Local Regularity Research Articles

Boundedness of Wolff-type potentials and applications to PDEs

We provide a short proof of a sharp rearrangement estimate for a generalized version of a potential of Wolff–Havin–Maz’ya type. As a consequence, we prove a reduction principle for that integral operators, that is, a characterization of those rearrangement invariant spaces between which the potentials are bounded via a one-dimensional inequality of Hardy-type.Since the special case of the mentioned potential is known to control precisely very weak solutions to a broad class of quasilinear elliptic PDEs of non-standard growth, we infer the local regularity properties of the solutions in rearrangement invariant spaces for prescribed classes of data.

Nondensely defined partial neutral functional integrodifferential equations with infinite delay under the light of integrated resolvent operators

Abstract In this work, we mainly focus on the local existence and regularity of integral solutions for a class of nondensely defined partial neutral functional integrodifferential equations with unbounded delay. We use the theory of integrated resolvent operators introduced by Oka [H. Oka, Integrated resolvent operators, J. Integral Equations Appl. 7 1995, 2, 193–232]. Finally, we provide an example to demonstrate the basic findings of our work.

Hierarchical graph augmented stacked autoencoders for multi-view representation learning

With recent success of deep neural networks, stacked autoencoder networks have received a lot of attention for robust unsupervised representation learning. However, recent autoencoder methods cannot make full use of multi-view information and thus fail to further improve many real-world applications by exploring the geometric structures of multi-view data. To address the above-mentioned issue, we introduce hierarchical graph augmented stacked autoencoders (HGSAE) for unsupervised multi-view representation learning. Specifically, a hierarchical graph structure is first adapted to stacked autoencoders to learn view-specific representations, aiming to preserve the geometric information of multi-view data through local and non-local graph regularizations. A general or common representation can then be learned by reconstructing each single view using fully connected neural networks. By doing this, the proposed method not only preserves the geometric information in multi-view data but also automatically balances the complementarity/consistency among different views. Extensive experiments on six popular unsupervised representation learning datasets demonstrate the effectiveness of our method when compared with recent state-of-the-art autoencoder methods.

Role of locality, fidelity and symmetry regularization in learning explainable representations

Despite their success deep neural networks still lack interpretability and are regarded as black boxes. This hampers a wider adoption in applications with societal, environmental or economical implications, and motivated a variety of techniques for explaining their outputs. Such explanations are however typically produced after model training so there is no guarantee that models learn faithful attributions, a goal they were not trained for. We evaluate the impact of different penalty terms in the loss function that promote explainable feature attributions, and that can be learned during training in an unsupervised way. We show that explainability-constrained models produce better saliency maps based on multiple metrics and tests. Regularizers imposing locality, fidelity and symmetry properties lead to the best performances in terms of MoRF and ROAR scores.

Unbounded Weak Solutions to Strongly q-Nonlinear Elliptic Systems. Local Regularity

Unbounded Weak Solutions to Strongly q-Nonlinear Elliptic Systems. Local Regularity

Quaternion Methods and Regular Models of Celestial Mechanics and Space Flight Mechanics: Local Regularization of the Singularities of the Equations of the Perturbed Spatial Restricted Three-Body Problem Generated by Gravitational Forces

Quaternion Methods and Regular Models of Celestial Mechanics and Space Flight Mechanics: Local Regularization of the Singularities of the Equations of the Perturbed Spatial Restricted Three-Body Problem Generated by Gravitational Forces

Numerical solution of one inverse problem for a diffusion model of a chemical-technological process

A chemical-technological process taking place in a chemical reactor with a chemical reaction of the second order is considered. A one-dimensional one-parameter diffusion model of the hydrodynamic flow in the reactor is proposed for the mathematical description of this process. Within the framework of the proposed model, the inverse problem is posed to determine the concentration of the selected reagent in the incoming flow, ensuring the implementation of a predetermined hydrodynamic regime at the reactor outlet. A discrete analogue of the inverse problem is constructed and the resulting difference problem is presented as a variational problem with local regularization. A special representation is proposed for the numerical solution of the variational problem. As a result, an explicit formula is obtained for determining the approximate concentration of the selected reagent in the incoming stream. The effectiveness of the proposed method is illustrated by numerical calculations for model problems

Local Regularity Near Boundary for the Stokes and Navier–Stokes Equations

Local Regularity Near Boundary for the Stokes and Navier–Stokes Equations

Semi-supervised local manifold regularization model based on dual representation for industrial soft sensor development

Data-driven soft sensors are crucial for predicting key quality variables in the process industry. In most processes, the dynamic characteristics are obvious, and the relationship between the primary and secondary variables is strongly nonlinear. Moreover, the worst thing is that labeled samples are usually scarce due to certain technical difficulties or measurement costs. To deal with these issues, this paper first proposes a semi-supervised manifold regularization model based on dual representation (SsMRM-DR). Then, a semi-supervised local manifold regularization model based on dual representation (SsLMRM-DR) is further proposed from the viewpoint of local manifold regularization. In the SsLMRM-DR, a space–time weighted similarity calculation method is designed to reduce the influence of measurement noise on similarity calculation. At the same time, a local manifold regularization method is developed to use unlabeled samples more efficiently and to improve the prediction accuracy and computational efficiency of the SsLMRM-DR. A numerical example and a real industrial process are used to evaluate the performance of the proposed schemes. The results show that SsLMRM-DR is effective and it has a good application prospect of soft sensors in the industry.

Quaternion Methods and Regular Models of Celestial Mechanics and Space Flight Mechanics: Local Regularization of the Singularities of the Equations of the Perturbed Spatial Restricted Three-Body Problem Generated by Gravitational Forces

The problem of local regularization of differential equations of a perturbed spatial restricted three-body problem is studied: elimination of singularities (dividing by zero) generated by gravity forces of differential equations of perturbed spatial motion of a material point M, which has a negligibly small mass, in the vicinity of two gravitating points M0 and M1 by writing equations of motion in rotating coordinate systems, the use of new regular variables and the regularizing transformation of time. Various systems of regular quaternion differential equations (RQDE) for this problem are obtained. The following groups of variables act as variables in these equations: (1) four-dimensional Kustaanheimo–Stiefel variables, Keplerian energies and time t, (2) distances from the point M to the points M0 and M1, modules of the vectors of the moment of velocities of the point M with respect to the points M0 and M1, Keplerian energy, time t and Euler (Rodrigues–Hamilton) parameters characterizing the orientations of the orbital coordinate systems in the inertial coordinate system; (3) two-dimensional Levi-Civita variables describing the motion of the point M in ideal coordinate systems, Keplerian energies, time t and Euler parameters characterizing the orientations of ideal coordinate systems in the inertial coordinate system and being osculating elements (slowly changing variables) for the motion of the point M in the neighborhood gravitating point M0 or M1, respectively. To construct the RQDE, the equations of the perturbed spatial restricted three-body problem, written either in nonholonomic (azimuthally free), or in orbital, or in ideal coordinate systems, were used as initial ones; “fictitious” times τ0 and τ1 are used as new independent variables (i.e., regularizing differential transformations of the Sundmann time are used) or angular variables φ0 and φ1, which are traditionally used in the study of orbital motion as part of polar coordinates. To match the two independent variables used in the vicinity of the gravitating points M0 and M1, additional differential equations are used. The obtained various locally regular quaternion differential equations of the perturbed spatial restricted three-body problem make it possible to develop regular analytical and numerical methods for studying the motion of a body of negligibly small mass in the vicinity of two other bodies with finite masses, and also make it possible to construct regular algorithms for the numerical integration of these equations. The equations can be effectively used to study the orbital motion of celestial and cosmic bodies and spacecraft, to predict their motion, as well as to solve problems of controlling the orbital motion of spacecraft and solving problems of inertial navigation in space.

Matrix weights and regularity for degenerate elliptic equations

We prove local boundedness, Harnack inequality and local regularity for weak solutions of quasilinear degenerate elliptic equations in divergence form. Degeneracy is via a non negative, symmetric, measurable matrix-valued function Q(x) and two suitable non negative weight functions. We setup an axiomatic approach in terms of suitable geometric conditions and local Sobolev–Poincaré inequalities. Data integrability is close to L1 and it is exploited in terms of suitable version of Stummel-Kato class that in some cases is also necessary to the regularity.

Cross-domain privacy-preserving broad network for fault diagnosis of rotating machinery

The broad learning system-based data processing method has been widely used in the field of industrial intelligent operation and maintenance, and has achieved impressive results. However, data island created by privacy concerns in real industrial scenarios makes centralized deep training extremely difficult. In addition, the fluctuating operating conditions of machines including speed and load significantly affect the distribution of monitoring data representing the operating state of the device. These two problems make it difficult to apply broad learning system to practical application scenarios. Therefore, in view of the above problems, a novel cross-domain privacy-preserving broad network (CDPPBN) for cross-domain fault classification of rotating machinery is presented. Specifically, a cross-domain interactive data encryption protocol feature extractor is designed firstly to ensure the privacy and security of both source and target domain data during feature extraction process. Then, a domain adaptation target classifier is constructed to achieve the fault classification task in the target domain without labels. In this classifier, a maximum mean discrepancy regularization term is constructed to match the mean projections of the source and target domains to adapt the source classifier to the target domain, and a semi-supervised local graph embedding regularization term is constructed to mine the structural features of the target domain data. CDPPBN not only has high performance in terms of data privacy protection, but also makes full use of both labeled source data and unlabeled target data to generate a domain adaptation target classifier. The effectiveness of the presented method is verified through bearing monitoring signal. Experimental analysis shows that the proposed CDPPBN has better data privacy protection capability and data cross-domain diagnosis ability compared with other methods.

On the Cauchy problem for a four‐component Novikov system with peaked solutions

Considered herein is the Cauchy problem for a four‐component Novikov system with peaked solutions. We first investigate the local Gevrey regularity and analyticity of the solutions by a generalized Ovsyannikov theorem. Then, based on the local well‐posedness of this problem, the results with respect to the nonuniformly continuous dependence on initial data of the solutions in Besov spaces and are established by constructing new approximate solutions and initial data.

A Discontinuous Galerkin and Semismooth Newton Approach for the Numerical Solution of Bingham Flow with Variable Density

Abstract This paper is devoted to the study of Bingham flow with variable density. We propose a local bi-viscosity regularization of the stress tensor based on a Huber smoothing step. Next, our computational approach is based on a second-order, divergence-conforming discretization of the Huber regularized Bingham constitutive equations, coupled with a discontinuous Galerkin scheme for the mass density. We take advantage of the properties of divergence-conforming and discontinuous Galerkin formulations to effectively incorporate upwind discretizations, thereby ensuring the stability of the formulation. The stability of the continuous problem and the fully discrete scheme are analyzed. Further, a semismooth Newton method is proposed for solving the obtained fully discretized system of equations at each time step. Finally, several numerical examples that illustrate the main features of the problem and the properties of the numerical scheme are presented.

On the Cauchy problem of the two-component Novikov-type system with peaked solutions and H1-conservation law

Considered herein is the Cauchy problem for the two-component Novikov-type system with peaked solutions and [Formula: see text]-conservation law. At first, we establish that the solutions maintain corresponding properties at infinity within the lifespan provided that the initial data decay exponentially and algebraically, respectively. Next, the local regularity and analyticity of the solutions to this problem in Sobolev–Gevrey spaces are discussed by a generalized Ovsyannikov theorem in detail.

BLoG: Bootstrapped graph representation learning with local and global regularization for recommendation

With the explosive growth of online information, the significant application value of recommender systems has received considerable attention. Since user–item interactions can naturally fit into graph structure data, graph neural networks (GNNs), by virtue of their strong ability in graph representation learning, have become the new state-of-the-art approach to recommender systems. Recently, GNN-based contrastive self-supervised learning (SSL) methods have received careful attention due to their superiority over graph-based recommendation under the typical supervised learning paradigm. However, to achieve state-of-the-art performance, GNN-based recommendation with SSL often needs a huge amount of negative examples and the model’s performance is heavily dependent on complex data augmentations. Also, the information interaction among various augmented views is often performed under a single perspective (e.g., structure/feature space or node/graph level). In this paper, we propose a novel bootstrapped graph representation learning with local and global regularization for recommendation, i.e., BLoG, which constructs positive/negative pairs based on the aggregated node features by referring to two alternate views of the original user–item graph structure. In particular, BLoG learns user–item representations by encoding two augmented versions of a user–item bipartite graph using two separate encoders: an online encoder and a target encoder. To facilitate the information interaction between these two distinct graph encoders, we introduce local and global regularization for recommendation, where a graph structural contrastive loss and a node-level semantic loss are defined for local regularization while a graph-level contrastive loss is used for global regularization. An alternative optimization approach is used to train the online encoder and the target encoder. Experimental studies on three benchmark datasets demonstrate that BLoG achieves better recommendation accuracy than several existing baselines.

LOCAL EXISTENCE AND REGULARITY OF SOLUTIONS FOR SOME SECOND ORDER DIFFERENTIAL EQUATION WITH INFINITE DELAY

In this work, we study the existence and regularity of solutions for some second order differential equations with infinite delay in Banach spaces. We suppose that the undelayed part admits a cosine operator in the sense given by Da Prato and Giusi, [ G. Da Prato and E. Giusi, \emph{Una caratterizzazione dei generatori di funzioni coseno astratte}, Bollettino dell'Unione Matematica Italiana, 22, 357-362, (1967)]. The delayed part is assumed to be locally Lipschitz. Firstly, we show the existence of the mild solutions. Secondly, we give sufficient conditions ensuring the existence of strict solutions.

On Conditions for Controllability and Local Regularity of A System of Differential Equations

We consider a system of differential equations on a Banach space X given by: x'(t) = Ax(t) + u(t)f(t, x(t)), x(0) = x0, where A is an infinitesimal generator of a C0-semigroup, f : R0+ × X → X is a locally Lipschitz function, and u ∈ Lp([0, T], R) is a control defined on [0, T] with 1 < p ≤ ∞. Using the Compactness Principle and the generalization of Gronwalls Lemma, the system is shown to be controllable for a γ-bounded function f. Another result of this study is the local existence and the uniqueness of the solution of the system for locally bounded function f through weighted ω-norm.

Adaptive and local regularization for data fitting by tensor-product spline surfaces

Adaptive and local regularization for data fitting by tensor-product spline surfaces

Comparison of local and nonlocal regularization approaches in Eulerian-based finite element analyses

Finite element (FE) results might suffer from significant mesh dependency, especially when modeling shear bands in strain-softening soils. On the other hand, the shear bands in the field are generally thin. The computational costs might dramatically increase when modeling such thin shear bands, especially for large-scale problems, such as progressive landslides. To overcome these issues, FE analyses are generally performed with larger shear band thicknesses by adopting ‘softening scaling’ rules based on local and nonlocal (averaged) strains at the shear bands. The present study compares the performances of local and nonlocal approaches, with a specific focus on softening scaling and large deformation. Analyses are performed using a Eulerian-based FE program, which can model large strains in the shear band without numerical issues related to mesh distortion. Implementing strain-softening behavior in local and nonlocal methods, two idealized cases are simulated: (i) biaxial compression test, (ii) slope failure due to upslope surface loading. FE simulations show that the macroscopic response (i.e., load–displacement behavior) can be modeled using both local and nonlocal regularization techniques. Shear band thickness increases with the progress of shearing. In local analysis, mesh orientation has a considerable effect on shear band thickness. The existence of neighboring shear bands could affect nonlocal strain calculation and the simulation results.