39 research outputs found
An entropy minimization approach to second-order variational mean-field games
We propose an entropy minimization viewpoint on variational mean-field games with diffusion and quadratic Hamiltonian. We carefully analyze the time discretization of such problems, establish Gamma-convergence results as the time step vanishes and propose an efficient algorithm relying on this entropic interpretation as well as on the Sinkhorn scaling algorithm
Local Volatility Calibration by Optimal Transport
The calibration of volatility models from observable option prices is a
fundamental problem in quantitative finance. The most common approach among
industry practitioners is based on the celebrated Dupire's formula [6], which
requires the knowledge of vanilla option prices for a continuum of strikes and
maturities that can only be obtained via some form of price interpolation. In
this paper, we propose a new local volatility calibration technique using the
theory of optimal transport. We formulate a time continuous martingale optimal
transport problem, which seeks a martingale diffusion process that matches the
known densities of an asset price at two different dates, while minimizing a
chosen cost function. Inspired by the seminal work of Benamou and Brenier [1],
we formulate the problem as a convex optimization problem, derive its dual
formulation, and solve it numerically via an augmented Lagrangian method and
the alternative direction method of multipliers (ADMM) algorithm. The solution
effectively reconstructs the dynamic of the asset price between the two dates
by recovering the optimal local volatility function, without requiring any time
interpolation of the option prices
Fast Optimal Transport Averaging of Neuroimaging Data
Knowing how the Human brain is anatomically and functionally organized at the
level of a group of healthy individuals or patients is the primary goal of
neuroimaging research. Yet computing an average of brain imaging data defined
over a voxel grid or a triangulation remains a challenge. Data are large, the
geometry of the brain is complex and the between subjects variability leads to
spatially or temporally non-overlapping effects of interest. To address the
problem of variability, data are commonly smoothed before group linear
averaging. In this work we build on ideas originally introduced by Kantorovich
to propose a new algorithm that can average efficiently non-normalized data
defined over arbitrary discrete domains using transportation metrics. We show
how Kantorovich means can be linked to Wasserstein barycenters in order to take
advantage of an entropic smoothing approach. It leads to a smooth convex
optimization problem and an algorithm with strong convergence guarantees. We
illustrate the versatility of this tool and its empirical behavior on
functional neuroimaging data, functional MRI and magnetoencephalography (MEG)
source estimates, defined on voxel grids and triangulations of the folded
cortical surface.Comment: Information Processing in Medical Imaging (IPMI), Jun 2015, Isle of
Skye, United Kingdom. Springer, 201
Gradient methods for problems with inexact model of the objective
We consider optimization methods for convex minimization problems under inexact information on the objective function. We introduce inexact model of the objective, which as a particular cases includes inexact oracle [19] and relative smoothness condition [43]. We analyze gradient method which uses this inexact model and obtain convergence rates for convex and strongly convex problems. To show potential applications of our general framework we consider three particular problems. The first one is clustering by electorial model introduced in [49]. The second one is approximating optimal transport distance, for which we propose a Proximal Sinkhorn algorithm. The third one is devoted to approximating optimal transport barycenter and we propose a Proximal Iterative Bregman Projections algorithm. We also illustrate the practical performance of our algorithms by numerical experiments
Ray mappings and the weighted least action principle
Abstract Two basic problems in optics are presented. The solutions to both problems are formulated in terms of the associated ray mappings. An alternative formulation based on a weighted sum of the actions along the rays is derived. Existence of solutions is established via the Weighted Least Action Principle. Numerical methods for computing the ray mappings are discussed. Finally, we demonstrate the theoretical considerations by presenting complete solutions to a phase retrieval problem and to a specific beam shaping lens design