Global Regularity Research Articles

Maximum Principle and Second-Order Optimality Conditions in Control Problems with Mixed Constraints

This article concerns the optimality conditions for a smooth optimal control problem with an endpoint and mixed constraints. Under the normality assumption, which corresponds to the full-rank condition of the associated controllability matrix, a simple proof of the second-order necessary optimality conditions based on the Robinson stability theorem is derived. The main novelty of this approach compared to the known results in this area is that only a local regularity with respect to the mixed constraints, that is, a regularity in an ε-tube about the minimizer, is required instead of the conventional stronger global regularity hypothesis. This affects the maximum condition. Therefore, the normal set of Lagrange multipliers in question satisfies the maximum principle, albeit along with the modified maximum condition, in which the maximum is taken over a reduced feasible set. In the second part of this work, we address the case of abnormal minimizers, that is, when the full rank of controllability matrix condition is not valid. The same type of reduced maximum condition is obtained.

Regularization tunnelling for full waveform inversion

SUMMARY Prior knowledge can be a powerful source of information in seismic inversion, leading to substantially more accurate models than those derived from seismic data alone. Prior knowledge about clustering of rock physics properties, supplied by appropriate rock physics models, or nearby well-logs, may be especially informative. However, incorporating general forms of this type of information into local optimization methods, such as those used in full waveform inversion (FWI), is problematic, since it will often call for model updates involving a departure from the basin of a local minimum. Here, we propose an optimization strategy for FWI, in which global regularization information is partially accounted for, and in which a model can, under the right circumstances, ‘tunnel’ between basins, to honour prior information. The process is based on standard FWI updates and begins with an initial model in which every model grid cell is in one of a range of pre-defined clusters. At each iteration, a standard update is computed, but it is then followed by a second update, in which each grid cell tunnels to a new cluster if the move is warranted by the update history of the model (its ‘momentum’) and the model regularization penalty (its ‘potential’). To limit the ability of the tunnelling procedure from introducing spatial variations on scales smaller than those contained in the seismic data, these conditions are considered in combination with a small mode filter. We implement the procedure by building on an existing 2-D viscoelastic FWI algorithm, limited such that only the P-wave velocity and density are updated, and evaluate it by subjecting it to a variety of basic tests involving simulated data. We conclude that the basic approach, and this specific implementation, using a simple initial model, provides a combination of well-resolved structures and grid cells occupying appropriate clusters that cannot be produced by standard means.

Global regularity for the 2D micropolar Rayleigh–Bénard problem with partial dissipation

This paper is devoted to the study of the global regularity problem to the 2D micropolar Rayleigh–Bénard problem with only partial velocity dissipation and micro-rotational dissipation in R2. By fully exploiting the special structure of the system and using the delicate energy methods, we establish the global well-posedness of classical solutions for this system.

On the KPZ equation with fractional diffusion: Global regularity and existence results

In this work we analyze the existence of solutions to the fractional quasilinear problem,(P){ut+(−Δ)su=|∇u|α+f in ΩT≡Ω×(0,T),u(x,t)=0 in (RN∖Ω)×[0,T),u(x,0)=u0(x) in Ω, where Ω is a C1,1 bounded domain in RN, N>2s and 12<s<1. We will assume that f and u0 are non negative functions satisfying some additional hypotheses that will be specified later on.Assuming certain regularity on f, we will prove the existence of a solution to problem (P) for values α<s1−s, as well as the non existence of such a solution when α>11−s. This behavior clearly exhibits a deep difference with the local case.

Global regularity and time decay for the 2D magneto-micropolar system with fractional dissipation and partial magnetic diffusion

Global regularity and time decay for the 2D magneto-micropolar system with fractional dissipation and partial magnetic diffusion

Global regularity for the 3D micropolar fluid flows

The aim of this note is to establish the global regularity of classical solutions of the 3D micropolar fluid equations for a family of large initial data with finite energy.

On regularity and stability for a class of nonlocal evolution equations with nonlinear perturbations

&lt;p style='text-indent:20px;'&gt;We study a class of nonlocal partial differential equations with nonlinear perturbations, which is a general model for some equations arose from fluid dynamics. Our aim is to analyze some sufficient conditions ensuring the global solvability, regularity and stability of solutions. Our analysis is based on the theory of completely positive kernel functions, local estimates and a new Gronwall type inequality.&lt;/p&gt;

Generalization on Unseen Domains via Model-Agnostic Learning for Intelligent Fault Diagnosis

Machine learning-based diagnosis methods have achieved remarkable success under the assumption that the training and test data are identically distributed. However, a critical requirement of these methods is the generalization capability to unseen domains when deployed to actual diagnosis scenarios. We introduce the challenging problem of domain generalization, i.e., learning from multiple source domains to produce a model that can directly generalize to unseen domains without target information. We adopt a model-agnostic learning produce that maximizes the dot product of gradients between the source domains. Such a gradient alignment objective encourages finding a common optimization path for all source domains, which helps to focus on invariant representations. Furthermore, we propose two feature regularizations that explicitly regularize the feature space. Global feature regularization aligns class relationships between different domains to preserve the domain-invariant knowledge. Local feature regularization encourages the model to learn domain-agnostic class-specific representations with intraclass compactness and interclass separability. The effectiveness of the proposed method is demonstrated with generalization experiments on two benchmarks.

ОСНОВНІ ЕКОНОМІЧНІ ЗАКОНОМІРНОСТІ ЗЕЛЕНИХ ТРАНСФОРМАЦІЙ СВІТОВОГО ВИРОБНИЦТВА

The article emphasizes that at today's stage of economic globalization, the fundamental qualitative renovation of the technological base of public production occurs, its efficiency and competitiveness are increasing, as well as a "greening" of the world production complex and global sales chains is dynamically developing. These trends have their own internal drivers and patterns of ensuring transformational changes, which are the purpose and content of this article. The article proposes the key regularities of modern transformational changes in world production that reflect their ecological imperative and systemic nature from the point of view of building a global green ecosystem. The peculiarities of sectoral diversification of the global production complex are characterized as one of the patterns of green transformations in global production. The presence of another global regularity, which proves the systemic nature of green transformational changes in world production – its digitization – has been substantiated in the article. It has been proved that in the complex characteristics of the key patterns of the development of the systemic green transformation of global production, a special place is occupied by the intellectualization of production management and organization systems based on the computerization of existing management systems. A characteristic of the regularity of networking as such, which finds its material embodiment in deep structural transformations of current organizational structures of global business, has been given. The author emphasizes that another objective regularity of its systemic green transformations is closely related to the global trend of networking of world production – there is horizontal structuring and deconcentration of production processes. It has been proved that among the objective regularities of the systemic green transformation in global production, the trend towards its dematerialization also deserves special attention. The author calls decarbonization with a deep structural crisis of the traditional energy sector the next economic regularity that testifies to the systemic nature of modern processes of green transformation in global production.

Global regularity of magneto-micropolar equations with logarithmically dissipation

Global regularity of magneto-micropolar equations with logarithmically dissipation

Some Remarks on the Issue of Second-order Optimality Conditions in Control Problems with Mixed Constraints

A smooth optimal control problem with mixed constraints is considered. Under the normality assumption, a proof of second-order necessary optimality conditions based on the Robinson stability theorem is provided. The main feature of the obtained result is that the local regularity with respect to the mixed constraints is assumed, that is, a regularity in an ε-tube along the minimizer, but not the conventional global regularity hypothesis. This impacts the maximum condition. Therefore, the normal set of Lagrange multipliers fulfills the Legendre-Clebsch condition and the maximum principle. At the same time, the maximum condition is modified since, now, the maximum is taken over a reduced feasible set. Furthermore, the case of abnormal minimizers is considered. The same type of reduced maximum condition is obtained along with a refined Legendre-Clebsch condition which is meaningful in the abnormal case.

Uniqueness of the partial travel time representation of a compact Riemannian manifold with strictly convex boundary

&lt;p style='text-indent:20px;'&gt;In this paper a compact Riemannian manifold with strictly convex boundary is reconstructed from its partial travel time data. This data assumes that an open measurement region on the boundary is given, and that for every point in the manifold, the respective distance function to the points on the measurement region is known. This geometric inverse problem has many connections to seismology, in particular to microseismicity. The reconstruction is based on embedding the manifold in a function space. This requires the differentiation of the distance functions. Therefore this paper also studies some global regularity properties of the distance function on a compact Riemannian manifold with strictly convex boundary.&lt;/p&gt;

Propagation of Regularity and Long Time Behavior of the 3D Massive Relativistic Transport Equation II: Vlasov–Maxwell System

Given any smooth, suitably small initial data, which decays polynomially at infinity, we prove global regularity for the 3D relativistic massive Vlasov–Maxwell system. In particular, the compact support assumption, which was widely used in the literature, is not imposed on the initial data. Our proofs are based on a combination of the Klainerman vector field method and the Fourier method, which allows us to exploit a crucial hidden null structure in the relativistic Vlasov–Maxwell system.

Global stability solution of the 2D incompressible anisotropic magneto-micropolar fluid equations

&lt;abstract&gt;&lt;p&gt;In this paper, we consider the two dimensional incompressible anisotropic magneto-micropolar fluid equations with partial mixed velocity dissipations, magnetic diffusion and horizontal vortex viscosity, and analyze the stability near a background magnetic field. At present, major works on the equations of magneto-micropolar fluid mainly focus on the global regularity of the solutions. While the stability of the solutions remains an open problem. This paper concentrates on establishing the stability for the linear and nonlinear system respectively. Two goals have been achieved. The first is to obtain the explicit decay rates for the solution of the linear system in $ H^s({\mathbb{R}}^2) $ Sobolev space. The second assesses the nonlinear stability by establishing the &lt;italic&gt;a priori estimate&lt;/italic&gt; and employing bootstrapping arguments. Our results reveal that any perturbations near a background magnetic field is globally stable in Sobolev space $ H^2({\mathbb{R}}^2) $.&lt;/p&gt;&lt;/abstract&gt;

On global regularity for a model of the regularized Boussinesq equations with zero diffusion

On global regularity for a model of the regularized Boussinesq equations with zero diffusion

Double phase problems: a survey of some recent results

We review some recent results on double phase problems. We focus on the relevant function space framework, which is provided by the generalized Orlicz spaces. We also describe the basic tools and methods used to deal with double phase problems, given that there is no global regularity theory for these problems.

Global regularity for a family of models of the axisymmetric Navier–Stokes system

Global regularity for a family of models of the axisymmetric Navier–Stokes system

Implicit auditory perception of local and global irregularities in passive listening condition

The auditory system detects differences in sounds at an implicit level, but data on this difference might not be sufficient to make explicit discrimination. The biomarkers of implicit auditory memory of ambiguous stimuli could shed light on unconscious auditory processing and implicit auditory learning. Mismatch negativity (MMN) and P3a, components of event-related potentials (ERPs) reflecting stimuli discrimination without direct attention, were previously detected in response to the local (short-term) irregularity in the auditory sequence even in an unconscious state. At the same time, P3b was elicited only in case of direct attention in response to the global (long-term) irregularity. In this study, we applied the local-global auditory paradigm to obtain possible electrophysiological signatures of implicit detection of hardly distinguishable auditory stimuli. ERPs were recorded from 20 healthy volunteers during active discrimination of deviant sounds in the old-ball sequence and passive listening of the same sounds in the sequence with local-global irregularity. The discrimination task consisted of two blocks with different deviant sounds targeted to respond. The sound discrimination accuracy was at an average of 40%, implying the difficulty of explicit sound recognition. Comparing ERPs to standard and deviant sounds, we found posterior negativity in ERP around 450–600 ms in response to targeted deviant sounds. MMN was significant only in response to non-target deviants. In the passive local-global paradigm, we observed an anterior positivity (284–412 ms), compatible with P3a, in response to a violation of local regularity. Violation of global regularity elicited an anterior negative response (228–586 ms), resembling the N400 component of ERPs. Importantly, the other indexes of auditory discrimination, such as MMN and P3b, were insignificant in ERPs to both regularity violations. The observed P3a and N400 components of ERPs may reflect prediction error signals in the implicit perception of sound patterns even if behavioral recognition was poor.

Weighted distribution approach to gradient estimates for quasilinear elliptic double-obstacle problems in Orlicz spaces

We construct an efficient approach to deal with the global regularity estimates for a class of elliptic double-obstacle problems in Lorentz and Orlicz spaces. Moreover, in the setting of the paper, we are led to the study of problems with nonlinearity is supposed to be partially weak BMO condition (is measurable in one fixed variable and only satisfies locally small-BMO seminorms in the remaining variables). This work also extends some abstract results proved in Nguyen and Tran (2021) [43] by using the weighted fractional maximal distributions (WFMDs). We further investigate a pointwise estimate of the gradient of weak solutions via fractional maximal operators and Riesz potential of data.

A review on arbitrarily regular conforming virtual element methods for second- and higher-order elliptic partial differential equations

The virtual element method is well suited to the formulation of arbitrarily regular Galerkin approximations of elliptic partial differential equations of order [Formula: see text], for any integer [Formula: see text]. In fact, the virtual element paradigm provides a very effective design framework for conforming, finite dimensional subspaces of [Formula: see text], [Formula: see text] being the computational domain and [Formula: see text] another suitable integer number. In this review, we first present an abstract setting for such highly regular approximations and discuss the mathematical details of how we can build conforming approximation spaces with a global high-order regularity on [Formula: see text]. Then, we illustrate specific examples in the case of second- and fourth-order partial differential equations, that correspond to the cases [Formula: see text] and [Formula: see text], respectively. Finally, we investigate numerically the effect on the approximation properties of the conforming highly-regular method that results from different choices of the degree of continuity of the underlying virtual element spaces and how different stabilization strategies may impact on convergence.