Scattered Date Interpolation from Principal Shift-Invariant Spaces

Under certain assumptions on the compactly supported function φC( ^d), we propose two methods of selecting a function s from the scaled principal shift-invariant space S^h(φ) such that s interpolates a given function f at a scattered set of data locations. For both methods, the selection scheme amounts to solving a quadratic programming problem and we are able to prove error estimates similar to those obtained by Duchon for surface spline interpolation.     

On the dual Stiefel-Whitney classes of some Grassmann manifolds

We present some non-vanishing dual Stiefel-Whitney classes of the Grassmann manifolds O(n)/O(4) × O(n − 4) for n = 2 ^s + 2 and n = 2 ^s + 3 (s ≧ 3), providing a supplement to results of Hiller, Stong, and Oproiu. Some applications are also mentioned. Part of this research was carried out while the first author was a member of three research teams supported in part by the grant agencies VEGA and APVV, and the second author was a member of a research team supported in part by VEGA.     

Suprema of Chaos Processes and the Restricted Isometry Property

We present a new bound for suprema of a special type of chaos process indexed by a set of matrices, which is based on a chaining method. As applications we show significantly improved estimates for the restricted isometry constants of partial random circulant matrices and time‐frequency structured random matrices. In both cases the required condition on the number m of rows in terms of the sparsity s and the vector length n is m ? s log² s log² n. © 2014 Wiley Periodicals, Inc.     

Two shape preserving lagrangeC 2-interpolants

Summary We present a LagrangeC ²-interpolant to scattered convex data which preserves convexity. We also present a LagrangeC ²-interpolant to uniformly spaced monotone data sites which preserves monotonicity. In both cases no further conditions are required on the data values. These interpolants are explicitely described and local. Error isO(h ³) when the function to be interpolated isC ³.     

New pairs of -sequences with 4-level cross-correlation

Let ω be a primitive element of GF(2ⁿ), where . Let d=(2^2k+2^s+1-2^k+1-1)/(2^s-1), where n=2k, and s is such that 2s divides k. We prove that the binary m-sequences s(t)=tr(ω^t) and s(dt) have a four-level cross-correlation function and give the distribution of the values.     

A Graham-Sloane Type Construction for s-Surjective Matrices

We give a construction of (n–s)-surjective matrices with n columns over using Abelian groups and additive s-bases. In particular we show that the minimum number of rows ms _q(n,n–s) in such a matrix is at most s ^s q ^n–s for all q, n and s.     

Derivative reproducing properties for kernel methods in learning theory

The regularity of functions from reproducing kernel Hilbert spaces (RKHSs) is studied in the setting of learning theory. We provide a reproducing property for partial derivatives up to order s when the Mercer kernel is C^2s. For such a kernel on a general domain we show that the RKHS can be embedded into the function space C^s. These observations yield a representer theorem for regularized learning algorithms involving data for function values and gradients. Examples of Hermite learning and semi-supervised learning penalized by gradients on data are considered.     

Decay properties of linear thermoelastic plates: Cattaneo versus Fourier law

In this article, we investigate the decay properties of the linear thermoelastic plate equations in the whole space for both Fourier and Cattaneo's laws of heat conduction. We point out that while the paradox of infinite propagation speed inherent in Fourier's law is removed by changing to the Cattaneo law, the latter always leads to a loss of regularity of the solution. The main tool used to prove our results is the energy method in the Fourier space together with some integral estimates. We prove the decay estimates for initial data U ₀?∈?H ^s (?)?∩?L ¹(?). In addition, by restricting the initial data to U ₀?∈?H ^s (?)?∩?L ^1,γ(?) and γ?∈?[0,?1], we can derive faster decay estimates with the decay rate improvement by a factor of t ^?γ/2.     

Multilevel Interpolation and Approximation

Interpolation by translates of a given radial basis function (RBF) has become a well-recognized means of fitting functions sampled at scattered sites in ^d. A major drawback of these methods is their inability to interpolate very large data sets in a numerically stable way while maintaining a good fit. To circumvent this problem, a multilevel interpolation (ML) method for scattered data was presented by Floater and Iske. Their approach involves m levels of interpolation where at the jth level, the residual of the previous level is interpolated. On each level, the RBF is scaled to match the data density. In this paper, we provide some theoretical underpinnings to the ML method by establishing rates of approximation for a technique that deviates somewhat from the Floater–Iske setting. The final goal of the ML method will be to provide a numerically stable method for interpolating several thousand points rapidly.     

Approximation in L p (R d ) from a space spanned by the scattered shifts of a radial basis function

A new multivariate approximation scheme on R ^d using scattered translates of the “shifted” surface spline function is developed. The scheme is shown to provide spectral L _p -approximation orders with 1 ≤ p ≤ ∞, i.e., approximation orders that depend on the smoothness of the approximands. In addition, it applies to noisy data as well as noiseless data. A numerical example is presented with a comparison between the new scheme and the surface spline interpolation method.     

SHAPEI PRESERVING PIECEWISE CUBIC INTERPOLATION

This paper presents a C^1-interpolation which preserves convexity to scattered convex data. The interpolant is local and explicitly described. The interpolating function si(x) is C^2 on each interval (xi, xi 1). Error will be O(h^2) when the function to he interpolated is C^3.     

Total Scalar Curvature and L 2 Harmonic 1-Forms on a Minimal Hypersurface in Euclidean Space

Let M ⁿ , n 3, be a complete oriented immersed minimal hypersurface in Euclidean space R ⁿ⁺¹. We show that if the total scalar curvature on M is less than the n/2 power of 1/C _s , where C _s is the Sobolev constant for M, then there are no L ² harmonic 1-forms on M. As corollaries, such a minimal hypersurface contains no nontrivial harmonic functions with finite Dirichlet integral and so it has only one end. This implies finally that M is a hyperplane.     

Sum of sequence spaces and matrix transformations mapping in s α 0 ((Δ ? λ I ) h ) + s β ( c ) ((Δ ? μ I ) l )

We deal with the sum of sequence spaces. Then we apply these results to characterize matrix transformations mapping between s ^h,l (λ, μ) = s _α ⁰((Δ − λI) ^h ) + s _β ^(c)((Δ − μI) ^l ) and s _γ . Among other things the aim of this paper is to reduce the set (s ^h,l (λ, μ), s _γ to a set of the form S _τ,γ .      

Random data Cauchy theory for supercritical wave equations I: local theory

We study the local existence of strong solutions for the cubic nonlinear wave equation with data in H ^s (M), s<1/2, where M is a three dimensional compact Riemannian manifold. This problem is supercritical and can be shown to be strongly ill-posed (in the Hadamard sense). However, after a suitable randomization, we are able to construct local strong solution for a large set of initial data in H ^s (M), where s≥1/4 in the case of a boundary less manifold and s≥8/21 in the case of a manifold with boundary. Mathematics Subject Classification (2000)  35Q55, 35BXX, 37K05, 37L50, 81Q20     

Low‐curvature image simplifiers: Global regularity of smooth solutions and Laplacian limiting schemes

We consider a class of fourth‐order nonlinear diffusion equations motivated by Tumblin and Turk's “low‐curvature image simplifiers” for image denoising and segmentation. The PDE for the image intensity u is of the form where g(s) = k²/(k² + s²) is a “curvature” threshold and λ denotes a fidelity‐matching parameter. We derive a priori bounds for Δu that allow us to prove global regularity of smooth solutions in one space dimension, and a geometric constraint for finite‐time singularities from smooth initial data in two space dimensions. This is in sharp contrast to the second‐order Perona‐Malik equation (an ill‐posed problem), on which the original LCIS method is modeled. The estimates also allow us to design a finite difference scheme that satisfies discrete versions of the estimates, in particular, a priori bounds on the smoothness estimator in both one and two space dimensions. We present computational results that show the effectiveness of such algorithms. Our results are connected to recent results for fourth‐order lubrication‐type equations and the design of positivity‐preserving schemes for such equations. This connection also has relevance for other related fourth‐order imaging equations. © 2004 Wiley Periodicals, Inc.     

Spectral Representation of Gaussian Semimartingales

The aim of the present paper is to characterize the spectral representation of Gaussian semimartingales. That is, we provide necessary and sufficient conditions on the kernel K for X _t =∫ K _t (s) dN _s to be a semimartingale. Here, N denotes an independently scattered Gaussian random measure on a general space S. We study the semimartingale property of X in three different filtrations. First, the ℱ ^X -semimartingale property is considered, and afterwards the ℱ ^X,∞-semimartingale property is treated in the case where X is a moving average process and ℱ _t ^X,∞=σ(X _s :s∈(−∞,t]). Finally, we study a generalization of Gaussian Volterra processes. In particular, we provide necessary and sufficient conditions on K for the Gaussian Volterra process ∫ _−∞ ^t K _t (s) dW _s to be an ℱ ^W,∞-semimartingale (W denotes a Wiener process). Hereby we generalize a result of Knight (Foundations of the Prediction Process, 1992) to the nonstationary case.     

Sampling Sets and Quadrature Formulae on the Rotation Group

In this paper, we construct sampling sets over the rotation group SO(3). The proposed construction is based on a parameterization, which reflects the product nature ² × ¹ of SO(3) very well, and leads to a spherical Pythagorean-like formula in the parameter domain. We prove that by using uniformly distributed points on ² and ¹, we obtain uniformly sampling nodes on the rotation group SO(3). Furthermore, quadrature formulae on ² and ¹ lead to quadratures on SO(3), as well. For scattered data on SO(3), we give a necessary condition on the mesh norm such that the sampling nodes possess nonnegative quadrature weights. We propose an algorithm for computing the quadrature weights for scattered data on SO(3) based on fast algorithms. We confirm our theoretical results with examples and numerical tests.     

Flat wavelet bases adapted to the homogeneous Sobolev spaces

We present a construction of “flat wavelet bases” adapted to the homogeneous Sobolev spaces ?^s (?ⁿ ). They solve the problem of the phenomenon of infrared divergence which appears for usual wavelet expansions in ?^s (?ⁿ ): these bases remove the divergence in the case s – n /2 ? ? since they are also bases of the realization of ?^s (?ⁿ ). In the critical case s – n /2 ∈ ?, they provide a confinement of the divergence in a “small” space. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The stable ATA-orthogonal s-step Orthomin(k) algorithm with the CADNA library

The major drawback of the s-step iterative methods for nonsymmetric linear systems of equations is that, in the floating-point arithmetic, a quick loss of orthogonality of s-dimensional direction subspaces can occur, and consequently slow convergence and instability in the algorithm may be observed as s gets larger than 5. In [18], Swanson and Chronopoulos have demonstrated that the value of s in the s-step Orthomin(k) algorithm can be increased beyond s=5 by orthogonalizing the s direction vectors in each iteration, and have shown that the A^TA-orthogonal s-step Orthomin(k) is stable for large values of s (up to s=16). The subject of this paper is to show how by using the CADNA library, it is possible to determine a good value of s for A^TA-orthogonal s-step Orthomin(k), and during the run of its code to detect the numerical instabilities and to stop the process correctly, and to restart the A^TA-orthogonal s-step Orthomin(k) in order to improve the computed solution. Numerical examples are used to show the good numerical properties. This revised version was published online in June 2006 with corrections to the Cover Date.     

Regularization in Sobolev Spaces with Fractional Order

We study the minimization of a quadratic functional where the Tichonov regularization term is an H ^s -norm with a fractional s > 0. Moreover, pointwise bounds for the unknown solution are given. A multilevel approach as an equivalent norm concept is introduced. We show higher regularity of the solution of the variational inequality. This regularity is used to show the existence of regular Lagrange multipliers in function space. The theory is illustrated by two applications: a Dirichlet boundary control problem and a parameter identification problem.