82 research outputs found
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 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
Mixed-integer linearity in nonlinear optimization: a trust region approach
Bringing together nonlinear optimization with mixed-integer linear
constraints enables versatile modeling, but poses significant computational
challenges. We investigate a method to solve these problems based on sequential
mixed-integer linearization with trust region safeguard, computing feasible
iterates via calls to a generic mixed-integer linear solver. Convergence to
critical, possibly suboptimal, feasible points is established for arbitrary
starting points. Finally, we present numerical applications in nonsmooth
optimal control and optimal network design and operation.Comment: 17 pages, 3 figures, 2 table
Continuous Methods for Elliptic Inverse Problems
Numerous mathematical models in applied and industrial mathematics take the form of a partial differential equation involving certain variable coefficients. These coefficients are known and they often describe some physical properties of the model. The direct problem in this context is to solve the partial differential equation. By contrast, an inverse problem asks for the identification of the variable coefficient when a certain measurement of a solution of the partial differential equation is available. A commonly used approach to inverse problems is to solve an optimization problem whose solution is an approximation of the sought coefficient. Such optimization problems are typically solved by discrete iterative schemes. It turns out that most known iterative schemes have their continuous counterparts given in terms of dynamical systems. However, such differential equations are usually solved by specific differential equation solvers. The primary objective of this thesis is to test the feasibility of differential equations based solvers for solving elliptic inverse problems. We will use differential equation solvers such as Euler\u27s Method, Trapezoidal Method, Runge-Kutta Method and Adams-Bashforth Method. In addition, these solvers will also be compared to built-in MATLAB ODE solvers. The performance and accuracy of these methods to solve inverse problems will be thoroughly discussed
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