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In this paper, we give a short proof of the Lorentz estimates for gradients of very weak solutions to the linear parabolic equations with the Muckenhoupt class A q -weights u t − div ( A ( x , t ) ∇ u ) = div ( F ) , in a bounded domain Ω × ( 0 , T ) ⊂ R N + 1 , where A has a small mean oscillation, and Ω is a Lipchistz domain with a small Lipschitz constant.

We aim to address the consensus tracking problem for multiple-input-multiple-output (MIMO) linear networked systems under directed switching topologies, where the leader is subject to some nonzero but norm bounded inputs. First, based on the relative outputs, a full-order unknown input observer (UIO) is designed for each agent to track the full states' error among neighboring agents. With the aid of such an observer, a discontinuous feedback protocol is subtly designed. And it is proven that consensus tracking can be achieved in the closed-loop networked system if the average dwell time (ADT) for switching among different interaction graph candidates is larger than a given positive threshold. By using the boundary layer technique, a continuous feedback protocol is skillfully designed and employed. It is shown that the consensus error converges into a bounded set under the designed continuous protocol. Second, as part of the full states' error can be constructed via the agents' outputs, a reduced-order UIO is thus designed based on which discontinuous and continuous feedback protocols are, respectively, proposed. By using the stability theory of the switched systems, it is proven that the consensus error converges asymptotically to 0 under the designed discontinuous protocol, and converges into a bounded set under the designed continuous protocol. Finally, the obtained theoretical results are validated through simulations.

In this paper, a general class of nonlinear initial value problems involving a Riemann-Liouville fractional derivative and a variable-order fractional derivative is investigated. An existence result of the exact solution is established by using Weissinger's fixed point theorem and Gronwall-Bellman lemma. An hp-version spectral collocation method is presented to solve the problem in numerical frames. The collocation method employs the Legendre-Gauss interpolations to conquer the influence of the nonlinear term and variable-order fractional derivative. The most remarkable feature of the method is its capability to achieve higher accuracy by refining the mesh and/or increasing the degree of the polynomial. The error estimates under the H 1 -norm for smooth solutions on arbitrary meshes and singular solutions on quasi-uniform meshes are derived. Numerical results are given to support the theoretical conclusions.

In this paper, energy-preserving time high-order average vector field (AVF) compact finite difference scheme is proposed and analyzed for solving two-dimensional nonlinear wave equations including the nonlinear sine-Gordon equation and the nonlinear Klein-Gordon equation. We first present the corresponding Hamiltonian system to the two-dimensional nonlinear wave equations, and further apply the compact finite difference (CFD) operator and AVF method to develop an energy conservative high-order scheme in two dimensions. The L p -norm boundedness of two-dimensional numerical solution is obtained from the energy conservation property, which plays an important role in the analysis of the scheme for the two-dimensional nonlinear wave equations in which the nonlinear term satisfies local Lipschitz continuity condition. We prove that the proposed scheme is energy conservative and uniquely solvable. Furthermore, optimal error estimate for the developed scheme is derived for the nonlinear sine-Gordon equation and the nonlinear Klein-Gordon equation in two dimensions. Numerical experiments are carried out to confirm the theoretical findings and to show the performance of the proposed method for simulating the propagation of nonlinear waves in layered media.

The geometric properties of the ensemble of numerical solutions obtained by the algorithms of different inner structure are addressed from the prospects for a posteriori error estimation. The numerical results are presented for the two-dimensional inviscid supersonic flows, containing shock waves. The truncation errors are computed using a postprocessor, the approximation errors are calculated by the subtraction of the numerical and the analytic solutions. The angles between the approximation errors are found to be far from zero that enables a posteriori estimation of the error norm. The correlation of the angles between the approximation errors and the corresponding angles between the computable truncation errors is observed in numerical tests and discussed from the viewpoint of the measure concentration effect and the algorithmic randomness. The analysis of the truncation errors’ geometry and the distances between solutions enables the estimation of the approximation error norm on the ensemble of numerical solutions obtained by the independent algorithms.

This letter presents a new data-driven controller synthesis based on the Koopman operator and the extended $\mathcal {H}_{2}$ norm characterization of discrete-time linear systems. We model dynamical systems as polytope sets which are derived from multiple data-driven linear models obtained by the finite approximation of the Koopman operator and then used to design robust feedback controllers combined with the $\mathcal {H}_{2}$ norm characterization. The use of the $\mathcal {H}_{2}$ norm characterization is aimed to deal with the model uncertainty that arises due to the nature of the data-driven setting of the problem. The effectiveness of the proposed controller synthesis is investigated through numerical simulations.

We develop two-scale methods for computing the convex envelope of a continuous function over a convex domain in any dimension.This hinges on a fully nonlinear obstacle formulation [A. M. Oberman, "The convex envelope is the solution of a nonlinear obstacle problem", Proc. Amer. Math. Soc. 135(6):1689--1694, 2007]. We prove convergence and error estimates in the max norm. The proof utilizes a discrete comparison principle, a discrete barrier argument to deal with Dirichlet boundary values, and the property of flatness in one direction within the non-contact set. Our error analysis extends to a modified version of the finite difference wide stencil method of [A. M. Oberman, "Computing the convex envelope using a nonlinear partial differential equation", Math. Models Meth. Appl. Sci, 18(05):759--780, 2008].

In this paper, based on the implicit Euler scheme in the temporal direction, the superconvergence property is investigated by using the special property of the bilinear element on the rectangular mesh for the Sobolev equation with Burgers' nonlinearity. The existence and uniqueness of the fully-discrete solution is proved. Further, the superconvergence error estimate in L ∞ ( H 1 ) -norm is established in terms of a novel approach, i.e., the technique of the combination of the interpolation operator and projection operator. Finally, a numerical experiment is carried out to confirm the theoretical analysis.

A fully discrete ADI scheme is proposed for solving the two-dimensional time-fractional diffusion equation with weakly singular solutions, where L1 scheme on graded mesh is adopted to tackle the initial singularity. An improved discrete fractional Gronwall inequality is employed to give an α-robust H 1 -norm convergence analysis of the fully discrete ADI scheme, where the error bound does not blow up when the order of fractional derivative α → 1 − . Numerical results show that the theoretical analysis is sharp.

We prove a quantitative regularity theorem and blowup criterion for classical solutions of the three-dimensional Navier-Stokes equations satisfying certain critical conditions. The solutions we consider have $\|r^{1-\frac3q}u\|_{L_t^\infty L_x^q}, Comment: The final version to appear in ARMA. 48 pages, 1 figure

We put forward a new method for obtaining quantitative lower bounds on the top Lyapunov exponent of stochastic differential equations (SDEs). Our method combines (i) an (apparently new) identity connecting the top Lyapunov exponent to a Fisher information-like functional of the stationary density of the Markov process tracking tangent directions with (ii) a novel, quantitative version of H\"ormander's hypoelliptic regularity theory in an $L^1$ framework which estimates this (degenerate) Fisher information from below by a $W^{s,1}_{\mathrm{loc}}$ Sobolev norm. This method is applicable to a wide range of systems beyond the reach of currently existing mathematically rigorous methods. As an initial application, we prove the positivity of the top Lyapunov exponent for a class of weakly-dissipative, weakly forced SDE; in this paper we prove that this class includes the Lorenz 96 model in any dimension, provided the additive stochastic driving is applied to any consecutive pair of modes., Comment: 62 pages, updated intro and appendix

An arbitrary order discontinuous Galerkin method in space and time is proposed to approximate the solution to hyperbolic systems of multi-dimensional conservation laws. Weak formulation is derived through the definition of weak divergence. Weak solution is given as a pair of weak functions on the element and the edge, respectively. Weak solution on the edge is characterized as the average of the solutions on the elements sharing the edge. Stability of the approximate solution is proved in a broken L 2 ( L 2 ) norm and also in a broken l ∞ ( L 2 ) norm. Error estimates of O ( h r + k n q ) with P r ( E ) and P q ( J n ) elements ( r , q > 1 + d 2 ) are then derived in a broken L 2 ( L 2 ) norm, where h and k n are the maximum diameters of the elements and the time step of J n , respectively, J n is the time interval, and d is the dimension of the spatial domain.

In recent years there is a vast application of conjugate gradient methods to restore the disturbed signals in compressive sensing. This research aims at developing a scheme, which is more effective for restoring disturbed signals than the popular PCG method (Liu & Li, 2015). To realize the desired goal, a new conjugate gradient approach combined with the projection scheme of Solodov and Svaiter [Kluwer Academic Publishers, pp. 355-369(1998)] for solving monotone nonlinear equations with convex constraints is presented. The main idea employed in this algorithm is to approximate the Jacobian matrix via acceleration parameter in order to propose an effective conjugate gradient parameter. In addition, the step length is calculated using inexact line search technique. The proposed approach is proved to converge globally under some mild conditions . The numerical experiment, depicts the efficacy our method. Apart from generating search directions that are vital for global convergence, a significant contribution of the new method lies in its applications to solve the l 1 -norm regularization problem in signal recovery. Experiments with the scheme and the effective PCG solver, existing in the previous literature, shows that the new method provides much better results.

Let Ω ⊂ R 3 be a Lipschitz domain. Consider a harmonic map v : Ω → S 2 with boundary data v | ∂ Ω = φ which minimises the Dirichlet energy. For p ≥ 2 , we show that any energy minimiser u whose boundary map ψ has a small W 1 , p -distance to φ is close to v in Holder norm modulo bi-Lipschitz homeomorphisms, provided that v is the unique minimiser attaining the boundary data. The index p = 2 is sharp: the above stability result fails for p 2 due to the constructions by Almgren–Lieb [2] and Mazowiecka–Strzelecki [15] .

In this paper, we study the numerical method for two dimensional fractional integro-differential equations, where the order of time fractional derivative α ∈ ( 1 , 2 ) and integral order γ ∈ ( 0 , 1 ) . To overcome the difficulty caused by the two fractional terms, we transform the original equation using the method of integration by parts. A novel high order compact alternating direction implicit (ADI) difference scheme is then proposed to solve the equivalent model. By some skills and detailed analysis, the unconditional stability and convergence in H 1 norm are proved, with the accuracy order O ( τ 2 + h 1 4 + h 2 4 ) , where τ , h 1 and h 2 are temporal and spatial step sizes, respectively. Finally, numerical results are presented to support the theoretical analysis.

We study global-in-time dynamics of the stochastic nonlinear wave equations (SNLW) with an additive space-time white noise forcing, posed on the two-dimensional torus. Our goal in this paper is two-fold. (i) By introducing a hybrid argument, combining the $I$-method in the stochastic setting with a Gronwall-type argument, we first prove global well-posedness of the (renormalized) cubic SNLW in the defocusing case. Our argument yields a double exponential growth bound on the Sobolev norm of a solution. (ii) We then study the stochastic damped nonlinear wave equations (SdNLW) in the defocusing case. In particular, by applying Bourgain's invariant measure argument, we prove almost sure global well-posedness of the (renormalized) defocusing SdNLW with respect to the Gibbs measure and invariance of the Gibbs measure., 33 pages. To appear in Internat. Math. Res. Not. Minor typos corrected

We introduce the concept of machine-learning minimal-residual (ML-MRes) finite element discretizations of partial differential equations (PDEs), which resolve quantities of interest with striking accuracy, regardless of the underlying mesh size. The methods are obtained within a machine-learning framework during which the parameters defining the method are tuned against available training data. In particular, we use a provably stable parametric Petrov–Galerkin method that is equivalent to a minimal-residual formulation using a weighted norm. While the trial space is a standard finite element space, the test space has parameters that are tuned in an off-line stage. Finding the optimal test space therefore amounts to obtaining a goal-oriented discretization that is completely tailored towards the quantity of interest. We use an artificial neural network to define the parametric family of test spaces. Using numerical examples for the Laplacian and advection equation in one and two dimensions, we demonstrate that the ML-MRes finite element method has superior approximation of quantities of interest even on very coarse meshes.

We summarize some general ideas regarding approximation of mixed variational problems using saddle point reformulation. We consider the concepts of optimal and almost optimal (or α ) test norm and provide estimates for the continuity and stability constants. A preconditioning strategy for solving the discrete mixed formulations is used in combination with the special test norms. We further provide a choice for a discrete trial space, that depends on the choice of a standard test space and leads to discrete stability, when using the appropriate test norm. Examples to illustrate how the stability of the saddle point discretization can be improved using special test norms are included.

For a wide range of PDEs, the discontinuous Petrov–Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan provides discrete stability starting from a coarse mesh and minimization of the residual in a user-controlled norm, among other appealing features. Research on DPG for transient problems has mainly focused on spacetime discretizations, which has theoretical advantages, but practical costs for computations and software implementations. The sole examination of time-stepping DPG formulations was performed by Fuhrer, Heuer, and Gupta, who applied Rothe’s method to an ultraweak formulation of the heat equation to develop an implicit time-stepping scheme; their work emphasized theoretical results, including error estimates in time and space. In the present work, we follow Fuhrer, Heuer, and Gupta in examining the heat equation; our focus is on numerical experiments, examining the stability and accuracy of several formulations, including primal as well as ultraweak, and explicit as well as implicit and Crank–Nicolson time-stepping schemes. We are additionally interested in communication-avoiding algorithms, and we therefore include a highly experimental formulation that places all the trace terms on the right-hand side of the equation.

We prove propagation of $\frac{1}{s}$-Gevrey regularity $(s\in(0,1))$ for the Vlasov-Poisson system on $\mathbb{T}^d$ using a Fourier space method in analogy to the results proved for the 2D Euler system in \cite{KV} and \cite{LO}. More precisely, we give a quantitative estimate for the growth in time of the $\frac{1}{s}$-Gevrey norm for the solution of the system in terms of the force field and the gradient in the velocity variable of the distribution of matter. As an application, we show global existence of $\frac{1}{s}$-Gevrey solutions ($s\in (0,1)$) for the Vlasov-Poisson system in $\mathbb{T}^3$. Furthermore, the propagation of Gevrey regularity can be easily modified to obtain the same result in $\mathbb{R}^d$. In particular, this implies global existence of analytic $(s=1)$ and $\frac{1}{s}$-Gevrey solutions ($s\in (0,1)$) for the Vlasov-Poisson system in $\mathbb{R}^3$., 18 pages

The main goal of this paper is to investigate the boundedness and essential norm of a class of product-type operators T u , v , φ m , m ∈ ℕ from Hardy spaces into n th weighted-type spaces. As a corollary, we obtain some equivalent conditions for compactness of such operators.

We review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on$$\textsf {CAT} (\kappa )$$CAT(κ)-spaces and prove that they can be characterized by the same differential inclusion$$y_t'\in -\partial ^-\textsf {E} (y_t)$$yt′∈-∂-E(yt)one uses in the smooth setting and more precisely that$$y_t'$$yt′selects the element of minimal norm in$$-\partial ^-\textsf {E} (y_t)$$-∂-E(yt). This generalizes previous results in this direction where the energy was also assumed to be Lipschitz. We then apply such result to the Korevaar–Schoen energy functional on the space of$$L^2$$L2and (0) valued maps: we define the Laplacian of such$$L^2$$L2map as the element of minimal norm in$$-\partial ^-\textsf {E} (u)$$-∂-E(u), provided it is not empty. The theory of gradient flows ensures that the set of maps admitting a Laplacian is$$L^2$$L2-dense. Basic properties of this Laplacian are then studied.

Gradient-based optimization algorithms can be studied from the perspective of limiting ordinary differential equations (ODEs). Motivated by the fact that existing ODEs do not distinguish between two fundamentally different algorithms---Nesterov's accelerated gradient method for strongly convex functions (NAG-SC) and Polyak's heavy-ball method---we study an alternative limiting process that yields high-resolution ODEs. We show that these ODEs permit a general Lyapunov function framework for the analysis of convergence in both continuous and discrete time. We also show that these ODEs are more accurate surrogates for the underlying algorithms; in particular, they not only distinguish between NAG-SC and Polyak's heavy-ball method, but they allow the identification of a term that we refer to as "gradient correction" that is present in NAG-SC but not in the heavy-ball method and is responsible for the qualitative difference in convergence of the two methods. We also use the high-resolution ODE framework to study Nesterov's accelerated gradient method for (non-strongly) convex functions, uncovering a hitherto unknown result---that NAG-C minimizes the squared gradient norm at an inverse cubic rate. Finally, by modifying the high-resolution ODE of NAG-C, we obtain a family of new optimization methods that are shown to maintain the accelerated convergence rates of NAG-C for smooth convex functions., 82 pages, 11 figures

Abstract We propose concepts to utilize basic mathematical principles for computing the exact mass properties of objects with varying densities. For objects given as 3D triangle meshes, the method is analytically accurate and at the same time faster than any established approximation method. Our concept is based on tetrahedra as underlying primitives, which allows for the object’s actual mesh surface to be incorporated in the computation. The density within a tetrahedron is allowed to vary linearly, i.e., arbitrary density fields can be approximated by specifying the density at all vertices of a tetrahedral mesh. Involved integrals are formulated in closed form and can be evaluated by simple, easily parallelized, vector-matrix multiplications. The ability to compute exact masses and centroids for objects of varying density enables novel or more exact solutions to several interesting problems: besides the accurate analysis of objects under given density fields, this includes the synthesis of parameterized density functions for the make-it-stand challenge or manufacturing of objects with controlled rotational inertia. In addition, based on the tetrahedralization of Voronoi cells we introduce a precise method to solve $$L_{2|\infty }$$ L 2 | ∞ Lloyd relaxations by exact integration of the Chebyshev norm. In the context of additive manufacturing research, objects of varying density are a prominent topic. However, current state-of-the-art algorithms are still based on voxelizations, which produce rather crude approximations of masses and mass centers of 3D objects. Many existing frameworks will benefit by replacing approximations with fast and exact calculations. Graphic abstract

We study an optimal control problem consisting in minimizing the L ∞ norm of a Borel measurable cost function, in finite time, and over all trajectories associated with a controlled dynamics which is reflected in a compact prox-regular set. The first part of the paper provides the viscosity characterization of the value function for uniformly continuous costs. The second part is concerned with linear programming formulations of the problem and the ensued by-products as e.g. dynamic programming principle for merely measurable costs.

This paper presents a modified graded mesh for singularly perturbed reaction–diffusion problems. The mesh we offer is generated recursively using Newton’s algorithm and some implicitly defined function. The problem is solved numerically using the finite element method based on polynomials of degree p ≥ 1 . We prove parameter uniform convergence of optimal order in ϵ -weighted energy norm. Test examples are taken, and we present a rigorous comparative analysis with other adaptive meshes. Moreover, we compare the proposed method with others found in the literature.

In this work, we propose a new quasi-optimal test norm for a discontinuous Petrov-Galerkin (DPG) discretization of the ultra-weak formulation of the convection-diffusion equation. We prove theoretically that the proposed test norm leads to bounds between the target norm and the energy norm induced by the test norm which have favorable scalings with respect to the diffusion parameter when compared with existing results for other test norms from the literature. We conclude with numerical experiments to confirm our theoretical results.

In this paper, the discontinuous Galerkin method (DGM) of nonconforming Wilson element is studied for the semi-linear parabolic problem. The global superconvergence with respect to the mesh size are derived in the modified H 1 -norm for the semi-discrete scheme and two fully discrete schemes, in which the usual extrapolation and interpolation post-processing approaches are not involved, and the error estimates are one order higher than that of the traditional Galerkin finite element method (FEM). Therefore, the corresponding results in the existing literature are improved. Finally, some numerical results are provided to confirm the theoretical analysis.

A positive operator T : E → E on a Banach lattice E with an order continuous norm is said to be B -Volterra with respect to a Boolean algebra B of order projections of E if the bands canonically corresponding to elements of B are left fixed by T . A linearly ordered sequence ξ in B connecting 0 to 1 is called a forward filtration. A forward filtration can be used to lift the action of the B -Volterra operator T from the underlying Banach lattice E to an action of a new norm continuous operator T ˆ ξ : M r ( ξ ) → M r ( ξ ) on the Banach lattice M r ( ξ ) of regular bounded martingales on E corresponding to ξ . In the present paper, we study properties of these actions. The set of forward filtrations are left fixed by a function which erases the first order projection of a forward filtration and which shifts the remaining order projections towards 0. This function canonically induces a norm continuous shift operator s between two Banach lattices of regular bounded martingales. Moreover, the operators T ˆ ξ and s commute. Utilizing this fact with inductive limits, we construct a categorical limit space M T , ξ which is called the associated space of the pair ( T , ξ ) . We present new connections between theories of Boolean algebras, abstract martingales and Banach lattices.

In this paper, superconvergence properties are presented for the nonlinear Benjamin-Bona-Mahony (BBM) equation with a mixed finite element method (MFEM). We propose an Euler scheme combined with the artificial Douglas-Dupont regularization terms, which guarantee the stability of the numerical scheme. Splitting technique is utilized to get rid of the ratio between the time step τ and the subdivision parameter h. Temporal error estimates and unconditional spatial error estimates are gained through some techniques, such as the function's monotonicity and the Green formula and so on. In turn, the regularities of the solutions about the time-discrete equations and the boundedness of the numerical solution are derived. Based on the above achievements, the unconditional superconvergent results of u n in H 1 -norm and q → n in L 2 -norm with order O ( h 2 + τ ) are obtained through the trigonometric inequality. The global superconvergent results are deduced by use of the interpolated postprocessing operators. Numerical example shows the validity of the theoretical analysis.

In [BS07] Breuil and Schneider formulated a conjecture on the equivalence of the existence of invariant norms on certain $p$-adically locally algebraic representations of $GL_n(F)$ and the existence of certain de-Rham representations of $Gal(\bar{F}/F)$, where $F$ is a finite extension of $\mathbb{Q}_p$. In [Bre03b, DI13] Breuil and de Ieso proved that in the case $n = 2$ and under some restrictions, the existence of certain admissible filtrations on the $\phi$-module associated to the two-dimensional de-Rham representation of $Gal(\bar{F}/F)$ implies the existence of invariant norms on the corresponding locally algebraic representation of $GL_2(F)$. In [Bre03b, DI13], there is a significant restriction on the weight - it must be small enough. In [CEG+13] the conjecture is proved in greater generality, but the weights are still restricted to the extended Fontaine-Laffaille range. In this paper we prove that in the case $n = 2$, even with larger weights, under some restrictions, the existence of certain admissible filtrations implies the existence of invariant norms.

We consider a singularly perturbed convection-diffusion boundary value problem whose solution contains exponential and characteristic boundary layers. The problem is numerically solved by the FEM and SDFEM method with bilinear elements on a graded mesh. For the FEM we prove almost uniform convergence and superconvergence. The use of a graded mesh allows for the SDFEM to yield almost uniform estimates in the SD norm, which is not possible for Shishkin type meshes. Numerical results are presented to support theoretical bounds.

We analyze a fully discrete finite element numerical scheme for the Cahn–Hilliard–Stokes–Darcy system that models two-phase flows in coupled free flow and porous media. To avoid a well-known difficulty associated with the coupling between the Cahn–Hilliard equation and the fluid motion, we make use of the operator-splitting in the numerical scheme, so that these two solvers are decoupled, which in turn would greatly improve the computational efficiency. The unique solvability and the energy stability have been proved in Chen et al. (2017, Uniquely solvable and energy stable decoupled numerical schemes for the Cahn–Hilliard–Stokes–Darcy system for two-phase flows in karstic geometry. Numer. Math., 137, 229–255). In this work, we carry out a detailed convergence analysis and error estimate for the fully discrete finite element scheme, so that the optimal rate convergence order is established in the energy norm, i.e., in the $\ell ^{\infty } (0, T; H^1) \cap \ell ^2 (0, T; H^2)$ norm for the phase variables, as well as in the $\ell ^{\infty } (0, T; H^1) \cap \ell ^2 (0, T; H^2)$ norm for the velocity variable. Such an energy norm error estimate leads to a cancelation of a nonlinear error term associated with the convection part, which turns out to be a key step to pass through the analysis. In addition, a discrete $\ell ^2 (0;T; H^3)$ bound of the numerical solution for the phase variables plays an important role in the error estimate, which is accomplished via a discrete version of Gagliardo–Nirenberg inequality in the finite element setting.

The study of convergence of time integrators, applied to linear discretized PDEs, relies on the power boundedness of the stability matrix R. The present work investigates power boundedness in the maximum norm for ADI-type integrators in arbitrary space dimension m. Examples are the Douglas scheme, the Craig–Sneyd scheme, and W-methods with a low stage number. It is shown that for some important integrators $$\Vert R^n\Vert _\infty $$ is bounded in the maximum norm by a constant times $$\min \bigl ( (\ln (1+n))^m, (\ln N)^m \bigr )$$ , where m is the space dimension of the PDE, and $$N\ge 2$$ is the space discretization parameter. For $$m\le 2$$ sharper bounds are obtained that are independent of n and N.

In this article, a boundary value problem for a nonlinear system of singularly perturbed two second order differential equations in which only the first equation is multiplied by a small positive parameter is considered. The first component of the solution exhibits boundary layers whereas the second component exhibits less-severe layers. A numerical method composed of a classical finite difference scheme applied on a piecewise uniform Shishkin mesh is suggested to solve the system. The method is proved to be essentially second order convergent in the maximum norm uniformly with respect to the perturbation parameter. Numerical illustration presented supports the proved theoretical results.

Let $${{\mathcal {A}}}$$ be a Banach algebra and let $$\varphi $$ be a non-zero character on $${{\mathcal {A}}}$$ . Suppose that $${{\mathcal {A}}}_M$$ is the closure of the faithful Banach algebra $${{\mathcal {A}}}$$ in the multiplier norm. In this paper, topologically left invariant $$\varphi $$ -means on $${{\mathcal {A}}}_M^*$$ are defined and studied. Under some conditions on $${{\mathcal {A}}}$$ , we will show that the set of topologically left invariant $$\varphi $$ -means on $${{\mathcal {A}}}^*$$ and on $${{\mathcal {A}}}_M^*$$ have the same cardinality. The main applications are concerned with the quantum group algebra $$L^1({\mathbb {G}})$$ of a locally compact quantum group $${\mathbb {G}}$$ . In particular, we obtain some characterizations of compactness of $${\mathbb {G}}$$ in terms of the existence of a non-zero (weakly) compact left or right multiplier on $$L^1_M({\mathbb {G}})$$ or on its bidual in some senses.

In this paper, we consider a class of singularly perturbed advanced-delay differential equations of convection-diffusion type. We use finite and hybrid difference schemes to solve the problem on piecewise Shishkin mesh. We have established almost first- and second-order convergence with respect to finite difference and hybrid difference methods. An error estimate is derived with the discrete norm. In the end, numerical examples are given to show the advantages of the proposed results (Mathematics Subject Classification: 65L11, 65L12, and 65L20).

We give an elementary proof of weighted resolvent estimates for the semiclassical Schrödinger operator $-h^2 \Delta + V(x) - E$ in dimension $n \neq 2$, where $h, \, E> 0$. The potential is real valued and $V$ and $\partial _r V$ exhibit long-range decay at infinity and may grow like a sufficiently small negative power of $r$ as $r \to 0$. The resolvent norm grows exponentially in $h^{-1}$, but near infinity it grows linearly. When $V$ is compactly supported, we obtain linear growth if the resolvent is multiplied by weights supported outside a ball of radius $CE^{-1/2}$ for some $C> 0$. This $E$-dependence is sharp and answers a question of Datchev and Jin.

In this paper, we deal with the following chemotaxis-Stokes model with non-Newtonian filtration slow diffusion (namely, p > 2 ) { n t + u ⋅ ∇ n = ∇ ⋅ ( | ∇ n | p − 2 ∇ n ) − χ ∇ ⋅ ( n ∇ c ) , c t + u ⋅ ∇ c − Δ c = − c n , u t + ∇ π = Δ u + n ∇ φ , div u = 0 in a bounded domain Ω of R 3 with zero-flux boundary conditions and no-slip boundary condition. Similar to the study for the chemotaxis-Stokes system with porous medium diffusion, it is also a challenging problem to find an optimal p-value ( p ≥ 2 ) which ensures that the solution is global bounded. In particular, the closer the value of p is to 2, the more difficult the study becomes. In the present paper, we prove that global bounded weak solutions exist whenever p > p ⁎ ( ≈ 2.012 ) . It improved the result of [21] , [22] , in which, the authors established the global bounded solutions for p > 23 11 . Moreover, we also consider the large time behavior of solutions, and show that the weak solutions will converge to the spatially homogeneous steady state ( n ‾ 0 , 0 , 0 ) . Comparing with the chemotaxis-fluid system with porous medium diffusion, the present convergence of n is proved in the sense of L ∞ -norm, not only in L p -norm or weak-* topology.

It is well-known that the Kadec-Klee property is an important property in the geometry of Banach spaces. It is closely connected with the approximation compactness and fixed point property of non-expansive mappings. In this paper, a criterion for Musielak-Orlicz function spaces equipped with the Orlicz norm to have the Kadec-Klee property are given. As a corollary, we obtain that a class of non-reflexive Musielak-Orlicz function spaces have the Fixed Point property.

The recently emerging technique of sparse reconstruction has received much attention in the field of photoacoustic imaging (PAI). Compressed sensing (CS) has large potential in efficiently reconstructing high-quality PAI images with sparse sampling signal. In this article, we propose a CS-based error-tolerant regularized smooth L0 (ReSL0) algorithm for PAI image reconstruction, which has the same computational advantages as the SL0 algorithm while having a higher degree of immunity to inaccuracy caused by noise. In order to evaluate the performance of the ReSL0 algorithm, we reconstruct the simulated dataset obtained from three phantoms. In addition, a real experimental dataset from agar phantom is also used to verify the effectiveness of the ReSL0 algorithm. Compared to three L0 norm, L1 norm, and TV norm-based CS algorithms for signal recovery and image reconstruction, experiments demonstrated that the ReSL0 algorithm provides a good balance between the quality and efficiency of reconstructions. Furthermore, the PSNR of the reconstructed image calculated by the introduced method was better than the other three methods. In particular, it can notably improve reconstruction quality in the case of noisy measurement.

We consider an optimal control problem for a time-dependent obstacle variational inequality involving fractional Liouville–Caputo derivative. The obstacle is considered as the control, and the corresponding solution to the obstacle problem is regarded as the state. Our aim is to find the optimal control with the properties that the state is closed to a given target profile and the obstacle is not excessively large in terms of its norm. We prove existence results and establish necessary conditions of obstacle problems via the approximated time fractional-order partial differential equations and their adjoint problems. The result in this paper is a generalization of the obstacle problem for a parabolic variational inequalities as the Liouville–Caputo fractional derivatives were used instead of the classical derivatives.

We study two dimensional Navier–Stokes equations driven by a multiplicative Gaussian noise in the Stratonovich form along with a constraint on the L 2 -norm of the solution. In the deterministic setting [5] , it was shown that the global solution exists only on a two dimensional torus and hence we focus on such a case here. The existence of a martingale solution is shown. Moreover, the pathwise uniqueness of the solution is proved by using Schmalfuss idea [24] , concluding the existence of a strong solution via a Yamada–Watanabe type result from Ondrejat [21] .

In this paper, Galerkin finite element (GFE) methods are extended to solve two-dimensional (2D) initial–boundary value problems of neutral delay-reaction–diffusion equations, where the spatial and temporal variables are discretized by the semi-discrete GEF methods and Crank–Nicolson method, respectively. By setting some appropriate conditions, it is proved that a fully discrete GFE method is uniquely solvable, stable and convergent of order 2 in time and order r (resp. r − 1 ) in space under the sense of L 2 -norm (resp. H 1 -norm), where r − 1 ( r ≥ 2 ) denotes the degree of piecewise polynomial in finite element space. Moreover, with some numerical experiments, we further illustrate the computational effectiveness and accuracy of the method.

Up to now, various matrix norms (e.g., $ {l_{1}}$ -norm, $ {l_{2}}$ -norm, $ {l_{2,1}}$ -norm, etc.) have been widely leveraged to form the loss function of different regression models, and have played an important role in image analysis. However, the previous regression models adopting the existing norms are sensitive to outliers and, thus, often bring about unsatisfactory results on the heavily corrupted images. This is because their adopted norms for measuring the data residual can hardly suppress the negative influence of noisy data, which will probably mislead the regression process. To address this issue, this paper proposes a novel $ {\delta }$ (delta)-norm to count the nonzero blocks around an element in a vector or matrix, which weakens the impacts of outliers and also takes the structure property of examples into account. After that, we present the $ {\delta }$ -norm-based robust regression (DRR) in which the data examples are mapped to the kernel space and measured by the proposed $ {\delta }$ -norm. By exploring an explicit kernel function, we show that DRR has a closed-form solution, which suggests that DRR can be efficiently solved. To further handle the influences from mixed noise, DRR is extended to a multiscale version. The experimental results on image classification and background modeling datasets validate the superiority of the proposed approach to the existing state-of-the-art robust regression models.

UDC 517.9 We present numerous refinements of the Young inequality by the Kantorovich constant. We use these improved inequalities to establish corresponding operator inequalities on a Hilbert space and some new inequalities involving the Hilbert – Schmidt norm of matrices.

We study the Cauchy problem associated with a quantum Zakharov-type system in three and higher spatial dimensions.Taking the quantum parameter to unit and developing Fourier restriction norm arguments, we establish local well-posedness property for wider range than the one known for the Zakharov system.

A robust adaptive filter is usually unaffected by spurious disturbances at the error sensor. In an endeavour to improve robustness of the adaptive filter, a novel modified Champernowne function (MCF) is proposed as a robust norm and the corresponding robust Champernowne adaptive filter (CMAF) is derived. To improve modelling accuracy and convergence performance for sparse systems along with being robust, a reweighted zero attraction (RZA) norm is incorporated in the cost function along with MCF and the corresponding RZA-CMAF algorithm is proposed. To further improve filter performance, the CMAF- $l_{0}$ algorithm is proposed where the $l_{0}$ -norm is approximated using the multivariate Geman-McClure function (GMF). Bound on learning rate for the proposed algorithms is also derived. Extensive simulation study shows the improved robustness achieved by the CMAF algorithm, especially when impulsive noises are present for a longer duration. On the other hand, RZA-CMAF and CMAF- $l_{0}$ can provide improved convergence performance under sparse and impulsive noise conditions, with CMAF- $l_{0}$ providing the best performance.

Ill-posed problems arise in many areas of science and engineering. Their solutions, if they exist, are very sensitive to perturbations in the data. To reduce this sensitivity, the original problem may be replaced by a minimization problem with a fidelity term and a regularization term. We consider minimization problems of this kind, in which the fidelity term is the square of the l 2 -norm of a discrepancy and the regularization term is the qth power of the l q -norm of the size of the computed solution measured in some manner. We are interested in the situation when 0 q ≤ 1 , because such a choice of q promotes sparsity of the computed solution. The regularization term is determined by a regularization matrix. Novati and Russo let q = 2 and proposed in (2014) [13] a regularization matrix that is a finite difference approximation of a differential operator applied to the computed approximate solution after reordering. This gives a Tikhonov regularization problem in general form. We show that this choice of regularization matrix also is well suited for minimization problems with 0 q ≤ 1 . Applications to image restoration are presented.

We prove the global well-posedness of the strong solution to the two-dimensional inhomogeneous incompressible primitive equations for initial data with small H 1 2 -norm, which also satisfies a natural compatibility condition. A logarithmic type Sobolev embedding inequality for the anisotropic L x ∞ L z 2 ( Ω ) norm is established to obtain the global in time a priori H 1 ( Ω ) ∩ W 1 , 6 ( Ω ) estimate of the density, which guarantee the local solution to be a global one.