Showing papers on "Dual norm published in 2023"

Dual natural-norm a posteriori error estimators for reduced basis approximations to parameterized linear equations

Abstract: In this work, the concept of dual natural-norm for parametrized linear equations is used to derive residual-based a posteriori error bounds characterized by a [Formula: see text] stability constant. We translate these error bounds into very effective practical a posteriori error estimators for reduced basis approximations and show how they can be efficiently computed following an offline/ online strategy. We prove that our practical dual natural-norm error estimator outperforms the classical inf–sup based error estimators in the self-adjoint case. Our findings are illustrated on anisotropic Helmholtz equations showing resonant behavior. Numerical results suggest that the proposed error estimator is able to successfully catch the correct order of magnitude of the reduced basis approximation error, thus outperforming the classical inf–sup based error estimator even for non-self-adjoint problems.

Counterexamples in rotundity of norms in Banach spaces

Abstract: We study several classical concepts in the topic of strict convexity of norms in infinite dimensional Banach spaces. Specifically, and in descending order of strength, we deal with Uniform Rotundity (UR), Weak Uniform Rotundity (WUR) and Uniform Rotundity in Every Direction (URED). Our first three results show that we may distinguish between all of these three properties in every Banach space where such renormings are possible. Specifically, we show that in every infinite dimensional Banach space which admits a WUR (resp. URED) renorming, we can find a norm with the same condition and which moreover fails to be UR (resp. WUR). We prove that these norms can be constructed to be Locally Uniformly Rotund (LUR) in Banach spaces admitting such renormings. Additionally, we obtain that in every Banach space with a LUR norm we can find a LUR renorming which is not URED. These results solve three open problems posed by A.J. Guirao, V. Montesinos and V. Zizler. The norms we construct in this first part are dense. In the last part of this note, we solve a fourth question posed by the same three authors by constructing a $C^\infty$-smooth norm in $c_0$ whose dual norm is not strictly convex.

The Norming Sets of Multilinear Forms on the Plane with a Certain Norm

Abstract: Let n ∈ ℕ. An element (x1, … , xn) ∈ En is called a norming point of if and , where denotes the space of all continuous symmetric n-linear forms on E. For , we defineNorm(T) is called the norming set of T.Let be the plane with a certain norm such that the set of the extreme points of its unit ball ext for some .In this paper, we classify Norm(T) for every . We also present relations between the norming sets of and .

On Operators Preserves in Normed Inner Product Spaces

Abstract: We consider that a finite dimensional real normed linear space X is an inner product space if for any linear operator T on X, T preserving its norm at e1,e2∈SX implies T attains its norm at span{e1,e2}∩SX . We prove by the convexity theorem.

Property (h) of Banach Lattice and Order-to-Norm Continuous Operators

Abstract: In this paper, we introduce the property (h) on Banach lattices and present its characterization in terms of disjoint sequences. Then, an example is given to show that an order-to-norm continuous operator may not be σ-order continuous. Suppose T:E→F is an order-bounded operator from Dedekind σ-complete Banach lattice E into Dedekind complete Banach lattice F. We prove that T is σ-order-to-norm continuous if and only if T is both order weakly compact and σ-order continuous. In addition, if E can be represented as an ideal of L0(μ), where (Ω,Σ,μ) is a σ-finite measure space, then T is σ-order-to-norm continuous if and only if T is order-to-norm continuous. As applications, we extend Wickstead’s results on the order continuity of norms on E and E′.

A note on the minimum‐norm in dual space approach to some classical linear optimal control problems

Abstract: Some classical optimal control problems of linear systems can be characterized as finding minimum-norm vectors from within linear varieties in appropriate dual spaces. Then, the solution to such an optimal control problem can be derived from the alignment between the optimal vector in the dual space and the optimal vector in the primal space, and the dual maximization problem of the minimum-norm problem. This note presents a detailed introduction to this minimum-norm in dual space approach by examples of minimum-supremum-norm, minimum-energy, and minimum-time optimal control problems of linear systems. Connections and differences between these problems in light of the introduced approach are discussed.

A variational multiscale method derived from an adaptive stabilized conforming finite element method via residual minimization on dual norms

Abstract: This paper interprets the stabilized finite element method via residual minimization as a variational multiscale method. We approximate the solution to the partial differential equations using two discrete spaces that we build on a triangulation of the domain; we denote these spaces as coarse and enriched spaces. Building on the adaptive stabilized finite element method via residual minimization, we find a coarse-scale approximation in a continuous space by minimizing the residual on a dual discontinuous Galerkin norm; this process allows us to compute a robust error estimate to construct an on-the-fly adaptive method. We reinterpret the residual projection using the variational multiscale framework to derive a fine-scale approximation. As a result, on each mesh of the adaptive process, we obtain stable coarse- and fine-scale solutions derived from a symmetric saddle-point formulation and an a-posteriori error indicator to guide automatic adaptivity. We test our framework in several challenging scenarios for linear and nonlinear convection-dominated diffusion problems to demonstrate the framework's performance in providing stability in the solution with optimal convergence rates in the asymptotic regime and robust performance in the pre-asymptotic regime. Lastly, we introduce a heuristic dual-term contribution in the variational form to improve the full-scale approximation for symmetric formulations (e.g., diffusion problem).

Mixed-norm Herz-slice Spaces and Their Applications

Abstract: We introduce mixed-norm Herz-slice spaces unifying classical Herz spaces and mixed-norm slice spaces, establish dual spaces and the block decomposition, and prove that the boundedness of Hardy-Littlewood maximal operator on mixed-norm Herz-slice spaces.

Maximum norm<i>a posteriori</i>error estimates for convection–diffusion problems

Abstract: Abstract We prove residual-type a posteriori error estimates in the maximum norm for a linear scalar elliptic convection–diffusion problem that may be singularly perturbed. Similar error analysis in the energy norm by Verfürth indicates that a dual norm of the convective derivative of the error must be added to the natural energy norm in order for the natural residual estimator to be reliable and efficient. We show that the situation is similar for the maximum norm. In particular, we define a mesh-dependent weighted seminorm of the convective error, which functions as a maximum-norm counterpart to the dual norm used in the energy norm setting. The total error is then defined as the sum of this seminorm, the maximum norm of the error and data oscillation. The natural maximum norm residual error estimator is shown to be equivalent to this total error notion, with constant independent of singular perturbation parameters. These estimates are proved under the assumption that certain natural estimates hold for the Green’s function for the problem at hand. Numerical experiments confirm that our estimators effectively capture the maximum-norm error behavior for singularly perturbed problems, and can effectively drive adaptive refinement in order to capture layer phenomena.

Disentanglement, multilinear duality and factorisation for nonpositive operators

Abstract: In previous work we established a multilinear duality and factorisation theory for norm inequalities for pointwise weighted geometric means of positive linear operators defined on normed lattices. In this paper we extend the reach of the theory for the first time to the setting of general linear operators defined on normed spaces. The scope of this theory includes multilinear Fourier restriction-type inequalities. We also sharpen our previous theory of positive operators. Our results all share a common theme: estimates on a weighted geometric mean of linear operators can be disentangled into quantitative estimates on each operator separately. The concept of disentanglement recurs throughout the paper. The methods we used in the previous work - principally convex optimisation - relied strongly on positivity. In contrast, in this paper we use a vector-valued reformulation of disentanglement, geometric properties (Rademacher-type) of the underlying normed spaces, and probabilistic considerations related to p-stable random variables.

A Note on the Chebyshev Set Problem in Normed Linear Spaces

Abstract: Best approximation (BA) is an interesting field in functional analysis that has attracted a lot of attention from many researchers for a very long period of time up-to-date. Of greatest consideration is the characterization of the Chebyshev set (CS) which is a subset of a normed linear space (NLS) which contains unique BAs. However, a fundamental question remains unsolved to-date regarding the convexity of the CS in infinite NLS known as the CS problem. The question which has not been answered is: Is every CS in a NLS convex?. This question has not got any solution including the simplest form of a real Hilbert space (HS). In this note, we characterize CSs and convexity in NLSs. In particular, we consider the space of all real-valued norm-attainable functions. We show that CSs of the space of all real-valued norm-attainable functions are convex when they are closed, rotund and admits both Gateaux and Fr\'{e}chet differentiability conditions.

A note on the n-norm of a normed space

Abstract: It is known that a normed space can be equipped with several -norms. Geometrically, -norm is interpreted as the volume of the -dimensional parallelepiped that is spanned by vectors. In this article, we propose a new -norm related to the semi-inner product in the normed space. The results are generalizations for and different from the existing -norms. We also investigate its relation to at least one of the -norms defined previously by Nur and Gunawan. Additionally, we show that the new -norm and the standard -norm are identical on an inner product space. Keywords: Normed Space, -Norm, Semi-Inner Product g DOI： https://doi.org/10.35741/issn.0258-2724.58.2.38

Poisson-orlicz norm and infinite ergodic theory

Abstract: Urbanik's theorem for a Poisson process on an infinite measure space (X, A, $\mu$) relates integrability of stochastic integrals to a particular Orlicz function space L$\Phi$ ($\mu$) on which the L1-norm of the Poisson process induces a norm (called Poisson-Orlicz in the sequel) that is shown to be equivalent to the classical gauge and Orlicz norms.We obtain a full characterization of stochastic integrals using difference operators that, together with a simple duality argument, allows to derive Urbanik's theorem as well as an optimal inequality between the Orlicz and the Poisson-Orlicz norm.In a second part, we show that the Poisson-Orlicz norm plays a role in infinite Ergodic Theory where it is seen as an alternative to the L1-norm to identify several dynamical invariants that the latter fails to identify. We also show that, whereas the L1-norm fully characterizes exact endomorphisms (Lin's theorem), Poisson-Orlicz norm fully characterizes remotely infinite endomorphisms.