arXiv subjects

Dinh Dung

Publications and source records attributed to Dinh Dung.

5 recordsLinked to original sources

arXiv2013

New explicit-in-dimension estimates for the cardinality of high-dimensional hyperbolic crosses and approximation of functions having mixed smoothness

We are aiming at sharp and explicit-in-dimension estimations of the cardinality of $s$-dimensional hyperbolic crosses where $s$ may be large, and applications in high-dimensional approximations of functions having mixed smoothness. In particular, we provide new tight and explicit-in-dimension upper and lower bounds for the cardinality of hyperbolic crosses. We apply them to obtain explicit upper and lower bounds for Kolmogorov $N$-widths and $\varepsilon$-dimensions of a modified Korobov class parametrized by positive $a$ of $s$-variate periodic functions having mixed smoothness $r$, as a function of three variables $N,s,a$ and $\varepsilon, s,a$, respectively, when $N,s$ may be large, $\varepsilon$ may be small and $a$ may range from 0 to infinity. Based on these results we describe a complete classification of tractability for the problem of $\varepsilon$-dimensions of the modified Korobov class. In particular, we prove the introduced exponential tractability of this problem for $a>1$. All of these methods and results are also extended to high-dimensional approximations of non-periodic functions by Jacobi polynomials with powers in hyperbolic crosses.

math.NA↗

arXiv2013

A splitting algorithm for system of composite monotone inclusions

We propose a splitting algorithm for solving a system of composite monotone inclusions formulated in the form of the extended set of solutions in real Hilbert spaces. The resluting algorithm is a an extension of the algorithm in [4]. The weak convergence of the algorithm proposed is proved. Applications to minimization problems is demonstrated.

math.OC↗

arXiv2013

Multivariate approximation by translates of the Korobov function on Smolyak grids

For a set $\mathbb{W} \subset L_p(\bT^d)$, $1 < p < \infty$, of multivariate periodic functions on the torus $\bT^d$ and a given function $φ\in L_p(\bT^d)$, we study the approximation in the $L_p(\bT^d)$-norm of functions $f \in \mathbb{W}$ by arbitrary linear combinations of $n$ translates of $φ$. For $\mathbb{W} = U^r_p(\bT^d)$ and $φ= κ_{r,d}$, we prove upper bounds of the worst case error of this approximation where $U^r_p(\bT^d)$ is the unit ball in the Korobov space $K^r_p(\bT^d)$ and $κ_{r,d}$ is the associated Korobov function. To obtain the upper bounds, we construct approximation methods based on sparse Smolyak grids. The case $p=2, \ r > 1/2$, is especially important since $K^r_2(\bT^d)$ is a reproducing kernel Hilbert space, whose reproducing kernel is a translation kernel determined by $κ_{r,d}$. We also provide lower bounds of the optimal approximation on the best choice of $φ$.

math.FA↗

arXiv2010

Proximity for Sums of Composite Functions

We propose an algorithm for computing the proximity operator of a sum of composite convex functions in Hilbert spaces and investigate its asymptotic behavior. Applications to best approximation and image recovery are described.

math.OC↗

arXiv2010

Dualization of Signal Recovery Problems

In convex optimization, duality theory can sometimes lead to simpler solution methods than those resulting from direct primal analysis. In this paper, this principle is applied to a class of composite variational problems arising in particular in signal recovery. These problems are not easily amenable to solution by current methods but they feature Fenchel-Moreau-Rockafellar dual problems that can be solved by forward-backward splitting. The proposed algorithm produces simultaneously a sequence converging weakly to a dual solution, and a sequence converging strongly to the primal solution. Our framework is shown to capture and extend several existing duality-based signal recovery methods and to be applicable to a variety of new problems beyond their scope.

math.OC↗