Consider the high-dimensional linear regression model y = Xβ∗ +w, where y ∈ R is an observation vector, X ∈ R is a design matrix with d > n, the quantity β∗ ∈ R is an unknown regression vector, and w ∼ N (0, σI) is additive Gaussian noise. This paper studies the minimax rates of convergence for estimating β∗ in either l2-loss and l2-prediction loss, assuming that β∗ belongs to an lq-ball Bq(Rq)...

We prove density estimates for level sets of minimizers of the energy ε‖u‖Hs(Ω) + ∫ Ω W (u) dx, with s ∈ (0, 1), where ‖u‖Hs(Ω) denotes the total contribution from Ω in the Hs norm of u, and W is a double-well potential. As a consequence we obtain, as ε → 0+, the uniform convergence of the level sets of u to either a Hs-nonlocal minimal surface if s ∈ (0, 1 2 ), or to a classical minimal surfac...

Residual norm estimates are derived for a general class of methods based on projection techniques on subspaces of the form K m + W, where K m is the standard Krylov subspace associated with the original linear system, and W is some other subspace. Thesèaugmented Krylov subspace methods' include eigenvalue deeation techniques as well as block-Krylov methods. Residual bounds are established which...

Note: The expression for L(s,W ) requires some care, since q v is a complex number acting on a p-adic vector space. What has to be proved is that the characteristic polynomial for the action of Frobv on W i has rational coefficients (and is independent of p), so evaluation using q v makes sense (and the Riemann Hypothesis ensures absolute convergence of the product in a suitable right half-plan...

Imaging of brain activity based on magnetoencephalogmphy (MEG) requires high resolution e s t f motes that closely approcimate the spatial distribution of the underlying currents. W e etamine the physics of the MEG problem t o motivate the development of a new algorithm that meets its unique requirements. The technique is a nonparametric, iterative, weighted norm minimization procedure with pos...

The study examines the concept of stochastic convergence in the EU28 countries over the 1994–2013 period. The convergence of individual countries’ GDP per capita towards the EU28 average per capita income level and the pairwise convergence between the GDP of individual countries are both analyzed. Additionally, we introduce our own concept of conditional stochastic convergence which is based on...

After F a s t [6] introduced the theory of statistical convergence of a real sequence, it has become popular among mathematicians ([2], [7]–[9], [17]). The ideas of statistical limit superior and limit inferior were first extensively studied by F r i d y and O r h a n [9]. After K o s t y r k o et al. [10] extended the idea of statistical convergence to I-convergence using the concept of an ide...

We study a class of symmetric discontinuous Galerkin methods on graded meshes. Optimal order error estimates are derived in both the energy norm and the L2 norm, and we establish the uniform convergence of V -cycle, F -cycle and W -cycle multigrid algorithms for the resulting discrete problems. Numerical results that confirm the theoretical results are also presented.

In the regression model with errors in variables, we observe n i.i.d. copies of (Y, Z) satisfying Y = fθ0(X) + ξ and Z = X + ε involving independent and unobserved random variables X, ξ, ε plus a regression function fθ0, known up to some finite dimensional θ. The common densities of the Xi’s and of the ξi’s are unknown whereas the distribution of ε is completely known. We aim at estimating the ...

The convergence to steady state solutions of the Euler equations for high order weighted essentially non-oscillatory (WENO) finite difference schemes with the Lax-Friedrichs flux splitting [Jiang G.-S. and Shu C.-W. (1996), J. Comput. Phys. 126, 202-228] is investigated. Numerical evidence in [Zhang S. and Shu C.-W. (2007), J. Sci. Comput. 31, 273-305] indicates that there exist slight post-sho...