5,704 research outputs found
Dynamical Optimal Transport on Discrete Surfaces
We propose a technique for interpolating between probability distributions on
discrete surfaces, based on the theory of optimal transport. Unlike previous
attempts that use linear programming, our method is based on a dynamical
formulation of quadratic optimal transport proposed for flat domains by Benamou
and Brenier [2000], adapted to discrete surfaces. Our structure-preserving
construction yields a Riemannian metric on the (finite-dimensional) space of
probability distributions on a discrete surface, which translates the so-called
Otto calculus to discrete language. From a practical perspective, our technique
provides a smooth interpolation between distributions on discrete surfaces with
less diffusion than state-of-the-art algorithms involving entropic
regularization. Beyond interpolation, we show how our discrete notion of
optimal transport extends to other tasks, such as distribution-valued Dirichlet
problems and time integration of gradient flows
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
Multiple Shape Registration using Constrained Optimal Control
Lagrangian particle formulations of the large deformation diffeomorphic
metric mapping algorithm (LDDMM) only allow for the study of a single shape. In
this paper, we introduce and discuss both a theoretical and practical setting
for the simultaneous study of multiple shapes that are either stitched to one
another or slide along a submanifold. The method is described within the
optimal control formalism, and optimality conditions are given, together with
the equations that are needed to implement augmented Lagrangian methods.
Experimental results are provided for stitched and sliding surfaces
Eulerian and Lagrangian solutions to the continuity and Euler equations with vorticity
In the first part of this paper we establish a uniqueness result for
continuity equations with velocity field whose derivative can be represented by
a singular integral operator of an function, extending the Lagrangian
theory in \cite{BouchutCrippa13}. The proof is based on a combination of a
stability estimate via optimal transport techniques developed in \cite{Seis16a}
and some tools from harmonic analysis introduced in \cite{BouchutCrippa13}. In
the second part of the paper, we address a question that arose in
\cite{FilhoMazzucatoNussenzveig06}, namely whether 2D Euler solutions obtained
via vanishing viscosity are renormalized (in the sense of DiPerna and Lions)
when the initial data has low integrability. We show that this is the case even
when the initial vorticity is only in~, extending the proof for the
case in \cite{CrippaSpirito15}
Lagrangian Numerical Methods for Ocean Biogeochemical Simulations
We propose two closely--related Lagrangian numerical methods for the
simulation of physical processes involving advection, reaction and diffusion.
The methods are intended to be used in settings where the flow is nearly
incompressible and the P\'eclet numbers are so high that resolving all the
scales of motion is unfeasible. This is commonplace in ocean flows. Our methods
consist in augmenting the method of characteristics, which is suitable for
advection--reaction problems, with couplings among nearby particles, producing
fluxes that mimic diffusion, or unresolved small-scale transport. The methods
conserve mass, obey the maximum principle, and allow to tune the strength of
the diffusive terms down to zero, while avoiding unwanted numerical dissipation
effects
Reconstruction of the primordial Universe by a Monge--Ampere--Kantorovich optimisation scheme
A method for the reconstruction of the primordial density fluctuation field
is presented. Various previous approaches to this problem rendered {\it
non-unique} solutions. Here, it is demonstrated that the initial positions of
dark matter fluid elements, under the hypothesis that their displacement is the
gradient of a convex potential, can be reconstructed uniquely. In our approach,
the cosmological reconstruction problem is reformulated as an assignment
problem in optimisation theory. When tested against numerical simulations, our
scheme yields excellent reconstruction on scales larger than a few megaparsecs.Comment: 14 pages, 10 figure
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