Research

Research

starting 2017 : mathematics applied to deep neural networks.
before 2017: convergent numerical methods for nonlinear Partial Differential Equations.

New

Scaleable input gradient regularization for adversarial robustness Chris Finlay, Adam Oberman arxiv github
Empirical confidence estimates for classification by deep neural networks Chris Finlay, Adam Oberman Presentation arxiv
The LogBarrier adversarial attack: making effective use of decision boundary information Chris Finlay, Aram-Alexander Pooladian, Adam Oberman arxiv
Stochastic Gradient Descent with Polyak's Learning Rate Mariana Prazeres, Adam Oberman arxiv
On proximal point-type algorithms for weakly convex functions and their connection to the backward Euler method. Tim Hoheisel, Maxime Laborde, Adam Oberman opt-online
Improved accuracy of monotone finite difference schemes on point clouds and regular grids Chris Finlay and Adam M. Oberman arxiv
Improved robustness to adversarial examples using Lipschitz regularization of the loss Chris Finlay, Adam Oberman, Bilal Abbasi; Oct 2018; arxiv
Lipschitz regularized Deep Neural Networks converge and generalize Adam M. Oberman and Jeff Calder August 2018; arxiv

2018

Parle: parallelizing stochastic gradient descent Pratik Chaudhari, Carlo Baldassi, Riccardo Zecchina, Stefano Soatto, Ameet Talwalkar, Adam Oberman SysML arxiv
Stochastic Backward Euler: An Implicit Gradient Descent Algorithm for k-means Clustering Penghang Yin, Minh Pham, Adam Oberman, Stanley Osher; Journal of Scientific Computing arxiv
Deep Relaxation: partial differential equations for optimizing deep neural networks Pratik Chaudhari, Adam M. Oberman, Stanley Osher, Stefano Soatto, Guillame Carlier, Research in Math Sciences arxiv
Approximate Homogenization of fully nonlinear elliptic PDEs Chris Finlay and Adam M. Oberman ; Journal of Scientific Computing; arxiv
Approximate Homogenization of convex nonlinear elliptic PDEs Chris Finlay and Adam M. Oberman; Comm Math Sci; arxiv

2017

Numerical methods for motion of level sets by affine curvature Adam M. Oberman and Tiago Salvador; IMA Journal of Numerical Analysis, 2017. journal arxiv
A partial differential equation for the rank one convex envelope Adam M. Oberman and Yuanlong Ruan; Archive for Rational Mechanics and Analysis arxiv ;
A partial differential equation for the uniformly quasiconvex envelope Bilal Abbasi and Adam M. Oberman 2017; IMA Journal of Numerical Analysis journal arxiv
Computing the quasiconvex envelope using a nonlocal solver Bilal Abbasi and Adam M. Oberman 2017; Journal of Scientific Computing arxiv
Anomaly detection and classification for streaming data using partial differential equations Bilal Abbasi, Jeff Calder and Adam M Oberman, 2017; SIAM journal of Applied Math arxiv

2016

Numerical Methods for the two-Hessian elliptic partial differential equation Brittany D. Froese, Adam M. Oberman, Tiago Salvador; IMA Journal of Numerical Analysis, May 2016 journal arxiv
A multigrid solver for the three dimensional Monge-Ampère equation Jun Liu, Brittany D. Froese, Adam M. Oberman, Mingqing Xiao; International Journal of Computer Mathematics, November 2016; journal arxiv

2015

Adaptive finite difference methods for nonlinear elliptic and parabolic partial differential equations with free boundaries Adam M. Oberman, Ian Zwiers; Journal of Scientific Computing, November 2015 arxiv JSC
Filtered schemes for Hamilton-Jacobi equations: a simple construction of convergent accurate difference schemes with Tiago Salvador; Journal of Computational Physics Volume 284, 1 March 2015, pages 367-388; arxiv JCP
Numerical methods for matching for teams and Wasserstein barycenters G. Carlier, A. Oberman and É. Oudet; ESAIM: Mathematical Modelling and Numerical Analysis (M2AN) 6 2015 1621-1642; arxiv M2AN
Nonlinear elliptic PDEs on graphs with Juan Manfredi and Alex Sviridov; Differential and Integral Equations Volume 28 Numbers 1-2, January/February 2015. DIE Journal version, arxiv

Research Notes

Finite difference methods for fractional Laplacians Adam M. Oberman and Yanghong Huang; arxiv
An efficient Linear Programming method for Optimal Transportation Adam M. Oberman, Yuanlong Ruan, arxiv
Approximate Convex Hulls: sketching the convex hull using curvature Robert Graham, Adam M. Oberman 2017; arxiv
Addtional notes on Numerical solution of the Optimal Transportation Problem using the Monge-Ampère equation arxiv

2014 and earlier

Numerical Methods for the Fractional Laplacian: A Finite Difference-Quadrature Approach Adam M. Oberman and Yanghong Huang; SIAM Journal of Numerical Analysis 2014 Volume 52, Issue 6, pp. 2599-3180 SINUM ; arxiv
(with Jean-David Benamou and Brittany Froese) Numerical solution of the Optimal Transportation Problem using the Monge-Ampère equation Journal of Computational Physics Volume 260, 1 March 2014, Pages 107–126 arxiv ; JCP version
(with Brittany Froese) Convergent Filtered Schemes for the Monge–Ampère Partial Differential Equation SIAM Journal on Numerical Analysis, 51(1):423–444, 2013. journal ; arxiv
Finite Difference Methods for the infinity Laplace and p-Laplace equations Journal of Computational and Applied Mathematics, Volume 254, 15 December 2013, Pages 65–80 arxiv ; journal
A numerical method for variational problems with convexity constraints. SIAM Journal on Scientific Computing Vol. 35, No. 1, A378-A396 arxiv ; SISC
(with Brittany Froese) Convergent finite difference solvers for viscosity solutions of the elliptic Monge-Ampère equation in dimensions two and higher SIAM Journal on Numerical Analysis, 49(4):1692–1714, 2012. arxiv ; journal ;
(with Rustum Choksi and Yves van Gennip) Anisotropic Total Variation Regularized L1-Approximation and Denoising/Deblurring of 2D Bar Codes Inverse Problems and Imaging, 5(3):591-617, 2011. arxiv.org ; Article
(with Brittany Froese) Fast finite difference solvers for singular solutions of the elliptic Monge-Ampère equation J. Comput. Phys. 230(3):818-834, 2011. Article ; arxiv.org
(with Stan Osher, Ryo Takei, and Richard Tsai) Numerical methods for anisotropic Mean Curvature flow based on a discrete time variational formulation Comm. Math. Sci. 9(3) :637-662, 2011 Cam Report ; Article
(with Luis Silvestre) The Dirichlet Problem for the Convex Envelope, Trans. Amer. Math. Soc 363(11) 5871-5886, 2011. Journal
Convergence Rates for Difference Schemes for Polyhedral Nonlinear Parabolic Equations, . J. Comput. Math. Vol. 28, No.4, 2010, 474–488. Journal Version
(with Jean-David Benamou and Brittany Froese) Two Numerical Methods for the Elliptic Monge-Ampère Equation, ESAIM: M2AN, Volume 44 (2010), Number 4, pp 737-758 Article ; journal link
(with Brittany Froese) Numerical Averaging of Non-Divergence Structure Elliptic Operators, Comm. Math. Sci, Vol. 7 (2009), No. 4, pp 785-804 journal link
(with Ryo Takei and Alexander Vladimirsky) Homogenization of Metric Hamilton-Jacobi Equations, SIAM Multiscale Modeling and Simulation, Vol. 8 Number 1 (2009) 269-295 Article ; journal link
(with Diogo Gomes) Viscosity Solutions Methods for Converse KAM Theory, ESAIM: M2AN, Vol. 42 (2008) 1047-1064. journal link ;
Computing the convex envelope using a nonlinear partial differential equation, Mathematical Models and Methods in Applied Sciences (M3AS), Vol. 18. No 5 (2008) 759-780 journal link
Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian, Discrete and Continuous Dynamical Systems series B (DCDS B) Volume 10, Number 1, July 2008 (221-238) Project Page ; journal link ; Eigenvalues.pdf
An explicit solution of the Lipschitz extension problem, Proc. Amer. Math. Soc. 136 (2008) 4329-4338. journal link ;
The convex envelope is the solution of a nonlinear obstacle problem, Proc. Amer. Math. Soc. 135 (2007), no. 6, 1689–1694 journal link
Convergent Difference Schemes for Nonlinear Elliptic and Parabolic Equations: Hamilton-Jacobi Equations and Free Boundary Problems, SIAM Journal on Numerical Analysis, Vol 44 (2006) No. 2 pp. 879-895. journal link
A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions, Mathematics of Computation, 74 (2005) Number 251, 1217-1230. journal link
A convergent monotone difference scheme for motion of level sets by mean curvature, Numerische Mathematik, Volume 99 (2004) Number 2 pages 365-379. journal link ;
(with Robert McCann) Exact semigeostrophic flows in an elliptical basin, Nonlinearity, Volume 17 pages 1891-1922 (2004). journal link
(with Diogo Gomes) Computing the effective Hamiltonian using a variational approach, SIAM Journal on Control and Optimization, Volume 43 (2004) Number 3 pages 793-812. journal link
(with T. Zariphopoulou) Pricing early exercise contracts in incomplete markets, Computational Management Science, Vol 1 Issue 1 December (2003) pages 75-107. journal link
(with P. Constantin, A. Kiselev, and L. Ryzhik) Bulk burning rate in passive-reactive diffusion, Archive for Rational Mechanics, 154 (2000), 53-91. journal link