Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence Theorems.- Uniqueness, Generation of Contraction Semigroups, Error Estimates.- Gradient Flow in the Space of Probability Measures.- Preliminary Results on Measure Theory.- The Optimal Transportation Problem.- The Wasserstein Distance and its Behaviour along Geodesics.- Absolutely Continuous Curves in p(X) and the Continuity Equation.- Convex Functionals in p(X).- Metric Slope and Subdifferential Calculus in (X).- Gradient Flows and Curves of Maximal Slope in p(X).

Gradient Flows: In Metric Spaces and in the Space of Probability Measures

A variational approach for filling-in regions of missing data in digital images is introduced. The approach is based on joint interpolation of the image gray levels and gradient/isophotes directions, smoothly extending in an automatic fashion the isophote lines into the holes of missing data. This interpolation is computed by solving the variational problem via its gradient descent flow, which leads to a set of coupled second order partial differential equations, one for the gray-levels and one for the gradient orientations. The process underlying this approach can be considered as an interpretation of the Gestaltist's principle of good continuation. No limitations are imposed on the topology of the holes, and all regions of missing data can be simultaneously processed, even if they are surrounded by completely different structures. Applications of this technique include the restoration of old photographs and removal of superimposed text like dates, subtitles, or publicity. Examples of these applications are given. We conclude the paper with a number of theoretical results on the proposed variational approach and its corresponding gradient descent flow.

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Filling-in by joint interpolation of vector fields and gray levels

This paper presents a modified regularized formulation of the Ambrosio–Tortorelli type to introduce the crack non-interpenetration condition in the variational approach to fracture mechanics proposed by Francfort and Marigo [1998. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (8), 1319–1342]. We focus on the linear elastic case where the contact condition appears as a local unilateral constraint on the displacement jump at the crack surfaces. The regularized model is obtained by splitting the strain energy in a spherical and a deviatoric parts and accounting for the sign of the local volume change. The numerical implementation is based on a standard finite element discretization and on the adaptation of an alternate minimization algorithm used in previous works. The new regularization avoids crack interpenetration and predicts asymmetric results in traction and in compression. Even though we do not exhibit any gamma-convergence proof toward the desired limit behavior, we illustrate through several numerical case studies the pertinence of the new model in comparison to other approaches.

Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments

A Model Problem.- Semiconcave Functions.- Generalized Gradients and Semiconcavity.- Singularities of Semiconcave Functions.- Hamilton-Jacobi Equations.- Calculus of Variations.- Optimal Control Problems.- Control Problems with Exit Time.

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Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control

Deep neural networks have achieved impressive experimental results in image classification, but can surprisingly be unstable with respect to adversarial perturbations, that is, minimal changes to the input image that cause the network to misclassify it With potential applications including perception modules and end-to-end controllers for self-driving cars, this raises concerns about their safety We develop a novel automated verification framework for feed-forward multi-layer neural networks based on Satisfiability Modulo Theory (SMT) We focus on safety of image classification decisions with respect to image manipulations, such as scratches or changes to camera angle or lighting conditions that would result in the same class being assigned by a human, and define safety for an individual decision in terms of invariance of the classification within a small neighbourhood of the original image We enable exhaustive search of the region by employing discretisation, and propagate the analysis layer by layer Our method works directly with the network code and, in contrast to existing methods, can guarantee that adversarial examples, if they exist, are found for the given region and family of manipulations If found, adversarial examples can be shown to human testers and/or used to fine-tune the network We implement the techniques using Z3 and evaluate them on state-of-the-art networks, including regularised and deep learning networks We also compare against existing techniques to search for adversarial examples and estimate network robustness

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Safety Verification of Deep Neural Networks

By Luigi Ambrosio, Nicolo Fucso and Diego Pallara: 434 pp., £55.00, isbn 0-19-850254-1 (Clarendon Press, Oxford, 2000).

FUNCTIONS OF BOUNDED VARIATION AND FREE DISCONTINUITY PROBLEMS (Oxford Mathematical Monographs)

For any non-negative, quasi-convex Hamiltonian H ∈ C 2(R n ), we consider the L ∞-functional F(u, ·) = ‖H(∇ u)‖ L ∞(·) over . We introduce the notion of comparison with generalized cones (abbreviated CGC) and prove that CGC, viscosity solutions of the Aronsson equation, and absolute minimizers of F(·) are equivalent. This extends an earlier result by Crandall et al. (2001).

Generalized Cone Comparison Principle for Viscosity Solutions of the Aronsson Equation and Absolute Minimizers

Here, A and B are respectively vector and scalar Borel functions defined on t2 x R 1 x R ~, where f2 is a bounded, open connected subset of Euclidean space RL The symbol Vu denotes the gradient of u=u(x) , where x = (xl . . . . . x~). Under certain structural assumptions on the coefficients of (1) it has been shown by several authors that a weak solution is H61der continuous in I2; cf. [LU], [S], IT]. The proof in [LU] relies on techniques introduced by DEGIORGI in his investigation of weak solutions of linear elliptic equations in divergence form [DG], whereas the proofs given in [S] and [T] are based on MOSER'S iteration method, [MO]. The main purpose of this paper is to show that if the upper capacitary density of R " f 2 at a point XoeSf2 is positive, then a bounded solution of (1) which takes on continuous boundary values in a weak sense assumes its value at x o continuously. The capacity used here is one whose null sets are the exceptional sets for Sobolev functions. In particular, if the capacity of a set is zero, then its Hausdorff dimension is at most n p , where p is a number determined by the structural assumptions. (In case p--2 , this capacity is equivalent to the classical Newtonian one.) It is also shown that the solution of (1) has an approximate limit at all points of the boundary of f2. Our proof employs the method of DEGIORGI, as presented in [LU].

Behavior at the boundary of solutions of quasilinear elliptic equations

Space analogues of some theorems for subharmonic and meromorphic functions

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Ronald Gariepy

Papers

FUNCTIONS OF BOUNDED VARIATION AND FREE DISCONTINUITY PROBLEMS (Oxford Mathematical Monographs)

Generalized Cone Comparison Principle for Viscosity Solutions of the Aronsson Equation and Absolute Minimizers

Behavior at the boundary of solutions of quasilinear elliptic equations

Space analogues of some theorems for subharmonic and meromorphic functions

A maximum principle with applications to subharmonic functions in n-space