accuracy and second-order temporal accuracy for solving the streamfunction-velocity formulation of the two-dimensional unsteady incompressible Navier–Stokes equations. The streamfunction and its first-order partial derivatives (velocities) are treated as unknown variables. Three types of compact difference schemes are employed to discretize the first-order partial derivatives of the streamfunction. Specifically, these schemes include the fourth-order symmetric scheme, the fifth-order upwind scheme, and the sixth-order symmetric scheme derived by combining the two parts of the fifth-order upwind scheme. As a result, the fourth-order spatial discretization schemes are established for the Laplacian term, the biharmonic term, and the nonlinear convective term, along with the Crank–Nicolson scheme for the temporal discretization. The unconditional stability characteristic of the scheme for the linear model is proved by discrete von Neumann analysis. Moreover, six numerical experiments involving three test problems with the analytic solutions, and three flow problems including doubly periodic double shear layer, lid-driven cavity flow, and dipole-wall interaction are carried out to demonstrate the accuracy, robustness, and efficiency of the present method. The results indicate that the present method not only has good numerical performance but also exhibits quite efficiency.

Phase transitions in the fractional three-dimensional Navier–Stokes equations

The fractional Navier–Stokes equations on a periodic domain [0,L]3 differ from their conventional counterpart by the replacement of the −νΔu Laplacian term by νsAsu , where A=−Δ is the Stokes operator and νs=νL2(s−1) is the viscosity parameter. Four critical values of the exponent s⩾0 have been identified where functional properties of solutions of the fractional Navier–Stokes equations change. These values are: s=13 ; s=34 ; s=56 and s=54 . In particular: (i) for s>13 we prove an analogue of one of the Prodi–Serrin regularity criteria; (ii) for s⩾34 we find an equation of local energy balance and; (iii) for s>56 we find an infinite hierarchy of weak solution time averages. The existence of our analogue of the Prodi–Serrin criterion for s>13 suggests the sharpness of the construction using convex integration of Hölder continuous solutions with epochs of regularity in the range 0<s<13 .

A thermal lattice Boltzmann model for evaporating multiphase flows

Modeling thermal multiphase flows has become a widely sought methodology due to its scientific relevance and broad industrial applications. Much progress has been achieved using different approaches, and the lattice Boltzmann method is one of the most popular methods for modeling liquid–vapor phase change. In this paper, we present a novel thermal lattice Boltzmann model for accurately simulating liquid–vapor phase change. The proposed model is built based on the equivalent variant of the temperature governing equation derived from the entropy balance law, in which the heat capacitance is absorbed into transient and convective terms. Then a modified equilibrium distribution function and a proper source term are elaborately designed in order to recover the targeting equation in the incompressible limit. The most striking feature of the present model is that the calculations of the Laplacian term of temperature, the gradient term of temperature, and the gradient term of density can be simultaneously avoided, which makes the formulation of the present model is more concise in contrast to all existing lattice Boltzmann models. Several benchmark examples, including droplet evaporation in open space, droplet evaporation on a heated wall, and nucleate boiling phenomenon, are carried out to assess numerical performance of the present model. It is found that the present model effectively improves the numerical accuracy in solving the interfacial behavior of liquid–vapor phase change within the lattice Boltzmann method framework.

Geometric inequalities of bi-warped product submanifold in generalized complex space form

In this article, we obtain some inequalities for the bi-warped product submanifold of nearly-Kähler manifold which is expressed in terms of Laplacian, gradient, slant coefficient, and second fundamental form. Our inequality generalizes the inequalities of [1,20]. We also compute the Dirichlet energy and first eigenvalue of the Laplacian of warping functions.

An efficient thermal lattice Boltzmann method for simulating three-dimensional liquid–vapor phase change

In this paper, a multiple-relaxation-time lattice Boltzmann (LB) approach is developed for the simulation of three-dimensional (3D) liquid–vapor phase change based on the pseudopotential model. In contrast to some existing 3D thermal LB models for liquid–vapor phase change, the present approach has two advantages: for one thing, some gradient terms, such as the gradient of volumetric heat capacity and the Laplacian term of temperature, that must be computed with finite-difference scheme in previous works is not needed for the present thermal model. For another, the current model is constructed based on the seven discrete velocities in three dimensions (D3Q7), making it more efficient and much easier to be implemented. Also, based on the scheme proposed by Zhou and He (1997), a pressure boundary condition for the D3Q19 lattice is proposed to model the multiphase flow in open systems. This model is then validated by considering the temperature distribution in a 3D saturated liquid–vapor system, the d2 law, the droplet evaporation on a heated surface and the nucleate boiling. It is observed that the numerical results fit well with the analytical solutions, and the results of the finite difference method or the experimental data. Our numerical results indicate that the present approach is reliable and efficient in dealing with the 3D liquid–vapor phase change.

Adaptation and assessement of projected Nesterov accelerated gradient flow to compute stationary states of nonlinear Schrödinger equations

The aim of the paper is to derive minimization algorithms based on the Nesterov accelerated gradient flow [Y. Nesterov, Gradient methods for minimizing composite objective function. Core discussion paper, (2007). Available at http://www.ecore.be/DPs/dp_1191313936.pdf; Y. Nesterov, A method of solving a convex programming problem with convergence rate O ( 1 / k 2 ) . In Doklady Akademii Nauk, Vol. 269, Russian Academy of Sciences, 1983, pp. 543–547; Y. Nesterov, Introductory Lectures on Convex Optimization: A basic course, Kluwer Academic Publishers, Massachusetts, 2004.] to compute the ground state of nonlinear Schrödinger equations, which can potentially include a fractional laplacian term. A comparison is developed with standard gradient flow formulations showing that the Nesterov accelerated gradient flow has some interesting properties but at the same time finds also some limitations due to the nature of the problem. A few simulations are finally reported to understand the behaviour of the algorithms and open the path to further complicate questions that require more advanced studies concerning the application of the Nesterov accelerated gradient flow to nonlinear Schrödinger equations.

A time splitting Chebyshev-Fourier spectral method for the time-dependent rotating nonlocal Schrödinger equation in polar coordinates

In this article, we propose a time splitting Chebyshev-Fourier spectral method to compute the dynamics of rotating nonlocal Schrödinger equation in polar coordinates. Since the rotation term is diagonalizable with Fourier spectral method in azimuthal direction, we split the Hamiltonian into a linear part, i.e., the Laplacian and rotation terms, and the nonlinear part. The linear part is discretized by Chebyshev-Fourier spectral method in space and integrated by Crank-Nicolson or exact matrix exponential. The nonlinear part is solved exactly in physical space, and the nonlocal potential, which is defined as convolution φ:=U⁎|ψ|2 with a singular kernel U, is computed via Kernel Truncation method with spectral accuracy. Then we construct a high order time splitting spectral method. Our scheme is spectrally accurate in space and of second/fourth order in the temporal direction. Extensive numerical results are presented to confirm the accuracy and efficiency for both the nonlocal potential and the wave function, together with one application to a two-component rotating dipolar Bose-Einstein condensates. In addition, we investigate the rotational symmetry preserving performance by a comprehensive comparison with existing scheme.

Collaborative representation based cross-domain semantic transfer for vehicle re-identification

Vehicle re-identification (re-ID) has received increasing attention due to its tremendous potential in practical intelligent transportation scenarios. The main difficulties of vehicle re-ID are generally induced by insufficient annotations, varying viewpoints and illuminations, and mismatched visual-semantic cues. Previous studies on vehicle re-ID have generally focused on exploring better visual representations of the target samples while ignoring semantic complementarity from external sources. In this paper, we propose an unsupervised semantic transfer method based on collaborative matrix representation (STCMR) for vehicle re-ID that obtains rich semantic representations by learning knowledge from external annotated data. STCMR contains three components: an asymmetrical collaborative matrix representation term, a low-rank regularized relaxed attribute term, and a dual-graph Laplacian regularization term. In particular, we first build two reliable feature matrices to better transfer external annotated semantics from the source domain to the target re-ID domain. We then propose a robust collaborative matrix representation term to decompose the feature matrices into user-defined, domain-shared, and domain-unique parts in an asymmetrical manner. To tackle samples with incomplete semantics, we design a low-rank regularized attribute relaxed term that fully exploits existing annotations to complete missing entries in a soft constraint manner. Finally, we exploit the local geometric structure consistency in source and target domains by a dual-graph Laplacian regularization term to adaptively estimate the relationships of source domain samples. In addition, we further propose an alternating iterative algorithm to solve the optimization problem. Experimental results conducted on the VehicleID and VeRi-776 datasets validate the effectiveness of our proposed method.

Mutual structure learning for multiple kernel clustering

Multiple kernel clustering (MKC) has garnered considerable attention in recent years, aiming to obtain an optimal partition from multiple base kernels. Existing MKC methods typically focus on either learning the pairwise structure by constructing an optimal kernel from the base kernels or learning the cluster structure by integrating multiple base partitions into a consensus one for clustering. However, these previous approaches overlook the potential for mutual structure learning between the two aspects, i.e., pairwise and cluster structure, of clustering. To address this limitation, we propose a novel multiple kernel clustering method, referred to as MSL-MKC, which simultaneously learns pairwise and cluster structure to achieve an improved consensus partition matrix for clustering. Specifically, MSL-MKC extends the adaptive neighbor graph learning approach into kernel space to construct a discriminative similarity graph from multiple base kernels for pairwise structure learning. The consensus partition is formulated by aligning it with multiple base partitions for cluster structure learning. We utilize a Laplacian regularization term to preserve the pairwise structure in the consensus partition and inject the cluster structure into the discriminative similarity graph. The proposed method integrates similarity graph learning, partition fusion, and Laplacian regularization into a unified framework, optimized using an iterative algorithm. Experimental results on various benchmark datasets demonstrate the efficacy of MSL-MKC. The demo code of this work is publicly available at https://github.com/guanyuezhen/MSL-MKC.

Small order limit of fractional Dirichlet sublinear-type problems

We study the asymptotic behavior of solutions to various Dirichlet sublinear-type problems involving the fractional Laplacian when the fractional parameter s tends to zero. Depending on the type on nonlinearity, positive solutions may converge to a characteristic function or to a positive solution of a limit nonlinear problem in terms of the logarithmic Laplacian, that is, the pseudodifferential operator with Fourier symbol ln (|xi |^2). In the case of a logistic-type nonlinearity, our results have the following biological interpretation: in the presence of a toxic boundary, species with reduced mobility have a lower saturation threshold, higher survival rate, and are more homogeneously distributed. As a result of independent interest, we show that sublinear logarithmic problems have a unique least-energy solution, which is bounded and Dini continuous with a log-Hölder modulus of continuity.

Regularity and attractors for the three‐dimensional generalized Boussinesq system

We study the three‐dimensional Boussinesq system with dissipation and diffusion generalized in terms of fractional Laplacians. First, we study the existence, uniqueness, and regularity of global weak solutions. Then, we investigate the asymptotic behavior of weak solutions via attractors. Since the system might not always have regular solutions, we use a new framework developed by Cheskidov and Lu which is called evolutionary system to obtain various attractors and their properties.

Control of Fractional Diffusion Problems via Dynamic Programming Equations

We explore the approximation of feedback control of integro-differential equations containing a fractional Laplacian term. To obtain feedback control for the state variable of this nonlocal equation, we use the Hamilton–Jacobi–Bellman equation. It is well known that this approach suffers from the curse of dimensionality, and to mitigate this problem we couple semi-Lagrangian schemes for the discretization of the dynamic programming principle with the use of Shepard approximation. This coupling enables approximation of high-dimensional problems. Numerical convergence toward the solution of the continuous problem is provided together with linear and nonlinear examples. The robustness of the method with respect to disturbances of the system is illustrated by comparisons with an open-loop control approach.

Row-Column Overcomplete Structured Dictionary Learning for Enhanced Fault Detection and Isolation

To improve the monitoring performance of dictionary learning based methods, this article proposes a row-column overcomplete structured dictionary learning (RCOSDL) method for fault detection and isolation of industrial processes. Unlike conventional dictionary learning approaches which are overcomplete column-wise, the proposed method involves a dictionary that is overcomplete both row- and column-wise. The introduction of row-column overcomplete dictionary results in monitoring statistics that are more sensitive to incipient faults. In order to incorporate structured information, two graph Laplacian regularization terms, namely, manifold graph Laplacian and full graph Laplacian are considered. Whilst the inherent local geometric structure in the data is preserved in the sparse representation by the manifold graph Laplacian term, the correlation structure between process variables is preserved by using the full graph Laplacian term. Hence violation in the geometric or correlation structure will be promptly detected. To pinpoint faulty variables, a fault isolation method is developed by imposing the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_{1}$</tex-math></inline-formula> / <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_{2,1}$</tex-math></inline-formula> -norm constraint on the sparse coefficients. In addition, theoretical property involving the condition for guaranteed fault isolation is presented in Theorem 1. The contributions of this article include the introduction of monitoring statistics based on RCOSDL that are suitable for incipient faults, a new fault isolation scheme as well as theoretical analysis on the condition of guaranteed fault isolation. The better performance of the proposed method is illustrated by applications to numerical studies and practical industrial cases.

Liouville-type theorem for a nonlinear sub-elliptic system involving $$\Delta _\lambda $$-Laplacian and advection terms

Liouville-type theorem for a nonlinear sub-elliptic system involving $$\Delta _\lambda $$-Laplacian and advection terms

Curved Versions of the Ovsienko–Redou Operators

Abstract We construct a family of conformally covariant bidifferential operators on pseudo-Riemannian manifolds. Our construction is analogous to the construction of Graham–Jenne–Mason–Sparling of conformally covariant differential operators via tangential powers of the Laplacian in the Fefferman–Graham ambient space. In fact, we completely classify the tangential bidifferential operators on the ambient space, which are expressed purely in terms of the ambient Laplacian. This gives a curved analogue of the classification, due to Ovsienko–Redou and Clerc, of conformally invariant bidifferential operators on the sphere. As an application, we construct a large class of formally self-adjoint conformally invariant differential operators.