Smooth Norm Research Articles

Smooth and polyhedral approximation in Banach spaces

We show that norms on certain Banach spaces X can be approximated uniformly, and with arbitrary precision, on bounded subsets of X by C∞ smooth norms and polyhedral norms. In particular, we show that this holds for any equivalent norm on c0(Γ), where Γ is an arbitrary set. We also give a necessary condition for the existence of a polyhedral norm on a weakly compactly generated Banach space, which extends a well-known result of Fonf.

Accelerating seismic interpolation with a gradient projection method based on tight frame property of curvelet

Seismic interpolation, as an efficient strategy of providing reliable wavefields, belongs to large-scale computing problems. The rapid increase of data volume in high dimensional interpolation requires highly efficient methods to relieve computational burden. Most methods adopt the L1 norm as a sparsity constraint of solutions in some transformed domain; however, the L1 norm is non-differentiable and gradient-type methods cannot be applied directly. On the other hand, methods for unconstrained L1 norm optimisation always depend on the regularisation parameter which needs to be chosen carefully. In this paper, a fast gradient projection method for the smooth L1 problem is proposed based on the tight frame property of the curvelet transform that can overcome these shortcomings. Some smooth L1 norm functions are discussed and their properties are analysed, then the Huber function is chosen to replace the L1 norm. The novelty of the proposed method is that the tight frame property of the curvelet transform is utilised to improve the computational efficiency. Numerical experiments on synthetic and real data demonstrate the validity of the proposed method which can be used in large-scale computing.

Image Reconstruction of Compressed Sensing Based on Improved Smoothed l0 Norm Algorithm

Image Reconstruction of Compressed Sensing Based on Improved Smoothed l0 Norm Algorithm

Non-negatively constrained least squares and parameter choice by the residual periodogram for the inversion of electrochemical impedance spectroscopy data

The inverse problem associated with electrochemical impedance spectroscopy requiring the solution of a Fredholm integral equation of the first kind is considered. If the underlying physical model is not clearly determined, the inverse problem needs to be solved using a regularized linear least squares problem that is obtained from the discretization of the integral equation. For this system, it is shown that the model error can be made negligible by a change of variables and by extending the effective range of quadrature. This change of variables serves as a right preconditioner that significantly improves the condition of the system. Still, to obtain feasible solutions the additional constraint of non-negativity is required. Simulations with artificial, but realistic, data demonstrate that the use of non-negatively constrained least squares with a smoothing norm provides higher quality solutions than those obtained without the non-negative constraint. Using higher-order smoothing norms also reduces the error in the solutions. The L-curve and residual periodogram parameter choice criteria, which are used for parameter choice with regularized linear least squares, are successfully adapted to be used for the non-negatively constrained Tikhonov least squares problem. Although these results have been verified within the context of the analysis of electrochemical impedance spectroscopy, there is no reason to suppose that they would not be relevant within the broader framework of solving Fredholm integral equations for other applications.

Reconstruction of the complex refractive index in nonlinear phase contrast tomography

A new nonlinear method to reconstruct the complex refractive index distribution with in-line phase tomography measurements is presented. The inverse problem is regularized with the Tikhonov smoothing norm of the index. The original nonlinear iterative approach is based on the Fréchet derivative of the intensity recorded at a single propagation distance and on a Simultaneous Algebraic Reconstruction technique. The reconstruction method requires no a priori knowledge about the materials. The algorithm is successfully applied to some simulated data with noise.

Non-negative sparse decomposition based on constrained smoothed ℓ0 norm

Sparse decomposition of a signal over an overcomplete dictionary has many applications including classification. One of the sparse solvers that has been proposed for finding the sparse solution of a spare decomposition problem (i.e., solving an underdetermined system of equations) is based on the Smoothed L0 norm (SL0). In some applications such as classification of visual data using sparse representation, the coefficients of the sparse solution should be in a specified range (e.g., non-negative solution). This paper presents a new approach based on the Constrained Smoothed L0 norm (CSL0) for solving sparse decomposition problems with non-negative constraint. The performance of the new sparse approach is evaluated on both simulated and real data. For the simulated data, the mean square error of the solution using the CSL0 is comparable to state-of-the-art sparse solvers. For real data, facial expression recognition via sparse representation is studied where the recognition rate using the CSL0 is better than other solver methods (in particular is about 4% better than the SL0).

SAR Image Compression and Reconstruction Based on Compressed Sensing

The appearance of multiplicative speckle noise in Synthetic Aperture Radar (SAR) imagery makes it very difficult to visualization and compression. This paper proposes a novel framework of Compressed Sensing (CS) to compress the SAR image. It combined the Nonsubsampled Contourlet Transform (NSCT) shrinkage method to SAR image sparse represent. And a modified Smoothed L0 norm (SL0) algorithm is used for SAR image accurate reconstruction. The experimental results show the proposed method is very effective and can get better reconstruction performances.

Using boundaries to find smooth norms

The aim of this paper is to present a tool used to show that certain Banach spaces can be endowed with $C^k$ smooth equivalent norms. The hypothesis uses particular countable decompositions of certain subsets of $B_{X^*}$, namely boundaries. Of interest is that the main result unifies two quite well known results. In the final section, some new corollaries are given.

𝐶^{𝑘}-smooth approximations of LUR norms

Let X X be a WCG Banach space admitting a C k C^{k} -smooth norm where k ∈ N ∪ { ∞ } k \in \mathbb {N} \cup \left \{\infty \right \} . Then X X admits an equivalent norm which is simultaneously, C 1 C^1 -smooth, LUR, and the limit of a sequence of C k C^{k} -smooth norms. If X = C ( [ 0 , α ] ) X=C([0,\alpha ]) , where α \alpha is any ordinal, then the same conclusion holds true with k = ∞ k=\infty .

SMOOTH APPROXIMATIONS OF NORMS IN SEPARABLE BANACH SPACES

Let X be a separable real Banach space having a k-times con- tinuously Fr echet dierentiable (i.e. C k -smooth) norm where k 2 f1;:::;1g. We show that any equivalent norm on X can be approximated uniformly on bounded sets by C k -smooth norms.

Ellipsoid characterization theorems

Abstract In this note we prove two ellipsoid characterization theorems. The first one is that if K is a convex body in a normed space with unit ball M, and for any point p ∉ K and in any 2-dimensional plane P intersecting intK and containing p, there are two tangent segments of the same normed length from p to K, then K and M are homothetic ellipsoids. Furthermore, we show that if M is the unit ball of a strictly convex, smooth norm, and in this norm billiard angular bisectors coincide with Busemann angular bisectors or Glogovskij angular bisectors, then M is an ellipse.

Three-space property for asymptotically uniformly smooth renormings

We prove that if Y is a closed subspace of a Banach space X such that Y and X/Y admit an equivalent asymptotically uniformly smooth norm, then X also admits an equivalent asymptotically uniformly smooth norm. The proof is based on the use of the Szlenk index and yields a few other applications to renorming theory.

Minsum hyperspheres in normed spaces

We study the minsum hypersphere problem in finite dimensional real Banach spaces: given a finite set D of (positively weighted) points in an n-dimensional normed space (n≥2), find a minsum hypersphere, i.e., a homothet of the unit sphere of this space that minimizes the sum of (weighted) distances between the hypersphere and the points of D.We show existence results of the following type: there are situations where minsum hyperspheres do not exist, no point-shaped hypersphere can be optimal, and for any norm there exists a set of points D such that a hyperplane is better than any proper hypersphere. We also prove that the intersection of a minsum hypersphere S and conv(D) is non-empty, that D⊆conv(S) implies |S∩conv(D)|≥2, and that |S∩conv(D)|<∞ implies S∩conv(D)⊆D. A certain halving criterion regarding the sums of weights inside and outside of S is verified, and various further results are obtained for large classes of norms, like strictly convex, smooth, and polyhedral norms.

Application of an idea of Voronoĭ to lattice zeta functions

A major problem in the geometry of numbers is the investigation of the local minima of the Epstein zeta function. In this article refined minimum properties of the Epstein zeta function and more general lattice zeta functions are studied. Using an idea of Voronoĭ, characterizations and sufficient conditions are given for lattices at which the Epstein zeta function is stationary or quadratic minimum. Similar problems of a duality character are investigated for the product of the Epstein zeta function of a lattice and the Epstein zeta function of the polar lattice. Besides Voronoĭ type notions such as versions of perfection and eutaxy, these results involve spherical designs and automorphism groups of lattices. Several results are extended to more general lattice zeta functions, where the Euclidean norm is replaced by a smooth norm.

Smooth renormings of the Lebesgue–Bochner function space L1(μ,X)

We show that, if $\mu$ is a probability measure and $X$ is a Banach space, then the space $L^1(\mu,X)$ of Bochner integrable functions admits an equivalent Gâteaux (or uniformly Gâteaux) smooth norm provided that $X$ has such a norm, and that if $X$ admit

Zone diagrams in Euclidean spaces and in other normed spaces

Zone diagrams are a variation on the classical concept of Voronoi diagrams. Given n sites in a metric space that compete for territory, the zone diagram is an equilibrium state in the competition. Formally it is defined as a fixed point of a certain “dominance” map. Asano, Matoušek, and Tokuyama proved the existence and uniqueness of a zone diagram for point sites in the Euclidean plane, and Reem and Reich showed existence for two arbitrary sites in an arbitrary metric space. We establish existence and uniqueness for n disjoint compact sites in a Euclidean space of arbitrary (finite) dimension, and more generally, in a finite-dimensional normed space with a smooth and rotund norm. The proof is considerably simpler than that of Asano et al. We also provide an example of non-uniqueness for a norm that is rotund but not smooth. Finally, we prove existence and uniqueness for two point sites in the plane with a smooth (but not necessarily rotund) norm.

Beamforming regularization matrix and inverse problems applied to sound field measurement and extrapolation using microphone array

For sound field reproduction using multichannel spatial sound systems such as Wave Field Synthesis and Ambisonics, sound field extrapolation is a useful tool for the measurement, description and characterization of a sound environment to be reproduced in a listening area. In this paper, the inverse problem theory is adapted to sound field extrapolation around a microphone array for further spatial sound and sound environment reproduction. A general review of inverse problem theory and analysis tools is given and used for the comparative evaluation of various microphone array configurations. Classical direct regularization methods such as truncated singular value decomposition and Tikhonov regularization are recalled. On the basis of the reviewed background, a new regularization method adapted to the problem at hand is introduced. This method involves the use of an a priori beamforming measurement to define a data-dependent discrete smoothing norm for the regularization of the inverse problem. This method which represents the main contribution of this paper shows promising results and opens new research avenues.

Curves in Banach spaces which allow a C 2 -parameterization

We give a complete characterization of those $f: [0,1] \to X$ (where $X$ is a Banach space which admits an equivalent Fr\'echet smooth norm) which allow an equivalent $C^2$ parametrization. For $X=\R$, a characterization is well-known. However, even in the case $X=\R^2$, several quite new ideas are needed. Moreover, the very close case of parametrizations with a bounded second derivative is solved.

ON THE REPRESENTATION OF MULTI-IDEALS BY TENSOR NORMS

AbstractA tensor norm β=(βn)∞n=1 is smooth if the natural correspondence where 𝕂=ℝ or ℂ, is always an isometric isomorphism. In this paper we study the representation of multi-ideals and of ideals of multilinear forms by smooth tensor norms.

Mechanical constraints on inversion of coseismic geodetic data for fault slip and geometry: Example from InSAR observation of the 6 October 2008Mw6.3 Dangxiong-Yangyi (Tibet) earthquake

[1] Modern geodetic techniques, such as the global positioning system (GPS) and Interferometric Synthetic Aperture Radar (InSAR), provide high-precision deformation measurements of earthquakes. Through elastic models and mathematical optimization methods, the observations can be related to a slip distribution model. The classic linear, kinematic, and static slip inversion problem requires specification of a smoothing norm of slip parameters and a residual norm of the data and a choice about the relative weight between the two norms. Inversions for unknown fault geometry are nonlinear and, therefore, the fault geometry is often assumed to be known for the slip inversion problem. We present a new method to invert simultaneously for fault slip and fault geometry assuming a uniform stress drop over the slipping area of the fault. The method uses a full Bayesian inference method as an engine to estimate the posterior probability distribution of stress drop, fault geometry parameters, and fault slip. We validate the method with a synthetic data set and apply the method to InSAR observations of a moderate-sized normal faulting event, the 6 October 2008 Mw 6.3 Dangxiong-Yangyi (Tibet) earthquake. The results show a 45.0 ± 0.2° west dipping fault with a maximum net slip of ∼1.13 m, and the static stress drop and rake angle are estimated as ∼5.43 MPa and ∼92.5°, respectively. The stress drop estimate falls within the typical range of earthquake stress drops known from previous studies.