Laplacian Term Research Articles

An approach to elliptic equations with nonlinear gradient terms via a modulation framework

We consider a class of nonhomogeneous elliptic equations with fractional Laplacian and nonlinear gradient terms, namely [Formula: see text] in [Formula: see text], where [Formula: see text], [Formula: see text] is the nonlinearity, [Formula: see text] the potential and [Formula: see text] is a forcing term. Some examples of nonlinearities dealt with are [Formula: see text], [Formula: see text] and [Formula: see text], covering large values of [Formula: see text], and particularly variational supercritical powers for [Formula: see text] and super-[Formula: see text] ones for [Formula: see text] (superquadratic if [Formula: see text]). Moreover, we are able to consider some exponential growths, [Formula: see text] belonging to certain classes of power series, or [Formula: see text] satisfying some conditions in the Lipschitz spirit. We obtain results on existence, uniqueness, symmetry, and other qualitative properties in a new framework, namely modulation-type spaces based on Lorentz spaces. For that, we need to develop properties and estimates in those spaces such as complex interpolation, Hölder-type inequality, estimates for product, convolution and Riesz potential operators, among others. In order to handle the nonlinearity, other ingredients are estimates for composition operators in our setting.

Quantum Physical Observables with Conjectural Modeling: Paradigm shifting Formalisms II: A Review

In continuation with the previous Review Force General Conjectural Modeling Transforms Formalism PHYSICS33 (Physics & Astronomy International Journal under publication), the current review article will try to develop quantum gravity gage transforms Algorithm Graphical Equation of micro-blackhole force to gauge fields-wavefunctions algorithm transforms equations. Theoretical derivations of experimental observable measurable parameters having scalar quantum gauge field as function in terms of Laplacian, Fourier, and the Legendre transform gaging the spin, rotation, revolution, and rotational angular velocity have been shown. These transforms appear only as a function of time, and that makes the formulation independent of assumptions of invariability of fundamental universally known constants. We consider examples of universal eonic parametric observables as well. The question of whether the time is a linear operator has been answered by employing Correspondence Principle’s linear operator four-vector time matrix spatial range algorithm equivalence and is estimated to be about ¼ mile. Theoretically derived algorithm physics designs enable prototype testing by utilizing experimental instrumentations measuring observables. A flowchart has been configured simplifying theoretical to experimental mathematical physical sciences to profile signal/noise intensity and the density matrix. Additionally, estimation of Hod PDP mechanistic probability, grand unifying physics operations, schematics of potential scalar gauge field alongside incorporating typical wavefunction general quantum computing signal/noise matrix graphing, simple lab-top prototype sound and light profiling intensity, and spectral density point-to-point matrix oscilloscopic observational measurement techniques have been schematically sketched out to enhance facilitation of future implementations of state-of-the-art physics techniques practically!!

Thermal lattice Boltzmann model for liquid-vapor phase change.

The lattice Boltzmann method is adopted to solve the liquid-vapor phase change problems in this article. By modifying the collision term for the temperature evolution equation, a thermal lattice Boltzmann model is constructed. As compared with previous studies, the most striking feature of the present approach is that it could avoid the calculations of both the Laplacian term of temperature [∇·(κ∇T)] and the gradient term of heat capacitance [∇(ρc_{v})]. In addition, since the present approach adopts a simple linear equilibrium distribution function, it is possible to use the D2Q5 lattice for the two-dimensional cases considered here. Thus, the present model is more efficient than previous models in which the lattice is usually limited to the D2Q9. The proposed model is first validated by the problems of droplet evaporation in open space and droplet evaporation on a heated surface, and the numerical results show good agreement with the analytical results and the finite difference method. Then it is used to model the nucleate boiling problem, and the relationship between detachment bubble diameter and gravitational acceleration obtained with the present approach fits well with previous works.

Dynamic nonlocal passive scalar subgrid-scale turbulence modeling

Extensive experimental evidence highlights that scalar turbulence exhibits anomalous diffusion and stronger intermittency levels at small scales compared to that in fluid turbulence. This renders the corresponding subgrid-scale dynamics modeling for scalar turbulence a greater challenge to date. We develop a new large eddy simulation (LES) paradigm for efficiently and dynamically nonlocal LES modeling of the scalar turbulence. To this end, we formulate the underlying nonlocal model starting from the filtered Boltzmann kinetic transport equation, where the divergence of subgrid-scale scalar fluxes emerges as a fractional-order Laplacian term in the filtered advection–diffusion model, coding the corresponding superdiffusive nature of scalar turbulence. Subsequently, we develop a robust data-driven algorithm for estimation of the fractional (noninteger) Laplacian exponent, where we, on the fly, calculate the corresponding model coefficient employing a new dynamic procedure. Our a priori tests show that our new dynamically nonlocal LES paradigm provides better agreement with the ground-truth filtered direct numerical simulation data in comparison to the conventional static and dynamic Prandtl–Smagorinsky models. Moreover, in order to analyze the numerical stability and assessing the model's performance, we carry out comprehensive a posteriori tests. They unanimously illustrate that our new model considerably outperforms other existing functional models, correctly predicting the backscattering phenomena and, at the same time, providing higher correlations at small-to-large filter sizes. We conclude that our proposed nonlocal subgrid-scale model for scalar turbulence is amenable for coarse LES and very large eddy simulation frameworks even with strong anisotropies, applicable to environmental applications.

Coupled Tensor Block Term Decomposition with Superpixel-Based Graph Laplacian Regularization for Hyperspectral Super-Resolution

Hyperspectral image (HSI) super-resolution aims at improving the spatial resolution of HSI by fusing a high spatial resolution multispectral image (MSI). To preserve local submanifold structures in HSI super-resolution, a novel superpixel graph-based super-resolution method is proposed. Firstly, the MSI is segmented into superpixel blocks to form two-directional feature tensors, then two graphs are created using spectral–spatial distance between the unfolded feature tensors. Secondly, two graph Laplacian terms involving underlying BTD factors of high-resolution HSI are developed, which ensures the inheritance of the spatial geometric structures. Finally, by incorporating graph Laplacian priors with the coupled BTD degradation model, a HSI super-resolution model is established. Experimental results demonstrate that the proposed method achieves better fused results compared with other advanced super-resolution methods, especially on the improvement of the spatial structure.

Fractional centralities on networks: Consolidating the local and the global.

We propose a new centrality incorporating two classical node-level centralities, the degree centrality and the information centrality, which are considered as local and global centralities, respectively. These two centralities have expressions in terms of the graph Laplacian L, which motivates us to exploit its fractional analog L^{γ} with a fractional parameter γ. As γ varies from 0 to 1, the proposed fractional version of the information centrality makes intriguing changes in the node centrality rankings. These changes could not be generated by the fractional degree centrality since it is mostly influenced by the local aspect. We prove that these two fractional centralities behave similarly when γ is close to 0. This result provides its complete understanding of the boundary of the interval in which γ lies since the fractional information centrality with γ=1 is the usual information centrality. Moreover, our computation for the correlation coefficients between the fractional information centrality and the degree centrality reveals that the fractional information centrality is transformed from a local centrality into being a global one as γ changes from 0 to 1.

Local KPZ Behavior Under Arbitrary Scaling Limits

One of the main difficulties in proving convergence of discrete models of surface growth to the Kardar-Parisi-Zhang (KPZ) equation in dimensions higher than one is that the correct way to take a scaling limit, so that the limit is nontrivial, is not known in a rigorous sense. To understand KPZ growth without being hindered by this issue, this article introduces a notion of "local KPZ behavior", which roughly means that the instantaneous growth of the surface at a point decomposes into the sum of a Laplacian term, a gradient squared term, a noise term that behaves like white noise, and a remainder term that is negligible compared to the other three terms and their sum. The main result is that for a general class of surfaces, which contains the model of directed polymers in a random environment as a special case, local KPZ behavior occurs under arbitrary scaling limits, in any dimension.

Variational Auto-Encoder for 3D Garment Deformation Prediction

Traditional garment animation workflow relies on the professional clothing simulator, which requires manual editing of artists or animators. There is no doubt that such a process is time-consuming and laborious. Synthesizing garment dynamics according to the input high-level parameters in a semi-automatic way not only helps dismiss the domain gap between inspiration and technical implementation, but also enables artists to focus on the authoring of animating contents. To that end, a variational auto-encoder-based garment animation synthesis method is presented. Firstly, a set of motion sequences composed of different poses are sampled to generate the human body dataset. Secondly, a variational auto-encoder network is constructed to learn the probabilistic distribution of clothing deformation from garment motions under different pose variations. Besides, a mesh Laplacian term on the loss function is introduced to preserve wrinkle details of the synthesized garments. After that, constraints on the latent space are imposed to control the garment shape to be generated. Finally, a refinement process is employed to resolve the penetration between the body surface and garment mesh, obtaining realistic clothing deformations. Proposed method is qualitatively and quantitatively evaluated on the AMASS dataset from different aspects: body motion/shape-driven garment synthesis, garment animation authoring. The experimental results demonstrate that proposed workflow is able to produce visually realistic garments without noticeable artifacts. Proposed method can produce temporally-consistent garment dynamics with shape and pose variations, which assists artists in authoring the desired clothing deformations.

Robust leader-following consensus control of one-sided Lipschitz multi-agent systems over heterogeneous matching uncertainties

This paper presents a novel consensus control methodology for the multi-agent systems (MASs) with one-sided Lipschitz (OSL) nonlinearities under heterogeneous matching uncertainties, non-zero input to the leader, and external perturbations. A leader-following consensus control scheme is considered, and a control law is provided to assure that the followers track the leader successfully to deal with the uncertainties. The present work considers heterogeneous matching uncertainties, and a new approach with different Lyapunov function, rather than using Laplacian matrix term in the Lyapunov function, has been considered for complete elimination of the uncertainties. The proposed strategy is extended to the case where exogenous disturbances are also acting (to deal with un-matching uncertainties) on the nonlinear dynamics, and a robust consensus control approach is revealed. To overcome the chattering problem, a continuous-time consensus control law is also presented to guarantee the convergence of the consensus error to a small bounded set in the presence of external disturbance. To the best of the authors’ knowledge, consensus control of a generalized form of OSL agents for the complete elimination of heterogeneous matching uncertainties and for considering non-zero input to the leader has been explored for the first time. The developed treatment is different from the conventional methods on the linear or Lipschitz nonlinear multi-agents under heterogeneous uncertainties. Finally, an application of the proposed methods to moving agents is furnished to illustrate the strength of the resultant consensualization protocol.

Discrete-time random walks and Lévy flights on arbitrary networks: when resetting becomes advantageous?

The spectral theory of random walks on networks of arbitrary topology can be readily extended to study random walks and Lévy flights subject to resetting on these structures. When a discrete-time process is stochastically brought back from time to time to its starting node, the mean search time needed to reach another node of the network may be significantly decreased. In other cases, however, resetting is detrimental to search. Using the eigenvalues and eigenvectors of the transition matrix defining the process without resetting, we derive a general criterion for finite networks that establishes when there exists a non-zero resetting probability that minimizes the mean first passage time (MFPT) at a target node. Right at optimality, the coefficient of variation of the first passage time is not unity, unlike in continuous time processes with instantaneous resetting, but above 1 and depends on the minimal MFPT. The approach is general and applicable to the study of different discrete-time ergodic Markov processes such as Lévy flights, where the long-range dynamics is introduced in terms of the fractional Laplacian of the graph. We apply these results to the study of optimal transport on rings and Cayley trees.

Brain Functional Connectivity Analysis via Graphical Deep Learning.

Graphical deep learning models provide a desirable way for brain functional connectivity analysis. However, the application of current graph deep learning models to brain network analysis is challenging due to the limited sample size and complex relationships between different brain regions. In this work, a graph convolutional network (GCN) based framework is proposed by exploiting the information from both region-to-region connectivities of the brain and subject-subject relationships. We first construct an affinity subject-subject graph followed by GCN analysis. A Laplacian regularization term is introduced in our model to tackle the overfitting problem. We apply and validate the proposed model to the Philadelphia Neurodevelopmental Cohort for the brain cognition study. Experimental analysis shows that our proposed framework outperforms other competing models in classifying groups with low and high Wide Range Achievement Test (WRAT) scores. Moreover, to examine each brain region's contribution to cognitive function, we use the occlusion sensitivity analysis method to identify cognition-related brain functional networks. The results are consistent with previous research yet yield new findings. Our study demonstrates that GCN incorporating prior knowledge about brain networks offers a powerful way to detect important brain networks and regions associated with cognitive functions.

Small order asymptotics for nonlinear fractional problems

We study the limiting behavior of solutions to boundary value nonlinear problems involving the fractional Laplacian of order 2s when the parameter s tends to zero. In particular, we show that least-energy solutions converge (up to a subsequence) to a nontrivial nonnegative least-energy solution of a limiting problem in terms of the logarithmic Laplacian, i.e. the pseudodifferential operator with Fourier symbol \(\ln (|\xi |^2)\). These results are motivated by some applications of nonlocal models where a small value for the parameter s yields the optimal choice. Our approach is based on variational methods, uniform energy-derived estimates, and the use of a new logarithmic-type Sobolev inequality.

DFT studies of camptothecins cytotoxicity II. Protonated lactone forms of camptothecin

Cytotoxicity of camptothecin (CPT) in neutral lactone form (I) and its most stable forms protonated at pyridine nitrogen (II) and at pyridone oxygen (III), which is much less stable, is investigated at the B3LYP/6-311G* level of theory. The structures of their hypothetical Cu(II) complexes were optimized. The reduced experimental cytotoxicity of protonated CPT salts is rationalized in terms of metal-ion affinities, Cu charges, and electron-density Laplacians of Cu-ligand bond critical points. Electron density transfer to Cu occurs predominantly from O and N atoms. The carbonyl oxygen atom at the lactone ring is the most reactive site in all CPT forms. Protonation may weaken hydrogen bonding of CPT with topoisomerase I and DNA as well.

Selecting the Best Part from Multiple Laplacian Autoencoders for Multi-view Subspace Clustering

The multi-view subspace clustering attracts much attention in recent years. Most methods follow the framework of fusing the affinity graph learned in each view. In this framework, both the fusion strategy and built graph of each view are very important. In this paper, we propose novel methods for multi-view subspace clustering to address these two aspects. On the one hand, we adopt the autoencoders with Laplacian regularization to construct the affinity graph in each view. Compared with previous work employing the autoencoders, the Laplacian term in our method can guide the learned latent representation favoring affinity extraction. Besides, we also discuss the reasons for adding Laplacian regularization. On the other hand, we propose a novel fusion strategy distinguished from the related literature. If the affinity graph of some view is not extracted well, the performance of previous fusion strategies will be seriously affected. Since our strategy can choose the best part from each affinity graph, it can overcome this limitation to some extent. Extensive experimental results on multiple benchmark data sets confirm the effectiveness of our method.

Laplacian-perturbed Lagrange-multiplier formulation for fluid-structure interaction

This paper presents a stabilized Lagrange-multiplier formulation for Euler-Lagrange type fluid-structure interaction (FSI) computations. The formulation adds a Laplacian perturbation term to the multiplier equation to relax over-constraint condition caused by the multimer formulation. The over-constraint occurs when the Euler mesh for fluid computation is finer than Lagrange meshes for structures around the FSI interfaces. We show a two-dimensional computation that demonstrates relaxation effect of the proposed method.

Optimal Bayesian smoothing of functional observations over a large graph

In modern contexts, some types of data are observed in high-resolution, essentially continuously in time. Such data units are best described as taking values in a space of functions. Subject units carrying the observations may have intrinsic relations among themselves, and are best described by the nodes of a large graph. It is often sensible to think that the underlying signals in these functional observations vary smoothly over the graph, in that neighboring nodes have similar underlying signals. This qualitative information allows borrowing of strength over neighboring nodes and consequently leads to more accurate inference. In this paper, we consider a model with Gaussian functional observations and adopt a Bayesian approach to smoothing over the nodes of the graph. We characterize the minimax rate of estimation in terms of the regularity of the signals and their variation across nodes quantified in terms of the graph Laplacian. We show that an appropriate prior constructed from the graph Laplacian can attain the minimax bound, while using a mixture prior, the minimax rate up to a logarithmic factor can be attained simultaneously for all possible values of functional and graphical smoothness. We also show that in the fixed smoothness setting, an optimal sized credible region has arbitrarily high frequentist coverage. A simulation experiment demonstrates that the method performs better than potential competing methods like the random forest. The method is also applied to a dataset on daily temperatures measured at several weather stations in the US state of North Carolina.

An optimal finite-difference method based on the elongated stencil for 2D frequency-domain acoustic-wave modeling

Frequency-domain finite-difference (FDFD) modeling plays an important role in exploration seismology. However, a major disadvantage of FDFD modeling is the computational cost, especially for large-scale models. By compactly distributing nonzero strips, the elongated stencil helps to generate a narrow-bandwidth impedance matrix, improving computational efficiency without sacrificing numerical accuracy. To further improve the accuracy and efficiency of modeling, we have developed an optimal FDFD method with an elongated stencil for 2D acoustic-wave modeling. The Laplacian term is approximated using the directional-derivative method and the average-derivative method. The dispersion analysis indicates that this elongated-stencil-based method (ESM) achieves higher accuracy than other finite-difference methods with the elongated stencil, and it is more suitable for large grid-spacing ratios. To keep the phase-velocity error within 1%, 15-point and 21-point schemes in the ESM only require approximately 2.28 and 2.19 grid points per wavelength, respectively, when the grid-spacing ratio, namely, the ratio of directional sampling intervals, is not less than 1.5. Moreover, we also adopt a variable-stencil-length scheme, in which the stencil length varies with the velocity, to further reduce the computational cost in frequency-domain modeling. Several numerical examples are presented to demonstrate the effectiveness of our ESM.

Knowledge Preserving and Distribution Alignment for Heterogeneous Domain Adaptation

Domain adaptation aims at improving the performance of learning tasks in a target domain by leveraging the knowledge extracted from a source domain. To this end, one can perform knowledge transfer between these two domains. However, this problem becomes extremely challenging when the data of these two domains are characterized by different types of features, i.e., the feature spaces of the source and target domains are different, which is referred to as heterogeneous domain adaptation (HDA). To solve this problem, we propose a novel model called Knowledge Preserving and Distribution Alignment (KPDA), which learns an augmented target space by jointly minimizing information loss and maximizing domain distribution alignment. Specifically, we seek to discover a latent space, where the knowledge is preserved by exploiting the Laplacian graph terms and reconstruction regularizations. Moreover, we adopt the Maximum Mean Discrepancy to align the distributions of the source and target domains in the latent space. Mathematically, KPDA is formulated as a minimization problem with orthogonal constraints, which involves two projection variables. Then, we develop an algorithm based on the Gauss–Seidel iteration scheme and split the problem into two subproblems, which are solved by searching algorithms based on the Barzilai–Borwein (BB) stepsize. Promising results demonstrate the effectiveness of the proposed method.

Drug–disease associations prediction via Multiple Kernel-based Dual Graph Regularized Least Squares

Predicting associations in drug–disease network provides effective information for the drug repositioning. Therefore, it is an important task to develop an effective drug–disease association prediction method. In this paper, we propose a model, called the Multiple Kernel-based Dual Graph Regularized Least Squares Model (MKDGRLS), to predict potential drug–disease associations. First, we calculate the multiple kernels of drug and disease spaces respectively. Moreover, we utilize multiple kernels and related Laplacian regularization terms to construct MKDGRLS. Finally, we solve the object function of MKDGRLS by the Alternating Least squares algorithm (ALSA), for predicting drug–disease associations. Simultaneously, we design a variant MKDGRLS-A, which is added one dimension as the bias for compensate the inexact solution of linear systems. Our proposed approach has better prediction performance than existing prediction tools under two types of cross validation on the three real drug–disease association network datasets. The results of the case studies of the two diseases prove that our model can effectively predict new associations. We also test MKDGRLS on six real-world networks and it achieves better performance than other methods. In conclusion, our research work can effectively discover potential drug–disease associations, and can provide effective help for related drug repositioning work.

Data-driven fractional subgrid-scale modeling for scalar turbulence: A nonlocal LES approach

Filtering the passive scalar transport equation in the large-eddy simulation (LES) of turbulent transport gives rise to the closure term corresponding to the unresolved scalar flux. Understanding and respecting the statistical features of subgrid-scale (SGS) flux is a crucial point in robustness and predictability of the LES. In this work, we investigate the intrinsic nonlocal behavior of the SGS passive scalar flux through studying its two-point statistics obtained from the filtered direct numerical simulation (DNS) data for passive scalar transport in homogeneous isotropic turbulence (HIT). Presence of long-range correlations in true SGS scalar flux urges to go beyond the conventional local closure modeling approaches that fail to predict the non-Gaussian statistical features of turbulent transport in passive scalars. Here, we propose an appropriate statistical model for microscopic SGS motions by taking into account the filtered Boltzmann transport equation (FBTE) for passive scalar. In FBTE, we approximate the filtered equilibrium distribution with an α-stable Lévy distribution that essentially incorporates a power-law behavior to resemble the observed nonlocal statistics of SGS scalar flux. Generic ensemble-averaging of such FBTE lets us formulate a continuum level closure model for the SGS scalar flux appearing in terms of fractional-order Laplacian that is inherently nonlocal. Through a data-driven approach, we infer the optimal version of our SGS model using the high-fidelity data for the two-point correlation function between the SGS scalar flux and filtered scalar gradient, and sparse linear regression. In an a priori test, the optimal fractional-order model yields a promising performance in reproducing the probability distribution function (PDF) of the SGS dissipation of the filtered scalar variance compared to its true PDF obtained from the filtered DNS data.