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This paper aims at developing two versions of the generalized Newton method to compute not merely arbitrary local minimizers of nonsmooth optimization problems but just those, which possess an important stability property known as tilt…
We consider nearly-integrable Hamiltonian systems defined over a non-resonant domain. In the neighborhood of resonances, we use Nekhoroshev-like estimates to provide effective stability bounds for the action variables over long time. The…
For solving large-scale non-convex problems, we propose inexact variants of trust region and adaptive cubic regularization methods, which, to increase efficiency, incorporate various approximations. In particular, in addition to approximate…
We develop R2N, a modified quasi-Newton method for minimizing the sum of a $\mathcal{C}^1$ function $f$ and a lower semi-continuous prox-bounded $h$. Both $f$ and $h$ may be nonconvex. At each iteration, our method computes a step by…
We analyse and compare several algorithms to compute numerically periodic solutions of high-dimensional dynamical systems and investigate their Floquet stability without building the monodromy matrix. The solution and its perturbation are…
It is well-known by now that the BFGS method is an effective method for minimizing nonsmooth functions. However, despite its popularity, theoretical convergence results are almost non-existent. One of the difficulties when analyzing the…
We re-analyze the quasi-linear self consistent dynamics for the beam-plasma instability, by comparing the theory predictions to numerical simulations of the corresponding Hamiltonian system. While the diffusive features of the asymptotic…
We study the geometrical characteristic of quasi-static fractures in disordered media, using iterated conformal maps to determine the evolution of the fracture pattern. This method allows an efficient and accurate solution of the Lam\'e…
We analyze stochastic algorithms for optimizing nonconvex, nonsmooth finite-sum problems, where the nonconvex part is smooth and the nonsmooth part is convex. Surprisingly, unlike the smooth case, our knowledge of this fundamental problem…
Optimizations of atomic positions belong to the most commonly performed tasks in electronic structure calculations. Many simulations like global minimum searches or characterizations of chemical reactions require performing hundreds or…
Travelling and rotating waves are ubiquitous phenomena observed in time dependent PDEs modelling the combined effect of dissipation and non-linear interaction. From an abstract viewpoint they appear as relative equilibria of an equivariant…
In this article, we derive an iterative scheme through a quasi-Newton technique to capture robust weakly efficient points of uncertain multiobjective optimization problems under the upper set less relation. It is assumed that the set of…
In this work, we investigate stochastic quasi-Newton methods for minimizing a finite sum of cost functions over a decentralized network. In Part I, we develop a general algorithmic framework that incorporates stochastic quasi-Newton…
Volume-preserving hyperelastic materials are widely used to model near-incompressible materials such as rubber and soft tissues. However, the numerical simulation of volume-preserving hyperelastic materials is notoriously challenging within…
Non-smooth regularization is widely used in image reconstruction to eliminate the noise while preserving subtle image structures. In this work, we investigate the use of proximal Newton (PN) method to solve an optimization problem with a…
We study an approximation scheme for a variational theory of quasi-static crack growth based on an eigendeformation approach. We consider a family of energy functionals depending on a small parameter $\varepsilon$ and on two fields, the…
In this paper, we present a generic framework to extend existing uniformly optimal convex programming algorithms to solve more general nonlinear, possibly nonconvex, optimization problems. The basic idea is to incorporate a local search…
A method is developed for solving quasilinear convection diffusion problems starting on a coarse mesh where the data and solution-dependent coefficients are unresolved, the problem is unstable and approximation properties do not hold. The…
We present novel algorithms for simulation optimization using random directions stochastic approximation (RDSA). These include first-order (gradient) as well as second-order (Newton) schemes. We incorporate both continuous-valued as well as…
The Whitham equation is a model for the evolution of small-amplitude, unidirectional waves of all wavelengths on shallow water. It has been shown to accurately model the evolution of waves in laboratory experiments. We compute…