Norm Estimates Research Articles

On the Coupling of the Curved Virtual Element Method with the One-Equation Boundary Element Method for 2D Exterior Helmholtz Problems

We consider the Helmholtz equation defined in unbounded domains, external to 2D bounded ones, endowed with a Dirichlet condition on the boundary and the Sommerfeld radiation condition at infinity. To solve it, we reduce the infinite region, in which the solution is defined, to a bounded computational one, delimited by a curved smooth artificial boundary and we impose on this latter a non reflecting condition of boundary integral type. Then, we apply the curved virtual element method in the finite computational domain, combined with the one-equation boundary element method on the artificial boundary. We present the theoretical analysis of the proposed approach and we provide an optimal convergence error estimate in the energy norm. The numerical tests confirm the theoretical results and show the effectiveness of the new proposed approach.

Variational Physics Informed Neural Networks: the Role of Quadratures and Test Functions

In this work we analyze how quadrature rules of different precisions and piecewise polynomial test functions of different degrees affect the convergence rate of Variational Physics Informed Neural Networks (VPINN) with respect to mesh refinement, while solving elliptic boundary-value problems. Using a Petrov-Galerkin framework relying on an inf-sup condition, we derive an a priori error estimate in the energy norm between the exact solution and a suitable high-order piecewise interpolant of a computed neural network. Numerical experiments confirm the theoretical predictions and highlight the importance of the inf-sup condition. Our results suggest, somehow counterintuitively, that for smooth solutions the best strategy to achieve a high decay rate of the error consists in choosing test functions of the lowest polynomial degree, while using quadrature formulas of suitably high precision.

Global attractivity of the equilibria of the diffusive SIR and SEIR epidemic models with multiple parallel infectious stages and nonlinear incidence mechanism

In this paper, we are concerned with the diffusive SIR and SEIR epidemic models with multiple parallel infectious stages and nonlinear incidence mechanism. We first establish a priori L∞-norm estimates for the solutions to the epidemic models, and then derive the global attractivity of disease-free equilibrium and endemic equilibrium for the two models in terms of the basic reproduction number. Our results show that the diversity of infectious stages, the infection mechanism and the diffusion do not affect the global dynamics of the infectious disease.

Toeplitz operator and the operator ∇ on the [formula omitted] space of a tree

Let T be a rooted, countable infinite tree without terminal vertices. Suppose that the degrees of the vertices of T are bounded. In this paper, we obtain a new upper bound for the norm of the operator ∇ on the Lp space of T. Moreover, we establish several sufficient conditions for Toeplitz operators to be compact on Lp with 1⩽p⩽∞. This answers two questions concerning the norm estimate of ∇ and the compactness of Toeplitz operators posed by Colonna and Martínez-Avendaño (2017) [10].

The $L^p$-$L^q$ Boundedness and Compactness of Bergman Type Operators

We investigate Bergman type operators on the complex unit ball, which are singular integral operators induced by the modified Bergman kernel. We consider the $L^p$-$L^q$ boundedness and compactness of Bergman type operators. The results of boundedness can be viewed as the Hardy–Littlewood–Sobolev (HLS) type theorem in the case unit ball. We also give some sharp norm estimates of Bergman type operators which in fact gives the upper bounds of the optimal constants in the HLS type inequality on the unit ball. Moreover, a trace formula is given.

Global well‐posedness for the generalized Navier‐Stokes‐Coriolis equations with highly oscillating initial data

We study the small initial date Cauchy problem for the generalized incompressible Navier‐Stokes‐Coriolis equations in critical hybrid‐Besov space with and . We prove that hybrid‐Besov spaces norm of a class of highly osillating initial velocity can be arbitrarily small, and we prove the estimation of highly frequency smoothing effect for generalized Stokes‐Coriolis semigroup with . At the same time, we prove space‐time norm estimation of hybrid‐Besov spaces for Stokes‐Coriolis semigroup. From this result, we deduce bilinear estimation in our work space. Our method relies upon Bony's high and low‐frequency decomposition technology and Banach fixed point theorem.

Adaptive dynamic surface output feedback control for a class of quadrotor aircraft with actuator faults

PurposeThe purpose of this study is to propose an adaptive fault-tolerant control approach based on output feedback for a class of quadrotor unmanned aerial vehicles system. In the event of a controlled actuator failure, a stable flying of the aircraft can be achieved by selecting an appropriate sliding mode surface.Design/methodology/approachAiming at the actuator failure of quadrotor aircraft during flight in the controllable range, a dynamic surface sliding mode passive fault-tolerant controller based on output feedback is designed based on the strong robustness of sliding mode method. Due to the unknown nonlinearity dynamics and parameter uncertainties in the system, a nonlinear observer is used to estimate them online.FindingsThe stability of the suggested algorithm is established using appropriate Lyapunov functions, and the performance of the proposed control approach is demonstrated using hardware-in-the-loop simulation.Originality/valueAn error performance function is introduced into the controller to ensure the convergence speed and accuracy of errors are within the predetermined range. By using the norm estimation method, there is only one parameter that needs to be updated in each step of the control process, which considerably minimizes the calculation burden. Finally, the validity of the proposed control scheme is verified on the hardware-in-the-loop simulation, and the results show that the proposed control method has achieved the desired results.

Coconvex multi approximation in the uniform norm

In this research, we study the results about coconvex multi approximation. Recently this subject received a lot of attention. The first result in the topic is a direct theorem for the error bound of the convex approximation for functions of two continuous derivatives that were introduced by Kopotun. In 2006 Kopotun, Leviatan and Shevchuk gave definitive answers to open problems of Jackson’s estimates by the moduli of smoothness of Ditzian- Totiktype, also in 2019 they stated many open problems in the coconvex case and mentioned recent developments on this subject. In convex multi approximation, we have a function f ε C[-1, 1] d change its convexity at a finite number of times in [-1, 1] d and we take care of error bound of approximation of the function f using multi algebraic polynomials, in other words, multi algebraic polynomials change its convex particularly at the same points where f is. We can get some of Jackson’s estimates in which the intended constants depend on the place of the convexity change points. In order to clarify the complexities, we discuss in this research invers inequality for the coconvex approximation of the function f using multi algebraic polynomials.

Optimal error bound for immersed weak Galerkin finite element method for elliptic interface problems

In Mu and Zhang (2019), an immersed weak Galerkin finite element method (IWG-FEM) is developed for solving elliptic interface problems and it is proved that this method has optimal a-priori error estimate in an energy norm under artificial smoothness assumption on the solution. In this study, we prove that IWG-FEM converges optimally in energy norm under natural smoothness assumption on solution. Furthermore, we show that IWG-FEM converges optimally in the L2 norm which did not present in Mu and Zhang (2019) because of the artificial H3 smoothness requirement. A series of numerical experiments are conducted and reported to verify the theoretical finding.

Multiple vector-valued, mixed norm estimates for Littlewood-Paley square functions

We prove that for any $L^Q$-valued Schwartz function $f$ defined on $\mathbb{R}^d$, one has the multiple vector-valued, mixed norm estimate $$ \| f \|_{L^P(L^Q)} \lesssim \| S f \|_{L^P(L^Q)} $$ valid for every $d$-tuple $P$ and every $n$-tuple $Q$ satisfying $0 < P, Q < \infty$ componentwise. Here $S:= S_{d_1}\otimes ... \otimes S_{d_N}$ is a tensor product of several Littlewood-Paley square functions $S_{d_j}$ defined on arbitrary Euclidean spaces $\mathbb{R}^{d_j}$ for $1\leq j\leq N$, with the property that $d_1 + ... + d_N = d$. This answers a question that came up implicitly in our recent works and completes in a natural way classical results of the Littlewood-Paley theory. The proof is based on the \emph{helicoidal method} introduced by the authors.

An optimal error estimate of the BDF-Galerkin FEM for the incompressible MHD system

In this article, a linearized backward differential formula type scheme for the numerical approximations of the magnetohydrodynamics equations is analyzed. The proposed fully discretized numerical scheme, which is comprised of a third order backward differential formula for time derivative terms, extrapolated treatments in linearization for nonlinear terms and mixed finite element approximation for spatial discretization. Based on the temporal-spatial error splitting technique, we demonstrate the almost unconditional optimal error estimate in L2 norm for the fluid velocity and the magnetic by getting boundedness of numerical solutions in certain norms. Finally, several numerical tests are carried out to verify the theoretical results.

Global solutions for the modified Camassa–Holm equation

AbstractThis paper is concerned with the global solutions for the modified Camassa–Holm equation. We first obtain the global existence and uniqueness of strong solutions for the initial datum with , and then give some norm estimates of strong solutions and its momentum density. Finally, by using the compensated compactness method and regularization technique, we establish the global existence and uniqueness of weak solutions for the initial datum .

Analysis of the numerical solution of the three-phase nonisothermal fluid flow problem

This paper is devoted to the construction and study of the stability and convergence of a numerical method for solving the problem of three-phase non-isothermal fluid flow in porous media, taking into account capillary forces, which was carried out within the framework of the grant project No. AP08053189 provided by the Ministry of Education and Science of the Republic of Kazakhstan. The model under consideration describes the processes occurring in oil reservoirs during the production of heavy oil by the method of thermal steam stimulation of the reservoir. The formulation of the differential problem is based on the introduction of a change of variables called global pressure, which makes it possible to exclude the capillary pressure gradient from the pressure equation. For the numerical solution, a numerical scheme was constructed. An a priori estimate in the energy norm is obtained, which expresses the stability of the constructed scheme with respect to the initial data and the right-hand sides of the equations. The theorem on the convergence of the solution of a numerical scheme to the solution of a differential problem is presented.

Inertial iterative method with self-adaptive step size for finite family of split monotone variational inclusion and fixed point problems in Banach spaces

Abstract In this paper, we propose and study a new inertial iterative algorithm with self-adaptive step size for approximating a common solution of finite family of split monotone variational inclusion problems and fixed point problem of a nonexpansive mapping between a Banach space and a Hilbert space. This method combines the inertial technique with viscosity method and self-adaptive step size for solving the common solution problem. We prove a strong convergence result for the proposed method under some mild conditions. Moreover, we apply our result to study the split feasibility problem and split minimization problem. Finally, we provide some numerical experiments to demonstrate the efficiency of our method in comparison with some well-known methods in the literature. Our method does not require prior knowledge or estimate of the operator norm, which makes it easily implementable unlike so many other methods in the literature, which require prior knowledge of the operator norm for their implementation.

Performance Optimization via Sequential Processing for Nonlinear State Estimation of Noisy Systems

We propose a framework for designing observers for noisy nonlinear systems with global convergence properties and performing robustness and noise sensitivity. This framework comes out from the combination of a state norm estimator with a chain of filters, adaptively tuned by the state norm estimator. The state estimate is sequentially processed through the chain of filters. Each filter contributes to improving, by a certain amount, the estimation error performances of the previous filter in terms of noise sensitivity, and this amount is quantitatively evaluated using a comparison criterion, which considers the ratio of the asymptotic error norm bounds of two consecutive filters in the chain. A recursive algorithm is given for implementing the chain of filters and guaranteeing a sequential error performance optimization process. Simulations show the effectiveness of these chains of filters.

Efficient Coalition Structure Generation via Approximately Equivalent Induced Subgraph Games.

We show that any characteristic function game (CFG) G can be always turned into an approximately equivalent game represented using the induced subgraph game (ISG) representation. Such a transformation incurs obvious benefits in terms of tractability of computing solution concepts for G . Our transformation approach, namely, AE-ISG, is based on the solution of a norm approximation problem. We then propose a novel coalition structure generation (CSG) approach for ISGs that is based on graph clustering, which outperforms existing CSG approaches for ISGs by using off-the-shelf optimization solvers. Finally, we provide theoretical guarantees on the value of the optimal CSG solution of G with respect to the optimal CSG solution of the approximately equivalent ISG. As a consequence, our approach allows one to compute approximate CSG solutions with quality guarantees for any CFG. Results on a real-world application domain show that our approach outperforms a domain-specific CSG algorithm, both in terms of quality of the solutions and theoretical quality guarantees.

Error estimates for deep learning methods in fluid dynamics

In this study, we provide error estimates and stability analysis of deep learning techniques for certain partial differential equations including the incompressible Navier–Stokes equations. In particular, we obtain explicit error estimates (in suitable norms) for the solution computed by optimizing a loss function in a Deep Neural Network approximation of the solution, with a fixed complexity.

Approximation of integrable functions by general linear matrix operators of their Fourier series

Abstract The pointwise estimates of the deviation T n , A f ( ⋅ ) − f ( ⋅ ) {T}_{n,A}f(\cdot )-f\left(\cdot ) in terms of pointwise moduli of continuity based on the points of differentiability of indefinite integral of f f , with application of the rth differences of the entries of A A , are proved. The similar results in case of the Lebesgue points are considered, too. Analogical results on norm approximation with remarks and corollaries are also given.

The local discontinuous Galerkin method on layer-adapted meshes for time-dependent singularly perturbed convection-diffusion problems

In this paper we analyze the error as well for the semi-discretization as the full discretization of a time-dependent convection-diffusion problem. We use for the discretization in space the local discontinuous Galerkin (LDG) method on a class of layer-adapted meshes including Shishkin-type and Bakhvalov-type meshes and the implicit θ-scheme in time. For piecewise tensor-product polynomials of degree k we obtain uniform or almost uniform error estimates with respect to space of order k+1/2 in some energy norm and optimal error estimates with respect to time. Our analysis is based on careful approximation error estimates for the Ritz projection related to the stationary problem on the anisotropic meshes used. We discuss also improved estimates in the one-dimensional case and the use of a discontinuous Galerkin discretization in time. Numerical experiments are given to support our theoretical results.

Power allocation in D2D enabled cellular network with probability constraints: A robust Stackelberg game approach

In this paper, we develop a Stackelberg game with a non-uniform pricing in a two-tier network environment, where multiple device-to-device (D2D) users reuse the uplink spectrum resources of the cellular user (CU). Assuming that the base station (BS) is protected by pricing the interference from D2D users, we propose a Stackelberg game framework where BS and D2D users are treated as the leader and followers respectively. Besides, the channel state information (CSI) uncertainty in the actual communication environment is considered to be suppressed the interference fluctuation effects. Specifically, the probability constraint is included to optimize the user utility. The complex probability constraint is transformed to the deterministic form of the convex second-order cone constraint by a novel method. Furthermore, a norm approximation method is proposed to reduce the computational complexity. Finally, the analytical solution of Nash equilibrium is acquired to generate the best response to the leader and all followers. Simulation results show that the proposed scheme is more effective comparing to the algorithms without the CSI probability constraint.