Weak Approximation Property Research Articles

The approximation property for locally compact quantum groups

We study the Haagerup–Kraus approximation property for locally compact quantum groups, generalising and unifying previous work by Kraus–Ruan and Crann. Along the way we discuss how multipliers of quantum groups interact with the C⁎-algebraic theory of locally compact quantum groups. Several inheritance properties of the approximation property are established in this setting, including passage to quantum subgroups, free products of discrete quantum groups, and duals of double crossed products. We also discuss a relation to the weak⁎ operator approximation property. For discrete quantum groups, we introduce a central variant of the approximation property, and relate this to a version of the approximation property for rigid C⁎-tensor categories, building on work of Arano–De Laat–Wahl.

Noninvariance of Weak Approximation Properties Under Extension of the Ground Field

For rational points on algebraic varieties defined over a number field K, we study the behavior of the property of weak approximation with Brauer–Manin obstruction under extension of the ground field. We construct K-varieties accompanied with a quadratic extension L|K such that the property holds over K (conditionally on a conjecture) whereas fails over L. The result is unconditional when K equals Q or certain quadratic number fields. We give an explicit example when K=Q.

Some Results on the $p$-Weak Approximation Property in Banach Spaces

In this study, some existing results dealing with the weak approximation property of Banach spaces are considered for the $p$-weak approximation property. Also, an observation on the bounded weak approximation and the $p$-bounded weak approximation properties is given. Moreover, the proof of the solution of the duality problem for the $p$-weak approximation property which exists in the literature is given in a shorter way as an alternative.

Amenable dynamical systems over locally compact groups

AbstractWe establish several new characterizations of amenable $W^*$ - and $C^*$ -dynamical systems over arbitrary locally compact groups. In the $W^*$ -setting we show that amenability is equivalent to (1) a Reiter property and (2) the existence of a certain net of completely positive Herz–Schur multipliers of $(M,G,\alpha )$ converging point weak* to the identity of $G\bar {\ltimes }M$ . In the $C^*$ -setting, we prove that amenability of $(A,G,\alpha )$ is equivalent to an analogous Herz–Schur multiplier approximation of the identity of the reduced crossed product $G\ltimes A$ , as well as a particular case of the positive weak approximation property of Bédos and Conti [On discrete twisted $C^*$ -dynamical systems, Hilbert $C^*$ -modules and regularity. Münster J. Math.5 (2012), 183–208] (generalized to the locally compact setting). When $Z(A^{**})=Z(A)^{**}$ , it follows that amenability is equivalent to the 1-positive approximation property of Exel and Ng [Approximation property of $C^*$ -algebraic bundles. Math. Proc. Cambridge Philos. Soc.132(3) (2002), 509–522]. In particular, when $A=C_0(X)$ is commutative, amenability of $(C_0(X),G,\alpha )$ coincides with topological amenability of the G-space $(G,X)$ .

The tight approximation property

AbstractThis article introduces and studies the tight approximation property, a property of algebraic varieties defined over the function field of a complex or real curve that refines the weak approximation property (and the known cohomological obstructions to it) by incorporating an approximation condition in the Euclidean topology. We prove that the tight approximation property is a stable birational invariant, is compatible with fibrations, and satisfies descent under torsors of linear algebraic groups. Its validity for a number of rationally connected varieties follows. Some concrete consequences are: smooth loops in the real locus of a smooth compactification of a real linear algebraic group, or in a smooth cubic hypersurface of dimension≥2{\geq 2}, can be approximated by rational algebraic curves; homogeneous spaces of linear algebraic groups over the function field of a real curve satisfy weak approximation.

Modifying AMG Coarse Spaces with Weak Approximation Property to Exhibit Approximation in Energy Norm

Algebraic multigrid (AMG) coarse spaces are commonly constructed so that they exhibit the so-called weak approximation property (WAP) which is a necessary and sufficient condition for uniform two-g...

Optimal Interpolation and Compatible Relaxation in Classical Algebraic Multigrid

In this paper, we consider a classical algebraic multigrid (AMG) form of optimal interpolation that directly minimizes the two-grid convergence rate and compare it with a so-called ideal interpolation that minimizes a weak approximation property of the coarse space. We study compatible relaxation type estimates for the quality of the coarse grid and derive a new sharp measure using optimal interpolation that provides a guaranteed lower bound on the convergence rate of the resulting two-grid method for a given grid. In addition, we design a generalized bootstrap AMG setup algorithm that computes a sparse approximation to the optimal interpolation matrix. We demonstrate numerically that the bootstrap AMG method with sparse interpolation matrix (and spanning multiple levels) converges faster than the two-grid method with the standard ideal interpolation (a dense matrix) for various scalar diffusion problems with highly varying diffusion coefficient.

The deflated conjugate gradient method: Convergence, perturbation and accuracy

Deflation techniques for Krylov subspace methods have seen a lot of attention in recent years. They provide means to improve the convergence speed of these methods by enriching the Krylov subspace with a deflation subspace. The most common approach for the construction of deflation subspaces is to use (approximate) eigenvectors, but also more general subspaces are applicable.In this paper we discuss two results concerning the accuracy requirements within the deflated CG method. First we show that the effective condition number which bounds the convergence rate of the deflated conjugate gradient method depends asymptotically linearly on the size of the perturbations in the deflation subspace. Second, we discuss the accuracy required in calculating the deflating projection. This is crucial concerning the overall convergence of the method, and also allows to save some computational work.To show these results, we use the fact that as a projection approach deflation has many similarities to multigrid methods. In particular, recent results relate the spectra of the deflated matrix to the spectra of the error propagator of twogrid methods. In the spirit of these results we show that the effective condition number can be bounded by the constant of a weak approximation property.

Progress concerning the local-global principle for zero-cycles on algebraic varieties

We consider zero-cycles on algebraic varieties defined over number fields. The Hasse principle and weak approximation property are obstructed by the Brauer group of the variety and it is conjectured to be the only obstruction for all proper smooth varieties by Colliot-Th\'el\`ene, Sansuc, Kato and Saito. We summarize the progress related to this conjecture, particularly for higher dimensional varieties such as fibrations and homogeneous varieties.

A two‐grid SA‐AMG convergence bound that improves when increasing the polynomial degree

SummaryIn this paper, we consider the convergence rate of a smoothed aggregation algebraic multigrid method, which uses a simple polynomial (1 − t)ν or an optimal Chebyshev‐like polynomial to construct the smoother and prolongation operator. The result is purely algebraic, whereas a required main weak approximation property of the tentative interpolation operator is verified for a spectral element agglomeration version of the method. More specifically, we prove that, for partial differential equations (PDEs), the two‐grid method converges uniformly without any regularity assumptions. Moreover, the convergence rate improves uniformly when the degree of the polynomials used for the smoother and the prolongation increases. Such a result, as is well‐known, would imply uniform convergence of the multilevel W‐cycle version of the algorithm. Numerical results, for both PDE and non‐PDE (graph Laplacian) problems are presented to illustrate the theoretical findings. Published 2016. This article is a U.S. Government work and is in the public domain in the USA.

Continuity Properties of the Inf-Sup Constant for the Divergence

The inf-sup constant for the divergence, or LBB constant, is explicitly known for only few domains. For other domains, upper and lower estimates are known. If more precise values are required, one can try to compute a numerical approximation. This involves, in general, approximation of the domain and then the computation of a discrete LBB constant that can be obtained from the numerical solution of an eigenvalue problem for the Stokes system. This eigenvalue problem does not fall into a class for which standard results about numerical approximations can be applied. Indeed, many reasonable finite element methods do not yield a convergent approximation. In this article, we show that under fairly weak conditions on the approximation of the domain, the LBB constant is an upper semi-continuous shape functional, and we give more restrictive sufficient conditions for its continuity with respect to the domain. For numerical approximations based on variational formulations of the Stokes eigenvalue problem, we also show upper semi-continuity under weak approximation properties, and we give stronger conditions that are sufficient for convergence of the discrete LBB constant towards the continuous LBB constant. Numerical examples show that our conditions are, while not quite optimal, not very far from necessary.

Resolution of the Wavefront Set Using General Continuous Wavelet Transforms

We consider the problem of characterizing the wavefront set of a tempered distribution $$u\in \mathcal {S}'(\mathbb {R}^{d})$$ in terms of its continuous wavelet transform, where the latter is defined with respect to a suitably chosen dilation group $$H\subset \mathrm{GL}(\mathbb {R}^{d})$$ . In this paper we develop a comprehensive and unified approach that allows to establish characterizations of the wavefront set in terms of rapid coefficient decay, for a large variety of dilation groups. For this purpose, we introduce two technical conditions on the dual action of the group H, called microlocal admissibility and (weak) cone approximation property. Essentially, microlocal admissibility sets up a systematic relationship between the scales in a wavelet dilated by $$h\in H$$ on one side, and the matrix norm of h on the other side. The (weak) cone approximation property describes the ability of the wavelet system to adapt its frequency-side localization to arbitrary frequency cones. Together, microlocal admissibility and the weak cone approximation property allow the characterization of points in the wavefront set using multiple wavelets. Replacing the weak cone approximation by its stronger counterpart gives rise to single wavelet characterizations. We illustrate the scope of our results by discussing—in any dimension $$d\ge 2$$ —the similitude, diagonal and shearlet dilation groups, for which we verify the pertinent conditions. As a result, similitude and diagonal groups can be employed for multiple wavelet characterizations, whereas for the shearlet groups a single wavelet suffices. In particular, the shearlet characterization (previously only established for $$d=2$$ ) holds in arbitrary dimensions.

Localization of Frames

Motivated by Balan, Casazza, Heil, and Landau's results on localization of frames in a separable Hilbert space ℋ, we investigate the transitivity of localization relation of the frame ℱ = {f i } i∈I and sequence ℰ = {e j } j∈G in ℋ with respect to the associated map a: I → G, where G is a discrete abelian group and I is a index set. We also study some properties of ℓ p -column decay, ℓ p -row decay, weak homogeneous approximation property, and strong homogeneous approximation property of frame ℱ = {f i } i∈I , and sequence ℰ = {e j } j∈G , with respect to the associated map a.

Impact of the Radon–Nikodým property on convex approximation properties

A classical result states that the approximation property (AP) of a dual space X* is metric whenever X* or X** has the Radon–Nikodým property (RNP). A version of the proof by Oja deals with the weak metric AP of the original space X and uses lifting techniques to return to the dual space. We streamline this proof, eliminating the need to enter the original space and show that the only essential component of the proof is Oja’s “RNP impact lemma”. We express the weak metric AP and notions of a similar kind in a concise language of systems of seminorms on operator spaces and provide results in the general setting of convex APs.

Weak approximation properties of subspaces

The paper is concerned with weak approximation properties which are weaker than the classical approximation property. For $\lambda \geq 1$, we prove that a Banach space $X$ has the $\lambda$-bounded weak approximation property ($\lambda$-BWAP) if and only if every locally $1$-complemented subspace of $X$ has the $\lambda$-BWAP, and that if $X$ has the $\lambda$-BWAP and $Z$ is a locally $\mu$-complemented subspace of $X$, then $Z$ has the $(2\mu+4)\mu\lambda$-BWAP. It also follows that $X$ has the weak approximation property (WAP) if and only if every locally complemented subspace of $X$ has the WAP.

Approximation faible et principe de Hasse pour des espaces homogènes à stabilisateur fini résoluble

Soit $K$ un corps global et $G$ un $K$-groupe fini r\'esoluble. Sous certaines hypoth\`eses sur une extension d\'eployant $G$, on d\'emontre que l'espace homog\`ene $V:=G'/G$ avec $G'$ un $K$-groupe semi-simple simplement connexe v\'erifie l'approximation faible. On utilise une version plus pr\'ecise de ce r\'esultat pour d\'emontrer le principe de Hasse pour des espaces homog\`enes $X$ sous un $K$-groupe $G'$ semi-simple simplement connexe \`a stabilisateur g\'eom\'etrique $\bar G$ fini et r\'esoluble, sous certaines hypoth\`eses sur le $K$-lien $(\bar G,\kappa)$ d\'efini par $X$. ----- Let $K$ be a global field and $G$ a finite solvable $K$-group. Under certain hypotheses concerning the extension splitting $G$, we show that the homogeneous space $V=G'/G$ with $G'$ a semi-simple simply connected $K$-group has weak approximation. We use a more precise version of this result to prove the Hasse principle for homogeneous spaces $X$ under a semi-simple simply connected $K$-group $G'$ with finite solvable geometric stabilizer $\bar G$, under certain hypotheses concerning the $K$-kernel (or $K$-lien) $(\bar G,\kappa)$ defined by $X$.

On distinguished polynomials and their projections

We study projections and injections between projective tensor products spaces or spaces of polynomials and we show that the example of a polynomial constructed in (4), that is neither p-dominated nor compact, can be identified with the projection map of the symmetric tensor product onto the space. Also we give a characterization of the weak and quasi approximation properties on symmetric tensor products.

Relaxation‐corrected bootstrap algebraic multigrid (rBAMG)

SUMMARYBootstrap algebraic multigrid (BAMG) is a multigrid‐based solver for matrix equations of the form Ax = b. Its aim is to automatically determine the interpolation weights used in algebraic multigrid by locally fitting a set of test vectors that have been relaxed as solutions to the corresponding homogeneous equation, Ax = 0. This paper studies an improved form of BAMG, called relaxation‐corrected bootstrap algebraic multigrid (rBAMG), that involves adding scaled residuals of the test vectors to the least‐squares equations.The basic rBAMG scheme was introduced in an earlier paper [1] and analyzed on a simple model problem. The purpose of the current paper is to further develop this algorithm by incorporating several new critical components and to systematically study its performance on an interesting model problem from quantum chromodynamics. Whereas the earlier paper introduced a new least‐squares principle involving the residuals of the test vectors, a simple extrapolation scheme is developed here to accurately estimate the convergence factors of the evolving algebraic multigrid solver. Such a capability is essential to the effective development of a fast solver, and the approach introduced here is shown numerically to be much more effective than the conventional approach of just observing successive error reduction factors. Another component of the setup process developed here is an adaptive cycling process. This component assesses the effectiveness of the V‐cycle constructed in the initial rBAMG phase by applying it to the homogeneous equation. When poor convergence is observed, the set of test vectors is enhanced with the resulting error, enabling the subsequent least‐squares fit of interpolation to produce an improved V‐cycle. A related component is the scaling and recombination Ritz process that targets the so‐called weak approximation property in an attempt to reveal the important elements of these evolving error and test vector spaces.The aim of the numerical study documented here is to provide insight into the various design choices that arise in the development of an rBAMG algorithm. With this in mind, the results for quantum chromodynamics focus on the behavior of rBAMG in terms of the number of initial test vectors used, the number of relaxation sweeps applied to them, and the size of the target matrices. Copyright © 2012 John Wiley & Sons, Ltd.

Weak Approximation Properties of Elliptic Projections with Functional Constraints

This paper is on the construction of energy-minimizing coarse spaces that obey certain functional constraints and can thus be used, for example, to build robust coarse spaces for elliptic problems with large variations in the coefficients. In practice they are built by patching together solutions to appropriate local saddle point or eigenvalue problems. We develop an abstract framework for such constructions, akin to an abstract Bramble–Hilbert-type lemma, and then apply it in the design of coarse spaces for discretizations of PDEs with highly varying coefficients. The stability and approximation bounds of the constructed interpolant are in the weighted norm and are independent of the variations in the coefficients. Such spaces can be used, for example, in two-level overlapping Schwarz algorithms for elliptic PDEs with large coefficient jumps generally not resolved by a standard coarse grid or for numerical upscaling purposes. Some numerical illustration is provided.

Jensen Measures in Potential Theory

It is shown that, for open sets in classical potential theory and—more generally—for elliptic harmonic spaces Y, the set J x (Y) of Jensen measures (representing measures with respect to superharmonic functions on Y) for a point x ∈ Y is a simple union of closed faces of the compact convex set $M_x(\mathcal P(Y))$ of representing measures with respect to potentials on Y, a set which has been thoroughly studied a long time ago. In particular, the set of extreme Jensen measures can be immediately identified. The results hold even without ellipticity (thus capturing also many examples for the heat equation) provided a rather weak approximation property for superharmonic functions holds. Equally sufficient are a certain transience property and a weak regularity property. More important, each of these properties turns out to be necessary and sufficient for obtaining (in the classical case) that J x (Y) coincides with the set of all compactly supported probability measures in $M_x(\mathcal P(Y))$ .