In this paper we present one approach to build optimal meshes for P1 interpolation. Considering classical geometric error estimates based on the Hessian matrix of a solution, we show it is possible to generate optimal meshes in H 1 semi-norm via a simple minimization procedure.

In the original Virtual Element space with degree of accuracy k, projector operators in theH-seminorm onto polynomials of degree ≤ k can be easily computed. On the other hand, projections in the L norm are available only on polynomials of degree ≤ k − 2 (directly from the degrees of freedom). Here we present a variant of VEM that allows the exact computations of the L projections on all polynom...

Let Lm,p(Rn) be the homogeneous Sobolev space for p?(n,?), ? a Borel regular measure on Rn, and Lm,p(Rn)+Lp(d?) of measurable functions with finite seminorm ?f?Lm,p(Rn)+Lp(d?):=inff1+f2=f?{?f1?Lm,p(Rn)p+?Rn|f2|pd?}1/p. We construct linear operator T:Lm,p(Rn)+Lp(d?)?Lm,p(Rn), that nearly optimally decomposes every function in sum space: ?Tf?Lm,p(Rn)p+?Rn|Tf?f|pd??C?f?Lm,p(Rn)+Lp(d?)p C dependent...

in this note we consider the notion of intuitionistic fuzzy (weak) dual hyper k-ideals and obtain related results. then we classify this notion according to level sets. after that we determine the relationships between intuitionistic fuzzy (weak) dual hyper k-ideals and intuitionistic fuzzy (weak) hyper k-ideals. finally, we define the notion of the product of two intuitionistic fuzzy (weak) du...

Tikhonov regularization is one of the most popular methods for solving linear systems of equations or linear least-squares problems with a severely ill-conditioned matrix A. This method replaces the given problem by a penalized least-squares problem. The present paper discusses measuring the residual error (discrepancy) in Tikhonov regularization with a seminorm that uses a fractional power of ...

In an earlier paper [SIAM J. Matrix Anal. Appl. vol. 30 (2008), 925–938] we gave sufficient conditions in terms of an energy seminorm for the convergence of stationary iterations for solving singular linear systems whose coefficient matrix is Hermitian and positive semidefinite. In this paper we show in which cases these conditions are also necessary, and show that they are not necesary in othe...

In this note we prove a trace theorem in fractional spaces with variable exponents. To be more precise, we show that if p : Ω × Ω → (1,∞) and q : ∂Ω→ (1,∞) are continuous functions such that (n− 1)p(x, x) n− sp(x, x) > q(x) in ∂Ω ∩ {x ∈ Ω: n− sp(x, x) > 0}, then the inequality ‖f‖Lq(·)(∂Ω) ≤ C { ‖f‖Lp̄(·)(Ω) + [f ]s,p(·,·) } holds. Here p̄(x) = p(x, x) and [f ]s,p(·,·) denotes the fractional semi...

We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by the problem of restoring noisy and blurry images via variational methods, specifically by using the BV seminorm. Although our procedure applies in quite general situations it was obtained by geometric...

By a quantum metric space we mean a C∗-algebra (or more generally an order-unit space) equipped with a generalization of the usual Lipschitz seminorm on functions which one associates to an ordinary metric. We develop for compact quantum metric spaces a version of Gromov–Hausdorff distance. We show that the basic theorems of the classical theory have natural quantum analogues. Our main example ...

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