A note on the dual treatment of higher order regularization functionals

In this paper, we apply the dual approach developed by A. Chambolle for the Rudin-Osher-Fatemi model to regularization functionals with higher order derivatives. We emphasize the linear algebra point of view by consequently using matrix-vector notation. Numerical examples demonstrate the differences between various second order regularization approaches

Quantization of the Nonlinear Sigma Model Revisited

We revisit the subject of perturbatively quantizing the nonlinear sigma model in two dimensions from a rigorous, mathematical point of view. Our main contribution is to make precise the cohomological problem of eliminating potential anomalies that may arise when trying to preserve symmetries under quantization. The symmetries we consider are twofold: (i) diffeomorphism covariance for a general target manifold; (ii) a transitive group of isometries when the target manifold is a homogeneous space. We show that there are no anomalies in case (i) and that (ii) is also anomaly-free under additional assumptions on the target homogeneous space, in agreement with the work of Friedan. We carry out some explicit computations for the $O(N)$-model. Finally, we show how a suitable notion of the renormalization group establishes the Ricci flow as the one loop renormalization group flow of the nonlinear sigma model.Comment: 51 page

Entropic Wasserstein Gradient Flows

This article details a novel numerical scheme to approximate gradient flows for optimal transport (i.e. Wasserstein) metrics. These flows have proved useful to tackle theoretically and numerically non-linear diffusion equations that model for instance porous media or crowd evolutions. These gradient flows define a suitable notion of weak solutions for these evolutions and they can be approximated in a stable way using discrete flows. These discrete flows are implicit Euler time stepping according to the Wasserstein metric. A bottleneck of these approaches is the high computational load induced by the resolution of each step. Indeed, this corresponds to the resolution of a convex optimization problem involving a Wasserstein distance to the previous iterate. Following several recent works on the approximation of Wasserstein distances, we consider a discrete flow induced by an entropic regularization of the transportation coupling. This entropic regularization allows one to trade the initial Wasserstein fidelity term for a Kulback-Leibler divergence, which is easier to deal with numerically. We show how KL proximal schemes, and in particular Dykstra's algorithm, can be used to compute each step of the regularized flow. The resulting algorithm is both fast, parallelizable and versatile, because it only requires multiplications by a Gibbs kernel. On Euclidean domains discretized on an uniform grid, this corresponds to a linear filtering (for instance a Gaussian filtering when $c$ is the squared Euclidean distance) which can be computed in nearly linear time. On more general domains, such as (possibly non-convex) shapes or on manifolds discretized by a triangular mesh, following a recently proposed numerical scheme for optimal transport, this Gibbs kernel multiplication is approximated by a short-time heat diffusion

Free vacuum for loop quantum gravity

We linearize extended ADM-gravity around the flat torus, and use the associated Fock vacuum to construct a state that could play the role of a free vacuum in loop quantum gravity. The state we obtain is an element of the gauge-invariant kinematic Hilbert space and restricted to a cutoff graph, as a natural consequence of the momentum cutoff of the original Fock state. It has the form of a Gaussian superposition of spin networks. We show that the peak of the Gaussian lies at weave-like states and derive a relation between the coloring of the weaves and the cutoff scale. Our analysis indicates that the peak weaves become independent of the cutoff length when the latter is much smaller than the Planck length. By the same method, we also construct multiple-graviton states. We discuss the possible use of these states for deriving a perturbation series in loop quantum gravity.Comment: 30 pages, 3 diagrams, treatment of phase factor adde