Research
  • mathematics and optimization of deep neural networks
  • numerical methods for nonlinear Partial Differential Equations and Optimal Transportation
New
  1. A Lyapunov analysis for accelerated gradient method: from deterministic to stochastic case, Maxime Laborde, Adam Oberman arxiv
  2. A principled approach for generating adversarial images under non-smooth dissimilarity metrics Aram-Alexandre Pooladian, Chris Finlay, Tim Hoheisel, Adam Oberman arxiv
  3. Scaleable input gradient regularization for adversarial robustness Chris Finlay, Adam Oberman arxiv github
  4. Empirical confidence estimates for classification by deep neural networks Chris Finlay, Adam Oberman  Presentation arxiv
  5. The LogBarrier adversarial attack: making effective use of decision boundary information Chris Finlay, Aram-Alexander Pooladian, Adam Oberman arxiv
  6. A Partial Differential Equation Obstacle problem for the Level Set Approach to Visibility Adam Oberman, Tiago Salvador arxiv
  7. Stochastic Gradient Descent with Polyak's Learning Rate Mariana Prazeres, Adam Oberman arxiv
2018
  1. Lipschitz regularized Deep Neural Networks generalize and are adversarially robust  Chris Finlay, Jeff Calder, Bilal Abbasi, and Adam M. Oberman  arxiv
  2. 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
  3. Improved accuracy of monotone finite difference schemes on point clouds and regular grids Chris Finlay, Adam M. Oberman; to appear in SIAM SISC arxiv
  4. Parle: parallelizing stochastic gradient descent Pratik Chaudhari, Carlo Baldassi, Riccardo Zecchina, Stefano Soatto, Ameet Talwalkar, Adam Oberman SysML arxiv
  5. 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
  6. 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
  7. Approximate Homogenization of fully nonlinear elliptic PDEs  Chris Finlay and Adam M. Oberman;  Journal of Scientific Computing;  arxiv
  8. Approximate Homogenization of convex nonlinear elliptic PDEs  Chris Finlay and Adam M. Oberman;  Comm Math Sci; arxiv
2017
  1. Numerical methods for motion of level sets by affine curvature Adam M. Oberman and Tiago Salvador; IMA Journal of Numerical Analysis, 2017.  journal  arxiv
  2. A partial differential equation for the rank one convex envelope Adam M. Oberman and Yuanlong Ruan; Archive for Rational Mechanics and Analysis arxiv ;
  3. A partial differential equation for the uniformly quasiconvex envelope Bilal Abbasi and Adam M. Oberman 2017; IMA Journal of Numerical Analysis journal arxiv
  4. Computing the quasiconvex envelope using a nonlocal solver Bilal Abbasi and Adam M. Oberman 2017; Journal of Scientific Computing  arxiv
  5. 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
  1. 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
  2. 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
  1. 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
  2. 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
  3. 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
  4. 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
  1. Finite difference methods for fractional Laplacians Adam M. Oberman and  Yanghong Huang; arxiv
  2. An efficient Linear Programming method for Optimal Transportation Adam M. Oberman, Yuanlong Ruan,  arxiv
  3. Approximate Convex Hulls: sketching the convex hull using curvature Robert Graham, Adam M. Oberman 2017; arxiv
  4. Addtional notes on  Numerical solution of the Optimal Transportation Problem using the Monge-Ampère equation  arxiv
2014 and earlier
  1. 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
  2. (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
  3. (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
  4. 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
  5. A numerical method for variational problems with convexity constraints. SIAM Journal on Scientific Computing Vol. 35, No. 1, A378-A396 arxiv ; SISC
  6. (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 ;
  7. (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
  8. (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
  9. (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
  10. (with Luis Silvestre) The Dirichlet Problem for the Convex Envelope, Trans. Amer. Math. Soc 363(11) 5871-5886, 2011.  Journal
  11. Convergence Rates for Difference Schemes for Polyhedral Nonlinear Parabolic Equations, . J. Comput. Math. Vol. 28, No.4, 2010, 474–488. Journal Version
  12. (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
  13. (with Brittany Froese) Numerical Averaging of Non-Divergence Structure Elliptic Operators, Comm. Math. Sci, Vol. 7 (2009), No. 4, pp 785-804   journal link
  14. (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
  15. (with Diogo Gomes) Viscosity Solutions Methods for Converse KAM Theory, ESAIM: M2AN, Vol. 42 (2008) 1047-1064. journal link ;
  16. 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 
  17. 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
  18. An explicit solution of the Lipschitz extension problem, Proc. Amer. Math. Soc. 136 (2008) 4329-4338. journal link ;
  19. The convex envelope is the solution of a nonlinear obstacle problem, Proc. Amer. Math. Soc. 135 (2007), no. 6, 1689–1694 journal link 
  20. 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
  21. 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
  22. 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 ;
  23. (with Robert McCann) Exact semigeostrophic flows in an elliptical basin, Nonlinearity, Volume 17 pages 1891-1922 (2004). journal link
  24. (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
  25. (with T. Zariphopoulou) Pricing early exercise contracts in incomplete markets, Computational Management Science, Vol 1 Issue 1 December (2003) pages 75-107. journal link
  26. (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