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Optimal transport (OT) is a powerful tool in mathematics and data science but faces severe computational and statistical challenges in high dimensions. We propose convex relaxation approaches based on marginal and cluster moment relaxations…
We present an algorithm for computing the nonlinear interface dynamics of the Mullins-Sekerka model for interfaces that are defined implicitly (e.g. by a level set function) using integral equations . The computation of the dynamics…
A local convergence rate is established for an orthogonal collocation method based on Gauss quadrature applied to an unconstrained optimal control problem. If the continuous problem has a sufficiently smooth solution and the Hamiltonian…
The study of combinatorial optimization problems with a submodular objective has attracted much attention in recent years. Such problems are important in both theory and practice because their objective functions are very general. Obtaining…
Scaling behavior is studied of several dominant eigenvalues of spectra of Markov matrices and the associated correlation times governing critical slowing down in models in the universality class of the two-dimensional Ising model. A scheme…
A main challenge in numerical simulations of moving contact line problems is that the adherence, or no-slip boundary condition leads to a non-integrable stress singularity at the contact line. In this report we perform the first steps in…
This paper studies the convergence properties of a family of Relaxed $\ell$-Minimal Gradient Descent methods for quadratic optimization; the family includes the omnipresent Steepest Descent method, as well as the Minimal Gradient method.…
Many high dimensional sparse learning problems are formulated as nonconvex optimization. A popular approach to solve these nonconvex optimization problems is through convex relaxations such as linear and semidefinite programming. In this…
We propose a new method for generating semidefinite relaxations of optimal power flow problems. The method is based on chordal conversion techniques: by dropping some equality constraints in the conversion, we obtain semidefinite…
The control of relaxation-type systems of ordinary differential equations is investigated using the Hamilton-Jacobi-Bellman equation. First, we recast the model as a singularly perturbed dynamics which we embed in a family of controlled…
Filaments are ubiquitous within the microscopic world. They occur frequently in both biological and industrial environments and display varied and rich dynamics. Their wide range of applications has spurred the development of a special…
This work studies the problem of maximizing a higher degree real homogeneous multivariate polynomial over the unit sphere. This problem is equivalent to finding the leading eigenvalue of the associated symmetric tensor of higher order,…
We introduce a comprehensive framework for analyzing convergence rates for infinite dimensional linear programming problems (LPs) within the context of the moment-sum-of-squares hierarchy. Our primary focus is on extending the existing…
This paper considers the optimal control for hybrid systems whose trajectories transition between distinct subsystems when state-dependent constraints are satisfied. Though this class of systems is useful while modeling a variety of…
We consider the sharp interface limit of the Allen-Cahn equation with homogeneous Neumann boundary condition in a two-dimensional domain $\Omega$, in the situation where an interface has developed and intersects $\partial\Omega$. Here a…
We consider a variational convex relaxation of a class of optimal partitioning and multiclass labeling problems, which has recently proven quite successful and can be seen as a continuous analogue of Linear Programming (LP) relaxation…
We consider a kinetic model of two species of particles interacting with a reservoir at fixed temperature, described by two coupled Vlasov-Fokker-Plank equations. We prove that in the diffusive limit the evolution is described by a…
For sufficiently low reservoir temperatures, it is known that open quantum systems subject to decoherent interactions with the reservoir relax towards their ground state in the weak coupling limit. Within the framework of quantum master…
In this paper we propose a solution strategy for the Cahn-Larch\'e equations, which is a model for linearized elasticity in a medium with two elastic phases that evolve subject to a Ginzburg-Landau type energy functional. The system can be…
We present a new approach for computing approximate global minimizers to a large class of non-local pairwise interaction problems defined over probability distributions. The approach predicts candidate global minimizers, with a recovery…