Publications (29)
Fast finite difference solvers for singular solutions of the elliptic Monge-Ampère equation
Brittany D. Froese, Adam M. Oberman
The elliptic Monge-Ampere equation is a fully nonlinear Partial Differential Equation which originated in geometric surface theory, and has been applied in dynamic meteorology, ela…
Finite difference methods for the Infinity Laplace and p-Laplace equations
Adam M. Oberman
We build convergent discretizations and semi-implicit solvers for the Infinity Laplacian and the game theoretical -Laplacian. The discretizations simplify and generalize earlier…
Approximate homogenization of fully nonlinear elliptic PDEs: estimates and numerical results for Pucci type equations
Chris Finlay, Adam M. Oberman
We are interested in the shape of the homogenized operator for PDEs which have the structure of a nonlinear Pucci operator. A typical operator is $H^{a_1,a_2}(Q,x)…
Filtered schemes for Hamilton-Jacobi equations: a simple construction of convergent accurate difference schemes
Adam M. Oberman, Tiago Salvador
We build a simple and general class of finite difference schemes for first order Hamilton-Jacobi (HJ) Partial Differential Equations. These filtered schemes are convergent to the u…
Anomaly detection and classification for streaming data using PDEs
Bilal Abbasi, Jeff Calder, Adam M. Oberman
Nondominated sorting, also called Pareto Depth Analysis (PDA), is widely used in multi-objective optimization and has recently found important applications in multi-criteria anomal…
A viscosity solution approach to the Monge-Ampere formulation of the Optimal Transportation Problem
Jean-David Benamou, Brittany D. Froese, Adam M. Oberman
In this work we present a numerical method for the Optimal Mass Transportation problem. Optimal Mass Transportation (OT) is an active research field in mathematics.It has recently…
Computing the quasiconvex envelope using a nonlocal line solver
Bilal Abbasi, Adam M. Oberman
Recently in a series of articles, Barron, Goebel, and Jensen \cite{barron2012functions} \cite{barron2012quasiconvex} \cite{barron2013quasiconvex} \cite{barron2013uniqueness} have s…
An efficient linear programming method for Optimal Transportation
Adam M. Oberman, Yuanlong Ruan
An efficient method for computing solutions to the Optimal Transportation (OT) problem with a wide class of cost functions is presented. The standard linear programming (LP) discre…
Fast Rates for Semi-Supervised Learning via Data-Augmentation Graph Regularization
Adam M. Oberman
The paper provides a theoretical analysis showing that self‑supervised learning with data augmentation can achieve a fast O(1/n_L) error rate in semi‑supervised settings, linking t…
The Dirichlet problem for the convex envelope
Luis Silvestre, Adam M. Oberman
The Convex Envelope of a given function was recently characterized as the solution of a fully nonlinear Partial Differential Equation (PDE). In this article we study a modified pro…
Nesterov's method with decreasing learning rate leads to accelerated stochastic gradient descent
Maxime Laborde, Adam M. Oberman
We present a coupled system of ODEs which, when discretized with a constant time step/learning rate, recovers Nesterov's accelerated gradient descent algorithm. The same ODEs, when…
The LogBarrier adversarial attack: making effective use of decision boundary information
Chris Finlay, Aram-Alexandre Pooladian, Adam M. Oberman
Adversarial attacks for image classification are small perturbations to images that are designed to cause misclassification by a model. Adversarial attacks formally correspond to a…
Nonlinear elliptic Partial Differential Equations and p-harmonic functions on graphs
Juan J. Manfredi, Adam M. Oberman, Alex P. Svirodov
In this article we study the well-posedness (uniqueness and existence of solutions) of nonlinear elliptic Partial Differential Equations (PDEs) on a finite graph. These results are…
Numerical solution of the Optimal Transportation problem using the Monge-Ampere equation
Jean-David Benamou, Brittany D. Froese, Adam M. Oberman
A numerical method for the solution of the elliptic Monge-Ampere Partial Differential Equation, with boundary conditions corresponding to the Optimal Transportation (OT) problem is…
Approximate homogenization of convex nonlinear elliptic PDEs
Chris Finlay, Adam M. Oberman
We approximate the homogenization of fully nonlinear, convex, uniformly elliptic Partial Differential Equations in the periodic setting, using a variational formula for the optimal…
A multigrid scheme for 3D Monge-Ampère equations
Jun Liu, Brittany D. Froese, Adam M. Oberman +1
The elliptic Monge-Ampère equation is a fully nonlinear partial differential equation which has been the focus of increasing attention from the scientific computing community. Fas…
Adaptive finite difference methods for nonlinear elliptic and parabolic partial differential equations with free boundaries
Adam M. Oberman, Ian Zwiers
Monotone finite difference methods provide stable convergent discretizations of a class of degenerate elliptic and parabolic Partial Differential Equations (PDEs). These methods ar…
Stochastic Gradient Descent with Polyak's Learning Rate
Adam M. Oberman, Mariana Prazeres
Stochastic gradient descent (SGD) for strongly convex functions converges at the rate $\bO(1/k)$. However, achieving good results in practice requires tuning the parameters (for ex…
A partial differential equation for the rank one convex envelope
Adam M. Oberman, Yuanlong Ruan
In this article we introduce a Partial Differential Equation (PDE) for the rank one convex envelope. Rank one convex envelopes arise in non-convex vector valued variational problem…
A numerical method for variational problems with convexity constraints
Adam M. Oberman
We consider the problem of approximating the solution of variational problems subject to the constraint that the admissible functions must be convex. This problem is at the interfa…
Numerical methods for motion of level sets by affine curvature
Adam M. Oberman, Tiago Salvador
We study numerical methods for the nonlinear partial differential equation that governs the motion of level sets by affine curvature. We show that standard finite difference scheme…
Avoiding unsafe sets when training with Langevin Dynamics
Adam M. Oberman
Training a model with noisy gradient descent can be idealized as overdamped Langevin dynamics, and a natural safety question is to bound the probability $ν_t(\mathcal{A}_H) = \mat…
Calibrated Top-1 Uncertainty estimates for classification by score based models
Adam M. Oberman, Chris Finlay, Alexander Iannantuono +1
While the accuracy of modern deep learning models has significantly improved in recent years, the ability of these models to generate uncertainty estimates has not progressed to th…
Approximate Convex Hulls: sketching the convex hull using curvature
Robert Graham, Adam M. Oberman
Convex hulls are fundamental objects in computational geometry. In moderate dimensions or for large numbers of vertices, computing the convex hull can be impractical due to the com…
Convergent finite difference solvers for viscosity solutions of the elliptic Monge-Ampère equation in dimensions two and higher
Brittany D. Froese, Adam M. Oberman
The elliptic Monge-Ampère equation is a fully nonlinear Partial Differential Equation that originated in geometric surface theory and has been applied in dynamic meteorology, elas…
A partial differential equation for the strictly quasiconvex envelope
Bilal Abbasi, Adam M. Oberman
In a series of papers Barron, Goebel, and Jensen studied Partial Differential Equations (PDE)s for quasiconvex (QC) functions \cite{barron2012functions, barron2012quasiconvex,barro…
Convergent filtered schemes for the Monge-Ampère partial differential equation
Brittany D. Froese, Adam M. Oberman
The theory of viscosity solutions has been effective for representing and approximating weak solutions to fully nonlinear Partial Differential Equations (PDEs) such as the elliptic…
No-collision Transportation Maps
Levon Nurbekyan, Alexander Iannantuono, Adam M. Oberman
Transportation maps between probability measures are critical objects in numerous areas of mathematics and applications such as PDE, fluid mechanics, geometry, machine learning, co…
Numerical Methods for the 2-Hessian Elliptic Partial Differential Equation
Brittany D. Froese, Adam M. Oberman, Tiago Salvador
The elliptic 2-Hessian equation is a fully nonlinear partial differential equation (PDE) that is related to intrinsic curvature for three dimensional manifolds. We introduce two nu…