We propose an algorithm-independent framework to equip existing optimization methods with primal-dual certificates. Such certificates and corresponding rate of convergence guarantees are important for practitioners to diagnose progress, in particular in machine learning applications. We obtain new primal-dual convergence rates e.g. for the Lasso as well as many L1, Elastic-Net and group-lasso-r...

In this paper, we define and study fuzzy generalized α-closed sets and r-open sets of a given L-fuzzy topological space and prime element r ∈ P (L) and coprime element α ∈ M(L). The concept of L-fuzzy ropen sets was introduced in [10], and it was proved that all r-open sets for L-fuzzy topological space form a new L-fuzzy topology, which is called stratiform L-fuzzy topology. Making use of the ...

Abstract Convergence of a finite element method with mass-lumping and flux upwinding is formulated for solving the immiscible two-phase flow problem in porous media. The approximates directly wetting phase pressure saturation, which are primary unknowns. Well-posedness obtained [J. Numer. Math., 29(2), 2021]. Theoretical convergence proved via compactness argument. numerical saturation converge...

We analyze a fully discrete leapfrog/Galerkin finite element method for the numerical solution of the space fractional order (fractional for simplicity) diffusion equation. The generalized fractional derivative spaces are defined in a bounded interval. And some related properties are further discussed for the following finite element analysis. Then the fractional diffusion equation is discretiz...

Abstract Based on the new approach to modular forms presented in [6] that uses rational functions, we prove a dominated convergence theorem for certain Eisenstein space. It states rearrangements of Fourier series will converge very fast near cusp $$\tau = 0$$ τ = 0 ...

Generalized adjoint with modified discretization and generalized coarse-grain adjoint are derived for a vector system of equations that contains parameterized on/off switches. With vector examples, it is shown that the conventional adjoint minimization may have a convergence problem in multidimensional space. The problem can be solved by the generalized adjoint with modified discretization or b...

We prove that the convergence of a sequence of functions in the space L0 of measurable functions, with respect to the topology of convergence in measure, implies the convergence μ-almost everywhere (μ denotes the Lebesgue measure) of the sequence of rearrangements. We obtain nonexpansivity of rearrangement on the space L∞, and also on Orlicz spaces LN with respect to a finitely additive extende...

we discuss in this paper the strong convergence for weighted sums of negatively orthant dependent (nod) random variables by generalized gaussian techniques. as a corollary, a cesaro law of large numbers of i.i.d. random variables is extended in nod setting by generalized gaussian techniques.