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Convex optimization problems with staged structure appear in several contexts, including optimal control, verification of deep neural networks, and isotonic regression. Off-the-shelf solvers can solve these problems but may scale poorly. We…
Motivated by Pan-Yang [PY] and Ma-Cheng [MC], we study a general linear nonlocal curvature flow for convex closed plane curves and discuss the short time existence and asymptotic convergence behavior of the flow. Due to the linear structure…
This paper addresses the problems of spline interpolation on smooth Riemannian manifolds, with or without the inclusion of least-squares fitting. Our unified approach utilizes gradient flows for successively connected curves or networks,…
This paper is concerned with the initial-boundary value problem for an evolutionary variational inequality complying with three intrinsic properties: complete irreversibility, unilateral equilibrium of an energy and an energy conservation…
In this paper we investigate regularity properties of weak solutions to a PDE system that arises in the study of biological transport networks. The system consists of a possibly singular elliptic equation for the scalar pressure of the…
We suggest simple implementable modifications of conditional gradient and gradient projection methods for smooth convex optimization problems in Hilbert spaces. Usually, the custom methods attain only weak convergence. We prove strong…
We develop a convex integration scheme for constructing nonunique weak solutions to the hydrostatic Euler equations (also known as the inviscid primitive equations of oceanic and atmospheric dynamics) in both two and three dimensions. We…
We consider a circulation system arising in turbulence modelling in fluid dynamics with unbounded eddy viscosities. Various notions of weak solutions are considered and compared. We establish existence and regularity results. In particular…
This article is devoted to questions concerning the existence of solutions for partial differential equation problems modeling granular flows. The models studied take into account the complex threshold rheology of these flows, as well as…
Certifying power flow solvability is important for reliable power system operations under volatile operating conditions, but solving power flow equations repeatedly can be costly and may encounter convergence issues. In this paper, we…
We show the existence and the regularity properties of the weak solutions to the two-dimensional stationary incompressible inhomogeneous Navier-Stokes equations with variable viscosity coefficient, by analyzing a fourth-order nonlinear…
Spaces of convex and concave functions appear naturally in theory and applications. For example, convex regression and log-concave density estimation are important topics in nonparametric statistics. In stochastic portfolio theory, concave…
We consider simple bilevel optimization problems where the goal is to compute among the optimal solutions of a composite convex optimization problem, one that minimizes a secondary objective function. Our main contribution is threefold. (i)…
Nonlinear variational methods have become very powerful tools for many image processing tasks. Recently a new line of research has emerged, dealing with nonlinear eigenfunctions induced by convex functionals. This has provided new insights…
In the first part of this doctoral thesis we develop a regularity theory for a polyconvex functional in compressible elasticity. In the second part, we will concentrate on uniqueness questions in various situations of finite elasticity.…
In this work, we present a systematic approach to investigate the existence, multiplicity, and local gradient regularity of solutions for nonlocal quasilinear equations with local gradient degeneracy. Our method involves an interactive…
The Cahn-Hilliard equation is the most common model to describe phase separation processes of a mixture of two components. For a better description of short-range interactions of the material with the solid wall, various dynamic boundary…
In this paper, we propose a unified convergence analysis for a class of generic shuffling-type gradient methods for solving finite-sum optimization problems. Our analysis works with any sampling without replacement strategy and covers many…
In this work, a novel approach for the solution of the inverse conductivity problem from one and multiple boundary measurements has been developed on the basis of the implication of the framework of BV - functions. The space of the…
This paper establishes the existence and uniqueness of classical solutions to the steady Triple-Deck equations, which describe incompressible boundary layer flow over localized roughness at high Reynolds numbers. The triple-deck theory was…