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The purpose of this paper is to prove ergodicity and provide asymptotic formulae for probabilities of threshold crossing related to smooth approximations of three fundamental nonlinear mechanical models: (a) an elasto-plastic oscillator,…
We estimate the error of the Deep Ritz Method for linear elliptic equations. For Dirichlet boundary conditions, we estimate the error when the boundary values are imposed through the boundary penalty method. Our results apply to arbitrary…
We study the equilibrium of hyperelastic solids subjected to kinematic constraints on many small regions, which we call perforations. Such constraints on the displacement $u$ are given in the quite general form $u(x) \in F_x$, where $F_x$…
We consider the spreading of a thin two-dimensional droplet on a solid substrate. We use a model for viscous fluids where the evolution is governed by Darcy's Law. At the triple point where air and liquid meet the solid substrate, the…
This paper develops an adaptive proximal alternating direction method of multipliers (ADMM) for solving linearly constrained, composite optimization problems under the assumption that the smooth component of the objective is weakly convex,…
First-order methods have been studied for nonlinear constrained optimization within the framework of the augmented Lagrangian method (ALM) or penalty method. We propose an improved inexact ALM (iALM) and conduct a unified analysis for…
The paper is devoted to the penalty Robin-Robin domain decomposition methods (DDMs), proposed by us for the solution of unilateral multibody contact problems of elasticity. These DDMs are based on the penalty method for variational…
This paper proposes and analyzes a dampened proximal alternating direction method of multipliers (DP.ADMM) for solving linearly-constrained nonconvex optimization problems where the smooth part of the objective function is nonseparable.…
We derive asymptotic properties of penalized estimators for singular models for which identifiability may break and the true parameter values can lie on the boundary of the parameter space. Selection consistency of the estimators is also…
A class of non-smooth and non-convex optimization problems with penalty constraints linked to variational inequalities (VI) is studied with respect to its shape differentiability. The specific problem stemming from quasi-brittle fracture…
In this paper we consider time-optimal control problems for systems with backlash. Such systems are described by second order differential equations coupled with restrictions modeling the inelastic shocks. A main feature of such systems is…
In this paper we consider a class of structured nonsmooth difference-of-convex (DC) constrained DC program in which the first convex component of the objective and constraints is the sum of a smooth and nonsmooth functions while their…
The inverse first-passage time problem determines a boundary such that the first-passage time of a Wiener process to this boundary has a given distribution. An approximation which is based on the starting value of the boundary to a smooth…
Tuning parameters are parameters involved in an estimating procedure for the purpose of reducing the risk of some other estimator. Examples include the degree of penalization in penalized regression and likelihood problems, as well as the…
In this paper we establish the convergence of a numerical scheme based, on the Finite Element Method, for a time-independent problem modelling the deformation of a linearly elastic elliptic membrane shell subjected to remaining confined in…
In this paper, we consider the problem of minimizing a smooth function, given as finite sum of black-box functions, over a convex set. In order to advantageously exploit the structure of the problem, for instance when the terms of the…
In this paper we address the convergence of stochastic approximation when the functions to be minimized are not convex and nonsmooth. We show that the "mean-limit" approach to the convergence which leads, for smooth problems, to the ODE…
We analyse the convergence of an approximate, fully inexact, ADMM algorithm under additive, deterministic and probabilistic error models. We consider the generalized ADMM scheme that is derived from generalized Lagrangian penalty with…
We propose an inexact proximal augmented Lagrangian method (P-ALM) for nonconvex structured optimization problems. The proposed method features an easily implementable rule not only for updating the penalty parameters, but also for…
We calculate arbitrarily tight upper and lower bounds on an unconstrained control, linear-quadratic, singularly perturbed optimal control problem whose exact solution is computationally intractable. It is well known that for the…