61 research outputs found
A numerical method for variational problems with convexity constraints
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 interface between convex analysis, convex optimization,
variational problems, and partial differential equation techniques.
The approach is to approximate the (non-polyhedral) cone of convex functions
by a polyhedral cone which can be represented by linear inequalities. This
approach leads to an optimization problem with linear constraints which can be
computed efficiently, hundreds of times faster than existing methods.Comment: 21 pages, 6 figures, 6 table
Finite difference methods for the Infinity Laplace and p-Laplace equations
We build convergent discretizations and semi-implicit solvers for the
Infinity Laplacian and the game theoretical -Laplacian. The discretizations
simplify and generalize earlier ones. We prove convergence of the solution of
the Wide Stencil finite difference schemes to the unique viscosity solution of
the underlying equation. We build a semi-implicit solver, which solves the
Laplace equation as each step. It is fast in the sense that the number of
iterations is independent of the problem size. This is an improvement over
previous explicit solvers, which are slow due to the CFL-condition.Comment: 22 pages, 10 figure
Filtered schemes for Hamilton-Jacobi equations: a simple construction of convergent accurate difference schemes
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 unique viscosity solution of the equation. The
schemes are accurate: we implement second, third and fourth order accurate
schemes in one dimension and second order accurate schemes in two dimensions,
indicating how to build higher order ones. They are also explicit, which means
they can be solved using the fast sweeping method or the fast marching
method.The accuracy of the method is validated with computational results for
the eikonal equation in one and two dimensions, using filtered schemes made
from standard centered differences, higher order upwinding and ENO
interpolation.Comment: 31 pages, 9 figures, 9 table
Scaleable input gradient regularization for adversarial robustness
In this work we revisit gradient regularization for adversarial robustness
with some new ingredients. First, we derive new per-image theoretical
robustness bounds based on local gradient information. These bounds strongly
motivate input gradient regularization. Second, we implement a scaleable
version of input gradient regularization which avoids double backpropagation:
adversarially robust ImageNet models are trained in 33 hours on four consumer
grade GPUs. Finally, we show experimentally and through theoretical
certification that input gradient regularization is competitive with
adversarial training. Moreover we demonstrate that gradient regularization does
not lead to gradient obfuscation or gradient masking
An efficient linear programming method for Optimal Transportation
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) discretization of the continuous problem becomes
intractible for moderate grid sizes. A grid refinement method results in a
linear cost algorithm. Weak convergence of solutions is stablished. Barycentric
projection of transference plans is used to improve the accuracy of solutions.
The method is applied to more general problems, including partial optimal
transportation, and barycenter problems. Computational examples validate the
accuracy and efficiency of the method. Optimal maps between nonconvex domains,
partial OT free boundaries, and high accuracy barycenters are presented.Comment: 25 pages, 11 figures, 2 table
The Dirichlet problem for the convex envelope
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 problem: the Dirichlet problem for the underlying
PDE. The main result is an optimal regularity result. Differentiability
( regularity) of the boundary data implies the corresponding
result for the solution in the interior, despite the fact that the solution
need not be continuous up to the boundary. Secondary results are the
characterization of the convex envelope as: (i) the value function of a
stochastic control problem, and (ii) the optimal underestimator for a class of
nonlinear elliptic PDEs.Comment: 16 pages, 2 figures. To be published in Trans. Amer. Math. So
Approximate Convex Hulls: sketching the convex hull using curvature
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 computational complexity of convex hull algorithms. In
this article we approximate the convex hull in using a scalable algorithm which
finds high curvature vertices with high probability. The algorithm is
particularly effective for approximating convex hulls which have a relatively
small number of extreme points.Comment: 16 pages, 8 figure
Adaptive finite difference methods for nonlinear elliptic and parabolic partial differential equations with free boundaries
Monotone finite difference methods provide stable convergent discretizations
of a class of degenerate elliptic and parabolic Partial Differential Equations
(PDEs). These methods are best suited to regular rectangular grids, which leads
to low accuracy near curved boundaries or singularities of solutions. In this
article we combine monotone finite difference methods with an adaptive grid
refinement technique to produce a PDE discretization and solver which is
applied to a broad class of equations, in curved or unbounded domains which
include free boundaries. The grid refinement is flexible and adaptive. The
discretization is combined with a fast solution method, which incorporates
asynchronous time stepping adapted to the spatial scale. The framework is
validated on linear problems in curved and unbounded domains. Key applications
include the obstacle problem and the one-phase Stefan free boundary problem.Comment: 19 pages, 12 figures, 2 table
Approximate homogenization of convex nonlinear elliptic PDEs
We approximate the homogenization of fully nonlinear, convex, uniformly
elliptic Partial Differential Equations in the periodic setting, using a
variational formula for the optimal invariant measure, which may be derived via
Legendre-Fenchel duality. The variational formula expresses as an
average of the operator against the optimal invariant measure, generalizing the
linear case. Several nontrivial analytic formulas for are obtained.
These formulas are compared to numerical simulations, using both PDE and
variational methods. We also perform a numerical study of convergence rates for
homogenization in the periodic and random setting and compare these to
theoretical results
A partial differential equation for the rank one convex envelope
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 problems \cite{BallElasticity, kohn1986optimal1,
BallJames87, chipot1988equilibrium}. More generally, we study a PDE for
directional convex envelopes, which includes the usual convex envelope
\cite{ObermanConvexEnvelope} and the rank one convex envelope as special cases.
Existence and uniqueness of viscosity solutions to the PDE is established. Wide
stencil elliptic finite difference schemes are built. Convergence of finite
difference solutions to the viscosity solution of the PDE is proven. Numerical
examples of rank one and other directional convex envelopes are presented.
Additionally, laminates are computed from the rank one convex envelope.Comment: 26 pages, 5 figures, 4 table
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