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We provide geometric conditions on the set of boundary points of infinite type of a smooth bounded pseudoconvex domain in $\mathbb{C}^{n}$ which imply that the $\bar{\partial}$-Neumann operator is compact. These conditions are formulated in…
We report on Domain Wall solution of Calabi-Yau compactifications with general fluxes and their application to the study of mirror symmetry in generalized backgrounds. We address, in particular, to the issue of magnetic NSNS fluxes. We show…
We consider the simplest gauge theories given by one- and two- matrix integrals and concentrate on their stringy and geometric properties. We remind general integrable structure behind the matrix integrals and turn to the geometric…
We study the high-frequency limit of non-autonomous gradient flows in metric spaces of energy functionals comprising an explicitly time-dependent perturbation term which might oscillate in a rapid way, but fulfills a certain Lipschitz…
Motivated by a constrained minimization problem, it is studied the gradient flows with respect to Hessian Riemannian metrics induced by convex functions of Legendre type. The first result characterizes Hessian Riemannian structures on…
We present a convex solution for the design of generalized accelerated gradient algorithms for strongly convex objective functions with Lipschitz continuous gradients. We utilize integral quadratic constraints and the Youla parameterization…
We construct topological string and topological membrane actions with a nontrivial 3-form flux H in arbitrary dimensions. These models realize Bianchi identities with a nontrivial H flux as consistency conditions. Especially, we discuss the…
A closure theory is developed for inhomogeneous turbulent flow, which enables a systematic derivation of the turbulence constitutive relations without relying on any empirical parameters. Renormalized-perturbation approximation is performed…
We analyze the global and local behavior of gradient-like flows under stochastic errors towards the aim of solving convex optimization problems with noisy gradient input. We first study the unconstrained differentiable convex case, using a…
We show that the Nernst-Planck-Euler system, which models ionic electrodiffusion in fluids, has global strong solutions for arbitrarily large data in the two dimensional bounded domains. The assumption on species is either there are two…
Gradient boundedness up to the boundary for solutions to Dirichlet and Neumann problems for elliptic systems with Uhlenbeck type structure is established. Nonlinearities of possibly non-polynomial type are allowed, and minimal regularity on…
We focus on nonconvex and nonsmooth minimization problems with a composite objective, where the differentiable part of the objective is freed from the usual and restrictive global Lipschitz gradient continuity assumption. This longstanding…
In this paper, we address stochastic optimization problems involving a composition of a non-smooth outer function and a smooth inner function, a formulation frequently encountered in machine learning and operations research. To deal with…
This thesis explores two important areas in the mathematical analysis of nonlinear partial differential equations: Generalized gradient flows and vector valued Orlicz spaces. The first part deals with the existence of strong solutions for…
R. Thom's gradient conjecture states that if a gradient flow of an analytic function converges to a limit, it does so along a unique limiting direction. In this paper, we extend and settle this conjecture in the context of infinite…
We consider the 1-harmonic flow of maps from a bounded domain into a submanifold of a Euclidean space, i.e. the gradient flow of the total variation functional restricted to maps taking values in the manifold. We restrict ourselves to…
Stability and error analysis remain challenging for problems that lack regularity properties near solutions, are subject to large perturbations, and might be infinite dimensional. We consider nonconvex optimization and generalized equations…
The aim of this paper is to prove the existence of multiple solutions for a family of nonlinear elliptic systems in divergence form coupled with a pointwise gradient constraint: \begin{align*} \left\{ \begin{array}{ll}…
Many large-scale and distributed optimization problems can be brought into a composite form in which the objective function is given by the sum of a smooth term and a nonsmooth regularizer. Such problems can be solved via a proximal…
Composite optimization offers a powerful modeling tool for a variety of applications and is often numerically solved by means of proximal gradient methods. In this paper, we consider fully nonconvex composite problems under only local…