arXiv subjects

Matt Jacobs

Publications and source records attributed to Matt Jacobs.

23 records · Page 2Linked to original sources

arXiv2020

The $L^1$-contraction principle in optimal transport

In this work, we use the JKO scheme to approximate a general class of diffusion problems generated by Darcy's law. Although the scheme is now classical, if the energy density is spatially inhomogeneous or irregular, many standard methods fail to apply to establish convergence in the continuum limit. To overcome these difficulties, we analyze the scheme through its dual problem and establish a novel $L^1$-contraction principle for the density variable. Notably, the contraction principle relies only on the existence of an optimal transport map and the convexity structure of the energy. As a result, the principle holds in a very general setting, and opens the door to using optimal-transport-based variational schemes to study a larger class of non-linear inhomogeneous parabolic equations.

math.AP↗

arXiv2020

Darcy's Law with a Source term

We introduce a novel variant of the JKO scheme to approximate Darcy's law with a pressure dependent source term. By introducing a new variable that implicitly controls the source term, our scheme is still able to use the standard Wasserstein-2-metric even though the total mass changes over time. Leveraging the dual formulation of our scheme, we show that the discrete-in-time approximations satisfy many useful properties expected for the continuum solutions, such as a comparison principle and uniform $L^1$-equicontinuity. Many of these properties are new even in the well-understood case where the growth term is absent. Finally, we show that our discrete approximations converge to a solution of the corresponding PDE system, including a tumor growth model with a general nonlinear source term.

math.AP↗

arXiv2020

A fast approach to optimal transport: The back-and-forth method

We present an iterative method to efficiently solve the optimal transportation problem for a class of strictly convex costs which includes quadratic and p-power costs. Given two probability measures supported on a discrete grid with n points, we compute the optimal map using O(n) storage space and O(n log(n)) operations per iteration, with an approximately exponential convergence rate. Our approach allows us to solve optimal transportation problems on spatial grids as large as 4096x4096 and 384x384x384 in a matter of minutes.

math.OC↗

arXiv2018

A partial differential equations approach to defeating partisan gerrymandering

We introduce a novel partial differential equations approach for addressing the problem of partisan gerrymandering. Our method is based on volume preserving curvature flow, a partial differential equation which we adapt to smooth voting district boundaries while preserving equal voting populations. We show that every step of the flow minimizes a "compactness energy", allowing us to demonstrate that our method produces more "compact" and reasonable district maps. We compute the flow using a variant of "auction dynamics" --- an efficient MBO type algorithm for computing volume preserving curvature flows. This "auction dynamics" approach can be used to generate hundreds of reasonable maps in a matter of seconds without parallelization. The compactness energy provides a way of comparing proposed districtings of a given state. We demonstrate both the map generation and map comparison features of our approach for several different states.

physics.soc-ph↗

arXiv2018

Solving Large-Scale Optimization Problems with a Convergence Rate Independent of Grid Size

We present a primal-dual method to solve L1-type non-smooth optimization problems independently of the grid size. We apply these results to two important problems : the Rudin-Osher-Fatemi image denoising model and the L1 earth mover's distance from optimal transport. Crucially, we provide analysis that determines the choice of optimal step sizes and we prove that our method converges independently of the grid size. Our approach allows us to solve these problems on grids as large as 4096 by 4096 in a few minutes without parallelization.

math.OC↗