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Convergence failure and slow convergence rates are among the biggest challenges with solving the system of non-linear equations numerically. Although mitigated, such issues still linger when using strictly small time steps and…

Separated flows past complex geometries are modelled by discrete vortex techniques. The flows are assumed to be rotational and inviscid, and a new technique is described to determine the streamfunctions for linear shear profiles. The…

We consider the problem of minimization of a convex function on a simple set with convex non-smooth inequality constraint and describe first-order methods to solve such problems in different situations: smooth or non-smooth objective…

Recovering high quality surfaces from noisy triangulated surfaces is a fundamental important problem in geometry processing. Sharp features including edges and corners can not be well preserved in most existing denoising methods except the…

The goal of this paper is to study approaches to bridge the gap between first-order and second-order type methods for composite convex programs. Our key observations are: i) Many well-known operator splitting methods, such as…

This paper investigates simple bilevel optimization problems where we minimize an upper-level objective over the optimal solution set of a convex lower-level objective. Existing methods for such problems either only guarantee asymptotic…

Two methods of level set type are proposed for solving the Cauchy problem for an elliptic equation. Convergence and stability results for both methods are proven, characterizing the iterative methods as regularization methods for this…

The study of provable adversarial robustness has mostly been limited to classification tasks and models with one-dimensional real-valued outputs. We extend the scope of certifiable robustness to problems with more general and structured…

Randomized smoothing is a widely adopted technique for optimizing nonsmooth objective functions. However, its efficiency analysis typically relies on global Lipschitz continuity, a condition rarely met in practical applications. To address…

Bilevel optimization is a fundamental tool in hierarchical decision-making and has been widely applied to machine learning tasks such as hyperparameter tuning, meta-learning, and continual learning. While significant progress has been made…

We present a second order accurate in time numerical scheme for curve shortening flow in the plane that is unconditionally monotone. It is a variant of threshold dynamics, a class of algorithms in the spirit of the level set method that…

The aim of this paper is to develop an efficient algorithm for solving a class of unconstrained nondifferentiable convex optimization problems in finite dimensional spaces. To this end we formulate first its Fenchel dual problem and…

We present a new high order finite element method for the discretization of partial differential equations on stationary smooth surfaces which are implicitly described as the zero level of a level set function. The discretization is based…

Pressure-robustness has been widely studied since the conception of the notion and the introduction of the reconstruction approach for classical mixed methods in [5]. Using discretizations capable of yielding velocity solutions that are…

For problems involving large deformations of thin structures, simulating fluid-structure interaction (FSI) remains challenging largely due to the need to balance computational feasibility, efficiency, and solution accuracy. Overlapping…

In this paper, we study a class of bilevel programming problem where the inner objective function is strongly convex. More specifically, under some mile assumptions on the partial derivatives of both inner and outer objective functions, we…

Normalizing flows are a promising tool for modeling probability distributions in physical systems. While state-of-the-art flows accurately approximate distributions and energies, applications in physics additionally require smooth energies…

This paper studies simple bilevel problems, where a convex upper-level function is minimized over the optimal solutions of a convex lower-level problem. We first show the fundamental difficulty of simple bilevel problems, that the…

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)…

Flexible sparsity regularization means stably approximating sparse solutions of operator equations by using coefficient-dependent penalizations. We propose and analyse a general nonconvex approach in this respect, from both theoretical and…